Polychromatic axial behavior of aberrated opticalsystems: Wigner distribution function approach

Walter D. Furlan, Genaro Saavedra, Enrique Silvestre, Pedro Andres, andMarıa J. Yzuel

We propose a new method for the computation of the tristimuli values that correspond to the impulseresponse along the optical axis provided by an imaging optical system working under polychromaticillumination. We show that all the monochromatic irradiance distributions needed for this calculationcan be obtained from the Wigner distribution function associated with a certain version of the pupilfunction of the system. The use of this single phase-space representation allows us to obtain the abovemerit function for aberrated systems with longitudinal chromatic aberration and primary sphericalaberration. Some numerical examples are given to verify the accuracy of our proposal. © 1997 OpticalSociety of America

1. Introduction

For assessing the performance of an optical imagingsystem working under broadband illumination, theclassical monochromatic merit functions, such as thepoint-spread function ~PSF! and the optical transferfunction, should be extended to the polychromaticdomain. This extension is usually carried out by theaddition of a suitable number of monochromatic com-ponents weighted by the spectral distribution of thesource and the color sensitivity of the receiver.

Often, the polychromatic axial response of the opti-cal system is of major interest as a measure of thetolerance to aberrations. In particular, for visual op-tical systems—i.e., those in which the human eye is thefinal detector—the tristimuli values along the opticalaxis are used as a figure of merit.1,2 These functionsare derived from the axial values of the different mono-chromatic irradiance PSF’s. Since analytical expres-sions for these PSF’s are achievable for only a fewsimple pupil functions, several numerical methods toevaluate them have been developed.3–5 However, thesequential calculation of the axial PSF’s for everywavelength in the polychromatic case leads to a time-

W. D. Furlan, G. Saavedra, E. Silvestre, and P. Andres are withthe Departament d’Optica, Universitat de Valencia, E-46100 Bur-jassot, Spain. M. J. Yzuel is with the Departament de Fısica,Universitat Autonoma de Barcelona, E-08193 Bellaterra, Spain.

Received 14 February 1997; revised manuscript received 18 July1997.

0003-6935y97y359146-06$10.00y0© 1997 Optical Society of America

9146 APPLIED OPTICS y Vol. 36, No. 35 y 10 December 1997

consuming procedure. On the other hand, somemonochromatic merit functions have been expressedsuccessfully in terms of different phase-space repre-sentations, such as the Wigner distribution function~WDF! or the ambiguity function.6–9

It is important to recognize that these approachesare especially useful when two circumstances takeplace together: First, a high number of monochro-matic merit functions must be calculated for the samepupil function with different amounts of aberration,and second, all the above merit functions can be ob-tained from a single phase-space representation. Inthis way a technique for the calculation of the poly-chromatic optical transfer function of systems suffer-ing from longitudinal chromatic aberration ~LCA!has been proposed recently.10

Bearing in mind the above statements, in thisstudy we propose a new method for the computationof the polychromatic axial response provided by anoptical system with an arbitrary exit-pupil transmit-tance suffering from LCA and primary spherical ab-erration ~SA!. This approach needs only the WDF ofthe azimuthally averaged pupil function of the sys-tem to obtain all the values of the monochromaticaxial irradiance PSF’s we require to construct thepolychromatic response along the optical axis. InSection 2 we give the theoretical basis of the method,and in Section 3 we present some numerical exam-ples to illustrate its performance.

2. Basic Theory

The behavior of an optical imaging system underbroadband illumination can be evaluated from the

polychromatic axial PSF, given the isoplanatism ofthe system. In particular, for visual systems theaxial response can be assessed from the tristimulivalues along the optical axis. These values are de-fined as a function of the defocus coefficient dv20 bythe weighted superpositions

X~dv20! 5 *l

I~dv20; l!x#lS~l!dl, (1a)

Y~dv20! 5 *l

I~dv20; l!y#lS~l!dl, (1b)

Z~dv20! 5 *l

I~dv20; l!z#lS~l!dl, (1c)

where I~dv20; l! is the axial monochromatic irradi-ance PSF, S~l! stands for the spectral distribution ofthe source, and x#l, y#l, and z#l denote the spectraltristimuli values. These latter parameters can beconsidered as the chromatic-sensitivity functions ofthe human eye taken as a receiver for a given selec-tion of the primary colors.

