The operation of a typical common emitter amplifier, including negative feedback, is studied taking into account the non-linearity characteristic of real-world transistors. This has been accomplished by employing a recently proposed Early modeling approach, which allowed the analytical equations to be obtained describing the current and voltage behavior in the adopted common emitter circuit. Average and dispersion (coefficient of variation) of the current and voltage gains can then be calculated and used to characterize the common emitter amplification while reflecting the transistor non-linearity. Several interesting results were obtained, including the fact that the negative feedback provided by the emitter resistance is not capable of completely eliminating effects of parameters differences exhibited by transistors. Importantly, transistors with larger Early voltage Va magnitudes tended to provide significantly enhanced linearity even when substantial negative feedback is used. These results motivates customized design, implementation and application approaches taking into account the parameters of the available devices.

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Delphic Proverb

I. INTRODUCTION

The common emitter design is one of the most fundamental amplifying circuit configurations, representing a reference in electronics circuits be it for its central role in research, application, or education. This circuit configuration can be found in virtually almost every textbook on electronics circuits (e.g. [START_REF] Gray | Electronic Principles: Physics, Models and Circuits[END_REF][START_REF] Gronner | Transistor Circuit Analysis[END_REF][START_REF] Horowitz | The Art of Electronics[END_REF][START_REF] Jaeger | Microelectronic Circuit Design[END_REF][START_REF] Boylestad | Electronic Devices and Circuit Theory[END_REF][START_REF] Pettit | Electronic Amplifier Circuits: Theory and Design[END_REF][START_REF] Sedra | Microelectronic circuits[END_REF][START_REF] Streetman | Solid State Electronic Device[END_REF]). Consequently, much has been said and written about this configuration, including its electronic properties as voltage gain, input and output resistance, linearity/distortion, the negative feedback action implemented by the emitter resistance, stability with component variations, frequency response, to name a few among a set of issues encompassing a great deal of the main interests in electronics. This circuit typically includes a voltage source attached through a capacitor a to voltage divider driving the amplifier input port, an emitter resistance for negative feedback shared by both the input and output mesh, and a resistance load in series with an external voltage supply. Either the input voltage or current are taken as the input signal, while the collector voltage, or that across the load, are often understood as the output signal.

The main purpose of the common emitter amplifier is to produce current and/or voltage gain over the load or output. The intrinsic parameter variability exhibited by real-world transistors, allied to the need to improve amplification linearity and stability, motive the incorporation of the emitter resistance as a means to implementa negative feedback. However, it is also possible (at the expense of some stability loss) to consider variations of the classic common emitter configuration devoid of the emitter resistance, which can be considered as a means to maximize gain and achieve wider voltage excursion across the load (i.e. avoid the voltage drop at the emitter resistance).

Commonly, studies of the common emitter circuit involve the adoption of a model and respectively implied equivalent circuit to represent the involved transistor. The most often employed approaches rely on two electronically meaningful parameters, namely the current gain β and the output resistance R o . This type of approach, however, cannot take into account the nonlinearities that are unavoidably found in any real-world transistor. These nonlinearities are a direct consequence of the fanned structure of the base current (I B ) indexed characteristic isolines of the transistor behavior along its operation space (V C , I C ), where V C ad I C stand for the collector voltage and current.

The above described geometrical organization of transistors action is a sole and direct consequence of the Early effect (e.g. [START_REF] Early | Effects of space-charge layer widening in junction transistors[END_REF][START_REF] Jaeger | Microelectronic Circuit Design[END_REF][START_REF] Streetman | Solid State Electronic Device[END_REF]), in which the base width varies in terms of the voltage applied to the base-collector junction. This effect implies mots of the I B -indexed characteristic isolines to cross at the same point (V C = V a , I C = 0) along the V C axis. An immediate implication is that the collector output resistance will depend on I B . Another critically induced property regards the fact that the local current gain β, as well as the output resis-tance R o become a function of both I B and V C . Consequently, transistor modeling approaches relying on β and R o as parameters need to consider their respective averages or point values. Frequently, it is assumed that the fanned isolines can be approximated by a series of equispaced, parallel isolines with the same inclination small m = 1/R o (I B ). As such, this simplified device is typically represented by an equivalent circuit involving a variable current source in parallel with a fixed output resistance. As an inevitable consequence, this transistor model becomes intrinsically linear, which is a severe simplification of what really happens in real-world transistors.

