Formas Diferentes

L A De Determinar

Del Velocidad

E L Movimiento En

Anónimo Tratado

SEX INCONVENIENTIBUS Sabine De

Rommevaux-Tani

THE PARADOXES PRODUCED BY THE DIFFERENT WAYS OF DETERMINING THE RAPIDITY OF MOTION IN THE ANONYMOUS TREATISE DE SEX INCONVENIENTIBUS LAS PARADOJAS CAUSADAS POR LAS

Keywords: Natural philosophy, Motion, Rapidity, 14th Century Resumen Filosofía natural, movimiento, velocidad, siglo XIV

The anonymous treatise De sex inconvenientibus is a good example of the calculatores' approach when dealing with motion. It is organized around four main questions relating to the determination of rapidity in four kinds of changes, i.e. in the generation of substantial forms, in alteration, in increase, and in local motion. In some arguments the author points out the paradoxes to which the two ways of determining the rapidity of a motion can lead: rapidity is determined by the effect produced (the degree of quality generated, the space covered, etc.) or it results from the ratio between the moving power and the resistance of the mobile or patient. While this twofold approach to determining rapidity appears in the majority of calculator texts, the two points of view -the analysis according to its effects and the analysis according to its causes -have rarely been confronted.

El tratado anónimo De sex inconvenientibus es un buen ejemplo del enfoque de los calculadores al tratar el movimiento. Está organizado en torno a cuatro cuestiones principales relativas a la determinación de la velocidad en los cuatro tipos de cambios, es decir, en la generación de formas sustanciales, en la alteración, en el aumento y en el movimiento local. En algunos argumentos, el autor señala las paradojas a las que pueden conducir las dos formas de determinar la velocidad

Introduction

In the 14th century, several masters at Oxford University, known as the Calculatores, introduced mathematical tools, notably the theory of proportions, mixed with logic, to study motion in the framework of Aristotelian physics (the most famous are Thomas Bradwardine, Richard Kilvington, William Heytesbury, John Dumbleton, and Richard Swineshead, and after them, many scholars in all European universities are taking up their ideas). 1 Aristotle, in Physics VII (250a 4-6), explained that if a motor moves a mobile with a certain rapidity, a motor of double power moves a mobile with double resistance with the same rapidity. It is tempting to deduce that velocity follows immediately from the ratio of the power to the resistance, but Aristotle himself shows what paradoxes such an identification leads to: for example, any motor can move a mobile of any resistance, however great it may be, or a man could move a boat that twenty men move, even if it is twenty times slower, while experience proves that he would not be able to move it (Aristotle, Physics, VII, v, 250 a 10-18).

Referring to Aristotle and seeking to determine velocity from the ratio between the power of the motor and the resistance of the mobile, Thomas Bradwardine sets out, in his famous treatise De proportionibus velocitatum in motibus, an alternative rule, which was generally accepted until the 17th century: 2

The ratio between the rapidities of motions derives from the ratio between motive powers and resistive powers, and conversely, or

The ratios between motive powers and resistive powers and the rapidities of motions are proportional in the same order, and conversely. According to this rule, the rapidities are proportional to the ratios between powers and resistances. I will not discuss here the mathematical problems posed by this formulation, which will be resolved by Nicole Oresme. 4 I notice that in the introduction to his treatise, Bradwardine says that he felt it necessary to present in a first chapter the little-known theory of proportions, which he uses for the study of motion, pointing out, following Boethius, that "whoever omits mathematical studies has destroyed the whole of philosophic knowledge." 5 In this same treatise, Bradwardine explains that the rapidity of any local motion is also determined by the greatest distance traveled in a given time, or that the rapidity of a locally moved body is determined by the rapidity of the fastest moving point or the fastest moving points in that body. 6 Unlike Bradwardine, who only studies local motion, William Heytesbury discusses, in the Regule solvendi sophismata, the three kinds of changes established by Aristotle: local motion, alteration, and increase or diminution. 7 Heytesbury takes up Bradwardine's rule of motion, 8 but he also explains, for each of the three kinds of changes, how to determine the rapidity from the space covered for local motion, from the latitude of the form acquired for alteration (e.g. the heat acquired during heating), from the quantity acquired for increase. 9 He then introduces the expressions "quo ad "Proportio velocitatum in motibus sequitur proportionem potentiarum moventium ad potentias resistivas, et etiam econtrario" or "Proportiones potentiarum moventium ad potentias resistivas, et velocitates in motibus, eodem ordine proportionales existunt, et similiter econtrario"; the English translation is mine.

4 See Sabine Rommevaux-Tani, Les nouvelles théories des rapports mathématiques du XIV e au XVI e siècle (Turnhout: Brepols, 2014), 15-34. 5 Crosby, Thomas of Bradwardine. His Tractatus de Proportionibus, 64: "quisquis scientias mathematicales praetermiserit, constat eum omnem philosophiae periddisse doctrinam"; English translation, 65. 6 Crosby, Thomas of Bradwardine. His Tractatus de Proportionibus, 130: "Cuiuslibet motus localis, velocitas secundum maximum spatium lineale ab aliquo puncto sui moti descriptum accipitur"; "Ideo videtur magis rationabiliter dici quod velocitas motus localis attenditur penes velocitatem puncti velocissime moti in corpore moto localiter".

7 William Heytesbury, Regule solvendi sophismata (Venice: Bonetus Locatelllus, 1494), see the sixth chapter "De tribus predicamentis", fols. 37ra-52rb. 8 Heytesbury, Regule solvendi sophismata, fol. 44vb: "(...) secundum proportionem potentie motoris ad potentiam resistivam generaliter attenditur velocitas in quocumque motu (...)". 9 Heytesbury, Regule solvendi sophismata, fol. 38vb: "In motu autem locali difformi in quocumque instanti attenditur velocitas penes lineam quam describeret punctus velocissime motus si per tempus moveretur uniformiter illo gradu velocitatis: quo movetur in eodem instanti: quocumque instanti dato"; fol. 45ra: "Ideo sequitur tertia positio quam inter alias in ista materia reputo veriorem scilicet quod universaliter omnis velocitas talis motus attenditur penes proportionem quantitatis de novo uniformiter acquirende in tanto tempore vel in tanto ad quantitatem prius habitam (...)"; fol. 51ra:"Ideo sequitur tertia positio et ultima quam magis probabiliter meo iudicio poterit sustineri: videlicet quod omnis velocitas in alteratione attenditur penes maximam latitudinem talis

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Revista Española de Filosofía Medieval, 29/1 (2022), ISSN: 1133-0902, pp. 97-111 https://doi.org/10.21071/refime.v29i1.15136

effectum", to qualify the determination of rapidity from the space covered, the heat or the quantity acquired, and "quo ad causam" to qualify the determination of rapidity from the ratio of power to resistance in the case of local motion. 10 It is indeed necessary that the power of the motor or the agent be greater than the resistance of the mobile or the patient for there to be motion, and the action of the motor or agent on the mobile or patient causes the change. Thus, Nicole Oresme, in his treatise De proportionibus proportionum, speaks of the ratio from which rapidity arises ("proportio a qua venit velocitas" 11 ) and of the rapidity that comes from such a ratio ("velocitas que a tali proportione oriatur" 12 ). Moreover, the effect produced is either the space covered, for local motion, or the form acquired, for alteration, or the quantity acquired, for increase.

In most of the treatises of the Calculatores tradition to which Bradwardine and Heytesbury belong, 13 where the question of determining rapidity for the different kinds of change is raised, the two ways of understanding rapidity, from the effect or from the cause, are treated separately. 14 Several remarks are called for at this point:

-It is anachronistic to identify these two ways of studying rapidity with the division between dynamics and kinematics in modern physics; I agree with Daniel A. Di Liscia, who has explained this point perfectly. 15 -To express the dependency between rapidity and cause or the ratio between power and resistance, Bradwardine uses the verb "sequitur", which Crosby translates as "varies in accordance with"; 16 and to express the dependency between rapidity and forme sue qualitatis que uniformiter acquiretur alicui subiecto maiori seu minori in tanto tempore vel in tanto correspondenter."

10 He ends the chapter on local motion with these words (Heytesbury, Regule solvendi sophismata, fol. 44rb): "Ideo viso iam generaliter penes quid tamquam quo ad effectum attendatur velocitas in motu locali: quia secundum proportionem potentie motoris ad potentiam resistitivam generaliter attenditur velocitas in quocumque motu tanquam quo ad eius causam sequitur secunda pars huius capituli in qua perscrutabit penes quid quo ad effectum attendatur velocitas in augmentatione et diminutione, communiter dictis que rarefactio et condensatio appellantur distance traveled, Bradwardine uses the expression "attenditur penes" translated by Crosby as "is to be determined by". Grant, in his edition of Nicole Oresme's De configurationibus qualitatum et motuum, translates "attenditur penes" as "is a function of" in most cases 17 or "is measured by" 18 or sometimes "is attended", specifying in the latter case "i.e. is measured by". 19 I note that in his treatise De configurationibus qualitatum et motuum, Nicole Oresme uses the verb "mensuratur" only in the case of determining the rapidity from the space traveled in the local motion. 20 Further study of these expressions and their uses in the Calculatores treatises would be necessary to determine their exact meanings; in particular, it would be necessary to see whether the term "mensura" is usually used in this context.

-I note that Bradwardine's rule and the rules which express rapidity in terms of space covered, form or quantity acquired, are rarely used in these texts in order to calculate rapidity (even though Bradwardine's rule is a mathematical statement). These rules are generally used, in the comparison of two motions, to explain that one is faster than the other, or to decide whether a motion is uniform or uniformly difform, for example. Thus, if the ratio of power to resistance for one change is greater than the ratio of the same type for the other change, the former is said to be faster than the latter. And if, during a change, the ratio of power to resistance remains unchanged, the change is said to be uniform; it is uniformly difform if this ratio increases or decreases continuously. Moreover, if more space is covered by the first mobile than by the second in the same time, the first will be said to be faster than the second. All these considerations are qualitative.

-To my knowledge, the link is never made by Bradwardine or Heytesbury between these two approaches to determining rapidity, quo ad effectum and quo ad causam. And this seems to be the case for the other members of the Calculatores tradition, 21 with the exception of the author of De sex inconvenientibus, as we shall now see.

