Fran De Aquino

Control of the Gravitational Energy by means of Sonic Waves

Keywords: Gravitational Energy Control, Gravitational Mass, Sonic Waves, Sound Pressure

published or not. The documents may come

Control of the Gravitational Energy by means of Sonic Waves

Fran de Aquino

The quantization of gravity showed that the gravitational mass m g and the inertial mass m i are correlated by means of the following factor [START_REF] De Aquino | Mathematical Foundations of the Relativistic Theory of Quantum Gravity[END_REF]:

( )

1 1 1 2 1 2 0 0 ⎪ ⎭ ⎪ ⎬ ⎫ ⎪ ⎩ ⎪ ⎨ ⎧ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ - ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ Δ + - = = c m p m m i i g χ
where is the rest inertial mass of the particle and is the variation in the particle's kinetic momentum; is the speed of light.

0 i m p Δ c
When an electromagnetic wave strikes an atom, it interacts electromagnetically with the atom, acting simultaneously on all its structure. Unlike a sonic wave that strikes the internal particles of the atom isolatedly, interacting mechanically with them. Thus, if a lamina of monoatomic material, with thickness equal toξ contains atoms/m n 3 , then the number of atoms per area unit is ξ n . Thus, if the sonic wave with frequency incides perpendicularly on an area of the lamina it reaches f S ξ nS atoms. Consequently, the wave strikes on ξ ZnS orbital electrons * ( Z is the atomic number of the atoms). Therefore, if it incides on the total area of the lamina, , then the total number of electrons reached by the radiation is

f S ξ f ZnS N = .
* Assuming that, all of them are reached by the sonic wave.

The number of atoms per unit of volume, , is given by n ( ) After these collisions, it carries out with the other orbital electrons (See Fig. 1).

2

electron S e

Wave

Thus, the total number of collisions in the volume ξ S is † The diameter of the electron and protons depends on the region where it is placed.

=

Since in each collision a momentum λ h ‡ is transferred to the atom, then the total momentum transferred to the lamina will be

( ) λ h N n p 1 = Δ
. Therefore, in accordance with Eq. ( 1), we can write that

( ) ( ) ( ) ( ) 6 1 1 2 1 1 1 2 1 2 0 2 0 1 0 ⎪ ⎭ ⎪ ⎬ ⎫ ⎪ ⎩ ⎪ ⎨ ⎧ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ - ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ + - = = ⎪ ⎭ ⎪ ⎬ ⎫ ⎪ ⎩ ⎪ ⎨ ⎧ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ - ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ + - = λ λ c m h N n c m h N n m m i collisions phonons total i l i l g Since Eq. (3) gives ξ S n N l collisions =
, we get

( ) ( 7
2 ξ S n hf P N n l collisions phonons total ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ =
) Substitution of Eq. ( 7) into Eq. ( 6) yields ( λ is the wavelength). Thus, the momentum carried out by a phonon is

( ) ( ) ( ) ( ) 8 1 1 2 1 2 0 2 0 ⎪ ⎭ ⎪ ⎬ ⎫ ⎪ ⎩ ⎪ ⎨ ⎧ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ - ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + - = λ ξ c m h S n hf P m m i l l i l g ‡ Phonon is a quantum of vibrational energy. The
λ λ ε h f hf v p = = =
. Thus, the expression of the momentum carried out by the phonon is similar to the expression for the momentum carried out by the photon [START_REF] Riessland | The Physics of Phonons[END_REF]. Substitution of P given by Eq. ( 4) into Eq. ( 8) gives

( ) ( ) ( ) ( ) 9 1 1 1 2 1 2 0 2 0 ⎪ ⎭ ⎪ ⎬ ⎫ ⎪ ⎩ ⎪ ⎨ ⎧ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ - ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + - = λ ξ c m S n f D S N m m l i l e f l i l g Substitution of ( ) e f l f S n Z N φ ≅
and into

Eq. ( 9) results

e f S N S = ( ) ( ) ( ) ( ) 10 1 1 1 2 1 2 2 0 2 2 2 3 2 0 ⎪ ⎭ ⎪ ⎬ ⎫ ⎪ ⎩ ⎪ ⎨ ⎧ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ - ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + - = λ ξ φ cf m D S S n Z m m l i e e f l l i l g where ( ) ( ) ( ) l l l i V m ρ = 0 .
The speed of the sound, v , as a function of frequency, , and wavelength, f λ , is given by f v λ = , (phase velocity) [START_REF] Ference | Analytical Experimental Physics[END_REF]. When the sonic wave propagates itself through the lamina its velocity is modified and becomes

( ) ( ) l r l r n f n v v λ = = mod
, where is the sonic refractive index of the lamina, which can be expressed by the following equation:

( ) l r n ( ) a la air l r v v n min = . Since f v mod mod λ =
, where mod λ is the modified wavelength, then we can write that

( ) ( ) ( ) 11 mod l r l r n f v n = = λ λ
Substitution of λ by mod λ into Eq. (10) yields where P is the pressure of the sonic radiation [START_REF] Ference | Analytical Experimental Physics[END_REF], then substitution of Eq. (15) into Eq. ( 14) gives

( ) ( ) ( ) ( ) ( ) 12 1 1 2 1 2 2 0 2 2 2 3 2 0 ⎪ ⎭ ⎪ ⎬ ⎫ ⎪ ⎩ ⎪ ⎨ ⎧ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ - ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + - = v f n cf m D S S n Z m m l r l i e e f l l i l g ξ φ Considering that ( ) ( ) ξ ρ α S m l l i = 0 , we obtain ( ) ( ) ( ) ( ) ( ) 13 1 1 2 1 2 2 2 2 2 2 4 4 4 6 2 2 0 ⎪ ⎭ ⎪ ⎬ ⎫ ⎪ ⎩ ⎪ ⎨ ⎧ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ - + - = = v f c S D S S n n Z m m l e e f l l r l i l g α ρ φ χ For α S S f = we obtain ( ) ( ) ( ) ( ) ( ) 14 1 1 2 1 2 2 2 2 2 4 4 2 6 2 2 0 ⎪ ⎭ ⎪ ⎬ ⎫ ⎪ ⎩ ⎪ ⎨ ⎧ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ - + - = = v f c D S S
( ) ( ) ( ) ( ) ( ) 16 1 4 1 2 1 4 2 2 4 4 4 4 2 6 2 2 0 ⎪ ⎭ ⎪ ⎬ ⎫ ⎪ ⎩ ⎪ ⎨ ⎧ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ - + - = = v f c P S S n n Z m m l e e l l r l i l g ρ φ χ α
This equation, deduced for phonons, is only valid for solids § , unlike the correspondent equation deduced for photons, which is valid for solid, liquid and gases.

The speed of the sound for pressure waves in solid materials is given by ( )

17 ρ Y v solid =
where Y is the Young's modulus.

Aerogels are solids with high porosity (<100 nm), with ultra low density (~3 Kg/m3 or less) and with ultra low sound speed (~110m/s) [START_REF] Thapliyal | Aerogels as Promising Thermal Insulating Materials: An Overview[END_REF]7,[START_REF] Jing | Thermal conductivity of a ZnO nanowire/silica aerogel nanocomposite[END_REF]. We can take Eq. ( 16) for a hypothetic aerogel with This shows that under these conditions, the weight of the lamina In this case, the weight of the lamina besides to be inverted, it is intensified 85.2 times. Thus, by controlling the magnitude of the gravitational mass is then possible to control the gravitational energy, gravity, etc.

Vulcanized Rubber can be as advantageous as Aerogels. In this case we have:

; ;

  matter density of the lamina (in kg/m 3 ) and A is the molar mass(kg/kmole).When the sonic wave incides on the lamina, it incides front electrons, where section area of one atom.

Fig. 1 -

 1 Fig. 1 -Collisions inside the lamina.

  The power density, , of the sonic radiation on the lamina can be expressed byDWe can express the total mean number of collisions in each orbital electron, , by means of the following equation 1

  bulk density nor the skeletal density it is the specific mass of the part solid) ; a phonon is a mechanical excitation that propagates itself through the crystalline network of a solid.

  Rubber is synthetic cispolyisoprene. Based on its chemical structure, we can calculate the value of . The result is sound waves cover sounds beneath 20 Hz down to 0.001 Hz.


Lois Mignard-Debise

Ivo Ihrke

Light-Field Microscopy with a Consumer Light-Field Camera

We explore the use of inexpensive consumer lightfield camera technology for the purpose of light-field microscopy. Our experiments are based on the Lytro (first generation) camera. Unfortunately, the optical systems of the Lytro and those of microscopes are not compatible, leading to a loss of light-field information due to angular and spatial vignetting when directly recording microscopic pictures. We therefore consider an adaptation of the Lytro optical system.

We demonstrate that using the Lytro directly as an ocular replacement, leads to unacceptable spatial vignetting. However, we also found a setting that allows the use of the Lytro camera in a virtual imaging mode which prevents the information loss to a large extent. We analyze the new virtual imaging mode and use it in two different setups for implementing light-field microscopy using a Lytro camera. As a practical result, we show that the camera can be used for low magnification work, as e.g. common in quality control, surface characterization, etc. We achieve a maximum spatial resolution of about 6.25µm, albeit at a limited SNR for the side views.

Introduction

Light-field imaging is a new tool in the field of digital photography. The increasing interest is shown by the recent development of several hardware systems on the consumer market (Lytro, Raytrix, Picam), and applications in the research domain (stereo-vision, panoramic imaging, refocusing). The commercial systems are reliable, functional and inexpensive. However, they are designed for the imaging of macroscopic objects. In this article, we explore the use of commercial light-field cameras for microscopic imaging applications.

The light-field microscope has been introduced and improved by Levoy et al. [9,[START_REF] Levoy | Recording and Controlling the 4D Light Field in a Microscope using Microlens Arrays[END_REF][START_REF] Broxton | Wave Optics Theory and 3-D Deconvolution for the Light Field Microscope[END_REF]. While its conceptual details are well understood, its practical implementation relies on the fabrication of a custom micro-lens array, which presents a hurdle for experimenting with the technology. In this article, we demonstrate the use of the Lytro camera, an inexpensive consumer-grade light-field sensor, for microscopic work. We achieve an inexpensive and accessible means of exploring light-field microscopy with good quality, albeit at a reduced optical magnification.

As we show, the major problem in combining the Lytro and additional magnification optics (in addition to f-number matching), is the loss of information due to spatial vignetting. Our main finding is the possibility of using the Lytro in what we term an inverse regime: in this setting the camera picks up a virtual object that is located far behind its imaging optics. To our knowledge, this is the first time that such a light-field imaging mode is described.