Recently [START_REF] Da | Characterizing complementary bipolar junction transistors by early modeling, image analysis, and pattern recognition[END_REF][START_REF] Da | Characterizing germanium junction transistors[END_REF][START_REF] Da | Towards a simple, and yet accurate, transistor equivalent circuit and its application to the analysis and design of discrete and integrated electronic circuits[END_REF][START_REF] Da | An Early model of transistors and circuits[END_REF], an alternative transistor model was proposed that is simple and yet can accurately account for the representation of the transistor nonlinearities induces by the fanned (or radiating) characteristic isolines. In addition, the two parameters used in this model, namely the Early voltage V a and a proportionality constant s, are constant for any given device, not depending of I C or V C . These distinguishing features are allowed by taking into account the very Early effect that causes the fanned isolines as the geometrical basis of the model. Actually speaking, this model also relies on another hypothesis, verified experimentally for hundreds of small signal BJTs [START_REF] Da | Characterizing germanium junction transistors[END_REF][START_REF] Da | An Early model of transistors and circuits[END_REF], that the angles θ of the radiating isolines with the horizontal (V C ) axis are directly proportional to I B through the proportionality constant s, i.e. θ = sI B , implying that R o = 1/tan(sI B ). As I B is usually very small, in the order to µA, it is often possible to make tan(sI B ) = 1/(sI B ) with minimal deviation, so that Ro(I B ) = 1/(sI B ).

Despite its recent introduction, the Early model has already been successfully applied to achieve more complete and accurate modeling and studies of: stability with respect to voltage supply oscillations [START_REF] Da | Towards a simple, and yet accurate, transistor equivalent circuit and its application to the analysis and design of discrete and integrated electronic circuits[END_REF], parallel combinations of transistors [START_REF] Da | Towards a simple, and yet accurate, transistor equivalent circuit and its application to the analysis and design of discrete and integrated electronic circuits[END_REF], characterization of the output resistance and current gain of silicon and germanium NPN and PNP small signal transistor models [START_REF] Da | Characterizing complementary bipolar junction transistors by early modeling, image analysis, and pattern recognition[END_REF][START_REF] Da | Characterizing germanium junction transistors[END_REF], study of a simplified common emitter configuration [START_REF] Da | Characterizing complementary bipolar junction transistors by early modeling, image analysis, and pattern recognition[END_REF], investigation of complementary transistor pairs in push-pull circuits [START_REF] Da | Characterizing complementary bipolar junction transistors by early modeling, image analysis, and pattern recognition[END_REF], as well as the characterization of the effects of reactive loads in amplification [4], among other results. The Early approach has also allowed the identification of the fact that the amplification in a simplified common emitter amplifier tends to be perfectly linear for very small resistive loads [4]. In addition, it has been found [START_REF] Da | Characterizing complementary bipolar junction transistors by early modeling, image analysis, and pattern recognition[END_REF]4] that the linearity of the transistors amplification devoid of feedback tends to depend only of s (directly), and not of V a . As a matter of fact, the simplicity and completeness of the Early model in being able to represent in an inherently suitable geometrical way the non-linearities of real-world transistors pave the way to a large number of implications for transistor amplification and even linear electronics in general.

The current work focuses on substantially extending previous analyses [4,[START_REF] Da | Towards a simple, and yet accurate, transistor equivalent circuit and its application to the analysis and design of discrete and integrated electronic circuits[END_REF] that had addressed a simplified version of the common emitter configuration. In those works, the emitter was connected directly to ground, and there was no voltage divider for polarization at the input port, so no negative feedback can be achieved. Though that configuration was still representative of several characteristics of transistor amplification, leading to the identification of a series of interesting effects as listed above, it did not account for the negative feedback afforded by the inclusion of an emitter resistor R E . This is very important because much of modern electronics has relied on negative feedback, introduced by H. S. Black [START_REF] Black | Stabilized feedback amplifiers[END_REF], in order to achieve improved linearity and stability to factors such as temperature and transistor parameter variations. These advantages are achieved at the expense of substantial gain expense. Yet, a recent study [START_REF] Da | Characterizing complementary bipolar junction transistors by early modeling, image analysis, and pattern recognition[END_REF][START_REF] Da | [END_REF] considering several types of commercial small signal BJTs suggested that negative feedback may not be enough, in several situations, to completely eliminate the relatively large parameter variations exhibited by real-world devices. It was also shown that negative feedback can introduce unwanted harmonics for certain types of non-linearities.

Th key role of negative feedback and the common emitter amplifier in electronics, allied to the above mentioned recent findings suggesting some respective limitations, motivate a reappraisal of the classic common emitter amplifier configuration. Of particular interest would be to investigate the efficacy of negative feedback in minimizing transistor non-linearity with respect to different transistor characteristics. For instance, will feedback work more effectively for transistor with large or small current gains β or by choosing different values for the external circuit components? If such differences can be found, how much can feedback action be improved by starting with a more linear transistor? These correspond to key issues in both discrete and integrated circuit analysis, design, implementation and applications. These questions provide the main motivation and objectives of the present work.