The anonymous treatise De sex inconvenientibus

In a treatise entitled De sex inconvenientibus, written between 1335 and 1339, the anonymous author, who is perfectly familiar with the works of Bradwardine, Heytesbury, whom he quotes, and undoubtedly Kilvington, even though he does not quote him, raises the question of determining the rapidity for the three usual kinds of changes, i.e., alteration, increase and local motion; but the author adds to these the generation of substantial forms. 22 The treatise thus consists of four main questions, to which are attached secondary questions or articles (three per main question). 23 The treatise is thus composed of Questiones. In this type of text, the quod non or quod sic arguments that open the Questio are not very developed, the most important part being the determinatio, in which the author develops his answer to the question asked. 24 This is not the case in De sex inconvenientibus. The determinationes are reduced; for the most part, the author simply says which opinion he agrees with, with a few exceptions (for example, for the article concerning the theorem of the middle degree, he gives several demonstrations of this theorem 25 ). On the other hand, he presents three opinions for each main question and produces for each of them a set of six inconvenientes or difficulties that would result from the acceptance of these opinions, even for the one (the third) that he finally accepts. And for each article, he produces six difficulties that would result from a positive or negative answer (as the case may be) to the question posed. Hence the title of the treatise is De sex inconvenientibus (About six difficulties). It is several of these inconvenientes that we will examine here, those in which the author confronts the points of view on the case treated.

Contradiction between the two ways of determining rapidity

The first main question of the treatise asks whether the rapidity of generation of substantial forms can be determined. The cases considered in this question are of two kinds: a hot body a warms another less hot or a cold body b, introducing its form of heat until b becomes as hot as a; a body becomes hotter by the destruction of its internal coldness by its internal heat. These are thus processes of alteration, but described as the generation of heat from other heat or by the destruction of cold.

The author records three positions regarding the determination of the rapidity of these processes. He rejects the first two and accepts the third, but presents six difficulties raised by each of these three positions. According to the first position, the rapidity of such a kind of generation would depend on the form or quality introduced by what generates it. 26 And the explanation that follows the statement of this position is the following:

(...) in heating, for example, where a more intense form of fire is introduced, the motion by which that form was introduced is faster than another motion in which a weaker form is introduced. 27 Then the author considers the case of an extreme heat a (a fire) approximated by a less hot body b. 28 He assumes that b is uniformly difformly hot, i.e. he considers a linear body whose heat is continuously distributed from one end to the other, from degree 0, or non-degree of heat, to an extreme degree, and he further assumes that the hottest end is of extreme heat exclusively (it has a degree of heat lower than the extreme degree by an indivisible).

On the one hand, the author notes that, since the heat of b increases due to the action of a on b, the resistance of b to heating by a diminishes continuously. Thus, the ratio between the power of a and the resistance of b increases continuously during the whole heating process. And since, according to Bradwardine´s rule, the rapidity on the ratio of the power of a to the resistance of b, then the rapidity increases; the motion is uniformly accelerated or uniformly difform.

But on the other hand, it may be noticed that a introduces during the whole period of the motion an extreme constant heat. Since the rapidity is, according to this first opinion, determined by the degree of heat introduced, the rapidity is constant; the motion is uniform.

Then the following difficulty arises, which leads the author to reject this first: "a, which generates, will continuously generate with a greater and greater ratio, and yet, continuously, it will generate uniformly." 29 The motion is uniform if the rapidity is determined by the generated form, i.e. by the effect produced, but the motion is uniformly difform if we consider it from the point of view of the cause, that is, the ratio between power and resistance (even if the author of De sex inconvenientibus does not use the terms 'effect' and 'cause' in this treatise).

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Revista Española de Filosofía Medieval, 29/1 ( 2022 The paradoxes caused by the two ways of determining rapidity are also at the heart of the difficulties raised by the second position presented by the author regarding the determination of the rapidity of generation: such rapidity would depend on the latitude of the form to be acquired and on the quantity in which the latitude of this form to be acquired is extended. 30 The term "latitude" refers to the variation in intensity of forms, qualities, quantities, rapidities, etc., during a change in a subject. 31 We also find in De sex inconvenientibus the term "degree", referring to the latitude. Thus, when a body is heated, its heat increases from degree a to degree b, passing through all intermediate all these degrees constitute the latitude ab.

Moreover, the statement should be understood as follows: the rapidity is determined by the quantity of the altered subject when equal forms are introduced, and it is determined by the latitude of the form introduced in equal subjects. The author never considers the case where latitudes of different forms are introduced in subjects of different quantities.

We are not going to examine all the difficulties arising from this second opinion, in which, as has been stated, the author compares the two modes of determining rapidity. Let us look at the fifth. 32 The author presents the case of two heats, a and b, acting respectively on c and d, introducing equal latitudes of heat. It is assumed that a is one hundred times hotter than b and that c and d are of equal quantities.

The two heats act with equal rapidities, since the rapidity is, according to this second opinion, determined from the latitudes introduced in equal subjects, and it was assumed that a and b introduced equal forms in their patients. Furthermore, it was assumed that a has a power to act one hundred times greater than b, and the patients c and d have equal resistances, so that a alters c one hundred times faster than b alters d.

In conclusion: "a and b, things that generate, alter their patients equally, and yet a does so a hundred times faster." 33 Again, equal effects imply equal rapidities, but unequal ratios lead to the conclusion that the changes are unequal. 30 Rommevaux-Tani, "De sex inconvenientibus", 142: "talis velocitas attenderetur penes latitudinem forme acquirende et penes quantitatem per quam extenditur ista latitudo istius forme acquirende, sicut ponit seconda positio." 31 It should be noted in this case that the numerical value "one hundred" is of no importance in the reasoning here. The only thing that matters is that a acts more strongly than b.

Let us now consider one of the arguments that the author opposes to the second opinion concerning the determination of the rapidity of alteration. According to this opinion, the rapidity would depend on the quantity or the extension of the subject in which the alteration takes place.

The author then considers the case of a fire a heating water b in such a way that a continuously introduces as much heat into any part of b during the process. The alteration is thus assumed to be uniform, according to this opinion. But the author notes that the resistance of b to the action of a decreases as b becomes hotter and hotter. So, the ratio increases between the power of a, which is constant, and the resistance of b, which decreases. Hence the difficulty that "something that alters will continually alter with an ever-increasing ratio and yet it will continually alter uniformly." 34 Here again, the author rejects this opinion because of this contradiction.

Distinction between "being moved or changed by faster motion or change" and "being moved or changed more quickly"

Let us return to the first opinion, considered in relation to the determination of the rapidity of generation: the rapidity would depend on the form or quality introduced. Let us consider the same case as above: the generation of heat produced by an extreme heat a in a uniformly difformly hot body b, the most intense end of which is in the extreme form of heat exclusively. Now compare it with the generation of heat produced by c, half as hot as a, on a uniformly difformly hot body d, similar to b but half as hot. 35 The heat introduced by is twice as low, therefore, according to this first opinion, as a generates; the time taken for the action of a in b is half the time taken for the action of c in d.

It may be observed, moreover, that the ratio of the power of a to the resistance of b is the same as the ratio of the power of c to the resistance of d. The author does not explicitly make this last remark, but it can be deduced from what he says: "a and c will generate their forms precisely as quickly ('eque cito'). Indeed, each of them immediately after that will introduce its form into the patient that resists it, and only by a motion of generation." 36 ________________________________________________________________ Revista Española de Filosofía Medieval, 29/1 ( 2022), ISSN: 1133-0902, pp. 97-111 https://doi.org/10.21071/refime.v29i1.15136

The difficulty is summarized as follows: "a and c are two things that generate similar forms in patients b and d, and a will generate its form in half the time of c and they begin to generate at the same time, yet c generates its form precisely as fast as a, all other things being equal."37 So, the motion is accelerated, if we consider the effect produced, but it is uniform if we consider the cause, i.e. the ratio of power to resistance.

Note that in this argument, the author uses two different terminologies: he speaks of a motion that is "faster" ("velocior") than another on the one hand, and on the other hand, he talks about mobiles that reach the final heat "as quickly" ("eque cito"); the first notion, "faster", refers to the determination of rapidity from the effect produced, and "as quickly" to the ratios between power and resistance, which are equal here.

Note, moreover, that for the reasoning to be valid, the author could have said that c is less hot than a, and b is less hot than d according to the same ratio, without specifying what ratio. And he would have come to the paradoxical conclusion that a and c are two things which generate similar forms in patients b and d, and a will generate its form in less time than c, and yet c generates its form precisely as quickly as a. That a will generate its form in half the time of c in the case under consideration is anecdotal.

Still, with regard to the generation, the author agrees with the third position, according to which the rapidity depends only on the latitude of the form to be acquired, without taking into consideration the extension of the subject of generation. 38 But here again, he proposes six difficulties that may arise if this position is accepted.

He thus considers two identical bodies, a and b, slightly hot. These two bodies are heated by the destruction of their intrinsic coldness by their intrinsic heat. And it is assumed that all the coldness is thus destroyed in b, so that the whole of b becomes extremely hot; in a, however, one half remains unchanged, while the other half becomes extremely hot by the destruction of the coldness in it. 39 The altered subjects are b half of Since b and a have been assumed to be identical, the altered subjects are in a double ratio. And the same extreme heat is acquired by both. It is deduced that the alteration by which b is changed is twice as fast as that by which a is changed. But the author notes also that the alterations in a and b begin and cease at the same time (the two halves of b begin and cease to heat up at the same time as the half of a).

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Revista Española de Filosofía Medieval, 29/1 (2022), ISSN: 1133-0902, pp. 97-111 https://doi.org/10.21071/refime.v29i1.15136

Thus we have the following difficulty (the fifth):

"in the intrinsic generation of an element, two uniform slightly hot things, of equal quantities and heats, are altered during precisely the same time, until each of them becomes extremely hot, in such a way that they begin to be altered as quickly and cease to be altered as quickly, and yet the entire alteration by which b will be continuously altered will be continuously twice as rapid as the alteration by which a will be continuously altered." 40 The author must respond to the argument because he accepts this opinion. 41 He rejects, of course, the argument that b moves twice as fast as a because the subject in which the alteration in b takes place is double that in which the alteration in a takes place. Indeed, according to this position, only the latitude of the acquired form is to be taken into consideration and not the extension of the subject where the alteration takes place.

