Three-dimensional polarization aberration functions in optical systembased on three-dimensional polarization ray-tracing calculus
Wenjun He, Yuegang Fu⁎, Zhiying Liu, Lei Zhang, Jiake Wang, Yang Zheng, Yahong Li
School of Opto-Electronic Engineering, Changchun University of Science and Technology, Changchun 130022, China
A R T I C L E I N F O
Keywords:Polarization aberration3D polarization aberration functionPolarization ray tracing matrix
A B S T R A C T
The polarization aberrations of a complex optical system with multi-element lens have been investigated using a3D polarization aberration function. The 3D polarization ray-tracing matrix has been combined with the opticalpath difference to obtain a 3D polarization aberration function, which avoids the need for a complicated phaseunwrapping process. The polarization aberrations of a microscope objective have been analyzed to include, thedistributions of 3D polarization aberration functions, diattenuation aberration, retardance aberration, andpolarization-dependent intensity on the exit pupil. Further, the aberrations created by the field of view and thecoating on the distribution rules of 3D polarization aberration functions are discussed in detail. Finally a novelappropriate field of view and wavelength correction is proposed for a polarization aberration function whichoptimizes the image quality of a multi-element optical system.
1. Introduction
Polarization aberration (PA) represents a non-uniform polarizationchange across wavefronts, was first proposed by Chipman in 1987 [1].The Fresnel equations state that when the light is obliquely incident onoptical interface, the variations of amplitude and phase between s- andp-polarized components are different, and lead to modifications ofpolarization states, i.e., non-normal incident light on optical interface isthe main cause of PA. Theoretical and experimental studies [2–5] showthat PA will affect the image resolution and accuracy of optical devices,especially for those with a high numerical aperture (NA), large field ofview (FOV), and a broad spectrum. Further optical system components,for example, diffraction gratings, folding mirrors, holograms, grazing-incident & polarization elements can also effect image quality andperformance through PA [6–12].
PA can be divided into diattenuation aberration (DA) [13], which ispolarization-dependent transmission; retardance aberration (RA) [14],which is polarization-dependent optical path difference (OPD); andskew aberration [15,16], which is an intrinsic rotation of polarizationstates due to the geometric transformation of local coordinates. Thereare several ways to represent PA, such as Jones pupil [1], Pauli-Zernike[17], orientation Zernike polynomials [18], field- orientation Zernikepolynomials [19], and polarization aberration function [20]. All ofthem are calculated from a 2D Jones matrix, but with differentmathematical forms. The Jones matrix can be decomposed into Paulimatrices, Pauli-Zernike representation and orientation Zernike poly-
nomials are essentially approximate analysis methods of polarizationaberration function. The main significance is that they describe PA withZernike polynomials which are commonly used to characterize wave-front aberration, and explain a part of the physical significance of PAwithin the traditional concept of geometric aberrations.
The polarization effects upon field calculations are of vital impor-tance for non-paraxial systems or high resolution imaging systems, aspointed out by D. Panneton [21]. But the 2D Jones matrices of non-paraxial rays with different FOVs have different local coordinatesystems. The traditional practice is to transform 2D PA into a globalcoordinate system by using mathematical coordinate transformation,rather than 3D PA defined by physics, which results in a lack ofprecision in the 3D imaging simulation [22]. In order to avoid the errorof a mathematical transformation, Yun proposed a 3D polarization ray-tracing matrix in global coordinate. Providing a 3D generalization ofthe Jones calculus, and extended the 3D PA to characterize thepolarization-dependent transformations of a wavefront [23,24]. Theapproach though of interest in addressing the 3D PA problem lacked adetailed analysis.
The goal of this paper is to further investigate 3D PA by introducingthe 3D polarization aberration function (PAF) to a real optical system,comprising of a multi-element lens, by analyzing various polarizationeffects in detail. A set of 3D PAFs contain 18 pupil functions (9 real and9 imaginary parts) to characterize the polarization behavior in opticalsystems. In order to gain a better understanding of the 3D PAFs, thePAs of a microscope objective are analyzed in detail, including DA and
http://dx.doi.org/10.1016/j.optcom.2016.11.046Received 14 July 2016; Received in revised form 14 November 2016; Accepted 19 November 2016
RA. And the effects of polarization on the image quality of opticalsystems investigated.
2. 3D polarization aberration functions
The mathematical representation of 3D PA is the basis of thispaper; we will briefly review the 3D polarization ray-tracing calculus inthis section, and supplement the algorithm of amplitude coefficients foroptical interfaces with coatings. In order to avoid the need for acomplicated phase unwrapping process, we combine the 3D polariza-tion ray-tracing matrix with the scalar OPD to get the 3D PAF.