We start by presenting the "classic" common emitter configuration, and then obtain a respective equivalent circuit by using the Early modeling methodology. Accurate mathematical expressions are obtained that characterize the voltages, currents and gains while taking into account the non-linear behavior of real-world transistors. By using these equations, it becomes possible to investigate in an objective, more complete, and accurate way how the transistor inherent non-linearity influences the efficiency of negative feedback in promoting linear amplification. This is done by considering the voltage gain average and coefficient of variation (closely related to the level of nonlinearity in the amplification) A number of interesting respective results are obtained and discussed, and the work is concluded by presenting some possibilities for further related research.

II. THE CLASSIC COMMON EMITTER CONFIGURATION

Figure 1 shows the configuration of what we will henceforth refer to as the "classic" common emitter circuit. A voltage divider (resistors R a and R b ) is used to bias the input port, which is driven by an external signal voltage source v i (t), attached through the decoupling capacitor C. The load is connected between the collector and the external voltage supply V CC , while resistor R f is connected between the emitter and ground in order to provide negative feedback. In this circuit, the base-collector junction of transistor T is inversely biased, while the junction base-emitter is directly biased.

FIG. 1: The "classic" common emitter configuration as adopted in this work. The base-collector junction of transistor T is inversely biased, while the junction base-emitter is forward biased. R is the resistive load, and R f provides negative feedback. The input port is biased into an operation (quiescent) point Q by the voltage divider implemented by Rx and Ry, receiving the input through the coupling capacitor C, which is chosen so as to have low impedance relatively to the input bandwidth.

First, we derive the equations for the input port, assuming that the forward-biased base-emitter junction is modeled as a diode with offset voltage V r and resistance R r . By using Thevenin's theorem, we obtain the respective equivalent circuit for the input voltage and divider as:

R b = R x R y R x + R y (1)
V b = R x V CC R x + R y (2)
Figure 2 shows the common emitter circuit in Figure 1 modified to show the input port equivalent circuit as well as the output mesh, with the transistor replaced by its Early equivalent circuit. Observe that the input voltage source v i (t) is not considered as it acts differentially in the amplification, oscillating around the operation point Q. As R b /(R b + R r ) is typically very close to 1 (R b is much larger than R r ), it becomes possible to include v i (t) in series along the input mesh, for simplicity's and generality's sake, at the expense of nearly negligible error. For simplicity's sake, the above circuit can be further simplified io the configuration shown in Figure 3. This has been achieved by subsuming the input and output series resistances and voltage sources, i.e.

R 1 = R b + Rr, R 2 = R + R o (I B ), V 2 = V CC -V a ,
and, in particular: Despite its seeming symmetry and simplicity, this circuit exhibits some relative complexity as a consequence of R 2 being a function of

V 1 = v i (t) + V b -V r (3)
I B , i.e. R 2 (I B ) = R + R o (I B ) = R + 1/tan(sI B ) ≈ R + 1/(sI B ).
The variables in this circuit are V , I B and I. By applying Kirchhoff's and Ohm's law, we have:

V 1 -R 1 I B -R f (I + I B ) = 0 V 2 -R 2 I -R f (I + I B ) = 0 (4)
The solution for I B and I in terms of v i (t) are given as I 1 and the DC current gain A i , as in Equations 5 and 6.

These equations describe the operation, in terms of V 1 (and hence v i (t), through Equation 3), of classic common emitter circuit as in Figure 1.

Proper operation of this circuit, without exceeding the minimum possible values of I, requires the conditions in Equation 8 and 9 to be obeyed at all times in order to avoid cut-off or saturation (for simplicity's aske, the Early geometric representation is here understood to cover the whole first quadrant, so that saturation and cut-off occurs at the V C and I C axes, respectively.

I ≥ 0 ( 8
)
I ≤ V CC R + R f (9)
As already observed, despite the seeming simplicity of the system of equations 4, the behavior of the common emitter configuration turns out to be relatively complex. However, Equations 5 and 6 can be conveniently used for obtaining almost any information about the common emitter circuit. In this work, we will focus on two particularly characteristics of this circuit, namely its AC voltage and current gain in terms of the average and coefficient of variation of these gains.

First, we obtain the expressions for the AC current gain. This is done by defining the reference I B and I values for the quiescent point Q as:

I B,Q = I B (v i (t) = 0) ( 10
)
I Q = I(v i (t) = 0) (11)
so that:

a i (v i (t)) = I(t) -I Q I B (t) -I B,Q (12)
where I B (t) and I(t) can be obtained from Equations 5 and 6.

The two indices of interest can now be determined by calculating the average and standard deviation of a i , i.e. a i and σ ai .

The voltage gain on the load can now be directly expressed as:

a v (v i ) = (I -I Q )R v i (13)
Similarly as developed above, this voltage gain can be characterized in terms of its average and standard distribution values a v and σ av . The respective coefficient are particularly relevant, and are henceforth adopted, as they provide a relative quantification of the gain variation, i.e.:

c ai = σ ai a i ( 14
)
c av = σ av a v (15)
The next sections study the behavior the current and voltage gains for different circuit (Section III) and transistor parameter (Section IV) configurations. It is henceforth adopted that V CC = 20V , R x = 50000Ω, R y = 100000Ω, R r = 30Ω, V r = 0.6V , and V = 4V .