We investigate two different setups based on this inverse regime that do not suffer from spatial vignetting: 1) Our first option enables the use of the Lytro camera in conjunction with an unmodified microscope by designing an optical matching system. 2) The second option uses the Lytro behind a standard SLR lens in a macrography configuration to achieve macro light-field photography.

The paper is organized as follows. In Section 3, we study the compatibility of both optical systems involved in the imaging process: the light-field camera and the microscope. We discuss the implications of the combination of both systems in terms of both spatial and angular resolution. We then explore the Lytro main optical system and show how to adapt it to the microscopic imaging context, Sects. 5 and 6. Finally, we evaluate and compare the different solutions and present application scenarios, Sect. 7.

Related Work

Light-field imaging requires the acquisition of a large number of viewpoints of a single scene. Two types of approaches exist, either using multiple sensors or a single sensor in conjunction with temporal or spatial multiplexing schemes.

Taking a picture with a conventional camera is similar to a 2D slicing of the 4D light-field. Repeating this operation with a planar array of cameras offers sufficient data to esti- mate the light-field function. Calibration, synchronization, as well as the available bandwidth of the camera hardware are the major determining features of this approach. More details can be found, e.g., in [START_REF] Yang | A real-time distributed light field camera[END_REF]23]. The strongest limitation of this approach for microscopic applications is the large size of the corresponding setups.

Photographs of specimens

An alternative method for light-field capture is to multiplex the different views onto a single sensor.

Temporal multiplexing is based on taking several pictures over time after moving the camera around static scenes. The movements can be a translation or rotation of the camera. Alternatively, mirrors [START_REF] Ihrke | Fast incident light field acquisition and rendering[END_REF][START_REF] Taguchi | Axial light field for curved mirrors: Reflect your perspective, widen your view[END_REF] can be moved to generate additional virtual viewpoints. Another alternative implementation are dynamic apertures [START_REF] Liang | Programmable aperture photography: Multiplexed light field acquisition[END_REF].

Spatial multiplexing allows to record dynamic scenes. Parallax barriers and integral imaging [12] are historically the first approaches to spatially multiplex the acquisition of a light-field, trading spatial resolution for angular resolution. A modern elaboration of this approach where the sensor and a micro-lens array are combined to form an incamera light-field imaging system is the Hand-Held Plenoptic Camera [START_REF] Ng | Light field photography with a handheld plenoptic camera[END_REF]. Alternatively, sensor masks [22,[START_REF] Veeraraghavan | Non-Refractive Modulators for Encoding and Capturing Scene Appearance and Depth[END_REF] or a light pipe [START_REF] Manakov | A Reconfigurable Camera Add-On for High Dynamic Range, Multispectral, Polarization, and Light-Field Imaging[END_REF] can be arranged such that in-camera lightfields can be recorded. Other methods use external arrays of mirrors instead of lenses [7], or external lens arrays [4].

Microscopy is a vast subject and many different illumination and observation schemes have been developed in the past. A general overview is given in [START_REF] Murphy | Fundamentals of Light Microscopy and Electronic Imaging[END_REF]; a comprehensive review of microscopy techniques, including light-field microscopy, for the neuro-sciences can be found in [24]. Light-field microscopy was introduced by Levoy et al. [9] and later augmented with light-field illumination [START_REF] Levoy | Recording and Controlling the 4D Light Field in a Microscope using Microlens Arrays[END_REF]. Recently, addressing the large spatial resolution loss implicit in LFM, the group has shown that computational superresolution can be achieved outside the focal plane of the microscope [START_REF] Broxton | Wave Optics Theory and 3-D Deconvolution for the Light Field Microscope[END_REF][START_REF] Cohen | Enhancing the performance of the light field microscope using wavefront coding[END_REF]. Another super-resolution scheme is combining a Shack-Hartmann wavefront sensor and a standard 2D image to compute a high-resolution microscopic lightfield [START_REF] Lu | High-resolution Light-Field Microscopy[END_REF]. LFM has been applied to polarization studies of mineral samples [18] and initial studies for extracting depth maps from the light-field data have been performed in microscopic contexts [START_REF] Lee | 3D Optical Microscopy Method based on Synthetic Aperture Integral Imaging[END_REF][START_REF] Thomas | Depth Map Extraction from Light Field Microscopes[END_REF]. Most of the work today uses the same optical configuration that was introduced in the original implementation [9]. With this article, we aim at providing an inexpensive means of experimenting with LFM.

Background & Problem Statement

Light-field Microscopy

The main function of an optical microscope is to magnify small objects so that they can be observed with the naked eye or a camera sensor. Light-field capabilities such as changing the viewpoint, focusing after taking the picture, and achieving 3D reconstruction of microscopic sam-ples rely on the number of view points that can be measured from a scene. This number is directly linked to the objectside numerical aperture NA o of the imaging system that is used as an image-forming system in front of the micro-lens array of the light-field sensor and the micro-lens f-number. The object-side numerical aperture is defined as

NA o = n sin(α), (1)
where n is the index of the material in object space (usually air, i.e. n = 1). The numerical aperture quantifies the extent of the cone of rays originating at an object point and being permitted into the optical system (see Fig. 2). Microscope objectives usually have a high NA o , because it is directly linked to better optical resolution and a shallower depth of field. A high NA o is also important for light-field microscopy as the base-line of the light-field views is directly linked to it. Details on how to design a light-field microscope can be found in [9]. The most important aspect is that the f-number of the micro-lens array matches the f-number of the microscope objective. The f-number F of any optical system is defined as the ratio of its focal length f over the diameter D of its entrance pupil (see Fig. 2)

F = f D . (2)
For a microscope of magnification M microscope and numerical aperture NA o , a more appropriate equation taking the finite image distance into account can be derived [START_REF] Born | Principles of optics[END_REF] from Eq. 2:

F microscope = M microscope 2NA o . ( 3
)
The majority of microscope objectives has an f-number between 15 and 40. In our experiments, we use a 10× objective with an f-number of F = 20.

Figure 2. Properties of an optical system. An optical system is defined by its focal length f , its principal planes H and H , and the diameter D of its entrance pupil. Two conjugate planes define a unique magnification M , the ratio of the image size y over the object size y. All optical equations can be found in [START_REF] Born | Principles of optics[END_REF].

Lytro Features

The Lytro camera is made of an optical system that is forming an image in the plane of a micro-lens array that is, in turn, redirecting the light rays to a sensor. It has a 3280 × 3280 pixel CMOS sensor with 12-bit A/D and 1.4µm × 1.4µm pixels [6]. Each micro-lens has a diameter of 14µm which is equivalent to 10 pixels. The microlenses are packed on a hexagonal lattice (see Fig. 3

(left)).

The effective spatial resolution is therefore 328×328 pixels whereas the angular sampling rate is 10 × 10 values. The Lytro's main objective lens has a fixed f-number of F = 2 and features an 8× optical zoom. We explore its potential as an imaging parameter in Sect. 5. Another important feature of the objective lens is that it can focus from 0mm to infinity.

F-number Mismatch

A prerequisite for non-vignetted imaging (see also Sect. 5) is that the f-number of the micro-lens array and that of the microscope objective match. This is the solution employed in conventional light-field microscopy [9,18,[START_REF] Levoy | Recording and Controlling the 4D Light Field in a Microscope using Microlens Arrays[END_REF]. A custom F = 20 micro-lens array (MLA) is typically employed as it is compatible with a large number of existing microscope objectives. However, this F = 20 MLA is not readily available and has to be custom manufactured.

Since the Lytro is designed for macroscopic imaging, its f-number (F = 2) is not adapted to the microscopic situation. If we were to use the Lytro micro-lens array as is, the angular sampling in one direction would be divided by the ratio of the f-number of the two systems and only one pixel would be lit under each micro-lens (instead of approximately one hundred) (see Fig. 5 (bottom)). This would remove any interest for light-field purposes since only a single view would be recorded and 99% of the sensor would remain unused, see Sect. 5 for examples.

Therefore, in order to successfully use the Lytro camera for microscopic imaging, the f-numbers of the two optical systems need to be adapted. There is, however, a tradeoff. The optical invariant, a fundamental law of optics (see e.g. [START_REF] Born | Principles of optics[END_REF]), states that for two conjugate planes, the product between the sine of the angle at which light rays reach a conjugate plane (α and α ) and the size of the object in this plane (y and y ) is equal at both planes.

y n sin(α) = y n sin(α ), (4) n and n are the optical index of the media on both sides of the optical system. For air, the index is equal to 1. We therefore opt for an optical demagnification scheme, decreasing y , to increase the angular size of the cone of light rays α that is incident on the Lytro's light-field sensor. Theoretically, we need to divide the microscope objective's f-number by 10 to reach the same f-number as the Lytro camera. An immediate consequence from Eq. 3 is that the combination of all optical elements must therefore have a magnification divided by 10, i.e. we are aiming to convert the system to unit-magnification. Due to the small size of the Lytro's micro-lenses, the optical resolution of the system is still satisfactory, even at this low magnification (see Sect. 7).

The magnification of the combined system M f inal can be written as the product of the magnifications of each individual system:

M f inal = M microscope M lens1 ... M lens N M Lytro , (5)
where M lensi, i=1..N indicates the magnification of N to-be-designed intermediate lens systems. The microscope objective has a fixed magnification of M microscope = 10, whereas the lowest magnification setting of the Lytro has a value of M Lytro = 0.5. The resulting M f inal = 5 without additional optical components (N = 0) is too large to prevent angular information loss.

We explore two different options (see Fig. 4) to implement the adapted system. The first option (see Sect. 5) is to demagnify the image of the microscope with additional lenses (N = 2). This solution lets us use the microscope and the Lytro camera unmodified. The second option is to remove the microscope, replacing it by an SLR lens in macro-imaging mode. Here, we compare a setup with and without the Lytro optics (see Sect. 6).

Sensor Coverage

Vignetting: When using two or more optical systems in conjunction, some light rays are lost because the pupils of the different systems do not match each other. This effect is called vignetting. Generally, there are two types of vignetting: spatial and angular vignetting.