In general, we are involved in a two-step procedurefor calculating numerically the polychromatic meritfunctions in Eqs. ~1!. The first step is to obtain, atdiscrete points along the axial interval of interest, thedifferent monochromatic irradiance PSF’s providedby the system for a suitably large number of wave-lengths. The second step is the computation of theaxial tristimuli values by means of the weighted su-perposition, evaluated in a discrete way, given byEqs. ~1!. Usually, the first step is a time-consumingprocedure, especially when a high accuracy is needed.Therefore an approach for reducing the computationtime of the monochromatic irradiance PSF’s is wel-comed.

The amplitude monochromatic PSF along the opti-cal axis for an imaging system is given for largeFresnel numbers by

p~dv20; l! 51l *

0

2p

*0

1

P~r, f; l!expSi2p

ldv20r

2Drdrdf

51l *

0

2p

*0

1

t~r, f!expHi2p

l

3 @v~r, f; l! 1 dv20r2#Jrdrdf, (2)

where P~r, f; l! is the generalized pupil function ofthe system, ~r, f! are the normalized polar coordi-nates at the exit-pupil plane, t~r, f! 5 uP~r, f; l!u, andv~r, f; l! is the wave-aberration function. We as-sume that the chromatic variations of the modulus ofthe generalized pupil function are negligible, i.e., thefunction t~r, f! has no dependence on l. If the sys-

tem suffers from only SA and LCA, the aberrationfunction is reduced to

v~r, f; l! 5 v20~l!r2 1 v40~l!r4, (3)

where v20~l! and v40~l! are the LCA and SA coeffi-cients, respectively. In this case, the axial mono-chromatic irradiance PSF can be expressed as afunction of a single radial integral, as follows:

I~dv20; l! 5 up~dv20; l!u2

51l2U *

0

1

t0~r!exp(i2p

l$@dv20 1 v20~l!#r2

1 v40~l!r4%)rdrU2

, (4)

where t0~r! is the azimuthal average of t~r, f!.11 Byperforming the change of variable m 5 r2, we obtain

I~dv20; l! 51l2U *

0

1

q0~m!exp(i2p

l

3 $@dv20 1 v20~l!#m 1 v40~l!m2%)dmU2

51l2 *

0

1

*0

1

q0~m!q0*~m9!exp(i2p

l

3 $@dv20 1 v20~l!#~m 2 m9! 1 v40~l!

3 ~m2 2 m92!%)dmdm9, (5)

where q0~m! 5 t0~r!. Using the transformation x 5~m 1 m9!y2 and x9 5 m 2 m9 causes Eq. ~5! to result,except for an irrelevant constant factor, in

I~dv20; l! 51l2 *

2`

1` (*2`

1`

q0Sx 1x9

2Dq0*Sx 2x9

2D3 expHi2pFdv20 1 v20~l!

l

12v40~l!

lxGx9Jdx9)dx, (6)

where the integration limits have been extended toinfinity, given the finite extension of the functionq0~x!. The inner integral in Eq. ~6! can be recog-nized as the WDF of the one-dimensional functionq0,12 namely

Wq0~x, n! 5 *

2`

1`

q0Sx 1x9

2Dq0*Sx 2x9

2D3 exp~2i2pnx9!dx9. (7)

Thus the axial monochromatic irradiance PSF can bewritten as a line integral of the above WDF, where

10 December 1997 y Vol. 36, No. 35 y APPLIED OPTICS 9147

the spatial-frequency variable n is related by a lineartransformation with the spatial variable x. Mathe-matically,

I~dv20; l! 51l2 *

2`

1`

Wq0@x, m~l!x 1 n~l!#dx, (8)

where

m~l! 5 22v40~l!

l, (9a)

n~l! 5 2dv20 1 v20~l!

l. (9b)

From Eqs. ~8! and ~9! we infer that all the values ofthe axial irradiance PSF for every wavelength and forany value of LCA and SA can be obtained from asingle two-dimensional representation, namely theWDF of the mapped pupil q0~x!. This result can beachieved by integration of the values of Wq0

alongstraight lines in the phase-space domain. The slopeand the y intersect of these lines are fixed by thevalue of the aberration coefficients and by the wave-length of the light @see Eqs. ~9!#. Moreover, the sameWDF provides all the information needed to assessthe axial behavior of the system for any scaled ver-sion of the pupil function, since they all lead to thesame q0~x!. If the pupil of the system presents noamplitude variations, the study of the system for dif-ferent values of its numerical aperture can be carriedout with the same phase-space function.