III. ANALYSIS IN TERMS OF CIRCUIT PARAMETERS

In this section we apply the obtained equations describing the operation of the classic common emitter amplifier while keeping the transistor parameters V a and s constant and varying the load resistance R and the feedback resistance R f . Recall that the gain variation provides an accurate quantification of the transistor amplification non-linearity, as larger relative variations necessarily imply larger distortions and more intense non-linearity.

IB(t) = sV1(R + R f ) -R1 -R f -sR f V2 -ℵ 2s(RR1 + RR f + R1R f ) (5) Ai(t) = I(t) IB(t) = 1 2R f (sR f V2 -sV1(R + R f ) -R1 -R f -ℵ) (6) with ℵ = 4sV1(RR1 + RR f + R1R f ) + (R1 + R f -sRV1 + sR f (V2 -V1)) 2 (7)
In order to derive interpretations for typical small signal NPN and PNP silicon BJT transistors, we use two respective transistor parameters configurations that can often be found experimentally in real-world silicon BJTs [START_REF] Da | Characterizing complementary bipolar junction transistors by early modeling, image analysis, and pattern recognition[END_REF], i.e. (V a,npn = -150, s npn = 2.5) and (V a,pnp = -50, s pnp = 10). As the current gain can be approximated as sV a , these two configurations yield about the same gain, but have widely varying parameters V a and s.

Figure 4 shows the average AC current gain a i (a) as well as the coefficient of variation of the AC voltage gain a v (b) with respect to the above mentioned NPN (a-b) and PNP (c-d) configurations.

As expected, the two considered parameterizations have nearly the same average gain (Figs. 4(a) and(c)).

Observe that the values given by the respective surfaces tend to agree with the less precise estimation of the gain as R/R f (note that the quiescent current has been chosen as half the V CC value). The coefficients of variation obtained for the NPN and PNP cases, however, show a marked difference of intensities, despite the fact that the respective surfaces (i.e. Fig. 4(b) and(d)) have very similar shapes. Indeed, the NPN device has over than 3 times less distortion, as a consequence of higher V a magnitudes typically found in NPN types.

The coefficient of variation of the voltage gain correlates strongly (positively) with the average voltage gain. Both the amplification magnitude and non-linearity increase steadily with R and R f , though tending to increase faster with the latter.

IV. ANALYSIS IN TERMS OF TRANSISTOR PARAMETERS

In this section, instead of investigating the behavior of the voltage gain with the circuit parameters R and R f , we fix R = 1000Ω, take two feedback configurations R f = 300Ω and 800Ω, and systematically vary the Early model parameters V a and s. Figure 5 presents the respectively obtained results regarding the averages and coefficients of variation of the voltage gain throughout the Early parameters space (-200 ≤ V a ≤ -5, 1 ≤ s ≤ 10).

The average voltage gain surfaces resulted with similar shapes, but showing different magnitudes (Figs. 5, with the amplification achieved for the NPN circuit being twice as much higher (approximately R/R f at the highest values). A closer inspection reveals that the average gain surface for R = 1000Ω and R f = 800Ω has a flatter top than for the other configuration. These surfaces show that the average voltage gain increases steeply with both V a magnitude and s, but then reach a plateau of greater gain uniformity moving toward (V a = -200, s = 10).

Interestingly, the observed dispersions of voltage gain (and hence larger non-linearity) verified for R = 1000Ω and R f = 300Ω, as shown in Fig. 5(b), present an opposite tendency to that of the average voltage gain. More specifically, the distortion decreases with both V a and s, with this decrease being markedly more accentuated with increase of the V a magnitude. A well-defined flat valley of linearity is obtained around (V a = -200, s = 10).

The coefficients of variation obtained for the case R = 1000Ω and R f = 300Ω (Fig. 5 (d)) resulted distinct in the sense that the distortion mostly decreases as s becomes larger. This has the interesting implication that transistors with larger values of s will tend, for such situations, to yield less distortion for this specific configuration of R and R f .

Perhaps the most important overall implication of the above discussed results is the fact that, even in presence of relatively high feedback levels (e.g. R f = 800Ω), when almost all gain is traded-off for linearity and stability, the linearity of the amplification still varied substantially with different values of the transistor parameters V a and s. This finding corroborates, for the considered configuration and ranges, the previously observed phenomenon [START_REF] Da | [END_REF][START_REF] Da | An Early model of transistors and circuits[END_REF] that negative feedback may not be capable of completely eliminating transistor parameter variation. Therefore, it becomes an interesting option to consider the transistor parameters V a and s for circuit design and implementation, even when adopting intense levels of negative feedback.

V. CURRENT GAIN

Though the common emitter amplifier is typically approached with respect to voltage gain, It is also interesting to consider the AC current gain as can be derived from the analytical Equations 5 and 6.