Spatial vignetting directly translates into a loss of field of view, which may reduce the image size at the sensor (see Fig. 5 (top)). Angular vignetting (see Fig. 5 (bottom)) occurs when the cone of rays permitted through one of the systems is smaller than for the other system, e.g. due to a stop positioned inside the system. Angular vignetting is not an issue in a standard camera : it only affects the exposure and is directly linked to the depth of field of the camera. In a light-field camera, however, it is crucial to minimize angular vignetting in order to prevent the loss of directional light-field information. We propose to measure the vignetting in terms of its adaptation to the recording light-field sensor. An ideal optical system that is adapted to a particular sensor would fully cover all its sensor elements. Since the raw pixel resolution is divided into spatial and angular parts, the sensor coverage c sensor can be approximately expressed as

c sensor [%] = c spatial [%] × c angular [%], (6)
where c spatial is the spatial coverage of the sensor in percent, and c angular is the angular coverage of one micro-lens sub-image, also in percent. In our experimental validation, we measure the spatial coverage in the center view and the angular coverage in the center lenslet. This choice is motivated by the simpler estimation of the relevant coverage areas as compared to using the side views/edges of the field. The measure can be considered to be an approximation of the upper bound of the system space-bandwidth product, i.e. the optical information capacity of the system [13].

Unmodified Use of the Lytro's Main Optics

The main optics of the Lytro camera, i.e. the optics without the micro-lens array, is designed to avoid angular vignetting when imaging onto the micro-lens array, i.e. the micro-lens array and the main optics have been designed with the same f-number of F = 2. We have observed that the main optical system can be used in two different ways with a microscope. These two imaging regimes can be used differently in designing an optical matching system.

Regular regime: The Lytro camera can image a plane as close as the first surface of its optics for a zoom level of 1×. This minimal focus distance increases with the zoom level. In order to use the Lytro with the microscope, it has to be positioned such that the near focus of the camera is placed at the image plane of the microscope objective. Since the image size y of the microscope objective is rather large (typically around 50mm × 50mm) whereas the Lytro's entrance pupil is only ≈ 20mm in diameter, spatial vignetting incurs a loss of sensor coverage of up to 94% as shown in Figure 7 (left). The angular vignetting is stronger with only 16% angular coverage.

Inverse regime: We discovered that, in a specific configuration where the camera is set to focus to the closest possible plane for a zoom level of 1×, the camera can enter into a virtual object regime. The camera is then able to image an object plane that is located behind its first lens, i.e. in the direction of the sensor (see Fig. 6). This configuration enables the positioning of the camera close to the microscope objective and therefore reduces the spatial vignetting since a larger number of rays can be captured by the lens surface (see Fig. 7 (right). This mode of operation inverts the image.

For both imaging regimes described above, the magnification M Lytro (≈ 0.5) is not low enough for achieving a good angular coverage. While the spatial vignetting problem can be successfully addressed with the inverse regime, the angular vignetting can only be dealt with by using a low magnification optical system. We therefore investigate the different options.

The Lytro Microscope

Our first option to achieve the matching of the f-numbers discussed in Section 3.3, consists in designing an optical de- The object is a blue LCD display. It consists of a square black grid that is separating the different pixels of 0.5mm × 0.5mm size. We hypothesize the white dots to be bubbles inside the liquid crystal. The large images show the equivalent camera image computed from the light-field, while the small images on the bottom right show a close-up on the micro-lens images of the raw sensor data for different zoom-levels. Spatial and angular vignetting are easily observed in the equivalent camera and micro-lens images, respectively. The zoom level setting is 1× (regular regime) on the left and 5× (inverse regime) on the right. The spatial vignetting is strong in the regular regime (6% of spatial coverage), while it is greatly improved in the inverse regime (100% of spatial coverage). Angular coverage is similar in both cases (c angular ≈ 25%).

magnification system (placed between the microscope objective and the Lytro) that increases the angular extent of the light (see Eq. 4). This solution keeps desirable properties like the large numerical aperture of the microscope and its fixed working distance, while at the same time, the Lytro camera can remain unmodified. The major task is to find a good trade-off between the vignetting and the magnification of the resulting light-field microscope.

Our best solution along this direction employs two lenses. This configuration serves two goals: 1) to decrease the magnification successively, simplifying the task of each individual lens, and 2) to move the image behind the Lytro camera so that it can be used in the inverse regime which offers a better spatial coverage c spatial . The ray-diagram in Fig. 8 illustrates the two-lens setting: an intermediate image that is slightly demagnified is created in front of the microscope. Then, the second lens creates a further demagnified image behind the Lytro camera. The Lytro camera is operated in its inverse imaging regime in order to pick up this virtual image. We determine the positions and focal lengths of the additional lenses using the following equation derived from the thin lens formula:

x = f M -1 M (7)
The focal length f and the position of the lens x are chosen according to the desired magnification M (negative because the image is inverted). In our implementation, the first additional lens has a focal length of 50mm and is put close to the microscope objective. The second additional lens has a focal length of 85mm and is put close to the Lytro. The effect of using two lenses is that the individual focal lengths are larger and the aberrations are reduced.

Figure 8. Ray-diagram of the system using two additional lenses. Each sub-system is indicated in a different color and has to be interpreted independently from the other sub-systems. Their operation can be understood in sequence: the objective images object AB to A B , the first additional lens images A B to A1B1 so that the second lens images A1B1 to A2B2 behind the Lytro camera. Finally, the Lytro's main optics in the inverse regime images this virtual object A2B2 to its sensor in the plane A B .

Macro Lytro

Our second option is to use a single SLR camera lens in front of the Lytro to achieve unit magnification. (M f inal = M SLR M Lytro = 1). As for the microscope, and because the focal length of the SLR lens is large (50mm and 100mm in our experiments), we want to use the Lytro in the inverse regime to keep the spatial coverage as high as possible. In practice, the Lytro is set as close as possible to the SLR lens. A variation of this setup is to remove the Lytro optics, and only use the SLR lens so that M f inal = M SLR = 1.

These designs have only one or two optical components and relieve the hurdle of undoing the work of the microscope objective with many lenses. However, the SLR lens is not specifically designed for the magnification of close objects and its aperture is not meant to be maintained at a constant value for the micro-lens array. The relations used to establish Eq. 3 are not valid in this macro-configuration. Instead, the relevant quantity is the working f-number F w [START_REF] Born | Principles of optics[END_REF] which is the f-number modified by the magnification M :

F w = 1 2NA i = (1 -M )F (8)
where NA i is the image-side numerical aperture NA i = n sinα . The minimal value of working f-number that can be achieved with a camera lens is close to F w = 2. It is reached for a limit f-number of F = 1 and a magnification of M = -1. This condition would be optimal for the Lytro micro-lens array. However, commercial lenses usually have a limit f-number between 1.4 and 3.5 increasing the working f-number to between 2.8 and 7.0. Compared to the light-field microscope, on the one hand, this setup is more versatile. The magnification can be set to the desired value by simply moving the object and the micro-lens array. It does not require the difficult alignment of several optics. On the other hand, this setup does not benefit from the structure of the microscope that already includes lighting and moving the sample through micrometer stages in three dimensions. In addition, the working distance is not fixed which changes the magnification as well as the object-side numerical aperture when moving the sample. However, the strong point of this design is its accessibility. Building a light-field macrography setup is done quickly and without the need for a deep understanding of the operating principles of a microscope.

Results

Before showing results, we describe and compare the different implementations. The different sensor coverages as well as their spatio-angular coverage values can be found in Fig. 9. It is clearly visible that directly using the Lytro camera in its regular imaging regime is unsuitable for microscopic light-field imaging. The inverse imaging regime improves on the spatial coverage, but the angular coverage is limited. The best combination is achieved with a 50mm SLR lens in front of the Lytro which yields the best overall sensor coverage c sensor .

Resolution Test Chart

In order to compare the resolution of the two different techniques, we use a 1951 USAF resolution test chart. The results of the experiments are summarized in Fig. 10.

The first option (Sect. 5) "Lytro Microscope" was implemented using a Canon 50mm SLR lens and a Nikon 85mm Figure 10. This table summarizes the results from the experiments of Sect. 5 and Sect. 6. Theoretical resolution is derived from the micro-lens pitch. The micro-lens footprint is the size of a microlens in object space.

SLR lens as additional lenses. Those lenses were put on top of a Leitz Ergolux microscope using an objective of magnification 10× with an object-side numerical aperture NA o = 0.2. This microscope has a lens tube with a magnification of 0.8× so the f-number F = 20. The images have been taken with a magnification of 2.88, i.e. a microlens covers 4.87µm in object space (see Fig. 11 (top)). The spatial coverage is above 99% but due to the large magnification the angular coverage is low (between 9% and 25%). The resolution is between 80 and 90 line pairs per mm.

The resolution indicated above is computed for the center view. It decreases with further distance from the center. A loss of image quality due to aberrations can be observed. They are introduced because the observed area is larger than usual for the microscope objective. Microscope objectives are typically designed so that only a reduced inner portion of the full field is very well corrected. In addition, our matching lenses introduce further aberrations. Since the angular vignetting is strong, the contrast of viewpoints far from the center is low. It should be noted that even viewpoints inside the vignetted area can be computed, albeit at a poor signal-to-noise ratio (see Fig. 11).

The second option (Sect. 6) was implemented in three ways: two times with the Lytro placed behind two different lenses, a 50mm and a 100mm Canon SLR lens (referred as SLR + Lytro), and with the Lytro micro-lens array without the Lytro main optics, see Fig. 3 (right), placed behind the 50mm lens (referred as SLR + MLA) (see Fig. 12). Magnifications from 1.26 to 2.34 were achieved. The spatial coverage is always 100% and angular coverage is good (up to 70%). In this case, chromatic aberrations are present which degrades the image. The aberrations are reduced in the SLR + Lytro case as compared to SLR + MLA, since the magnification of the SLR lens is lower in this setting. It is most noticable in the side views since, for these views, imaging is performed through the outer pupil regions of the SLR lens. We suspect that the chromatic aberration is introduced by the SLR lens because it is not intended for macro-imaging. Using a dedicated macro-lens instead would likely remove this effect.

Microscopic Sample

The most direct application of the light-field microscope is the study of microscopic samples. The low magnification and the large field of view allow us to see in detail an object area that is between 1.5mm×1.5mm and 3.5mm×3.5mm with a magnification of 2.88× and 1.3× respectively. Cell tissues or rough surfaces of different materials have a structure close to the millimeter so high magnification is not always necessary to analyze them.

We illustrate this technique in Fig. 1 (right). Several images of microscopic specimen were taken with the same settings as in the previous section. The magnification is 2.88× and we can clearly see the structure of different kind of surfaces that are invisible to the naked eye.