But he also takes advantage of this case to distinguish between "be altered by faster motion" ("velociori motu alteratur") and being "altered more quickly" ("velocius alteratur): a is altered by a motion twice as fast, when the ratio of power to resistance is twice as great, and a is said to be altered twice as fast, if the latitude acquired by a is double. And the author accepts that an alteration can be said to be twice as fast, while the thing altered is not twice as fast. He therefore rejects the following implication: "b is altered by a motion twice as fast as the motion by which a is altered, so b is altered twice as fast as a." He thus acknowledges that the two ways of determining rapidity can lead to different conclusions, at least for this type of alteration.

Note again that the fact that b is altered precisely twice as fast as a is unnecessary; the paradox is that b is altered faster than a. 40 Rommevaux-Tani, "De sex inconvenientibus", 147: "QUINTO, quod in generatione intrinseca elementi, aliqua sunt duo calida remissa uniformia equalis quantitatis et eque calida, que alterabuntur per idem tempus precise, quousque utrumque istorum fuerit calidum in summo, ita quod eque cito incipiunt alterari et eque cito desinent alterari, et tamen tota alteratio qua b continue alterabitur erit continue in duplo velocior quam alteratio qua a continue alterabitur." This distinction between "having a faster motion" and "being moved more quickly" can be found in the third main question concerning increase in the arguments against the third opinion, which is the one accepted by the author. According to this opinion, the rapidity of the increase results from the ratio between the latitudes of rarity (i.e. the quantities of bodies at the beginning and at the end of the increase), and one increase is faster if the linear space described by the fastest moving point or points is longer in the same time. 42 In the case of increase, in the absence of an identified agent, rapidity cannot result from the ratio of the power of the agent to the resistance of the patient. The difficulties raised relate to the apparent contradiction that may arise between the two ways of determining rapidity stated by this position.

Difficulties may arise in particular from comparing the increase of a finite body a and an infinitesimally small body b, for example, an infinitesimally small part of a (one can imagine that and b are linear quantities). Since the body b is infinitesimally small, it begins to move infinitely slowly, because the distance covered by its fastest moving point is infinitesimally small. This statement is justified as follows: if one body begins to rarefy along a certain distance with a certain rapidity, another body would begin to rarefy along half that distance with half the rapidity, and another along a third of that distance with three times the rapidity, etc., because the ratio between the rapidities depends on the ratio between the distances travelled by the fastest moving point. So, since b has been assumed to be infinitesimally small, it starts to become infinitely rarefied. If, on the other hand, it is assumed that the finite body a begins to rarefy with the same degree of rapidity, i.e., with the same ratio between the latitude of final rarity and the latitude of initial rarity, we have the first difficulty: "a and b begin rarefied with the same ratio, and yet b begins rarefied infinitely more slowly than a." 43 Moreover, it follows from the same case that, whatever the degree of rapidity with which this same continuous a becomes rarefied (i.e. whatever the ratio between the latitudes of rarity), it begins to become rarefied infinitely slowly. Indeed, it is supposed that a is continuously rarefied, therefore part by part. Now, given a part of a, it is always possible to find one that is infinitely smaller than it. And according to the previous case, this infinitely small part will increase infinitely slowly. So, a will start to increase infinitely slowly. Thus, we have the second difficulty: "whatever ratio a begins to rarefy, it begins to rarefy infinitely slowly." 44 42 Rommevaux-Tani, "De sex inconvenientibus", 267: "velocitas in tali motu augmentationis attendetur penes proportionem latitudinum raritatis, et ipsum velocius penes proportionem quantitatum linealium a puncto velocissime moto vel a punctis velocissime motis in tanto vel in tanto tempore descriptarum."

43 Rommevaux-Tani, "De sex inconvenientibus", 267: "a et b ab eadem proportione incipiunt rarefieri, et tamen in infinitum tardius incipit b rarefieri quam a."; proof on pages 267-268. 44 Rommevaux-Tani, "De sex inconvenientibus", 267: "quecumque fuerit proportio a qua a incipit rarefieri, in infinitum tarde ipsum incipit rarefieri."; proof on page 268.

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To these two difficulties the author responds by admitting the conclusions, and he adds: "as far as I can see, they have been clearly demonstrated and are not in contradiction with the rules of the ratios." 45 He then explains that the rapidity or slowness of an increase results from the ratio between the latitudes of rarity, and whether one increase is faster or slower than another depends on the ratio between the distances travelled by the fastest moving point or points in the bodies, in the same time.

He thus distinguishes two ways of qualifying the increase, which can lead to different conclusions, without this being a difficulty. And he adds that, in so doing, augmentation is distinguished from local motion: And although, perhaps, this is against the rules of ratios for local motion, it is not against these demonstrative rules for the rarefaction and increase, since these motions differ specifically from local motion.46

Conclusion

The author of the De sex inconvenientibus is well aware that there are different ways of determining the rapidity of a motion or change. In particular, rapidity can be determined by the effect produced, and any motion or change is more or less rapid according to the ratio between the power of the agent or motor and the resistance of the mobile or patient. The author makes evident this double way of considering rapidity by using different expressions to qualify them. He distinguishes between "having a faster motion or change" and "being moved or changing faster." The motion or change is said to be faster when the ratio of power to resistance is greater, and a is said to be moved or change faster than b, if the effect of the motion of change for a is more intense than for b. In the first case, the object considered is the motion or change which results from the ratio between power and resistance. In the second case, the object is the subject that is moved or changed, and the effect of the motion or the change on this subject tells us how quickly the subject is affected. The confrontation of these two points of view leads the author to reject certain opinions regarding generation, alteration, and increase.

Furthermore, we have noticed on several occasions that the numerical values present in the different cases are of no importance. The paradoxes raised arise from the fact that the rapidities are unequal, whereas the data of the problem should have led to

22

  See J. Papiernik, "How to measure different movements? The 14th-century treatise De sex inconvenientibus", Przegląd Tomistyczny 25 (2019): 445-460. 23 See Sabine Rommevaux-Tani, "De sex inconvenientibus". Un traité anonyme de philosophie naturelle au XIV e siècle (Paris: Vrin, 2022); this book contains the critical edition of the De sex inconvenientibus and a doctrinal analysis of the treatise. 24 For the genre of the Questio see, for example: Olga Weijers, La 'disputatio' dans les facultés des arts au moyen âge (Turnhout: Brepols, 2002) and more recently Olga Weijers, In Search of the Truth. A History or Disputation Techniques from Antiquity to Early Modern Times (Turnhout: Brepols, 2013). 25 See Sabine Rommevaux-Tani, "The influence of the Oxford Calculatores on the Understanding or Local Motion: The Example of the Tractatus de sex inconvenientibus", in Quantifying Aristotle. The Impact, Spread and Decline of the Calculatores Tradition, edited by D. A. Di Liscia and E. Sylla (Leiden: Brill, 2022), 153-185.________________________________________________________________Revista Española de Filosofía Medieval, 29/1 (2022),ISSN: 1133-0902, pp. 97-111 https://doi.org/10.21071/refime.v29i1.15136

  41 Rommevaux-Tani, "De sex inconvenientibus", 201: "AD QUINTUM ET SEXTUM, que in modico discrepant, dicitur concedendo illas conclusiones contra, quarum primam arguitur sic: « a et b iam sunt per omnia similia et utrumque alterabitur uniformiter, quousque ipsum fuerit summum; et b continue alterabitur in duplo velocius; igitur b erit citius summum quam a », conceditur consequentia et negatur antecedens pro ista parte: « b continue alterabitur in duplo velocius a ». Sed contra: « duplo velociori motu alteratur b quam a », conceditur. « Igitur in duplo velocius alteratur b quam a », non sequitur. Et si adhuc arguitur: « a maiori proportione alteratur b quam a, igitur velocius alteratur b quam a », adhuc non sequitur. Sed illud sequitur quod velociori motu alteratur b quam a, quoniam motus sequitur proportionem, ipsum autem velocius, tardius vel equaliter sequitur magnitudinem spatii in eodem tempore vel equali descripti, sicut clarius patebit in questione de augmentatione." ________________________________________________________________ Revista Española de Filosofía Medieval, 29/1 (2022), ISSN: 1133-0902, pp. 97-111 https://doi.org/10.21071/refime.v29i1.15136

  Method and Order in Renaissance Philosophy of Nature. The Aristotle Commentary Tradition, edited by D. A. Di Liscia, E. Kessler and Ch. Methuen (Aldershot: Ashgate, 1997), 143-176.

	."
	11 Nicole Oresme, De proportionibus proportionibus and Ad pauca respicientes, edited with
	Introductions, English Translations and Critical Notes by E. Grant (Madison: University of Wisconsin
	Press, 1966), 288, l. 357.
	12 Nicole Oresme, De proportionibus proportionibus and Ad pauca respicientes, 290, l. 374.
	13 There are many works on this tradition. See for example: Edith Dudley Sylla, "The Oxford
	calculators", in The Cambridge History of Later Medieval Philosophy. Vol. 1: From the Rediscovery of Aristotle
	to the Disintegration of Scholasticism 1100-1600, edited by N. Kretzmann, A. Kenny, J. Pinborg and E.
	Stump (Cambridge: Cambridge University Press, 1982), 540-563.
	14 See Daniel A. Di Liscia, "Velocidad quo ad effectus y velocidad quo ad causas: la tradición de los
	calculadores y la metodología aristotélica", in

15

Di Liscia, "Velocidad quo ad effectus y velocidad quo ad causas", 173-176.

16

Crosby, Thomas of Bradwardine. His Tractatus de Proportionibus, 112 and 113.