The obtained results are qualitatively similar (though with distinct intensities), so we restrict our discussion to the analysis of the average current gain throughout the space (V a , s), as observed for R = 1000Ω and R f = 300Ω, which is depicted in Figure 6. This result shows that the average current gain for the mentioned configuration increases in an almost linear way with both V a and s, reaching its peak of nearly 2000 at (V a = -200, s = 10). Though a direct consequence of the circuit operation dynamics, it is interesting to observe that the average current and voltage gains depend in such distinct ways on the parameters V a and s.

VI. CONCLUDING REMARKS

Amplification is a key issue in analog/linear electronics because of the non-linearity that is inherent to any real-world amplifying devices such as transistors. As such, much effort has been invested in devising circuits and techniques capable of achieving satisfactory performance under specific circumstances. The common emitter circuit is probably the "quintessential" amplification approach in electronics, acting as a reference for other designs and development of novel techniques aimed at improving amplification, as is the case of negative feedback.

The non-linearity inherent to junction transistors stems mostly from the Early effect, which implies the base current-indexed isolines characterizing the typical amplification to converge into a single point along the V C axis. The recent introduction of the Early modeling approach [START_REF] Da | Characterizing complementary bipolar junction transistors by early modeling, image analysis, and pattern recognition[END_REF][START_REF] Da | Characterizing germanium junction transistors[END_REF][START_REF] Da | Towards a simple, and yet accurate, transistor equivalent circuit and its application to the analysis and design of discrete and integrated electronic circuits[END_REF][START_REF] Da | An Early model of transistors and circuits[END_REF] paved the way to more systematic, analytical investigations of the behavior of amplifying circuits, including the common emitter configuration. Preliminary approaches were reported recently [4,[START_REF] Da | Towards a simple, and yet accurate, transistor equivalent circuit and its application to the analysis and design of discrete and integrated electronic circuits[END_REF], but those approaches took into account a simplified common emitter configuration devoid of negative feedback. Even so, it was possible to achieve several interesting results, including the relatively severe effect of the transistor nonlinear amplification, as well as its great dependence on the transistor Early parameters V a and s.

The present work addressed the complete, classic common emitter configuration, incorporating voltage dividerbased input biasing and, more importantly, negative feedback as provided by a resistance R f attached between the collector and the ground. This resistance has to be kept at relatively small values in order not to reduce too much the current and voltage amplification too much.

The reported analysis was developed as follows. First, the equivalent circuit of the classic common emitter configuration was derived by using electrical circuit laws and the Early voltage equivalent circuit for a transistor in amplifying mode. The base current I B and collector current I were treated as functions of the input voltage v i (t). Analytical solutions were obtained for I B and I, providing the complete electrical description of the considered common emitter amplifier. The average, standard deviation, and coefficient of variation of the voltage and, to a smaller extent, current amplification (gains) were obtained and used for investigating the effect of several circuit and transistor parameter configurations. As gain variations are closely related to non-linearities, it was possible to address both the overall gain, as expressed in terms of the average of the gains obtained along v i (t) excursions, as well as to infer the respective linearities.

Several interesting results have been reported. First, we have that the choices of the load and feedback resistance (circuit components) influence strongly both average gain (as expected) and linearity. Importantly, these gains were found to vary with v i (t) even in presence of relatively intense negative feedback, implying amplifying distortion. Perhaps more importantly, the transistor inherent characteristics V a and s were found to strongly influence the linearity of the amplfiication even in presence of relatively high negative feedback. Similar results had been experimentally observed recently [START_REF] Da | Characterizing complementary bipolar junction transistors by early modeling, image analysis, and pattern recognition[END_REF][START_REF] Da | [END_REF]. By providing a complete and accurate analytical framework to characterize the common emitter circuit, the present work not only confirmed those preliminary experimental indications, but also showed in a more systematic and conclusive way that, though certainly remaining an interesting option, negative feedback is not enough to completely eliminate effects of transistor parameter variations. This important fact implies that, especially in particularly critical circumstances, it may be worth taking into account the specific characteristics of available realworld transistors. Another interesting result concerns the fact that the linearity can increase or decrease according to s for different R and R f configurations.

The implications of the above results are several and have great potential for motivating many related further investigations, including the study of higher power amplifiers, consideration of reactive loads, as well as the systematic study and characterization of other important electronic circuits such as current mirror, phase splitters, and differential amplifiers, up to complete operational amplifier configurations, to name but a few possibilities.

FIG. 2 :

 2 FIG. 2: The equivalent circuit of the common emitter configuration inFigure1obtained by replacing the voltage divider in the input port by its Thevenin equivalent circuit, and by replacing the output mesh of transistor T by the respective Early equivalent circuit.