The light-field data allows for the reconstruction of the depth of the sample when the number of views is sufficient. We took a picture of a ground truth aluminium staircase with stairs of 1.00mm width and 0.50mm height with an accuracy of ±5µm with the 50mmSLR + Lytro setup. We obtained the depth map in Fig. 13 using a modified variational multi-scale optical flow algorithm for light-field depth estimation [START_REF] Manakov | A Reconfigurable Camera Add-On for High Dynamic Range, Multispectral, Polarization, and Light-Field Imaging[END_REF]. Although only a small slice of the staircase is in focus (the depth of field is 1mm), the depth can be computed outside of this area. Essentially, out-offocus regions are naturally considered as a different scale by the algorithm, so, the estimation of the depth of the closest and furthest steps is correct. This behavior nicely interacts with the scene properties since the parallax is larger in out-of-focus regions. The optical system can be seen as supporting the part of the algorithm that is handling large dis- Top: Images taken with the "Lytro Microscope" (Sect. 5). The magnification is 2.88. Bottom: Images taken with the "50mmSLR + Lytro" (Sect. 6). A red-green color shift due to strong chromatic aberrations can be observed in the side views. Note that the top row has a higher resolution: it shows the pattern that is visible in the center of the bottom row (level 4 and 5). The contrast of the images of the same row have been set to a similar level for comparison.

placements. Since the detailed properties of the Lytro main optics are unknown, it is necessary to adjust the scale of the computed depth. We use the aluminium staircase as reference. We find the affine transformation between the depth map and the staircase model by fitting planes to match the stairs. After the transformation, we measure an RMS error of 75µm for vertical planes and 17µm for horizontal planes. The difference is due to the different slopes of the horizontal and vertical steps with relation to the camera view direction. The magnification is equal to 1.32. We also applied this depth reconstruction to a daisy head (see Fig. 13). In practice, the depth inside a cube of about 3.5 × 3.5 × 3.5mm 3 can be estimated, which is rather large for a microscopic setting.

Discussion and Conclusion

We have developed and tested several adaptations of the Lytro consumer light-field camera to enable an entrylevel experimentation with light-field microscopy. While the fixed f-number of the Lytro's micro-lens array prevents its direct use with a standard microscope (regular regime), it is possible to trade the overall system magnification for light-field features and to avoid spatial vignetting with the inverse imaging regime. Lytro microscopy is therefore an option for low-magnification work as is common in industrial settings, or for investigations into the meso-and largescale micro-structure of materials. Even though an optical magnification between 1 and 3, as achieved in this work, appears to be low, the small size of the micro-lenses still yields a decent optical resolution of up to 6.25µm in object space which already shows interesting optical structures that are imperceivable by the naked eye. For the future, we would like to investigate image-based denoising schemes for the vignetted side-views, as well as algorithmic developments for structure recovery. In terms of applications, the imaging of micro-and meso-BRDFs and their relation to macroscopic BRDF models appears to be an interesting development.

Figure 1 .

 1 Figure 1. Left: Microscope setup with the Lytro camera on top of two additional SLR lenses. Images of several samples have been taken under different illumination conditions. Magnification is 3.0. Top and bottom left: brushed steel from scissors blade with lighting from the right and bottom. Top middle left: Scratched surface of a piece of metal. Bottom middle left: Plastic surface with highly retro-reflective properties. The material is made of micro bubbles of transparent plastic that are invisible to the naked eye. Top middle right: Fabric with a hexagonal structure. The lighting is coming from the side and casts shadows and strong highlights on the three-dimensional structure of the fabric. Bottom middle right: Highly retro-reflective material from security reflective tape. Top right: Tonsil tissue with bright field illumination. Bottom right: Pins of a electronic component on a circuit board.

Figure 3 .

 3 Figure 3. Left: Image of the micro-lens array taken with a microscope with top illumination. We can see the hexagonal structure of the micro-lens array. The bright dot in the middle of each hexagon is an image of the light source reflected by the surface of the microlens. Right: The Lytro camera without its optics.

Figure 4 .

 4 Figure 4. The diagram shows our two proposed solutions to achieve low magnification. The first option (first row) keeps the camera and the microscope intact. The second option (second row) replaces the microscope with an SLR lens.

Figure 5 .

 5 Figure 5. Top: Spatial vignetting occurs when light rays from the object (in red) do not pass through the second lens. For nonvignetted imaging, light rays (in blue) converging to a point at the sensor edge should include all the red light rays emerging from the object. Bottom: Representation of the sampling of a cone of light rays by a F = 2 micro-lens. Green rays symbolize the angular cone that can be acquired by the micro-lens, while blue rays emerge from an F = 20 optical system. As can be seen from the figure, angular vignetting prevents an effective sensor utilization.

Figure 6 .Figure 7 .

 67 Figure 6. Top: Setup for the inverse regime configuration. Bottom: Distance of Lytro focus plane to its front lens with the variation of zoom. An abrupt change of position of the zoom lens group occurs for a zoom level of 3.7×, enabling to switch between the regular and inverse regime.

Figure 9 .

 9 Figure 9. Combined results of the experiments from Sect. 5 and Sect. 6.

Figure 12 .

 12 Figure 12. Setup from the SLR + Lytro experiment using a 100mm SLR lens (100mm SLR + Lytro).

Figure 11 .

 11 Figure 11. Set of different viewpoints of the resolution target with the center view in the middle (the red dot in the top left inset indicates the position of the view).Top: Images taken with the "Lytro Microscope" (Sect. 5). The magnification is 2.88. Bottom: Images taken with the "50mmSLR + Lytro" (Sect. 6). A red-green color shift due to strong chromatic aberrations can be observed in the side views. Note that the top row has a higher resolution: it shows the pattern that is visible in the center of the bottom row (level 4 and 5). The contrast of the images of the same row have been set to a similar level for comparison.

Figure 13 .

 13 Figure 13. Sub-view (top left) and computed depth map (top middle) of a daisy flower. Sub-view (bottom left) and computed depth map (bottom middle) of the aluminium staircase. The bottom right picture is a 3D visualization of the depth map after the calibrating and the top right picture is a projection of a region in the middle of the calibrated depth map onto the xz plane showing the plane fits in red.

Acknowledgements

We would also like to thank Patrick Reuter for his helpful comments. This work was supported by the German Research Foundation (DFG) through Emmy-Noether fellowship IH 114/1-1 as well as the ANR ISAR project of the French Agence Nationale de la Recherche.


John Restrepo

Pawel J Stoerck

Journal of the Optical Society of America A 1 Ray and Wave Aberrations Revisited: A Huygens-Like Construction Yields Exact Relations

The aberrations of an optical system can be described in terms of the wave aberrations, defined as the departure from the ideal spherical wavefront; or the ray aberrations, which are in turn the deviations from the paraxial ray intersections measured in the image plane. The classical connection between the two descriptions is an approximation, the error of which has, so far, not been quantified analytically. We derive exact analytical equations for computing the wavefront surface, the aberrated ray directions, and the transverse ray aberrations in terms of the wave aberrations (OPD) and the reference sphere. We introduce precise conditions for a function to be an OPD function, show that every such function has an associated wavefront, and study the error arising from the classical approximation. We establish strict conditions for the error to be small. We illustrate our results with numerical simulations. Our results show that large numerical apertures and OPD functions with strong gradients yield larger approximation errors.

INTRODUCTION

The geometrical theory of aberrations adjusts the predictions of paraxial optics to a more realistic depiction of how a real lens performs. Two of its descriptors are the ray aberrations and the wave aberrations. Both concepts are directly related, as has been shown in the classic literature [START_REF] Born | Principles of Optics[END_REF][START_REF] Welford | Aberrations of the Symmetrical Optical System[END_REF][START_REF] Mahajan | Optical imaging and aberrations. Part 1. , Ray geometrical optics[END_REF][START_REF] Malacara | Handbook of lens design, Optical engineering[END_REF]. The ray aberrations describe the deviation between the aberrated rays and the paraxial/ideal rays as a transverse distance measured in the image plane. The wave aberration describes the deviation of the aberrated wavefront as compared to an ideal spherical wavefront that produces a perfect image point. Equivalently, one can consider the wave aberrations as the differences in time of flight of the light along an aberrated ray with respect to the time it would take to reach the image along a paraxial ray, hence the alternative use of the name optical path differences (OPD).

Both concepts, the ray aberrations and wave aberrations are commonly related by means of [START_REF] Born | Principles of Optics[END_REF][START_REF] Wyant | Basic Wavefront Aberration Theory for Optical Metrology[END_REF]

∂W ∂x ≈ - x r
, and

∂W ∂y ≈ - y r , (1)
where W represents the wave aberration (OPD), x and y are the ray aberrations, and r is the radius of the ideal wavefront, also known as the reference sphere.

The approximation in Eqs. [START_REF] Born | Principles of Optics[END_REF] is commonly held to be valid for small numerical apertures and small aberrations. In spite of this, they are widely used for instance in the analysis of the Hartmann-Shack sensor [START_REF] Platt | History and principles of Shack-Hartmann wavefront sensing[END_REF][START_REF] Neal | Shack-hartmann wavefront sensor precision and accuracy[END_REF] or in optical design software.

While exact relations between quantities indirectly related to the aberrations [START_REF] Mejia | Exact relations between wave aberration and the sagitta difference, and between ray aberration and the slope difference[END_REF], or based on re-definitions of the optical path difference [START_REF] Rayces | Exact relation between wave aberration and ray aberration[END_REF], have appeared in the literature, there is, so far, no such relation for the standard definition of the optical path difference as a phase difference along the aberrated ray.

In this paper, we derive exact analytical equations to compute wavefront points, aberrated ray directions and the transverse ray aberrations in terms of the OPD function along the aberrated ray. Whereas the classical equations, Eqs. [START_REF] Born | Principles of Optics[END_REF], are only an approximation, the new equations are applicable to large numerical aperture settings and for arbitrary differentiable OPD functions. We prove the exactness of the equations by validating the defining properties of the wavefront, i.e. the distance to the reference sphere and the orthogonality with the aberrated rays. We show that the classical equations for the ray aberrations are a special case of our equations and detail the conditions for a good approximation. Finally, we evaluate the approximation error quantitatively.

OVERVIEW

The main tool for our derivation is a Huygens-like interpretation of the wavefront as an envelope of spheres with a varying radius that is given by the OPD function, Fig. 2a.

This conception enables us to perform limit considerations that are most suitably studied in the tangent space of the reference sphere. The derivations are initially performed in this local space rather than in exit pupil coordinates. In the new coordinate system, we arrive at exact analytic equations for the wavefront and the aberrated ray directions, which, by means of a suitable transformation can be related to the original exit pupil coordinates. In this scheme, the aberrated ray directions can be computed without differentiating the wavefront.