________________________________________________________________

Revista Española de Filosofía Medieval, 29/1 (2022),

ISSN: 1133-0902, pp. 97-111

https://doi.org/10.21071/refime.v29i1.15136

  Nicole Oresme and the Medieval Geometry of Qualities and Motions. A Treatise an the Uniformity and Difformity of Intensities known as Tractatus de configurationibus qualitatum et motuum (Madison, Milwaukee and London: University of Wisconsin Press, 1968), 169, 221, 223, 225, 245.18 Clagett, Nicole Oresme and the Medieval Geometry of Qualities and Motions, 215, 277.19 Clagett, Nicole Oresme and the Medieval Geometry of Qualities and Motions, 279.20 Clagett, Nicole Oresme and the Medieval Geometry of Qualities and Motions, 168, l. 9-11: "Item sicut velocitas in motu locali secundum longitudinem spatii mensuratur, ita in alteratione velocitas attenditur penes intensionem."

17

Marshall

Clagett, 21

See Daniel A. Di Liscia, "Velocidad quo ad effectus y velocidad quo ad causas…". ________________________________________________________________ Revista Española de Filosofía Medieval, 29/1 (2022), ISSN: 1133-0902, pp. 97-111 https://doi.org/10.21071/refime.v29i1.15136

  For this notion see: Edith Sylla, "Medieval concepts of the latitude of forms. The Oxford calculators", Archives doctrinales et littéraire du Moyen Age 40 (1973) : 223-283, in particular 251-257.

32

Rommevaux-Tani, "De sex inconvenientibus", 145. 33 Ibid., 143: "a et b generantia equaliter alterant sua passa, a tamen in centuplo velocius." ________________________________________________________________ Revista Española de Filosofía Medieval, 29/1 (2022), ISSN: 1133-0902, pp. 97-111 https://doi.org/10.21071/refime.v29i1.15136

Rommevaux-Tani, "De sex inconvenientibus", 136: "secundum sectam positionis prime sequitur quod talis velocitas attenditur penes formam inductam vel inducendam a generante."

Rommevaux-Tani, "De sex inconvenientibus", 135: "quando generans inducit vel incipit inducere formam suam, verbi gratia in calefactione, ubicumque inducitur forma ignis intensior, motus iste, quo inducitur, est velocior aliquo alio motu, quo forma remissior inducitur."

See the second difficulty raised by this position: Rommevaux-Tani, "De sex inconvenientibus", 138.

Rommevaux-Tani, "De sex inconvenientibus", 136: "a generans continue generabit a proportione maiori et maiori, et tamen ipsum continue uniformiter generabit."

Ibid, 208: " aliquod alterans alterabit continue a maiori proportione et maiori, et tamen continue uniformiter alterabit"; proof on page 209.

See the fifth inconveniens: Rommevaux-Tani, "De sex inconvenientibus", 140-141.

Rommevaux-Tani, "De sex inconvenientibus", 141: "eque cito precise generabunt a et c formas suas, quia utrumque istorum immediate post hoc inducet formam suam in passum sibi resistens, et solum motu generationis."

Rommevaux-Tani, "De sex inconvenientibus", 137: "a et c sunt duo generantia que generabunt ex b et d passis formas illis consimiles, et a in duplo minori tempore generabit formam suam quam c, et simul incipiunt generare, et tamen eque cito precise generabit c formam suam sicut a formam suam, ceteris paribus".

Rommevaux-Tani, "De sex inconvenientibus", 147: "si in generatione formarum sit certa ponenda velocitas, igitur talis velocitas attenderetur solum penes latitudinem forme acquirende, sicut ponit tertia positio et tenet tota scola oxoniensis."

Rommevaux-Tani, "De sex inconvenientibus", 151.

Rommevaux-Tani, "De sex inconvenientibus", 295: "ET TUNC AD ARGUMENTA IN OPPOSITUM dico quod conclusiones adducte pro maiori parte sunt vere, videlicet prima, secunda, quarta et quinta, que, sicut mihi apparet, sunt clarius demonstrate, nec ille sunt contra regulas proportionum, quoniam, licet motus et ipsa velocitas et tarditas proportionem sequantur, ipsum tamen velocius et tardius sequitur proportionem spaciarum linealium in eodem tempore descriptarum."

Rommevaux-Tani, "De sex inconvenientibus", 295: "Et licet forte hoc foret contra regulas proportionum de motu locali, non tamen foret hoc contra regulas demonstrativas in motu rarefactionis et augmentationis, cum illi motus specifice differant a motu locali."


S Rozen

S Larroque

N Dudovich

Y Mairesse

B Pons

Enhanced chiral-sensitivity of Coulomb-focused electrons in strong field ionization

Strong-field light-matter interactions initiate a wide range of phenomena in which the quantum paths of electronic wavepackets can be manipulated by tailoring the laser field. Among the electrons released by a strong laser pulse from atomic and molecular targets, some are subsequently driven back to the vicinity of the ionic core by the oscillating laser field. The trajectories of these returning electrons are bent towards the core by the ionic potential, an effect known as Coulomb focusing. This process, studied over the past two decades, has been associated with the long range influence of the Coulomb potential. Here we explore the structural properties of the Coulomb focusing phenomenon. Specifically, we numerically study the sensitivity of the returning electron dynamics to the anisotropy of the ionic potential. We employ orthogonally polarized two-color strong fields and chiral molecules, whose asymmetric features lead to unambiguous fingerprints of the potential on the freed electrons. The Coulomb-focused electrons show an enhanced sensitivity to chirality, related to an asymmetric attoclock-like angular streaking stemming from field-assisted scattering of the electrons onto the chiral ionic potential. Anisotropic features of the ionic potential thus monitor the motion of Coulomb-focused electrons throughout their returning paths, shedding light on the structural properties of the interaction.

INTRODUCTION

The interaction of strong laser fields with atoms or molecules triggers coherent electronic dynamics on the attosecond timescale, which can be used to probe the structure and dynamics of the irradiated target with unique spatial and temporal resolutions. Electron wavepackets are released by tunnel-ionization around the maxima of the laser field oscillation, and are subsequently accelerated by the laser field. The electrons born before the field maxima directly escape the ionic potential to form above-threshold ionization (ATI) peaks [1]. These electrons are identified as "direct" electrons. By contrast, the electrons born after the field maxima are driven back to the vicinity of their parent ion, where they can be subject to: (i) elastic forward scattering onto the ionic core, yielding the "indirect" contribution to the ATI distribution; (ii) elastic backward scattering, leading to high order ATI and laser-induced electron diffraction [2][3][4]; (ii) inelastic scattering, producing nonsequential double ionization [5]; (iii) radiative recombination with the ion to produce bursts of attosecond extreme ultraviolet radiation, forming high order harmonics [6][7][8].

Stating the important role of the ionic potential in the scattering process seems like a truism. Nevertheless some of its aspects are often neglected in the description and interpretation of the strong-field spectroscopic techniques listed above. Indeed, the strong-laser field generally dominates the interaction, such that the Strong-Field Approximation (SFA) is an excellent approach to describe many properties of the electron dynamics [9,10]. One important correction to this description is Coulomb focusing [11]. The Coulomb force exerted by the ionic potential bends the returning electron trajectories towards the ionic core, focusing the wavepacket. This has important consequences on high-harmonic generation (HHG, [12]) and non-sequential double ionization [11]. However the influence of Coulomb focusing on the indirect electrons in ATI is more elusive. It has been shown that the focusing reduces the transverse momentum distribution of the ionized electrons [13][14][15][16]. Recently, a spectacular signature of Coulomb focusing has been observed in strong-field ionization of argon atoms by orthogonally polarized two-color (OTC) laser fields [17]. In this configuration, a weak second-harmonic field is added to the fundamental one to manipulate the electron trajectories. By measuring the photoelectron angular distribution in the laser polarization plane, Richter et al. found out that some of the electrons remained rather insensitive to the second-harmonic field, seemingly ignoring the lateral momentum streaking resulting from the trajectory manipulation. These "unstreakable" electrons were identified as being indirect electrons that have experienced Coulomb focusing. The OTC field configuration is thus particularly adequate to reveal the underlying spatial and temporal properties of the Coulomb focusing.

In the framework of HHG, Coulomb focusing was found to influence mainly the long trajectories [12]. Therefore it was interpreted in terms of a long-range effect, dictated by the asymptotic, Coulombic, part of the ionic potential. The influence of the detailed target structure and associated potential geometry on Coulomb focusing has thus never been considered so far.

In this paper, we numerically investigate the sensitivity of Coulomb-focused electrons to the structure of the ionized target. To emphasize the influence of the ionic potential, we use a target with a specific geometrical property: chiral molecules, which are not superimposable to their mirror image. When chiral molecules are photoionized by circularly polarized light, the photoelectron momentum distribution is forward/backward asymmetric with respect to the light propagation axis. This effect, called PhotoElectron Circular Dichroism (PECD), produces large chirosensitive signals, resulting from pure electric-dipole interaction. PECD was initially predicted [18,19] and measured [20][21][22] in single-photon ionization by extreme ultraviolet radiation. It was interpreted as resulting from the influence of the chiral potential on the scattering of the outgoing electrons in the rotating electromagnetic field. Classically, this asymmetry can be seen as analogue to the conversion of the rotation motion of a nut into directional translation on a bolt [START_REF] Powis | Advances in Chemical Physics[END_REF]. Quantum mechanically, the PECD is dictated by the phase shifts of the continuum scattering wavefunctions. For more information we refer the reader to the tutorial paper by Powis [START_REF] Powis | Advances in Chemical Physics[END_REF]. PECD was later extended to the multiphoton [START_REF] Lux | Angewandte Chemie International Edition[END_REF][START_REF] Lehmann | [END_REF] and strong-field [26] regimes. Strong-field PECD combines the advantages of strongfield spectroscopy -orbital selectivity, attosecond temporal resolution and Angström spatial resolution -with the unambiguous structural sensitivity of chiroptical measurements. It was used to measure the attosecond delays between ionization of a molecule and its mirror image [27], to relate PECD to different fragmentation pathways associated with different molecular alignments [28], to investigate the chirosensitivity of angular streaking [29,30], and to probe the influence of the instantaneous chirality of light to chiral light-matter interaction [31].