2.1. 3D polarization ray-tracing calculus
3D polarization ray-tracing matrix characterizes the change in thethree-element electric field vector due to interaction with an opticalelement, a sequence of optical elements, or an entire optical system[22]. In this method, the polarization states of entrance and exit light,Ein and Eout , are related by the 3D polarization ray-tracing matrix Ptotal,and the relevant equations are listed as follows,
E P E= ⋅out totol in (1)
∏P P=totolq
Q
q=1 (2)
⎛
⎝
⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟
⎛
⎝⎜⎜⎜
⎞
⎠⎟⎟⎟
⎛
⎝
⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟P O J O
s p k
s p k
s p k
αα
s s s
p p p
k k k
= ⋅ ⋅ =
′
′
′
0 00 00 0 1
q out q q in q
x q x q x q
y q y q y q
z q z q z q
s q
p q
x q y q z q
x q y q z q
x q y q z q
, ,−1
⎯→⎯,
⎯ →⎯⎯
,
⎯→⎯,
⎯→⎯,
⎯ →⎯⎯
,
⎯→⎯,
⎯→⎯,
⎯ →⎯⎯
,
⎯→⎯,
,
,
⎯→⎯,
⎯→⎯,
⎯→⎯,
⎯ →⎯⎯,
⎯ →⎯⎯,
⎯ →⎯⎯,
⎯→⎯, −1
⎯→⎯, −1
⎯→⎯, −1
(3)
sk k
k kp k s s s p k s→ =
→×
→
→×
→ , → =→
× →, →′ = →, →′ =→
× →q
q q
q qq q q q q q q q
−1
−1−1
(4)
where, Pq is the 3D polarization ray-tracing matrix for the q th opticalinterface, and it can be calculated by 3D Jones matrix Jq and two realunitary matrices Oin q, and Oout q, , which describe rotations of orthogonal
coordinate systems. s→ and p→ are the eigenpolarization for the Fresnel
equations, k→
q and k→
q−1 are the propagation vectors. αs q, and αp q, in thediagonal matrix Jq are s- and p-amplitude reflection or transmissioncoefficients for the interaction in a local coordinate system, s is for s-polarization, and p is for p-polarization.
2.2. Amplitude coefficients of multilayer film
For an optical surface without coating, the amplitude coefficients αs
and αp can be calculated directly from the Fresnel equations. But for anoptical interface with multilayer film coating, the calculation methodfor αs and αp is complicated, not mentioned in [23]. As shown in Fig. 1,multilayer film is composed of G layers homogenous dielectric med-iums, Nsub is the complex refractive index of substrate, Ng represents thecomplex refractive index of g th layer, dg is the thickness, and θg is therefraction angle. When the light beam incidents the multilayer filmwith the incident angle θin, the combination eigenmatrix of multilayerfilm and substrate is expressed in Eqs. (5) and (6).
⎛⎝⎜
⎞⎠⎟
⎛
⎝⎜⎜
⎞
⎠⎟⎟
⎛⎝⎜
⎞⎠⎟∏B
Cδ δ
i η δ δ η=cos − sin
− ⋅ ⋅ sin cos1
g
G giη g
g g gsub
=1
g
(5)
⎧⎨⎪⎩⎪
δ πλ
N d θ ηN θ for s
for p= 2 cos , =
cos , →
, →g g g g g
g gN
θcosg
g (6)
For the reflection rays, the amplitude coefficient α α= r; and for the
refraction rays, α α= t.
αη B Cη B C
αη
η B C=
−+
, =2
+r t0
0
0
0 (7)
2.3. Definition of 3D polarization aberration functions
The 3D polarization ray-tracing matrix is a function of wavelength,pupil coordinate and object coordinate,
⎛
⎝
⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟P h ρ λ
P h ρ λ P h ρ λ P h ρ λ
P h ρ λ P h ρ λ P h ρ λ
P h ρ λ P h ρ λ P h ρ λ
(→
, →, ) =(→
, →, ) (→
, →, ) (→
, →, )
(→
, →, ) (→
, →, ) (→
, →, )
(→
, →, ) (→
, →, ) (→
, →, )
11 12 13
21 22 23
31 32 33 (8)
where, P h ρ λ(→
, →, )jk denotes the matrix element of 3D polarization ray-
tracing matrix, the subscripts j k, = 1, 2, 3. h→
is the object coordinatevector, ρ→ is the pupil coordinate vector and λ represents the wave-length.