As a result, we obtain a set of rays with origins at the wavefront that can be propagated to the image plane to compute the exact analytic expressions for the transverse ray aberrations.

We perform the derivation in several steps. First, Sect. 4, we perform the basic geometric construction of a wavefront tangent in one dimension in the canonical setting afforded by the tangent space construction outlined above.

We leave the detailed definition of the required coordinate transformation for the discussion of the two-dimensional case, Sect. 5. We introduce the transformation between global and local coordinate frames, putting special emphasis on the transformation of functions defined in global exit pupil coordinates to the local tangent frame systems.

We then generalize the one-dimensional geometric argument to two dimensions, Sect. 6, and derive local expressions for the wavefront and the aberrated ray directions. We continue by linking these expressions back to exit pupil coordinates, both in their arguments and in their values. We arrive at the key results, Eqs. ( 35) and (37).

Finally, we make a connection to the transverse ray aberrations and the classical approximation, Eqs. (1), elucidating the conditions for a valid approximation. Sect. 8 demonstrates exemplary applications and quantitative properties of the equations.

The Appendix contains a proof of the wavefront properties of the derived quantities and establishes that every OPD function corresponds to a wavefront.

THE WAVE ABERRATION FUNCTION (OPD)

A. OPD Definitions

In the classic literature, there are two different recurrent definitions for the wave aberration function (OPD).

The standard definition [START_REF] Born | Principles of Optics[END_REF][START_REF] Mahajan | Optical imaging and aberrations. Part 1. , Ray geometrical optics[END_REF][START_REF] Wyant | Basic Wavefront Aberration Theory for Optical Metrology[END_REF] considers the path length between points on the aberrated wavefront and on the reference sphere, connected along the direction of the propagation of the aberrated ray. The OPD value is reported for the coordinate of the reference sphere point in the exit pupil plane. Its relation to the ray aberration is stated in Eqs. [START_REF] Born | Principles of Optics[END_REF].

The alternative definition [START_REF] Mahajan | Optical imaging and aberrations. Part 1. , Ray geometrical optics[END_REF][START_REF] Rayces | Exact relation between wave aberration and ray aberration[END_REF] considers the wave aberration to be measured along a radius of the reference sphere, again between two points on the wavefront and on the reference sphere, but with the OPD value assigned to the exit pupil location of the wavefront point.

Both definitions are illustrated in Fig. 1 for a point u on the wavefront. For simplicity, we refer to the first definition as W ray since the path lengths are measured along the ray, while the second one is denoted as W radius . The pupil coordinate x t represents the pupil coordinate position for W ray , while x u is the same for the alternative wave aberration definition. The wave aberration value is given a sign that depends on the delay relation between the wavefront and the reference sphere. For the particular case of point u in Fig. 1, the wavefront is delayed which implies a negative sign of the OPD.

For W radius the resulting relation between ray and wave aberration is different from Eqs. (1) and is reported [START_REF] Rayces | Exact relation between wave aberration and ray aberration[END_REF] as:

∂W radius ∂x = - x r -W radius , and
∂W radius ∂y = - y r -W radius . ( 2
)
The advantage of the W radius definition is a simple computation of the wavefront surface via triangle relationships. However, Eqs. (2) are differential equations as compared to the classical Eqs. (1) and therefore, even though they are exact [START_REF] Rayces | Exact relation between wave aberration and ray aberration[END_REF], difficult to solve.

On the other hand, for the standard definition, W ray , there is no obvious construction of the wavefront surface from the OPD values, since the aberrated ray directions are unknown.

Most authors prefer the W ray definition since it provides a direct connection with the pupil function and the calculation of the point spread function (PSF) [START_REF] Welford | Aberrations of the Symmetrical Optical System[END_REF], but often the distinction is not clearly made.

In this article, we derive exact equations for the wavefront and the aberrated ray directions for the standard OPD definition W ray and use them to make an exact link with the ray aberrations.

B. Properties of the OPD Function

Since the derivations critically depend on the exact properties of the OPD definition, we discuss the interpretation underlying our derivations in detail.

We use the standard definition of the OPD function W ray as given above, simply denoting it as W in the following. In contrast to most of the literature, we interpret the domain of the OPD function to be the reference sphere. The usual parameterization in terms of exit pupil coordinates is, in this sense, one parameterization of the function's domain. Other parameterizations are possible and we will use this insight to define the OPD function in local coordinate systems, Sect. 5. These differently parameterized OPD functions all describe the same quantity, i.e. the phase delay along an aberrated ray intersecting the reference sphere at the position of intersection, just with a different frame of reference.

A direct consequence of the above considerations and the fact that the OPD is single-valued is that no wavefront point can be at a closer distance to the reference sphere than the OPD value. This implies that there is an open ball around any point on the reference sphere that does not contain wavefront points. The radius of this open ball is equal to the OPD value. A sphere with this radius, i.e. the closure of the open ball, is tangent to the wavefront. The wavefront can therefore be considered as the envelope of a set of spheres with a varying radius that is described by the OPD function, a statement of Huygens principle. The concept is illustrated in Fig. 2 a), where R(x) is the reference sphere, W(x) the OPD function, and ϕ(x) the resulting wavefront.

A misconception that is often found in the literature is an ambiguity between the slope of the wavefront and the slope of the OPD function. We emphasize that these two concepts must be distinguished.

A key property of an OPD function is that its gradient magnitude cannot exceed one. To appreciate this point, consider a 1D setting, Fig. 2 b) and c). As mentioned above, the OPD value W(x) implies an open ball without wavefront points surrounding a particular point x on the reference sphere. If the norm of the OPD derivative is greater than one |dW/dx| > 1, the radius of this ball changes more quickly than the evaluation position , i.e. |dW| > |dx|. It follows that one of the balls completely contains the other -which is a contradiction since both balls, by definition, do not contain wavefront points, but are simultaneously tangent to it. It follows that the norm of the OPD derivative cannot exceed one. An intuitive interpretation of this property is that the OPD would be required to be multivalued in this situation. An alternative interpretation is that the wavefront can only be represented as an envelope of balls if the condition on the OPD norm is satisfied.

The illustration in Fig. 2 b) and c) shows the two cases for a finite displacement ∆x. The constraint that one ball does not contain the other yields the triangle inequalities illustrated in the Figure . Passing to the limit as ∆x → 0 yields the condition |dW/dx| ≤ 1.

With these prerequisites, we introduce the following The existence of the second derivatives is a technical requirement for subsequent developments. As implied by the definition, the gradient is to be taken on the reference sphere.

In contrast to the above discussion, the wavefront derivative can have arbitrary values. It follows that there are wavefronts that cannot be represented by an OPD function. However, we show in the Appendix that all OPD functions, satisfying the above conditions, describe wavefronts by deriving explicit construction rules. The specification of an OPD function is therefore a sufficient condition for a wavefront to exist.

WAVEFRONT POINTS AND ABERRATED RAY DIREC-TIONS IN A 1D CANONICAL SETTING

We develop the major geometric reasoning of our derivation in a canonical one-dimensional setting. Without loss of generality, we consider an orthonormal local coordinate system with its origin on the reference sphere and spanning its tangent space.

To complete the basis, we use the tangent plane normal, oriented towards the paraxial image point s. Such a local coordinate system is depicted in Fig. 3 for the current one-dimensional setting, for the point p. In local coordinates p = (0, 0) T , i.e. it coincides with the local coordinate origin.

As detailed in Sect. 5 for the full 2D case, these local coordinate systems can be obtained by a rigid body transformation of the exit pupil coordinate system.

All quantities relating to the local system are denoted with circumflex symbols. In particular, p is the local coordinate origin, X and Ẑ are the local coordinate axes and Ŵ( x) is the OPD function parameterized in local coordinates x. The specifics of this parameterization are also covered in Sect. 5 for the 2D-setting. In addition, we denote the local representation of the reference sphere as R( x) := r -(r 2 -x2 ) 1/2 and the local aberrated ray directions as n( x). Given these quantities, we may express a wavefront point as

φ( xq ) = q( xq ) + Ŵ( xq ) • n( xq ), (3)
where q( xq ) = ( xq , R( xq )) T is a point on the reference sphere corresponding to the local tangent space coordinate xq , see Fig.

A. Constructing a Tangent to the Wavefront

Considering the fundamental definition of the wavefront as the surface which is normal to the aberrated rays, we can consider a circle with radius r 1 = Ŵ(0) that is centered in the point p = (0, 0) T , i.e. in the origin of the local coordinate system. This circle must be tangent to the wavefront since the wavefront is, by definition, located at a distance Ŵ(0) from point p. We wish to determine the point of intersection between the circle and the wavefront.

For this, we introduce a neighboring second circle with its center at point q = ( xq , R( xq )) T , also centered on the reference sphere and also tangent to the wavefront with radius r 2 = Ŵ( xq ). Both circles are shown in Fig. 4.

We now consider the circle at q to be approaching the circle at the center of the local coordinates p. In the limit, as the horizontal distance ∆ x = xq between the centers tends to zero, the two circles coincide and the tangent to both becomes the tangent of the wavefront. Expressing this intuition mathematically leads to equations for the ray direction and the wavefront itself.

Considering in more detail the geometry of Fig. 4, the tangent to both circles intersects the line connecting their centers at the point ĉ, where distances between the points are related by the simple triangle relation

r 1 | pĉ| = r 2 | qĉ| . ( 4
)
Using | qĉ| = | pĉ| + | p q| we arrive at the following equation

| pĉ| = r 1 | p q| r 2 -r 1 . (5)
The point ĉ is then given by:

ĉ = p - ( q -p)r 1 r 2 -r 1 . (6)
We now introduce the explicit expressions for the vector quantities

= (0, 0) T , q = (∆ x, R(∆ x)) T = ∆ x, r -(r 2 -∆ x2 ) 1/2 T , r 1 = Ŵ(0), r 2 = Ŵ(∆ x) = Ŵ(0) + d Ŵ d x 0 ∆ x + O(∆ x2 ), (7)
and insert the definitions into Eq. ( 6). We use a Taylor expansion for Ŵ at the point q for studying the limit as point q approaches point p. Since the OPD function Ŵ is differentiable, the relation is exact in this limit. We thus obtain an explicit equation for the vector ĉ = -

Ŵ(0)   ∆ x r -(r 2 -∆ x2 ) 1/2   d Ŵ d x | 0 ∆ x + O(∆ x2 ) , (8)
which, as ∆ x tends to zero becomes lim

∆ x→0 ĉ =    -Ŵ(0) d Ŵ d x | 0 0    , (9)
where the limit has been determined using the rule of l H ôpital. The previous equation is illustrated in Fig. 5. The limit behavior of the point ĉ is finite if d Ŵ d x | 0 = 0 and it is located on the X axis. For the case that d Ŵ d x | 0 = 0, the point ĉ is located at infinity. However the calculation of the wavefront point is now trivial since it is located on the axis Ẑ at a distance Ŵ(0) from the origin of the local coordinates.