Here, strong-field PECD is used to reveal the chiral sensitivity of Coulomb-focused electrons. We employ a toy-model chiral molecule which was shown to produce chiroptical signals similar to genuine chiral molecules [31]. The chiral interaction is driven by OTC fields which yield an unambiguous fingerprint of indirect electrons, Coulomb-focused along the fundamental component of the field [17]. These electrons exhibit enhanced sensitivity to chirality. We show that molecular chirality is encoded in the Coulomb-focused signal in terms of a forward/backward asymmetric angular streaking of the electrons, due to laser-assisted scattering of the electrons onto the chiral ionic potential.

IDENTIFYING COULOMB FOCUSING IN PHOTOELECTRON ANGULAR DIS-TRIBUTIONS

PECD measurements are conventionally performed using circularly polarized radiation.

In the strong-field regime, the rotating laser field lowers the potential barrier of the molecule, continuously releasing electrons by tunnel ionization. These electrons are accelerated by the field and directly escape the molecular potential, without coming back to the vicinity of the ion. The forward/backward asymmetry in the electron angular distribution arises from both the electron scattering off the ionic potential [START_REF] Powis | Advances in Chemical Physics[END_REF] and the sensitivity of tunneling dynamics to the chirality of the barrier [30]. As soon as the polarization state is not circular, the magnitude of the electric field oscillates in time and tunneling releases multiple families of electron trajectories in the continuum. Manipulating the vectorial properties of the laser field oscillations provides a unique way to finely control the chiroptical interaction [31], by shaping the electron trajectories [32] as well as the time-dependent optical chirality [33] of the ionizing radiation.

In this work, we use an OTC field E defined as:

E(t) = E 0 cos (ωt) ŷ + E 0 √ r cos 2ωt + ϕ ω/2ω x (1)
where E 0 is the amplitude of the fundamental field with frequency ω, r = I 2ω /I ω < 1 is the ratio between the second harmonic and fundamental field intensities, and ϕ ω/2ω is the relative phase between the two frequency components. Varying ϕ ω/2ω allows to tune the OTC field shape which switches from an "8", when ϕ ω/2ω = π/2 ± π, to a "C" when ϕ ω/2ω = 0 ± π.

The strong-field ionization dynamics in an OTC fields are very close to those in linearly polarized fields. The electrons emitted before the maxima of the laser field oscillation directly escape the molecular potential -these are the direct electrons. By contrast, the electrons ionized after the maxima of the field escape in one direction before being driven back to the vicinity of the ion when the field reverses -they form the indirect electron family.

The two electron families naturally show up in the SFA where the photoionization process is described in terms of quantum paths exclusively monitored by the strong driving field [9,34].

The phase of each quantum path is given by the action integral S accumulated along the electron trajectory. For a given final electron momentum p, minimizing the action integral provides the dominant quantum paths j, with j = 1 and 2 for direct and indirect electrons, respectively. The photoelectron momentum distribution is thus

P(p) = j e i S(p,t j 0 ) 2 ,
where t j 0 is the ionization time of electron j. The influence of the ionic potential on the escaping electrons is neglected in the SFA framework. As a result, the final momentum p of the electrons is completely determined by the value of the vector potential A at the time of ionization t 0 : p = -A(t 0 ). Direct and indirect electron paths thus asymptotically lead to identical momentum distributions when ϕ ω/2ω = π/2. This is illustrated in the inset of Fig. 1(a) for ionization of a target with ionization potential I P = 7.2 eV by an OTC field with I ω = 5×10 13 W.cm -2 and r = 0.1. The shape of direct and indirect momentum distributions indeed conforms to the "C"-shape of -A, and their overlap results in interference patterns along the multiple concentric shells associated to the ATI peaks (see Fig. 1(a)). When the relative phase ϕ ω/2ω is switched to 0, the direct and indirect electrons are streaked towards opposite directions along the x-axis since the vector potential exhibits an "8"-shape in that case. However, the direct and indirect asymptotic wavepackets still significantly overlap, and produce clear interference patterns along the ATI rings, as shown in Fig. 1(d).

The SFA calculations provide an intuitive picture of the interaction, but they neglect the influence of the ionic potential on the ionization dynamics. They are thus unable not only to describe Coulomb focusing but also to produce any PECD signal, as demonstrated in [35]. To investigate the impact of the potential, we solved the Time-Dependent Schrödinger Equation (TDSE) for the toy-model chiral system introduced in [30,31]. The calculations strictly followed the numerical recipes described in [30,31]. Briefly, the toy-model system consists of a single-electron evolving in the (chiral) field of four fictitious nuclei. Its ionization potential is I P = 9 eV. We mimic the random orientation of the molecular sample by solving the TDSE for many molecular orientations defined in the laboratory frame. The TDSE is solved in the velocity gauge by expanding the total electron wavefunction onto an underlying because of its insensitivity to the second harmonic field [17]. By using classical calculations, they showed that this component resulted from the Coulomb focusing of indirect electrons.

In order to confirm the origin of the sharp component in the momentum distribution, we damped the long-range part of the potential of our target molecule and repeated the TDSE calculations. The screening was introduced by multiplying the molecular potential by an isotropic cut-off Yukawa term exp -(r-r 0 ) for radial distances r larger than r 0 = 3.5 a.u., where r 0 represents the spatial extent of our molecular system. Since the ionic potential remains unchanged in the r < r 0 inner range, the Yukawa term makes the ionization potential of the screened system slightly smaller than its unscreened counterpart (I P = 7.2 eV).

The screened potential vanishes for r > ∼ 8 a.u.. The screening procedure thus cancels the influence of the long-range potential on electron scattering, as recently established by Torlina et al. in atomic attoclock simulations [36]. The momentum distributions from the screened molecules are presented in Fig. 1(c) and Fig. 1(f) for ϕ ω/2ω = π/2 and 0, respectively. They are remarkably similar to the SFA distributions of Fig. 1(a) and Fig. 1(d), confirming that the localized vertical component observed in the unscreened calculations results from the influence of the long-range potential in the ionization dynamics.

The fact that Coulomb focusing disappears when the ionic potential is screened indicates that it is a long-range effect. This is consistent with the conclusions drawn from HHG driven by elliptical fields, where Coulomb focusing was found to influence mainly the long trajectories [12]. Here, long range explicitly means r > ∼ 8 a.u., which is the boundary beyond which the screened potential vanishes. Coulomb focusing is thus dictated by this outer part of the potential which tends towards its isotropic Coulombic limit as r → ∞. However, below this limit the unscreened potential is chiral, so that the focusing dynamics could present asymmetric features. Can we observe a chiral response of the Coulomb-focused electrons?

CHIRAL SENSITIVITY OF COULOMB-FOCUSED ELECTRONS

To reveal the influence of molecular chirality on photoionization, we rely on the rotation of the OTC field in time. This rotation reverses every half cycle of the fundamental field.

Therefore, the OTC field thus carries zero net chirality. However its instantaneous chirality C(t) can be defined as [33]:

C(t) = C 0 (E x (t)∂ t E y (t) -E y (t)∂ t E x (t)) (2)
where C 0 is a normalization factor defined such that C(t) = 1 for a right circularly polarized field. This quantity describes the rotation speed of the electric field. As shown theoretically in [31,37] and experimentally in [31], even if their instantaneous chirality reverses every half cycle, OTC fields can produce chiroptical signals in photoionization. This is due to the fact that the chiral response is accumulated while the departing electron is in the vicinity of the molecule, i.e. over a few hundreds of attoseconds during which the instantaneous chirality can keep a well defined sign. The variations of C(t) induce a temporal gating of the chiral response, to which we referred to as ESCARGOT (Enantiosensitive SubCycle Antisymmetric Response Gated by electric-field rOTation) [31].

The use of OTC fields in chiral photoionization induces several characteristic symmetry properties. First, the chiral response appears as a forward/backward asymmetry in the photoelectron angular distribution, as in conventional PECD. Secondly, as shown in Figure 2, the chirality of the field reverses every half cycle, for all two color phases. This leads to an up/down antisymmetry of the chiral response, observed both in calculations and experiments [31,37].

To investigate the chirosensitive part of the signal, we extract the forward/backward antisymmetric component of the photoelectron momentum distribution:

∆P f /b (p x , p y , p z ) = [P (p x , p y , p z ) -P (p x , p y , -p z )]/P max (3)
that we normalize to the maximum of the 3D distribution, P max = max(P (p)).

The contribution of Coulomb-focused indirect electrons to P (p) maximizes about the p z = 0 polarization plane [13][14][15]. We thus present in Figure 2 the evolution of the chiral signal as a function of ϕ ω/2ω . In the upper hemisphere (p y > 0), the asymmetry is positive for all phases on the left edge (p x < 0) of the momentum distribution and negative on the right edge. The situation is opposite in the lower hemisphere.

Similarly to what we observed for the photoelectron momentum distributions, the whole chiral response of Coulomb-focused electrons is thus almost insensitive to the second harmonic component of the OTC field. This is a puzzling result since one would expect that electrons driven by an almost linearly polarized field will show a low chiral response. It is particularly remarkable that switching the relative phase from ϕ ω/2ω = π/2 to ϕ ω/2ω = 3π/2 reverses the helicity of the field but keeps the chiral signal positive in the upper left quadrant and negative in the upper right one. In the following we show that this can be intuitively understood as an effect of the angular streaking by the rotating laser field. (m-p) Same schematic representation than (i-l) but for a two-color phase δφ chiral = 0.

ANGULAR STREAKING IN ORTHOGONAL TWO-COLOR LASER FIELDS

Angular streaking occurs when atoms or molecules are ionized by a rotating strong laser field, and has been extensively used to perform attoclock measurements [38,39]. The angular streaking effect in a counter-clockwise rotating laser field is schematically depicted in Fig. 3(a). The electrons tunnel out in a direction set by the laser field, and are angularly streaked by the rotation of the field. Within the strong field approximation, the final momentum of the electrons is dictated by the vector potential at the time of ionization. Deviations from this direction can be induced by delays in the tunneling process, or by scattering in the ionic potential at long range [39,40]. The latter effect generally dominates the interaction and induces an angular shift φ 0 of the electron momentum distribution in the rotation direction of the field, as schematized in Fig. 3(a) [36,41]. Reversing the helicity of the laser field switches the direction of the angular streaking, as shown in Fig 3(e).

The phenomenon of angular streaking is not restricted to circular or elliptical polarization.