FIG. 3 :

 3 FIG. 3: The common emitter equivalent circuit in Figure 2 can be further simplified into the depicted configuration by making R1 = R b + Rr, R2 = R + Ro(I B ), V1 = vi(t) + V b -Vr and V2 = V CC -Va.

FIG. 5 :

 5 FIG. 5: The voltage gain average and coefficient of variation in terms of Va and s for two configurations R = 1000Ω and R f = 300Ω (a,b) and R f = 800Ω (c,d). See text for discussion.

  Analysis of the Common Emitter Amplifier Taking into Account Transistor Non-Linearity Luciano da F. Costa * São Carlos Institute of Physics, IFSC-USP, São Carlos, SP, Brazil (Dated: February 19, 2018)

Acknowledgments. Luciano da F. Costa thanks CNPq (grant no. 307333/2013-2) for sponsorship. This work has benefited from FAPESP grants 11/50761-2 and 2015/22308-2.


Antoine Bierret

Grégory Vincent

Julien Jaeck

Jean-Luc Pelouard

Riad Haïdar

Fabrice Pardo
email: fabrice.pardo@c2n.upsaclay.fr

Field extension inside guided-mode-resonance filters under a focused beam

We present a theoretical study of mid-infrared guided mode resonance spectral filters made of two subwavelength metallic gratings and a dielectric waveguide, under a focused beam with a finite spot size. Study shows that, at the resonant wavelength, the lateral extension of the electromagnetic field in the waveguide is close to the width of the beam. We compare the performance of filters using gratings with a one-slit pattern and gratings with a two-slit pattern and we show that the latter gratings (biatom gratings) provide a higher transmission and a better limitation of field extension, due to an improved angular acceptance. These results open new perspectives for pixel-sized infrared filters.

Spectral filtering in the infrared wavelength range can be achieved through resonant sub-wavelength periodic structures [1][2][3]. These structures have a limited number of layers, which is technological advantage as compared to standard multilayer devices like Bragg filters. This makes them particularly suited for the realization of mosaic of reduced-size filters, to be integrated into imaging systems [4]. However, the compatibility of sub-wavelength gratings filters with dimensions as small as the actual pixel size of infrared detectors still needs to be demonstrated. Most of the time, components are studied under plane wave illumination, possibly for variable incidence angles, and the maximum of transmission can be theoretically almost perfect, depending on the use of lossy [5] or lossless materials [6,7]. For practical infrared imaging systems, it is no longer possible to consider that incoming light is a plane wave but rather a focused beam with a limited spot size.

We consider a component that belongs to the family of guided mode resonance filters (GMR filters), initially proposed by Magnusson and Wang [1], and then derived in numerous studies [8]. Such a component is based on the coupling of the free-space incoming and outgoing light with a guided mode inside a dielectric layer. We will focus on the study of a free-standing metallo-dielectric GMR filter, made of a dielectric waveguide sandwiched by symmetric metallic gratings deposited on each sides, to achieve spectral filtering in the midwave infrared (3 -5 µm) because they lead to better angular tolerances than full dielectric GMR filters [9]. The coupling is ensured by the two sub-wavelength gratings and this device leads to a spectrally selective resonant transmission [3]. The rejection of the component is rather high thanks to a perfect extinction mechanism at high wavelengths, as demonstrated by Estruch [10] and Tardieu [11]. More precisely in this paper, we consider on the one hand gratings with a one-slit pattern, called a monoatom filter, and on the other hand gratings with a two-slit pattern called a biatom filter [9,12]. As described in reference [12], the biatom structure uses a second slit to tune the resonant wavelength without changing the period of the grating. By choosing a resonant wavelength under normal incidence close to the crossing wavelength of the dispersion bands of the grating, we obtain a flatter transmission band, which results on a maximized angular acceptance. This acceptance plays a major role when used under focused beams as the angular independence is a necessary condition for focused light to be efficiently coupled into the waveguide. However, the narrow extent of the electro-magnetic field inside the waveguide, which is essential to create pixel-sized filters, is not definitely ensured by the angular independence of the filter. Several studies have been lead on dielectric-only GMR filters under focused beams in the literature [13][14][15]. However, the beams were always very loosely focused.

In this letter, we investigate through simulations the optical properties of guided mode resonance filters under a focused illumination, and we show that the lateral extension of the electromagnetic field in the waveguide at resonant wavelength is close to the width of the incident beam.