For the limiting case where ĉ is finite, we determine the tangent point with the wavefront utilizing the triangle between the tangent point, the origin of the local coordinate system, and the limiting point ĉ. This triangle is illustrated in Fig. 5. Using the geometry, the tangential point, with coordinates ( x, z) T is calculated as

x = c × cos(α), z = (c 2 -x2 ) 1/2 . ( 10
)
Trigonometry yields

x = c 2 b . ( 11
)
Note that there may be two solutions since two tangent lines to the circles exist. However, since the sign of the OPD function indicates the direction of the wavefront with respect to the reference sphere, we can unambiguously select the correct solution.

B. Wavefront Point and Aberrated Ray

Replacing the auxiliary variables with their definitions in terms of the OPD function,

c = Ŵ(0), b = - Ŵ(0) d Ŵ d x | 0 , (12)
we arrive at explicit expressions for the local coordinates of the wavefront point:

x = -Ŵ(0) d Ŵ d x 0 , z = Ŵ(0)   1 - d Ŵ d x 0 2   1/2 . ( 13
)
We denote it in vector notation as φ(0) = ( x, z) T :

φ(0) = Ŵ(0)    -d Ŵ d x 0 1 -d Ŵ d x 0 2 1/2    . ( 14
)
Here we notice the significance of the condition that the norm of the OPD derivative be smaller or equal to one, Sect 3.

Eq. ( 14) allows to simultaneously obtain the normalized ray direction. As seen from the equation, the vectorial part is a) a unit vector, and b) multiplied by the OPD value Ŵ(0). Therefore, this vector component represents exactly the normalized ray direction:

n(0) =    -d Ŵ d x 0 1 -d Ŵ d x 0 2 1/2    . ( 15
)
We interpret the result as showing that, in the canonical situation, the OPD derivative equals the (negative) direction cosine of the ray.

Eqs. ( 14) and (15) allow us to obtain the exact wavefront point and aberrated ray direction knowing only the local function for the reference sphere R( x) and the local OPD function Ŵ( x).

LOCAL COORDINATE SYSTEMS AND FUNCTIONS THEREIN

So far, we have been describing the canonical situation in one dimension. For a generalization of the result, we need to 1) detail the construction of the local coordinate system as well as the transfer of the relevant functions, and 2) expand the results of the previous section towards two dimensions. As mentioned in Sect. 4, we require an orthonormal tangent frame to the reference sphere and a transformation T -1 p from this local system to the global exit pupil coordinate system. The underlying reason for requiring orthonormality is a preservation of distance measures in relation to the global coordinate system.

The tangent frame of the reference sphere and its associated transformation are parameterized by an evaluation point p = (x p , y p , R(x p , y p )) T on it. The reference sphere in global exit pupil coordinates is given by

R(x p , y p ) = z s -(r 2 -(x p -x s ) 2 -(y p -y s ) 2 ) 1/2 . ( 16
)
The point s = (x s , y s , z s ) is the paraxial image point, also in global coordinates, and r = (x s 2 + y s 2 + z s 2 ) 1/2 is the radius of the reference sphere. The exit pupil is centered in the global origin, and the positive Z-axis is pointing towards the image plane, which is located at a distance z s .

A. Local Orthonormal Tangent-Frame to the Reference Sphere

We construct an orthonormal local tangent frame coordinate system at point p by resorting to spherical coordinates around the paraxial image point s. Suitably normalized derivatives with respect to the spherical coordinates then provide us with the local linear approximation of the reference sphere, i.e. with its tangent plane. We choose to position the poles of the spherical coordinate system in the image plane, in particular along the Y-axis, in order to avoid singularities of the reference sphere parameterization in the space between the image plane and the exit pupil.

In particular, we use the following assignment of Euler angles θ(x p , y p ) = tan -1 R(x p , y p )z s x px s , φ(x p , y p ) = cos -1 y py s r , and

ρ(x p , y p ) = const. = r, (17)
the inverse equations of which are given by x = ρ cos(θ) sin(φ) + x s , y = ρ cos(φ) + y s , and z = ρ sin(θ) sin(φ) + z s .

(18)

The variables ρ, θ and φ all depend on the evaluation position (x p , y p ) in global coordinates, Eq. ( 17). The unit vectors of the tangent frame are then obtained via

u x (x p , y p ) = 1 ρ sin φ •      ∂x ∂θ | (x p ,y p ) ∂y ∂θ | (x p ,y p ) ∂z ∂θ | (x p ,y p )      =      -sin(θ) 0 cos(θ)      , u y (x p , y p ) = 1 ρ •      ∂x ∂φ | (x p ,y p ) ∂y ∂φ | (x p ,y p ) ∂z ∂φ | (x p ,y p )      =      cos(θ) cos(φ) -sin(φ) sin(θ) cos(φ)      , u z (x p , y p ) = -      ∂x ∂ρ | (x p ,y p ) ∂y ∂ρ | (x p ,y p ) ∂z ∂ρ | (x p ,y p )      =      -cos(θ) sin(φ) -cos(φ)
sin(θ) sin(φ)

     . ( 19
)
Here, the vectors have been normalized and u z has been inverted to point towards the paraxial image point s. Since the unit vectors depend on (x p , y p ), the transformation from the local coordinate systems to the global one is parameterized by the point of evaluation p. The associated matrix is given by

T -1 p =   u x u y u z p 0 0 0 1   , (20)
and its spatial derivatives are denoted as

∂T -1 p ∂x p = ∂ ∂x p   u x u y u z p 0 0 0 1   , (21)
and ∂T -1 p ∂y p respectively. We use homogeneous coordinates in order to describe rigid body transformations, including their translational part, as matrix-vector products.

B. Transforming Exit Pupil Functions into Local Coordinates

Key to the transfer of functions into the local coordinate systems is the realization that they are defined with respect to a common surface that is known in both systems. In particular, the reference sphere is given in global coordinates by Eq. 16, whereas in any

local system it is R( x, ŷ) = r -(r 2 -x2 -ŷ2 ) 1/2 , ( 22
)
due to the symmetry of the sphere. Further, as mentioned in Sect. 3, we define the domain of the OPD function W as the surface of the reference sphere. The exit pupil coordinate representation that is commonly used is then a parameterization of this function on the sphere. Let us denote this parameterization as W(x, y).

Fig. 6. (Color online) Transfer of function values between coordinate systems:

The center point for the transformation is p and the point q is an arbitrary point on the reference sphere. A function f defined on the reference sphere whose value is given in exit pupil coordinates by f (x q ) must return the same value for the local coordinates f ( xq ), i.e. f (x q ) = f ( xq ).

The OPD function value for a point q = (x q , y q , R(x q , y q )) T on the reference sphere is therefore obtained by evaluating W(x q , y q ).

In a local coordinate system, the point q = T -1 p q has a different set of coordinates q. However, the function values of the OPD function Ŵ( xq , ŷq ) = W(x q , y q ) in both systems must be the same since q and q are only different coordinates of the same point, see Fig. 6. We therefore define Ŵ( x, ŷ) := W(x( x, ŷ), y( x, ŷ)),

where the functions

       
x( x, ŷ; x p , y p ) y( x, ŷ; x p , y p ) z( x, ŷ; x p , y p )

1         = T -1 p         x ŷ R( x, ŷ) 1         , (24)
perform a remapping of the two different parameterizations of the reference sphere. For clarity, we have explicitly denoted the dependence on the evaluation point p. For the transfer of functions between the two systems, only the functions x( x, ŷ) and y( x, ŷ) are significant.

The construction above serves as a general tool to transfer all quantities of interest into a local coordinate system, i.e. in addition to the OPD function W(x, y), the wavefront ϕ(x, y) and the normalized ray direction of the aberrated ray n(x, y) are defined locally via φ( x, ŷ) := ϕ(x( x, ŷ), y( x, ŷ)), and n( x, ŷ) := n(x( x, ŷ), y( x, ŷ)),

by linking them to their global definition in exit pupil coordinates.

EXACT 2D CALCULATION OF THE WAVEFRONT FROM THE WAVE ABERRATION A. Constructing a Tangent Plane to the Wavefront

The derivation for the 2D case utilizes the 1D result of Sect. 4 and proceeds along similar lines. To restate the problem: knowing the reference sphere R(x, y) and the OPD function W(x, y) we want to calculate the corresponding wavefront ϕ(x, y) : R 2 → R 3 , which is now a function that takes a 2D coordinate to a point in three-dimensional space. According to the wavefront and OPD definitions, ϕ(x p , y p ) = p(x p , y p ) + W(x p , y p )n(x p , y p ),

where n(x p , y p ) is again the unknown normalized direction of the aberrated ray and p(x p , y p ) = (x p , y p , R(x p , y p )) T represents the evaluation point on the reference sphere. The derivation is similar to the 1D case, namely the combination of a coordinate transformation and the calculation of a limit. The geometry of the setting as well as the global and local coordinate systems involved are illustrated in Fig. 7.

We continue the derivation in local coordinates. We now study two spheres with their centers located on the surface of the reference sphere. Both spheres are tangent to the wavefront. One of them is fixed at the origin of the local coordinate system while the other one approaches it. Ultimately, the two spheres coincide at the limit. Since the second sphere can be approaching from an arbitrary direction, we instead consider two pairs of spheres moving along the local coordinate axes X and Ŷ, respectively. In doing so, we obtain equations where the X and Ŷ components are independent. To each pair of spheres, there is a tangent cone with its vertex located in the X Ẑ and Ŷ Ẑ planes respectively.

The two pairs of spheres are illustrated in Fig. 8. In this Figure, the wavefront surface and the reference sphere previously plotted in Fig. 7, are now replaced by single profiles. In the side views we see the two pairs of spheres. The objects tangent to each are now cones with vertices ĉ X and ĉ Ŷ . Employing the same analysis as in Fig. 5, we find the equations for the coordinates of the vertices.