Attoclock measurements can be performed with more complex laser fields, for instance combining counter-rotating or co-rotating fundamental and second harmonic beams [30,42].

Angular streaking should thus play a role in the photoionization by OTC fields. The most intuitive situation in terms of angular streaking is the one where the relative phase between the two components of the OTC field is ϕ ω/2ω = π (Fig. 3(i)). The vector potential describes an 8 shape. In the upper hemisphere, the potential rotates counter-clockwise. In the absence of interaction with the ionic potential, the electron distribution would maximize along p x = 0 according to SFA. The Coulomb-focused signal is clearly shifted to the left, i.e. towards positive attoclock streaking angles. The attoclock offset angle φ 0 decreases from lower to higher ATI peaks. This indicates a lower influence of the potential on high kinetic energy electrons. In the lower hemisphere, the potential rotates clockwise. The momentum distribution is again shifted to the left, which also corresponds to a positive streaking angle with respect to the field rotation direction, but to an opposite shift -φ 0 in the absolute lab-frame coordinates. Thus, the ionic potential not only induces a focusing of the indirect electrons, considerably narrowing their momentum distribution, but also an angular streaking of the distribution. Does the interplay of focusing and angular streaking dynamics, which are both due to the ionic potential, explain the large asymmetry observed in the Coulomb-focused signal induced by the OTC fields?

The attoclock streaking is determined by the scattering of the outgoing electrons in the ionic potential. In chiral molecules, this scattering induces an asymmetry between the number of electrons ending up in the forward and backward hemispheres -the well known PECD effect. It was recently shown that the attoclock streaking angle is also forward/backward asymmetric [29,30]. The chirality of the potential causes a shift δφ chiral of the forward electrons, and an opposite shift of the backward electrons. This is schematized in Fig. 3(b,c), which presents the result of a rotation by ±4 • of the momentum distribution from Fig. 3(a).

Note that this figure does not present the results of actual photoionization calculations, but aims at pedagogically describing the underlying process. The chiral attoclock shift produces a ∆P f /b asymmetry, obtained by subtracting the forward and backward distributions (Fig

3(d))

. Note that this asymmetry appears even if the total number of electrons ejected forward and backward is the same. The δφ chiral shift, being a chirosensitive quantity, it reverses when switching from one enantiomer to its mirror image, or when inverting the helicity of the ionizing radiation. However, in the latter case, one has to keep in mind that this reversal is relative to the laser field rotation direction. This can be intuitively understood by considering the angular shift as a delay. In the schematic representation of Fig. 3(b-c), the forward electrons appear at later angles along the rotation of the vector potential. By analogy with a clock, these electrons are positively delayed. The situation must reverse when switching the light helicity [29,30]. The forward electrons must appear at earlier angles along the rotation of the vector potential. As a consequence, the chiral angular shift δφ chiral does not change sign in the absolute coordinate system when reversing the light helicity (see Figs. 3(f-g)).

Comparing the differential distributions ∆P f /b displayed in Figs. 3(d,h) then shows that with both helicities, the forward electrons are deviated in the counter-clockwise direction with respect to the backward electrons. This shift would change sign upon switching of the enantiomer. Indeed, while switching the light or molecular handedness is equivalent in chiral photoionization by circularly polarized radiation, these two operations have different effects in elliptical (or more complex) fields [43]. Resolving the forward/backward asymmetry in the (p x , p y )-plane thus produces a chirosensitive signal which enables the determination of the absolute configuration of the enantiomer, independently of the helicity of the ionizing radiation.

How should we expect the chiral attoclock shift to play a role in the OTC field case? The streaking dynamics are displayed in Figs. (i-l) for ϕ ω/2ω = π. In the upper hemisphere, the situation is similar to the one described in Fig. 3(a-d). The vector potential rotates counterclockwise and we assume that a ±δφ chiral shift is induced between electrons ejected forward and backward by molecular chirality. In the lower hemisphere, the vector potential rotates clockwise. As we have just seen, whereas the non-chiral part of the streaking angle reverses between the upper and lower hemispheres, the chiral angular shift goes in the same direction.

Chirality induces a rotation of the whole angular momentum distribution (Fig. 3(j,k)), just like in the single helicity case of Figs. 3(b,c). As a consequence, the differential distribution ∆P f /b shows four main quadrants, whose sign depends on the absolute configuration of the enantiomer. Adding π to the selected two-color phase reverses the OTC field rotation.

As shown in Figs. 3(m-p), φ 0 is accordingly changed to -φ 0 in both hemispheres while δφ chiral remains unchanged. The streaking of forward electrons in the ϕ ω/2ω + π case is then symmetric to the one of backward electrons for ϕ ω/2ω with respect to the y-axis, and vice-versa. We thus understand that when changing ϕ ω/2ω to ϕ ω/2ω + π, the interplay of isotropic (φ 0 ) and chiral (δφ chiral ) streakings leads to asymmetric distributions which are negative mirror-images of each other with respect to the Coulomb focusing direction.

The analysis presented in Fig. 3 is qualitative. It assumes that the only effect of molecular chirality is to induce a forward/backward asymmetric rotation of the photoelectron angular momentum distribution. This description presents a simplistic picture of the interaction. In general, the number of electrons ejected forward and backward differs -this is the essence of PECD. Furthermore, the interference between direct and indirect electrons can play an important role in the chiroptical signal. Last, all these effects generally strongly depend on the kinetic energy of the ejected electrons. In spite of these multiple effects, the differential distribution obtained in Fig. 3(l) is remarkably similar to the actual result of the TDSE calculation, as can be seen in Fig. 4. The peak value of ∆P f /b can be adjusted in the model by setting the value of the δφ chiral rotation. A good agreement is obtained for δφ chiral = 0.07 • .

The chiral signal is almost exclusively located along the Coulomb-focused component. The model predicts some chiral signal along the interference fringes in the first ATI peak , but they are absent from the TDSE calculation (see Fig. 1). This strengthens the conclusion that the chirality of the potential mostly affects the Coulomb-focused indirect electrons, which are too localized to produce significant interference patterns.

These results demonstrate that the indirect electrons, which are Coulomb-focused, are angularly streaked by the rotating electric field, and that this process is sensitive to the We have found that these electrons present an enhanced sensitivity to chirality, which is almost independent to the two-color phase of the OTC field. We traced back the root of this sensitivity to a forward/backward asymmetric angular streaking, induced by the laser-assisted scattering of the electrons onto the chiral ionic potential.

We ascertained from a quantum mechanical point of view that Coulomb focusing operates at long range, out of the molecular core. On the other hand, large chiral features are expected to be imprinted in the photoelectron signal at short range. However one must refrain from interpreting our results in terms of a two-step process composed of focusing by the isotropic long-range part of the potential, and chiral response at the vicinity of the ionic core. Chiral patterns survive in the ionic potential far away from the core. Even if they decrease faster than the isotropic term as the radial distance r increases, these asymmetric components influence continuously the electron motion. Coulomb focusing and chiral scattering are thus intertwined, and the returning electrons are focused by the whole electrostatic force exerted by the ion core, including its isotropic (Coulombic) and anisotropic (eventually chiral) components.

We have shown that electrons experiencing laser-assisted elastic scattering onto a chiral ionic core exhibit a large chiroptical response. This could be demonstrated from an experimental point of view, using OTC driving fields and 3D photoelectron momentum imaging as in [17]. More generally, one can expect Coulomb focusing to play a role in the chiral signal whenever the strong-field radiation is not circular. For instance, it could influence the electron dynamics driven by locally and globally chiral fields, which provide gigantic chiral responses in high-harmonic generation [44].

In this work we focused on electrons with rather low asymptotic energies, belonging to the first ATI peaks. Future works should investigate the chiral response of higher energy electrons, including those who backscatter onto the ionic core and present asymptotic energies up to ∼ 10U P , where U P is the ponderomotive energy of the driving pulse [2]. Among these electrons, some of them have de Broglie wavelengths appropriate to self-image the target structure by diffraction. This would enlarge the scope of laser-induced electron diffraction [3,4] to chiral imaging.

Figure 1 :

 1 Figure 1: (a) Photoelectron angular distribution (PAD) in the polarization plane of a 800nm/400nm OTC field with I ω = 5 × 10 13 W.cm -2 , r = 0.1 and ϕ ω/2ω π/2, obtained by SFA when ionizing a target with ionization potential I P = 7.2 eV. The concentric radial ATI shells are angularly structured because of the interference of direct and indirect wavepackets schematically represented in the inset. (b) Result of TDSE calculations for ionization of the toy-model chiral molecule by the same OTC field. (c) Result of TDSE calculations for ionization of the long-range Yukawa-screened molecule. (d-f) Same as (a-c) but for an OTC field with ϕ ω/2ω = π.

  prototypical cuts of ∆P f /b in the p z = 0.055 a.u. plane, for different values of ϕ ω/2ω . For all shapes of the OTC field, the forward/backward asymmetry remains localized around the polarization direction ŷ of the fundamental laser field, in the area identified as originating from Coulomb-focused indirect trajectories. These results demonstrate the large influence of molecular chirality on Coulomb-focused electrons. A deeper insight into the process can be obtained by monitoring

Figure 2 :

 2 Figure 2: TDSE calculations of chiral photoionization by an OTC field. Top: shape of the electric field and opposite of the vector potential, as a function of the relative phase between the fundamental and second harmonic component of the OTC field. The color scale depicts the instantaneous optical chirality. (a-e) Cuts of the photoelectron angular distribution in the p z = 0.055 a.u. plane, and (f-j) cuts of the forward backward asymmetry ∆P f /b , obtained by TDSE calculations in a toy-model chiral molecule ionized by an OTC field with 5×10 13 W.cm -2 and r = 0.1.