As we know, wavefront aberration in optical design only describesthe OPD between the actual wavefront and the ideal wavefront. 3Dpolarization ray-tracing matrix is only used to describe the polarizationproperties of the optical system. In order to characterize the completeinformation of the wavefront, we use the 3D PAF to represent thewavefront aberration and the polarization effect of the optical systemsimultaneously:
⎛⎝⎜
⎞⎠⎟PAF h ρ λ P h ρ λ i π
λOPD h ρ λ(
→, →, ) = (
→, →, ) ⋅ exp − 2 (
→, →, )
(9)
In Eq. (9), optical path difference function OPD h ρ λ(→
, →, ) can becalculated by the geometric ray tracing method; the exponent portion
represents the phase change caused by OPD; P h ρ λ(→
, →, ) represents thepolarization properties and it can be used to correct wavefrontaberration by coating or other polarization effects. Therefore, the 3DPAF is the generalization of the wavefront aberration and includes the
wavefront aberration as a subset. P h ρ λ(→
, →, ) is a complex matrix andthe argument of a complex number is always less than π2 , therefore,
the physical meaning of adding OPD h ρ λ(→
, →, ) in Eq. (9) is to avoidphase wrapping, and we can keep track of the OPD greater than π2 by
calculating OPD h ρ λ(→
, →, ) separately from P h ρ λ(→
, →, ).The 3D PAF in Eq. (9) just represents one of the rays in optical
system. In general, a limited number of rays on pupil are traced tocalculate the PA. An optical system with M N× rays has been traced,yielding a M N× grid of 3D PAFs for each ray position,
Fig. 1. Multilayer film and its parameters.
W. He et al. Optics Communications 387 (2017) 128–134
129
⎛
⎝
⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟PAF
PAF PAF PAFPAF PAF PAF
PAF PAF PAF
=
⋯⋯
⋮ ⋮ ⋱ ⋮⋯
total
N
N
M M MN
11 12 1
21 22 2
1 2 (10)
Eq. (10) describes the wavefront aberrations and polarizationproperties of the optical system. In order to explain the physicalsignificance of the 3D PAFs in detail, we will analyze the 3D PAFs in areal optical system as an example in next section.
3. Example and discussions
As an example shown in Fig. 2, there are seven lenses in the opticalsystem layout of a microscope objective with high NA, and three filedangles are shown. This microscope objective is U.S. patent 2,604,013.The system has F/0.832, NA of 0.59, FOV of 9°, wavelengths of 656,546 and 435 nm, and it is rotationally symmetric. The global coordi-nate system is constructed on the entrance pupil, and z- axis is alongthe optical axis.
3.1. Three-dimensional polarization aberration functions of U.S.patent 2,604,013
In order to evaluate the polarization properties of the U.S. patent2,604,013, the 3D PAF is calculated for each ray path using 3Dpolarization ray tracing method and Eqs. (8)–(10). Then the distribu-tions of 3D PAFs, diattenuation, retardance, and pupil apodization onthe exit pupil for different field angles are obtained to characterize thepolarization effects of the optical system.
Fig. 3 shows the 3D PAFs on the exit pupil of the U.S. patent2,604,013 for the on-axis FOV without coating. Fig. 3(a) shows the realpart and Fig. 3(b) shows the imaginary part. For an ideal optical systemwithout polarization aberration, the transmissions of s- and p-polarized
components are equal, P h ρ λ(→
, →, ) is simplified to a unit matrix with acommon amplitude factor; and then the 3D PAFs should be like this:the diagonal coefficients of real part (sub-graphs 1, 5, 9 in Fig. 3(a)) areequal to one and evenly distributed on the pupil, and others are zero.All the coefficients of imaginary part on the pupil are zero or π . Thecenter points of all sub-graphs in Fig. 3 represent the on-axis ray.When the on-axis ray propagates through each surface of the opticalsystem, the amplitude transmission coefficients of p-polarization and s-polarization components are always equal according to the Fresnelequations, therefore, only normal incident rays do not introducepolarization aberration.
As shown in Fig. 3(a), the center points in sub-graphs which on thediagonal are equal to one, and the center points in the other sub-graphsare equal to zero. For the rays in other positions, they have differentincident angles for different positions, and as a consequence non-uniform distributions of real parts appeared on the pupil. If the realparts of the 3D PAFs of an optical system have non-zero off-diagonalelements, it represents that the optical system will change thepolarization state of incident light beam, and the three orthogonalcomponents (x, y, z) of the electric field produce cross-influence. InFig. 3(a), the real part coefficients of sub-graphs 2 and 4 are very small(close to zero), but the real part coefficients in the other sub-graphswhich outside the diagonal are large, especially on the edges of thepupil. This indicates that there is no cross-influence for x-, y-compo-nents, but there exist cross-influences for x-, z-components and y-, z-components.
Fig. 4 shows the 3D PAFs on the exit pupil of the U.S. patent2,604,013 for the edge FOV without coating. As shown in Fig. 4(a), the
Fig. 2. Optical system layout of U.S. patent 2,604,013 has seven lenses.
Fig. 3. (a) Real part and (b) Imaginary part of the 3D PAFs on the exit pupil for the on-axis FOV without coating.