Consequently, when the spheres are coinciding at the limit, we find the coordinates for the vertices of the cones to be lim

∆ x→0 ĉ X =       -Ŵ(0, 0) ∂ Ŵ ∂ x (0,0) 0 0       , ( 27
)
and lim

∆ ŷ→0 ĉ Ŷ =       0 -Ŵ(0, 0) ∂ Ŵ ∂ ŷ (0,0) 0       , ( 28
)
where ∆ x and ∆ ŷ represent the distances between centers, along the corresponding axis, for each pair of approaching spheres. The coinciding spheres are shown in Fig. 9. The limit points ĉ X and ĉ Ŷ are located on the axes X and Ŷ respectively. The tangential circles are located in planes parallel to the X Ẑ and Ŷ Ẑ planes.

In the case that either limit approaches infinity, the solution reduces to the 1D case. If both limits are infinite, the wavefront point is at a distance Ŵ(0, 0) along the Ẑ axis.

Similar to the 1D case where we obtained the wavefront point Ŵ(0) using Eqs. [START_REF] Glassner | On the transformation of surface normals[END_REF], we compute the wavefront point Ŵ(0, 0) as the point of intersection of the limiting tangent circles, as shown in Fig. 9. The X and Ŷ coordinates are obtained using triangle relations as in Fig. 5. The Ẑ component is determined by observing that both circles in Fig. 9 are part of the limit sphere. The wavefront in the local coordinate system is therefore given by φ(0, 0) = Ŵ(0, 0)

      -∂ Ŵ ∂ x (0,0) -∂ Ŵ ∂ ŷ (0,0) 1 -∂ Ŵ ∂ x | (0,0) 2 -∂ Ŵ ∂ ŷ | (0,0) 2       . ( 29
)
The normalized ray direction in local coordinates is the vectorial part of the wavefront point and is given by

n(0, 0) =       -∂ Ŵ ∂ x (0,0) -∂ Ŵ ∂ ŷ (0,0) 1 -∂ Ŵ ∂ x | (0,0) 2 -∂ Ŵ ∂ ŷ | (0,0) 2       . ( 30
)

B. Explicit Expressions in Global Coordinates

In the previous sections we have derived equations for the wavefront point and the aberrated ray direction in local coordinates.

We now relate them back to the global coordinate system. Since both the parameters and the values of the functions in Eqs. ( 29) and (30) are in local coordinates, two steps are necessary:

1. a replacement of local wavefront and wavefront derivative terms by their corresponding expressions in global coordinates, followed by 2. a back-transformation (using Eq. ( 24)) of the still local result into global coordinates.

Before carrying out the two steps, we note that the local OPD function Ŵ(0, 0) evaluated at the local origin is equal to the global OPD function W(x p , y p ) evaluated at the exit pupil coordinates of point p by Eq. ( 24).

Second, the 2D expression for the local wavefront derivatives in global coordinates is

  ∂ Ŵ ∂ x | (0,0) ∂ Ŵ ∂ ŷ | (0,0)   = ∇ Ŵ| (0,0) = T ∇W| (x p ,y p ) , (31)
where

T =   T -1 11 T -1 21 T -1 12 T -1 22   , ( 32
) with T -1 11 , T -1 21 , T -1 12 , T -1
22 being the elements of the upper left 2 × 2 sub-matrix from Eq. ( 20). The relation can be found by differentiating the definition of the function Ŵ, Eq. ( 23); the Jacobian of the coordinate transformation, Eq. ( 24), is found to be T and Eq. (31) follows by the chain rule. Note that T is the transpose of the upper left 2 × 2 sub-matrix of T -1 p and therefore also dependent on p.

With these prerequisites we can now perform the transformation in two steps.

Wavefront Point:

Performing step 1 yields

φ(0, 0) =      -Ŵ(0, 0)∇ Ŵ| (0,0) Ŵ(0, 0) • 1 -∇ ŴT • ∇ Ŵ| (0,0) 1      =      -W(x p , y p ) T ∇W| (x p ,y p ) W(x p , y p ) 1 -∇W T T T • T ∇W| (x p ,y p ) 1      . ( 33
)
Defining

n f = (1 -∇W T T T • T ∇W| (x p ,y p ) ) 1/2 (34)
and performing step 2 yields

ϕ(x p , y p ) = T -1 p φ(0, 0) = T -1 p      -W • T ∇W W • n f 1      . (35)
which is the 3D wavefront point in global (exit pupil) coordinates as a function of the exit pupil coordinates. Note that the Xand Y-components of ϕ(x p , y p ) will typically not be (x p , y p ) unless W(x p , y p ) = 0, Eq. (24).

Ray Direction:

The aberrated ray direction is similarly transformed into global coordinates. Step 1 results in

n(0, 0) =      -T ∇W| (x p ,y p ) n f 0      . ( 36
)
Applying step 2, we obtain

n(x p , y p ) = T -1 p      -T ∇W| (x p ,y p ) n f 0      . ( 37
)
Since only three components of the homogeneous direction vector n are non-zero, only the upper left 3 × 3 matrix of T -1 p is effective. For general transformations M, normals need to be transformed via M -T [START_REF] Glassner | On the transformation of surface normals[END_REF]. In our case, the upper left 3 × 3 matrix of T -1 p is orthonormal and therefore its own inverse transpose.

Eqs. ( 35) and (37) are the key results of this paper. An analytical proof that the ray directions n are orthogonal to the wavefront ϕ at every point and that the wavefront is located at the OPD distance of W is given in the Appendix.

CONNECTION TO THE RAY ABERRATIONS

Once the wavefront point and the aberrated ray direction are known, they can be used to compute the transverse ray aberrations. We make a connection to the classical approximation, Eqs. 1, that relates wave and ray aberrations and derive the exact conditions for the approximation to be valid.

The transverse ray aberrations are obtained by computing the aberrated image point i and subtracting the paraxial image point s = (x s , y s , z s ) T from it, i.e.

= i -s. ( 38
)
The aberrated image point i is obtained by computing the ray intersection of the aberrated ray with the image plane situated at z s . The aberrated ray passes, by definition, through the wavefront point ϕ(x p , y p ), Eq. ( 35), and has the direction n(x p , y p ), Eq. (37). We compute the path length t to the image plane by

t = z s -[ϕ] z [n] z = (z s -R(x p , y p )) [n] z -W(x p , y p ) (39)
where

([ϕ] x , [ϕ] y , [ϕ] z ) T are the components of ϕ(x p , y p ), ([n] x , [n] y , [n] z )
T the components of n(x p , y p ), and Eq. ( 26) has been used. The aberrated image point can now be written as

i = ϕ(x p , y p ) + tn(x p , y p ) = p + W(x p , y p )n(x p , y p ) + tn(x p , y p ) = p + (z s -R(x p , y p )) [n] z n(x p , y p ). =        x p + (z s -R(x p , y p )) [n] x [n] z y p + (z s -R(x p , y p )) [n] y [n] z z s        . ( 40
)
We see that the point i is indeed in the image plane. Using Eq. (38) and ignoring the zero z-component, the transverse ray aberrations become

x = (x p -x s ) + (z s -R(x p , y p )) [n] x [n] z y = (y p -y s ) + (z s -R(x p , y p )) [n] y [n] z (41)
which, with the help of a computer algebra package, can be simplified to a closed form in terms of the OPD derivative and the paraxial image position s:

x = - r 2 ∂W ∂x A -B y = - r 2 ∂W ∂y A -B (42)
with

B = ∂W ∂x ∆x + ∂W ∂y ∆y, A = r 2 1 - ∂W ∂x 2 - ∂W ∂y 2 + B 2 (43)
and ∆x = (x px s ) and ∆y = (y py s ). Eqs. (42) are the exact equations of the transverse ray aberrations in terms of the OPD function. We now show that the classic approximation can be obtained as a special case.

Taking the limit as ∆x → 0, ∆y → 0 results in lim It is interesting to note that the best approximation is not obtained in the origin of the exit pupil, but in the exit pupil position closest to the paraxial image.

EXAMPLES OF WAVEFRONT AND RAY ABERRATION CALCULATIONS

In the following, we provide three representative examples of computations with our analytic expressions. We provide a comparison with results obtained from the classical approximation in Eqs. [START_REF] Born | Principles of Optics[END_REF]. The comparisons are performed for both the transverse ray aberrations and the reconstructed wavefront. The error between our computation and the classical approximation is studied for all three examples. The results are summarized in Fig. 10. Each column is dedicated to a different test case and each row contains the same type of plot for each example. First, we briefly describe the information found in each row and later discuss the results in the context of each of the optical systems.

The first row shows a scaled system diagram, together with the lens prescription and the imaging conditions, reporting all quantities in millimeters. We obtain the first order system properties and a Zernike decomposition of the OPDs from OSLO, a commonly known Optics design software. The number of polynomials for the fit are set to the maximum available, yielding 37 Zernike coefficients. We use the Zernike expansions as ground truth OPD functions, shown in the second row. Additionally, we provide the peak-to-valley (P-V) and RMS values to emphasize the degree of the aberrations.

In the third row we plot the transverse ray aberrations for both directions, which can be considered as a re-centered spot diagram in the paraxial image plane. We simultaneously plot the exact ray aberrations y vs. x and the classical ray aberrations, now re-named to ¯ y vs. ¯ x .

The fourth row presents a comparison between the exact ray aberrations and the classical approximation, by plotting the error in the norms of the transverse ray aberrations || || -|| ¯ ||, with || || = ( x 2 + y 2 ) 1/2 being the norm of the exact transverse ray aberrations, Eqs. (42), and || ¯ || = ( ¯ 2

x + ¯ 2 y ) 1/2 the norm for the classic equations, Eqs. [START_REF] Born | Principles of Optics[END_REF]. This surface error is scaled in wavelengths and plotted on a logarithmic Z axis to enhance its dynamic range. Finally, the last row presents a comparison of the exactly computed wavefront, Eq. ( 35), with an approximation obtained from the classical equations. For computing the latter, we propose the following procedure: we connect the reference sphere evaluation point p with the approximation of the aberrated image point ī, obtained by adding the classical transverse ray aberrations ( ¯ x , ¯ y ) T to the paraxial image point s. The direction of this 3D line is an approximation to the true aberrated ray direction n, Eq. ( 37). We use it in conjunction with Eq. ( 26) to construct an approximated wavefront ψ. The difference in 3D positions ||ϕ(x, y)ψ(x, y)|| is shown in the last row, again scaled in wavelengths and with a logarithmic Z axis. We include this additional comparison, since the wavefront shape is important for an exact computation of wave-optical point spread functions using Huygens' principle. We now discuss the individual test cases.