Figure 3 :

 3 Figure 3: Influence of angular streaking on the chiroptical response of Coulomb-focused electrons. (a) Schematic representation of a photoelectron momentum distribution in the polarization plane of a counter-clockwise rotating laser field. The electron distribution is shifted with respect to the Coulomb-focusing y-direction by an angle φ 0 , which is determined by the scattering in the potential. (b,c) If the molecule is chiral, the distributions of electrons ejected in the backward (b) and forward (c) hemispheres are shifted by the additional, and opposite, angles ∓δφ chiral . (d) Asymmetric signal obtained by subtracting the forward (c) and backward (b) momentum distributions. (e-h) Same schematic representation for a clockwise rotating field. (i) TDSE calculations of the photoelectron momentum distribution in an OTC field with I ω = 5 × 10 13 W.cm -2 , r = 0.1, ϕ ω/2ω = π. The time-dependent rotation of the vector potential, depicted by the color-coded line, induces opposite attoclock streaking angles ±φ 0 in the upper and lower hemispheres, while molecular chirality leads to opposite chiral streaking angles ∓δφ chiral in the forward (j) and backward (k) hemispheres. (l) Resulting forward/backward asymmetry.(m-p) Same schematic representation than (i-l) but for a two-color phase δφ chiral = 0.

Figure 4 :

 4 Figure 4: Forward/backward asymmetry of the electron momentum distribution, obtained from the simple model assuming a chiral-induced rotation of the distribution (a) and from full TDSE calculations with an OTC field at 5×10 13 W.cm -2 , r = 0.1 and ϕ ω/2ω = π. The central part of the distributions is not shown to draw attention to the momentum range where Coulomb focusing operates on indirect electrons.

ACKNOWLEDGEMENTS

We acknowledge S. Beaulieu, V. Blanchet and E. Bloch for fruitful discussion. This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program no. 682978 -EXCITERS, 864127 -ATTOGRAM and from 871124 -Laserlab-Europe. N. D. is the incumbent of the Robin Chemers Neustein Professorial Chair. N. D. acknowledges the Minerva Foundation, the Israeli Science Foundation.


Arman Melkonyan

Etienne Tigranyan

Mushegh Rafayelyan

Henrik Melkonyan

Arman Tigranyan

Etienne Brasselet
email: etienne.brasselet@u-bordeaux.fr

Optical cloaking of macroscopic objects by geometric-phase vortex processing

Keywords: geometric phase, cloaking, optical vortices

published or not. The documents may come

Optical cloaking of macroscopic objects by geometric-phase vortex processing

Mushegh Rafayelyan, Henrik

Introduction

To see an object using waves, the observer must essentially detect the changes made by the object to the illuminating wave as waver-matter interaction takes place, or detecting the wave emitted by the object itself. Preventing the scattering of a probing wave is a usual option for stealth technologies with the aim at escaping detection from radar in electromagnetism and sonar in acoustics. Concealment management can also consist to absorb the radiation from an emitting object covered by an absorbing material, which is relevant for thermal radiation. The ability of an object to escape attempted observation under irradiation can also be achieved by redirecting the field around the object so that there is no field-object interaction, allowing the object to remain hidden from an observer. Such strategy has been originally proposed in 2006 and relies on specific dielectric and magnetic properties of metamaterials enabling the redirection of the electromagnetic flow around an object * Author to whom any correspondence should be addressed. embedded in a stealth medium acting as an invisibility cloak, as shown by Leonhardt [1] and by Pendry et al [2]. In that case, the effective behavior of the 'cloak + object' is that of free-space, which does not require the cloak to adapt to the object to be concealed, in contrast to earlier attempt [3]. Noteworthy, although the former approach deals with geometrical optics whereas the latter accounts for the Maxwell's equations, both of them basically rely on coordinate transformations [4] and transformation optics is nowadays a standalone research topic [5]. So far, numerous devices to hide objects have been proposed [6][7][8] and efforts have been paid to the development of cloaking devices that are robust against wavelength, polarization or coherence variations for the probing field [9][10][11][12][13][14][15][16][17][18].

In particular, cloaking devices for macroscopic objects that operate in the optical domain arouse interest for applications. Conventional isotropic or anisotropic homogeneous optical elements such as optical lenses [19], calcite crystals [20], or even simple water-filled reservoirs [21] have been shown suitable for concealing macroscopic objects in the visible range. Noticeably, these geometrical optics based approaches do not rely on the fabrication of large-scale devices based on optical metamaterials requiring nanoscale resolution. It should be noted that the preceding examples of electromagnetic concealment of macroscopic objects that can be described and understood from ray-tracing analysis are not unique to the optical domain. See, for example [22,23], which report on graded index lensing concealment strategies in the microwave and radio wave domains, respectively, however, with objects whose size is comparable to the wavelength.

Wave-optics cloaking device relying on isotropic refractive optical elements has also been demonstrated by using the nodal line associated with the optical phase singularity of an optical vortex beam as a polarization-independent 1D concealing region [24]. To turn it into a cloaking device, Sun et al used a pair of refractive spiral phase plates having opposite topological charges ±1 and illuminated by a Gaussian laser beam. The first one generates a single-charge on-axis optical vortex beam whose propagation is barely affected when placing a metallic needle with small enough diameter along the propagation axis while the second restores a quasi-Gaussian beam by cancelling the optical phase singularity.

Here, by combining the use of refractive regular optics (standard lenses) with singular geometric-phase optical elements (optical vortex phase masks), we extend the previous use of a point of null intensity in the transverse plane [24] to the use of a nodal surface. This allows considering polarizationindependent concealment of three-dimensional (3D) objects. This is made possible by using two cascaded stages of optical vortex Fourier processing, each of them consisting to impart an optical phase singularity to the zero spatial frequency component of the light field. The first stage allows generating a nodal area-a remarkable wave-optics phenomenon introduced in 2005 in the context of high-contrast astronomical imaging [25,26]-while the second stage restores the illuminating light source. The quasi-nodal volume surrounding the nodal area subsequently behave as a region of space that inhibits light scattering from objects placed inside.

The paper is organized as follows. The section 2 introduces the cloaking device made of an aperture, four lenses and two geometric phase vortex masks. Analytical description of the cloaking region is then given and main figures of merit are discussed. A numerical optical cloaking experiment is presented in section 3 assuming a spherical perfect conductor as the object to conceal. Moreover, the operation principle is explored as a function of its position inside the cloaking region. The section 4 then report on the experimental assessment of the quasi-nodal volume as well as optical cloaking demonstration using a 2 mm diameter stainless steel sphere and the section 5 summarizes the study.

Geometric phase optical cloak device

The proposed optical cloaking device is depicted in figure 1(a). It consists of two stages of vortex Fourier processing. Each of them is made of a pair of lenses with focal length f placed at a distance 2 f from one another. Two phase masks having complex conjugated transfer function exp(±2iϕ), where ϕ is the polar angle in the cylindrical coordinate system (r, ϕ, z) and z refers to the optical axis of the system, is placed at equal distance from the two lenses. By placing a circular clear aperture with radius R at a distance f before the first lens, a nodal area with radius R is obtained at a distance f from the second lens, which defines the plane z = 0, provided that the aperture is illuminated by a plane wave [25,26]. The second optical vortex Fourier processing stage allows recovering the uniformly illuminated entrance pupil aperture at a 4 f distance downstream the nodal area. The optical cloaking principle relies on the fact that the nodal area is surrounded by a volume with low-level intensity. This quasi-nodal volume defines the cloaking region, as indicated in figure 1(a).

Quantitatively, the intensity distribution between the two vortex Fourier filters can be obtained from the knowledge of the effect of a thin spherical lens in the paraxial regime, which is mathematically expressed by an integral relationship between the field before and after a lens in free space [27]. In the present case, introducing the normalized transverse and longitudinal coordinates r = r/R and z = z/f, and the characteristic diffraction length Λ = f/(kR) where k = 2π/λ is the wavenumber with λ the wavelength, one gets after calculations, for -1 < z < 1,

I(r, z) = I 0 2 ´∞ 0 J 1 (ξ)J 2 (ξr)e i ξ 2 2 Λ R (1-z) dξ 2 ´∞ 0 ´∞ 0 J 1 (ξ)J 2 (ξr)e i ξ 2 2 Λ R (1-z) dξ 2 rdr , (1)
where I 0 is the uniform optical intensity irradiating the circular aperture and J n is the Bessel function of the first kind of order n. In the simulations and experiments reported in sections 3 and 4, we use R = (1.5, 2, 2.5) mm, f = 200 mm and λ = 632.8 nm. Also, we use a collimated continuouswave He-Ne laser beam whose diameter is large enough to consider the aperture as uniformly illuminated. The corresponding intensity distributions calculated from equation (1) for R = 2 mm at z = (-1, -1/2, 0, 1/2, 1) are displayed in the panels (c)-(g) of figure 1, respectively. The intensity distribution in the aperture plane at z = -4 and the re-images one at z = 4 are shown in panels (b) and (h). The cloaking experiments are conducted by placing a camera at z = 4, called from now on the observation plane. If the cloaking scheme works, we thus expect to observe a uniformly illuminated disk of radius R with or without an object placed in the cloaking region.

Although the described principle of operation of the proposed cloaking device does not depend on the nature of the two conjugated vortex phase masks, we argue that choosing geometric phase vortex masks based on inhomogeneous half-wave plates offers enhanced performance. A description of their behavior can be found in [25] and can be summarized by the following transformation of an incident circularly polarized light,

c σ → c -σ exp(i2σmϕ), (2)
where c σ = (x + iσy)/ √ 2 is the unit vector describing a circular polarization, σ = ±1 is the helicity of light and (x, y, z) the orthonormal Cartesian basis and m = ±1 refers to the azimuthally varying optical axis orientation angle ψ = mϕ in the (x, y) plane for the inhomogeneous half-wave plate. Assuming an input circularly polarized state c σ , the first geometric phase mask is associated with a transfer function exp(i2σϕ) whereas the circular polarization flipping to the c -σ state makes the second mask having a conjugated transfer function exp(-i2σϕ). Such a device is therefore inherently polarization insensitive. Indeed, an incident light with an arbitrary polarization state can be described as a superposition of orthogonally polarized states c ±1 to which the previous reasoning individually applies. Also, arbitrarily broadband light could be used provided the use a pair of orthogonal achromatic circular polarizers placed, for instance, at z = -1 (selection of the c -σ state) and z = 3 (selection of the c σ state). Nevertheless the latter gain in bandwidth comes with an optical throughput trade-off as the circular polarizers will reduce the overall transmitted optical power. Using an achromatic space-variant halfwave plate, whose creation is much more challenging, would waive the latter drawback. Finally, we note that the geometric phase approach allows considering the folding of the setup, by placing a mirror at z = 0 (hence using a single geometric phase vortex mask) and adapting the observation to a reflection mode, which cannot be done using a dynamical phase vortex phase mask.