The present study can be extended in a straighfor-ward manner to cases in which other aberrations—apart from the LCA and SA—with negligiblewavelength dependence are present in the wave-aberration function. In this situation the additionalphase variations of the generalized pupil P~r, f; l!should be joined with the function t~r, f! to obtain thefunction q0~x!.

Note that, because of the change of variable m 5 r2,the axial values of the polychromatic irradiance PSFare determined by the one-dimensional mapped func-tion q0. Hence a two-dimensional phase-space rep-resentation contains the whole of the information tosolve our problem, despite the fact that our startingfunction is a two-dimensional pupil. This is a sin-gular result since the WDF doubles the number ofvariables of the original function as any other phase-space representation. Several relevant properties ofthe WDF are presented in Refs. 12–14 and the refer-ences cited therein.

3. Numerical Examples

For testing the method and its performance we eval-uate the polychromatic axial response produced by anoptical system with a clear circular pupil that suffersfrom two different aberration functions. In the firstcase we consider that the system is affected by onlyLCA, whose v20~l! coefficient is shown in Fig. 1 ~case1!. In the second situation, in addition to the previ-

9148 APPLIED OPTICS y Vol. 36, No. 35 y 10 December 1997

ous LCA the system also suffers from SA with a con-stant coefficient of v40~l! 5 360 nm ~case 2!.

As was pointed out in Section 2, we can discussboth situations starting from the same WDF associ-ated with the function q0~x! of the system, which, inthis case, is simply a rectangle function. The func-tion Wq0

~x, n! is digitally obtained by a sequence offast Fourier transformations with a resolution of4096 3 4096 points in the phase-space domain. Agray-scale picture of the modulus of this bidimen-sional function is shown in Fig. 2. From this resultthe axial values of the monochromatic PSF’s are cal-

Fig. 1. Variation of the LCA coefficient that affects the systemunder study with l.

Fig. 2. Modulus of the WDF of the mapped pupil q0 under con-sideration. A schematic representation of the integration linescited in the main text is also shown, with m~l! and n~l! given byEqs. ~9!.

culated by numerical integration along the straightlines fixed by the parameters given by Eqs. ~9!.

For the system at issue it is possible to obtain ananalytical expression for the axial monochromatic ir-radiance PSF’s for both aberration functions. Thistheoretical result is used to assess the accuracy of theproposed technique for the calculation of the mono-chromatic components. For case 1 @v40~l! 5 0#, thetheoretical result is

I~dv20; l! 51l2 sinc2Fdv20 1 v20~l!

l G ,

where sinc~x! 5 sin~px!y~px!. To show the agree-ment between the theoretical prediction and the dataobtained with the technique we propose, we selectedthree different wavelengths within the visible spec-trum. Both results are compared in Fig. 3 and showperfect matching between them.

We also perform the calculation of axial monochro-matic irradiance for case 2. Since the mapped pupilfunction q0~x! remains unchanged, no further calcu-lation of the WDF is needed. The result is obtainedby a simple change, according to Eqs. ~9!, of thestraight lines along which the WDF should be inte-grated. Here, the monochromatic axial irradiancePSF’s also have an analytical expression in terms ofthe Fresnel integrals.3 The theoretical result pre-dicts, for each wavelength, an intensity maximumlocated at dv20 5 2@v20~l! 1 v40~l!#. Figure 4 showsthe perfect correspondence between the analytical re-sult and the data obtained with our method.

Fig. 3. Normalized monochromatic axial irradiance for threewavelengths of the optical system with a clear circular pupil andLCA shown in Fig. 1. The curves and symbols correspond to thetheoretical and the computed results, respectively. The normal-ization is such that the maximum value of the irradiance for l 5550 nm is unity.