(i) Singlet:

This system is one of the simplest optical set-ups with correspondingly large aberration values, especially at lower f-numbers. We chose a working f /# = 4, focal length f ≈ 60mm at a wavelength λ = 587.56nm. In the OPD plot, a large astigmatic component can be identified, along with a large P-V value. The spot diagram indicates that the classical approximation has a large error for the most external intersection points. The ray aberration error plot further emphasizes this fact, presenting a maximum in the range of 10 3 λ. The corresponding wavefront error has a maximum value on the order of 8λ.

(ii) Cooke Triplet:

We continue, by studying a more complex and well known system: the Cooke Triplet. The lens prescription is given in the plot. The imaging conditions closely resemble the singlet example, with a working f /# = 4, focal length f ≈ 50mm at a wavelength λ = 587.56nm. The OPD plot indicates a better corrected system, with again a large astigmatic component. The spot diagram indicates that the intersection points from the classical approximation closely resemble the exact calculations. The errors of the classical approximation in both, the ray aberration and the wavefront, are now significantly smaller, with maximum values in the range of 7λ and 2 × 10 -3 λ, respectively.

(iii) Post-Surgery Cornea:

We select this example as a test case presenting aberrations of higher order, in contrast to the previous examples where the OPD functions were slowly varying. The Zernike coefficients are extracted from a post-LASIK-surgery corneal topography [START_REF] Fricker | Analyzing lasik optical data using Zernike functions[END_REF]. In the first row of Fig. 10, we present the list of Zernike coefficients, measured in microns, were the values are listed using the Noll index ordering [START_REF] Noll | Zernike polynomials and atmospheric turbulence[END_REF]. To maintain the context of an eye as the optical system, we select a working f /# = 2.6, focal length f = 17mm at a wavelength λ = 587.56nm. The OPD plot shows a more complex structure giving rise to an unusual spot diagram. The maximum error of the classical approximation in the ray aberrations is now in the order of 70λ and the corresponding error for the wavefront comparison has a maximum value on the order of 3 × 10 -2 λ.

Comparing the errors between the cornea and the Cooke triplet examples, we observe the dependence of the approximation error of the classical Eqs. (1) on the derivatives of the wave aberration function. As stated before, Eq. ( 42), the classical approximation becomes more accurate for smaller gradients of the OPD function. In the post-surgery cornea example, even though the magnitude of the aberrations is smaller than in the Cooke triplet case, we observe approximation errors that are an order of magnitude larger.

CONCLUSIONS

We have derived exact analytic expressions for the wavefront surface, the aberrated ray directions, and the transverse ray aberrations for the standard definition of the optical path difference as a phase delay along the aberrated ray.

A transition to the local tangent frames of the reference sphere enables a Huygens-like geometric construction of the wavefront as an envelope of spheres and yields additional constraints on the OPD function. We show in the Appendix, that every OPD function satisfying the constraints has an associated wavefront and that the constructed wavefronts and rays fulfill the wavefront properties exactly.

The exact aberrated rays yield exact equations for the transverse ray aberrations. We have identified the precise conditions for the classical approximation to hold. The conditions differ from commonly held assumptions. The relevant factors are small OPD gradients, as opposed to the OPD magnitude, in conjunction with evaluation positions close to the paraxial image coordinates, as opposed to the pupil center.

We presented numerical simulations to illustrate the errors arising from the classical approximation for typical scenarios. The simulations provide a quantitative background for the theoretical results.

APPENDIX: VALIDATING THE WAVEFRONT PROPER-TIES

We now validate the wavefront properties of the wavefront derived in Eq. (35) and the ray direction from Eq. (37). Two conditions must be met for these quantities to be compatible: (i) The wavefront point ϕ(x p , y p ) obtained via Eq. (35) must have a distance of W(x p , y p ) from the point p, and

(ii) The aberrated ray direction n(x p , y p ), Eq. (37), must be orthogonal to the wavefront in the point ϕ(x p , y p ).

The two conditions constitute the definition of a wavefront. In showing that they are met by the quantities in Eqs. ( 35) and (37), we prove their correctness. In addition, the proof also shows that every OPD function, according to the definition of Sect. 3, has a corresponding wavefront. We need to show n(x p , y p ) T • ∂ϕ(x p , y p ) ∂x p = 0, and n(x p , y p ) T • ∂ϕ(x p , y p ) ∂y p = 0.

We proof the equality for the x-tangent vector ∂ϕ ∂x p , the derivation for the y-tangent vector being strictly similar. It is important to perform the proof in global coordinates since the local coordinate system changes when changing the evaluation position p.

Denoting the homogeneous local coordinate origin as Ô = (0, 0, 0, 1) T , the dot-product between the tangent vector and the aberrated ray direction is given by

n T ∂ϕ ∂x p = nT T -1 p T T -1 p ∂ ∂x p W n + Ô + nT T -1 p T ∂T -1 p ∂x p W n + Ô = nT ∂ ∂x p (W n) (I) yields ∂W ∂xp + nT T -1 p T ∂T -1 p ∂x p W n + Ô (I I) yields -∂W ∂xp = 0. (47)
Simplification of the complete Eq. (47) with a computer algebra package is, unfortunately, not tractable. As indicated above, we decompose the equation into term (I) and term (II). Term (I) will be shown to equal ∂W ∂x p ( ∂W ∂y p for the y-tangent). Term (II) can be shown to equal -∂W ∂x p (-∂W ∂y p for the y-tangent) with the help of a computer algebra software.

Term (I): contributes the majority of cross-terms when multiplying out Eq. (47). It can be conveniently treated in a vectorial fashion:

nT ∂ ∂x p (W n) = nT • ( ∂W ∂x p n + W ∂ n ∂x p ) = nT n • ∂W ∂x p + W nT ∂ n ∂x p . = ∂W ∂x p . ( 48
)
The last equality is due to the fact that nT n = 1, as shown in Eq. (45), and nT ∂ n

∂x p = ∇W T T T ∂ T ∂x p ∇W + ∇W T T T T   ∂ 2 W ∂x p 2 ∂ 2 W ∂y p ∂x p   + n f ∂n f ∂x p = 0. ( 49
)
The latter equality is due to

∂n f ∂x p = - ∇W T T T ∂ T ∂x p ∇W + ∇W T T T T   ∂ 2 W ∂x p 2 ∂ 2 W ∂y p ∂x p   n f , (50)
which can be verified by multiplying out the expressions.

Term (I I): is tractable with computer algebra software. Defining

r x = ∂T -1 p ∂x p φ(0, 0), (51)
we find that

n T • r x = - ∂W ∂x p . ( 52
)
Summarizing, we have shown that the wavefront point and the aberrated ray directions derived in Sect. 6 are compatible with the wavefront and the OPD properties. Since the proof has been constructive, we have shown that every OPD function according to the definition of Sect. 3 has indeed an associated wavefront.

Fig. 1 .

 1 Fig. 1. (Color online) Alternative definitions of the wave aberrations. The exit pupil is located along the X axis, while the image plane is located at a distance z S along the Z axis. The wavefront is the curve ϕ(x). The reference sphere R(x) is centered at the paraxial image point s and has a radius of r. The aberrated ray passing through the point u intersects the image plane at the aberrated image point i. The transverse ray aberration is the difference is.

Fig. 2 .

 2 Fig. 2. (Color online) a) Huygens-like wavefront construction illustrated -the wavefront ϕ(x) is the envelope of spheres with radii defined by the OPD function W(x). b) and c) Illustrating the condition that the gradient norm of the OPD function is smaller or equal to one. Applying a Taylor expansion to the term W(x + ∆x) and simplifying the triangle inequality yields the condition. The colors of the mathematical terms correspond to the colored segments indicating the distances that constitute the triangle.

Fig. 3 .

 3 Fig. 3. (Color online) Geometry considered. The axes XZ represent the original coordinate system or global coordinates and the axes X Ẑ represent the local coordinates for the point p(x p ). We utilize circumflex symbols to distinguish quantities of the local system from those of the global one.

  3. The local wavefront function φ : R → R 2 yields the local 2D coordinates of the wavefront point. Its horizontal coordinate may be different from the evaluation position xq .

Fig. 4 .

 4 Fig. 4. (Color online) Approaching circles. As point q moves towards p along the reference sphere, the circles get closer. The sub-images (a) and (b) represent a decrease in ∆ x.

Fig. 5 .

 5 Fig. 5. (Color online) Limit when circles coincide. The point ĉ is now located on the axis.

Fig. 7 .

 7 Fig. 7. (Color online) Geometry considered. XYZ are the global coordinates. X Ŷ Ẑ are the local coordinate axes at point p. The wavefront W(x, y) and the reference sphere R(x, y) are plotted along with the tangent space for the point p.

Fig. 8 .

 8 Fig. 8. (Color online) Approaching spheres. The wavefront surface and the reference sphere are represented as single profiles to simplify the graphics. The two approaching spheres are centered on the reference sphere and their tangent cones have vertices ĉ X and ĉ Ŷ .

Fig. 9 .

 9 Fig. 9. (Color online) Coinciding spheres. The vertices are now located along the corresponding axis. The tangent circles intersect at two points, for two solutions of the wavefront φ(0, 0). We select the solution whose sign fits the OPD.

2 ,

 2 (44) from which we obtain the classical form, Eqs. (1), in the case of ||∇W|| 1. The conditions for the validity of the classical approximation are therefore: (i) the evaluation position (x p , y p ) in the exit pupil approaches the paraxial image coordinate (x s , y s ), and (ii) the gradient norm of the OPD ||∇W|| 1.

Fig. 10 .

 10 Fig. 10. (Color online) Multiple computations for a singlet, a Cooke triplet and a post-surgery corneal aberration map.

n 2 f= 2 f= 1 . ( 45 )

 22145 Eqs. (35) and (37), we verify that Eq. (26) holds.We need to show that ||n(x p , y p )|| = 1. For, in this case ||ϕ(x p , y p ) -p|| = |W(x p , y p )|. Using Eq. (36) in local coordinates, the result is readily obtained:||n(x p , y p )|| 2 = || n(0, 0)|| 2 = -T ∇W 2 + ∇W T T T • T ∇W + n ∇W T T T • T ∇W + 1 -∇W T T T • T ∇W = Since the upper left 3 × 3 sub-matrix of transformation T -1p is orthonormal, it does not change the length of the normal vector when changing to global coordinates.(ii) Orthogonality of Wavefront and Aberrated Ray Direction:

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ACKNOWLEDGMENTS

We thank the anonymous reviewers for their helpful advice and attention to detail. This work has been supported by the German Research Foundation (DFG) through Emmy-Noether grant IH 114/1-1.