Among the many technological options available for space-variant waveplates such as dielectric nanostructured solid-state media and patterned polymer liquid crystals or liquid crystals, we choose in our experiments the polymer liquid crystal technology (from Beam Engineering) and we select m = 1.

Numerical simulations

The proposed strategy is first tested numerically by using a spherical perfect conductor as the object to conceal. The simulations are made using free-space Fourier transform propagation in between each optical element of the optical system presented in figure 1 that are treated by complex transfer functions for the amplitude t X with X = (A, L, GPM 1 , GPM 2 ):

t A (r < R) = 1 and t A (r > R) = 0, (3)
t L (r) = exp -i kr 2 /2 f , ( 4
)
t GPM1 (ϕ) = exp(i2σϕ), (5)
t GPM2 (ϕ) = exp(-i2σϕ), (6)
where the two last equations refer to a circularly polarized field with helicity σ impinging on the aperture A. In between two transmittance masks the propagation of the field from z1 to z2 is evaluated as

E(x, y, z 2 ) = F -1 [F [E(x, y, z 1 )] exp[ik z (z 2 -z 1 )]] , (7)
where F and F -1 refer to fast Fourier and inverse fast Fourier transforms, and object to conceal, is carried out by discretizing the spherical object into 100 opaque disks with varying radius that match the spherical envelope of radius R o .

k z = (k 2 -k 2 x -k 2 y ) 1/2 with k = k x x + k y y + k z z)
The optical cloaking demonstration is displayed in figure 2 where the object is placed at z o = 0 with (panels (a) and (b)) and without (panels (c) and (d)) the presence of the geometric phase masks. On the one hand, the panels (a) and (c) show the intensity distributions in the meridional plane of the cloaking region. We choose a logarithmic color bar in order to better appreciate how dark is the quasi-nodal volume when the optical cloak is activated. On the other hand, the panels (b) and (d) show the transverse intensity distributions in the observation plane z = 4, which allow appreciating the high level of optical concealment. Indeed, the contrast of the image, defined as (I max -I min )/(I max + I min ) for r < R, is reduced from 1 to ∼0.05 as the cloak is activated.

Then we evaluate the robustness of the cloaking event with respect to the position (r o , z o ) of the object inside the cloaking region both along the longitudinal and transverse direction. The results are shown in figures 3 and 4, which respectively correspond to (r o , z o ) = (0, ±f ) and ([0, 0.5, 1, 1.5], 0). On-axis longitudinal displacements barely affect the propagation of light in the cloaking region (figures 3(a) and (c)) and the intensity distribution in the observation plane accordingly remains almost uniform (figures 3(b) and (d)). The optical cloaking is also virtually immune to transverse displacements

Experimental implementation

The preceding proof of principle analysis is validated experimentally using the same parameters for the optical system, see section 2. The structure of the cloaking region is retrieved by direct imaging, scanning a camera along the z axis. A representative set of intensity distribution in the transverse plane is shown in figure 5, which can be compared to the numerical experiments reported in figures 1(d)-(f). A 3D reconstruction of the quasi-nodal cloaking volume with tubelike boundaries is reconstructed from a collection of several tens of intensity cross-sections images using Mayavi tool [START_REF]Mayavi is a general purpose, cross-platform tool for 2D and 3D scientific data visualization in Python. The code used to obtain figure 5(d) can be[END_REF]. We also observe the re-imaged aperture at z = 4 without the object when the cloak is deactivated and activated, see figures 6(a) and (b), respectively. Non-uniform intensity distribution becomes clearly visible when the geometric phase masks are inserted in the optical system. This is quantitatively ascertained from the ratio between the standard deviation and the mean value of the intensity distribution for r < R, which increases by a factor of two as the vortex masks are inserted, from ∼0.085 to ∼0. 16. We ascribe such a behavior to instrumental limitations such as the relative placement of the optical elements and the lack of resolution of structured birefringent masks, whose central part suffers from broken axisymmetric distribution of the optical axis, as illustrated in figure 6(c).

Then, we select a 2 mm diameter stainless steel metallic sphere for the cloaking demonstration, as shown in figure 7(a). The object is placed at (r o , z o ) = (0, 0) and its cloaking is reported in figures 7(b) and (c). These experimental results validate the numerical simulations shown in figures 2(b) and (d). The robustness of the optical cloak is highlighted by displacing the sphere along and perpendicular to the optical axis, as summarized in figures 8 and 9, which validate their numerical counterparts, figures 3 and4.

Finally, the quantitative analysis of the cloaking performance is made by evaluating the Pearson correlation coefficient C between the recorded images in the observation plane when

Conclusion

A wave-optics approach to optical cloaking based on a quasinodal volume shaped from optical vortex Fourier filtering has been proposed, which is based on the use of a pair of pure phase masks imparting optical phase singularities with topological charge ±2 to the light field. Numerical analysis and experimental demonstration have been reported in the framework of the concealment of macroscopic objects in the visible domain. The proposed approach is polarization independent and can be extended to any operating wavelength by using appropriate vortex beam shapers. In particular, the use of geometric phase vortex masks instead of dynamical phase ones provide enhanced polychromatic robustness, however at the expense of transmission losses and polarization-optics add-on when considering ultra-broadband fields.

Although the present work deals with axisymmetric cloaking regions, its extension to non-axisymmetric situation can be considered by using computer-assisted design toward the fabrication of appropriate singular phase masks [START_REF] Ruane | [END_REF]. Furthermore, we note that the generation of a nodal surface is not limited to the use of vortex phase masks of topological charge two but extends to all even charges [25], which opens for exploring the generalization of the present work to high-order vortex masks. E B acknowledges funding from Conseil Régional de Nouvelle Aquitaine (Project HELIXOPTICS). M R and H M acknowledge funding from Science Committee of Armenia under the Grant Agreement No. 21T-1C331.

Figure 1 .

 1 Figure 1. (a) Sketch of the experimental setup for the implementation of geometric phase optical cloaking at the macroscopic scale. A sufficiently expanded continuous-wave collimated laser beam impinges on a circular aperture (A) with radius R. Two 4 f optical systems made of a pair of lenses with focal length f act on the light field as optical vortex Fourier processors owing to geometric phase masks (GPM 1,2 ) placed at z = ±2, which respectively impart a pure phase singular amplitude profile proportional to exp(±2iϕ) to the light impinging on it, see text for details. A camera re-images the aperture A at z = 4, to which we refer as the observation plane. Panels (b)-(h) show the calculated normalized transverse intensity distributions at various position along the optical axis z, highlighting the quasi-nodal nature of the 3D cloaking region located between the planes z = ±1.

  being the wavevector. Moreover the propagation between the planes z o ± R o , where z o and R o = 1 mm respectively refer to the longitudinal position and the radius of the

Figure 2 .

 2 Figure 2. Numerical demonstration of geometric phase optical cloaking. (a), (b) The cloak is deactivated by removing the geometric phase masks. (c), (d) The cloak is activated. In both cases the spherical object to conceal is placed in the center of the cloaking region, at (ro, zo) = (0, 0). The normalized intensity distributions I/Imax inside the cloaking region (panels (a) and (c)) are calculated by using 300 elementary steps along the z axis with 100 of them being dedicated to the object that lies in the range -Ro < z < Ro. Note that the sphere (represented in white color) has an elliptical shape in panels (a) and (c) due to the different scaling used for the transverse and longitudinal axes. The normalized intensity distributions I obs /Imax in the observation plane at z = 4 (panels (b) and (d)) highlight the performance of the cloaking event.

Figure 3 .

 3 Figure 3. Numerical assessment of the influence of on-axis longitudinal displacement of the object inside the cloaking region. (a), (b) zo = -f. (c), (d) zo = f. The four panels are displayed in a similar manner as in figure 2.

Figure 4 .

 4 Figure 4. Numerical assessment of the influence of transverse displacement of the object inside the cloaking region at zo = 0 for ro = 0 (a), ro = 0.5 (b), ro = 1 (c) and ro = 1.5 (d).

Figure 5 .

 5 Figure 5. (a)-(c) Experimental measurement of the transverse intensity profiles at different positions inside the cloaking region, z = (-0.5, 0, 0.5), in the absence of the object. (d) Experimental 3D construction of the tube-like intensity boundaries that define the cloaking region based on a collection of 31 equidistant transverse intensity profiles in the range -0.94 < z < 0.94.

Figure 6 .

 6 Figure 6. (a), (b) Normalized intensity distribution in the observation plane in the absence of the object when the cloak is deactivated (panel (a)) and activated (panel (b)). (c) Crossed linear polarizers imaging of the central part of the geometric phase mask with m = 1.

Figure 7 .

 7 Figure 7. (a) Picture of the stainless steel sphere with radius Ro = 1 mm used as the object to conceal. (b), (c) Experimental observations at z = 4 in presence of the object placed in the center of the cloaking region (ro, zo) = (0, 0) when the cloak is deactivated (panel (b)) and activated (panel (c)).

Figure 8 .

 8 Figure 8. Experimental cloaking robustness versus the position of the object along the optical axis. Normalized intensity imaging in the observation plane when the cloak is deactivated (panels (a)-(c)) and activated (panels (d)-(f)) for zo = -0.5 (first column), zo = 0 (second column) and zo = 0.5 (third column).

Figure 9 .

 9 Figure 9. Experimental cloaking robustness versus the position of the object perpendicular to the optical axis. Normalized intensity imaging in the observation plane at zo = 0 for ro = 0 (a), ro = 0.5 (b), ro = 1 (c) and ro = 1.5 (d).

Figure 10 .

 10 Figure 10. Dependence of the Pearson correlation coefficient C versus zo at ro = 0 for three different apertures A with radius R = 1.5 mm (blue), 2 mm (green) and 2.5 mm (red).

Funding

Data availability statement

The data that support the findings of this study are available upon reasonable request from the authors.