Finally, the polychromatic response of the systemis also computed for both cases 1 and 2. Thespectral-sensitivity functions used in the computa-tion of the axial tristimuli values correspond to thoseassociated with the CIE 1931 standard observer, andwe identify the spectral distribution of the sourcewith the standard illuminant C. The integrals inEqs. ~1! are numerically evaluated with a sampling of400 equally spaced wavelengths in the range 400nm # l # 750 nm.

For comparison purposes we use the results pre-sented in Ref. 15 for the same systems. In that re-search the monochromatic irradiance PSF’s wereobtained by use of the classical method of Hopkinsand Yzuel,3 and Eqs. ~1! were evaluated by directnumerical computation. For comparison with theseresults, an alternative description of the polychro-matic axial behavior is used. In fact, instead of thetristimuli values we employ a conventional set of pa-rameters derived from them, namely the normalizedaxial illuminance YN and the axial chromaticity co-ordinates x, y. These magnitudes are defined as afunction of the defocus coefficient by

YN~dv20! 5Y~dv20!

*l

~1yl2!y#lSldl

, (10)

x~dv20! 5X~dv20!

X~dv20! 1 Y~dv20! 1 Z~dv20!, (11)

y~dv20! 5Y~dv20!

X~dv20! 1 Y~dv20! 1 Z~dv20!. (12)

Fig. 4. Same as in Fig. 3, but here the system suffers from anadditional SA with a constant coefficient of v40~l! 5 360 nm.

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The results reported in Ref. 15 concerning the axialilluminance YN for cases 1 and 2 are plotted togetherwith our numerical data in Fig. 5. A quantitativecomparison shows that the relative difference, on av-erage, for the computed axial positions in the centrallobe is less than 2.5% for each curve. The axial chro-maticity coordinates are shown graphically in a sim-ilar way in Fig. 6. As one can see from Fig. 6, theagreement between both results is again evident.

Fig. 5. Normalized axial illuminance as a function of dv20 for thetwo systems under study. The solid and dotted curves representthe results shown in Ref. 15 obtained by use of the Hopkins andYzuel method,3 whereas the symbols correspond to the values ob-tained with the method we propose.

Fig. 6. Axial chromaticity diagram for the two systems understudy. The notation used is the same as that for Fig. 5.

9150 APPLIED OPTICS y Vol. 36, No. 35 y 10 December 1997

As was expected a considerable reduction in thecomputation time of the tristimuli values is achievedwith the technique we propose. This is because allthe monochromatic components needed for the calcu-lation of the above parameters for both aberrationtypes are obtained from a single representation of thepupil, whereas in the classical methods the entirecalculation must be repeated for each axial position,wavelength, and aberration function. Although thecalculation of Wq0

~x, n! must be taken into account inthe computation time, the ratio of this contribution tothe total time is less and less important when a highnumber of axial positions, wavelengths, or both areinvolved.

4. Conclusions

An efficient method for the computation of the axialpolychromatic impulse response provided by imagingsystems suffering from LCA and SA has been pro-posed. The large amount of calculation time neededwith the classical methods is reduced substantiallyby our approach because of the use of a single phase-space representation for obtaining all the data for thecomputation. This representation is the WDF of acertain mapped version of the azimuthally averagedpupil function of the system.

Numerical examples show that the accuracy of themethod is high when compared with both analyticaland numerical results obtained by other approaches.A detailed comparative analysis between the presentmethod and others already reported requires the op-timization of the corresponding algorithms, andtherefore additional research is being carried out toquantify the savings in computation time yielded byour proposed technique for the same degree of accu-racy.

Another outstanding feature of our approach isthat two of the most common aberrations affectingoptical systems, the LCA and the SA, become param-eters, and consequently our formulation can be ap-plied successfully for designing optical systems withlow sensitivity to defocus, LCA, and SA.

This research has been supported by the Universitatde Valencia ~Projectes Precompetitius d’Investigacio1994!. G. Saavedra was supported during this re-search by the Conselleria d’Educacio i Ciencia de laGeneralitat Valenciana, Spain. E. Silvestre grate-fully acknowledges the financial support of the Direc-cion General de Investigacion Cientıfica y Tecnica~grant PB93-0354-C02-01!, Ministerio de Educacion yCiencia, Spain.

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