In order to address the specific case of GMR filters under a focused beam, we first need to design such filters with suitable spectral and angular characteristics under a plane wave illumination. Optical properties of the filters are computed using rigorous coupled-wave analysis (RCWA) [16]. The simulations are carried for 2-dimensional structures, meaning that the grating is periodic along x-axis and invariant along y-axis. This config- uration was chosen to provide a good insight on the properties of our components while maintaining a short computation time. Because we consider metallo-dielectric GMR made of infinitely long slits, the simulations only take into account transversemagnetic (TM) polarized light, which means that magnetic field is parallel to the slits (y-axis). Monoatom and biatom filter geometries are respectively described in figure 1.a) and figure 1.b). Chosen dimensions are as follows, for monoatom filter: period is d mono = 2.075 µm and slits width is a mono = 0.16 µm; for biatom filter: d bi = 2.8 µm, a 1,bi = 0.19 µm, a 2,bi = 0.67 µm. The distance between the center of each slit in the bi-atom patern is d 2 . Thicknesses are t m = 100 nm, t d = 650 nm for both filters. Dielectric layer is made of silicon nitride, whose refractive index is taken as n SiNx = 2.15. The refractive index of gold is given by a Drude model:

m (λ) = 1 -( λ p λ + iγ) λ p λ -1
with λ p = 159 nm and γ = 0.0048 matching recent experimental results [17,18]. Figure 1.c) shows the transmission spectra for both filters. The dimensions of monoatom and biatom filters were optimized to obtain similar spectral shapes, with resonant wavelength λ mono = λ bi = 3.905 µm, and spectral full width at half maximum FW HM mono = FW HM bi = 110 nm. The transmissions at the resonant wavelengths could not be exactly matched with these conditions and are respectively T max,mono = 72.7 % and T max,bi = 78.3 %. Figure 1.d) gives the evolution of the transmission when the angle of incidence of the plane wave increases for both filters at λ = 3.905 µm. It reveals that, as shown by Sakat [9], the biatom filter offers a much larger angular acceptance, Δθ, defined as the angle where transmission reaches half of the normal incidence value: Δθ bi = 20 • for biatom component and Δθ mono = 7 • for monoatom component. Now that the opto-geometrical parameters of the components have been properly defined, we can then study the response of the filters to a focused beam. Since our simulations are performed with RCWA software, the focused beam can be described as a sum of plane waves, by means of a de-composition in Fourier series as proposed by Chevalier [19]. Depending on the beam focus, the Fourier series can be truncated of its highest orders because of their low magnitude. This method allows choosing the shape and size of the beam at will and reducing the computation time. In accordance with the 1-dimensional geometry of the grating, the incoming beam is thus invariant along the y-axis. Along the x-axis we consider a periodic repetition of the beam, since we use a RCWA algorithm dedicated to the study of periodic structures: the incoming wavefront is here repeated every 112 µm. This value is way higher than the period of the gratings and way higher than the width of the spots. To ensure independent responses to each beam, the pattern of the amplitude of the beam is null except inside the interval [0; 2.466 w 0 ] where it is defined as an approximated Blackman equation [20]:

B(x) = 1 -a 2 - 1 2 cos 2πx 2.466 w 0 + a 2 cos 4πx 2.466 w 0
where x ∈ [0; 2.466 w 0 ] and w 0 is the desired FWHM, with the usual parameter a = 0.16. The Blackman beam shape was chosen because of the quickly decreasing magnitudes of the terms of its Fourier decomposition. High-order terms, which correspond to the most tilted plane waves, are truncated when their magnitude is a thousand times lower than the zero-order magnitude. This introduces a maximum angle for the plane waves taken into account in the computation. This angle, called θ ap , is the angular aperture of our beam (see figure 2.b). The simulation is carried for each plane wave on a single period of the grating, and then the results can be propagated to any number of periods. This is faster than the direct computation of an incoming plane wave on a large number of periods. Last, the response of the component to the focused beam is a sum of all the previous responses, weighted by the Fourier coefficients of each plane wave.

At first, we consider a Blackman function with a spatial FWHM of w 0 = 9 µm, meaning that the full width at 10% of the maximum (spot size) is 16 µm. In this case, the angular aperture of the beam is θ ap = 20 • which is consistent with the angular acceptance of the biatom filter (see fig. In figure 2.a, we show the spectra for a biatom (green lines) and a monoatom (blue lines) filter either under focused beam (plain lines) or under normally-incident plane wave (dotted lines). Resonance is barely modified for biatom structuration under focused beam illumination with respect to the resonance under plane wave illumination. On the contrary, under focused beam for monoatom structuration the resonant wavelength is slightly blue-shifted to λ mono = 3.885 µm and the maximum of transmission drops to T max = 62%. These results are directly linked to the angular behavior of the filters: with an angular aperture of θ ap = 20 • , the beam remains close to the angular acceptance of the biatom filter, Δθ bi , but is well above that of the monoatom filter, Δθ mono . In this last case, it leads to a weaker coupling of the beam with the waveguide. On figure 3 we represent the magnetic field intensity in the filters at their respective resonant wavelengths λ mono = 3.885 µm and λ bi = 3.905 µm, under the focused beam described previously. On figures 3.a (monoatom filter) and 3.b (biatom filter) we show cross-sections of the intensity inside the waveguide in linear color level. In 3.c and 3.d, we give another representation of the same magnetic field intensities with plots (blue lines) along a horizontal cut in the middle of the waveguide.These intensities were normalized in order to compare their lateral extension with that of the intensity of the incident beam (green lines), because the maximum intensity inside the waveguide is way higher than in the incoming beam (67 times for a monoatom filter and 114 times for a biatom filter).

It is noteworthy that the field confinement is better in a biatom filter than for a monoatom. If w 0 = 9 µm for the incident beam, then the field extension (spatial FWHM of the envelope of the magnetic intensity) inside the waveguide for both filters are w bi = 9.6 µm and w mono = 13.0 µm. Similar calculations have been led for various values of w 0 , ranging from 5 µm to 30 µm, at resonant wavelength. The values of w bi and w mono have been calculated for each beam, and are plotted on figure 4.a), respectively in green and blue, while w 0 the spatial FWHM of the incident beam is plotted in red. Normalized incoming beam intensity is represented in inset, for two different values of w 0 : 9 µm (value considered in figures 2 and 3) and 25 µm. For w 0 = 5 µm, the field extension for the two filters are w mono = 11.5 µm and w bi = 6.3 µm. Both field extensions converge asymptotically towards w 0 as the beam widens. While the responses of both filters exhibit a hyperbolic behavior as a function of w 0 in the studied range [5; 30] µm, the monoatom filter exhibits a wider field extension than the biatom. This is due to its lower angular acceptance: when θ ap > Δθ, there is a widening of the field extension inside the waveguide because the the highest orders of the decomposition (corresponding to the most tilted plane waves) are not efficiently coupled into the waveguide. On the contrary, when θ ap < Δθ every plane wave is coupled into the waveguide and contributes to the interfer-ence leading to a limited field extension.

In figure 4.b) we study the spectral properties of the filters by investigating the transmission at the resonant wavelength for various values of w 0 . For w 0 = 5 µm, the transmissions are T mono = 40.4 % and T bi = 72.8 %. Both transmissions converge towards their respective plane-wave value as the beam widens. Furthermore, the maximum transmission of the biatom filter starts to decrease for a narrower beam than that of the monoatom filter. Indeed, for the same reason that led to widening of the field extension, the transmission only decreases when the aperture of the beam becomes greater than the acceptance of the filter. Consequently, both filters are adequate for loosely focused beam, but biatom filters are more efficient for tighter beams.

In summary, we demonstrated that GMR filters keep their filtering abilities under focused beam. In addition, we have demonstrated that field extension in the waveguide is comparable to the spatial FWHM of the incoming beam, provided that its angular aperture is compatible with the angular acceptance of the filter (this is the reason why the biatom structure should be used preferably for tightly focused beams). The study was lead with an incoming focused beam whose spot size is close to the size of an infrared pixel (roughly 15 µm to 30 µm). We showed that for such a small spot size, the magnetic field is contained in the pixel area too. This opens new possibilities for mosaics of pixel-sized infrared spectral filters. Because it overcomes technological limits shown by mosaic of thin-film pixelsized filters [21,22], subwavelength gratings could be a new technological solution to achieve colored pixels in the infrared, thus leading to multispectral infrared focal plane arrays.

  Δθ mono = 7°Δθbi = 20°F ig. 1. a) Cross-section of a single pattern of a monoatom filter. b) Cross-section of a single pattern of a biatom filter. c) Spectra corresponding to infinite-sized filters of patterns described in 1.a) and 1.b) with the parameters: d mono = 2.075 µm, a mono = 0.16 µm, d bi = 2.8 µm, a 1,bi = 0.19 µm, a 2,bi = 0.67 µm,t m = 100 nm, t d = 650 nm illuminated by a TM plane wave. d) Evolution of the transmission with the angle of incidence of the plane wave at resonant wavelength λ = 3.905 µm.

  1.d).

Fig. 2 .

 2 Fig. 2. a) Spectra of the mono-atom and bi-atom Dual Metallic Gratings described in figure 1, under a plane wave illumination and for a focused beam. Plane wave and focused beam spectra are overlapping for the biatom filter. b) Schematic view of the periodic structure and the focused beam. The angle θ ap corresponds to the angular aperture of the beam.

Fig. 3 .

 3 Fig. 3. a) and b) Cross section of the magnetic field intensity in a monoatom (a) or biatom (b) filter under a monochromatic focused beam of waist w 0 = 9 µm and wavelength 3.885 µm (a) or 3.905 µm (b). c) and d) In blue: |H y | 2 along the x-axis for a monoatom (c) or biatom (d) filter. Incident beam is represented in green. Curves are normalized at x = 0.

Fig. 4 .

 4 Fig.4. a) Magnetic field extension inside the waveguide at the resonant wavelength for the monoatom (blue) and biatom (green) filters described in figure1, under focused beam illumination, as a function of w 0 . The inset shows the shape of the incoming beam at w 0 = 9 µm and 25 µm b) Transmission maximum of filters.