W. He et al. Optics Communications 387 (2017) 128–134
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real part coefficients of sub-graphs on the diagonal appear asymmetricdistributions, i.e., tilt in the direction of FOV (along the y- direction).The real part coefficients of sub-graphs 2, 4, 6, 8 also appear obviouslyasymmetric distributions in the y- direction on the exit pupil, thesesub-graphs are amplitude terms associated with y- component ofelectric field. Comparing Fig. 4(a) and Fig. 3(a), we can find that thepeak values of sub-graphs 2 and 4 in Fig. 4(a) are much larger thanthem in Fig. 3(a); but the distributions of sub-graphs 3 and 7 have tinychange, since the amplitude terms are uncorrelated with the directionof FOV in these sub-graphs. Thus, the 3D PAFs are closely related withthe FOV of the optical system.
As shown in Figs. 3(b) and 4(b), the imaginary part of the 3D PAFsfor an uncoated optical system have only two values, i.e., zero or π ,either for the on-axis FOV or for the edge FOV. When the rays arerefracted, the amplitude transmission coefficients of uncoated opticalsurfaces are real numbers. Therefore, the polarization effects of theoptical system only take the forms of pupil apodization and diattenua-tion, but without retardance. This situation also applies to the reflectingrays on uncoated surfaces, unless the reflecting rays satisfy the totalreflection condition, then the amplitude reflection coefficients of raysare complex numbers, and the polarization effects contain retardance.
However, the amplitude coefficients are generally complex numbersfor both refracted and reflected rays on the optical surfaces withcoating, the optical system will change the amplitude and the phase ofthe incident rays at the same time, and the polarization aberrations willcontain both diattenuation and retardance. Fig. 5 shows the 3D PAFsfor the edge FOV with simple antireflection (AR) coating. ComparingFig. 5(a) and Fig. 4(a), we can find that the distributions of sub-graphs1 and 5 in pupil changed, but sub-graph 9 remains unchanged; boththe shape and the magnitude of the real part coefficients in sub-graphs2 and 4 are changed obviously, and there are small changes in theshape of sub-graphs 6 and 8, i.e., the contour lines slightly curved alongy- direction; the distributions of sub-graphs 3 and 7 change little. Itindicates that coating introduce small effect on the real part coefficientsof amplitude terms which correlated with z- direction.
We can find that there are complex changes on the imaginary partof the 3D PAFs after coating by comparing Fig. 5(b) and Fig. 4(b). Anotable feature in Fig. 5(b) is that the imaginary part coefficients of thesub-graphs on the diagonal are distributed continuously in pupil, butthe imaginary part coefficients in the other sub-graphs appear muta-tions in some positions, and these discontinuous points might cause
polarization singularity in the output light field of the optical system.Thus, coating has a great contribution to the PA of the optical
system. When coating on the optical lens changes, the PA will alsochanges. It may be a viable approach to reduce the PAs of opticalsystem, if we consciously design the polarization properties of thecoating on a particular surface, and make the PA caused by this surfaceto balance the PAs caused by the other surfaces.
3.2. Distributions of diattenuation and retardance on exit pupil
Diattenuation aberration (DA) and retardance aberration (RA) aredescribed by the distributions of diattenuation and retardance on exitpupil respectively. As shown in Fig. 6, AR coating can reduce DAs forboth the center and edge of FOV, which does not change thedistribution trends of DA. As shown in Fig. 7, RAs are obviouslyincreased with AR coating, and the distribution trends of RA arechanged at the same time. For an ordinary optical system withoutretarders and diffractive elements, the optical film is the major sourceof RA.
As shown in Figs. 6 and 7, when the FOV is zero, both DA and RAon exit pupil are distributed rotationally symmetric about the opticalaxis, and there are much larger DA and RA in the edge of the pupil.When the FOV=9°, the symmetry of DA and RA distributions in the y-direction is destroyed.
Fig. 8 shows polarization-dependent intensity distributions of theU.S. patent 2,604,013 with AR coating on exit pupil. We can find thatthe distributions of intensity are different for input beams withdifferent polarization states; the value of DA on the center point iszero in Fig. 6(b), and the intensity values of the center points inFig. 8(a)–(d) are the same (equal to 91.117), then the intensitydifference equal to zero, i.e., there is no transmission difference foron-axis rays with different polarization states. The points on the edge ofthe pupil in Fig. 6(b) have the maximum DA values which are equal to0.1, and there are the maximum intensity differences (equal to 9.518)for the points on the edge of the pupil in Fig. 8(a)–(d). This confirmsthat the DAs of the optical system have a great influence on intensitydistribution. Thus, it is a practical approach to optimize the intensitydistribution of the output light field or improve the optical imagingquality of optical system by reasonable designing the polarization statesof the input beam, especially for lithography objectives and lasercutting systems.
Fig. 4. (a) Real part and (b) Imaginary part of the 3D PAFs on the exit pupil for the edge FOV without coating.
W. He et al. Optics Communications 387 (2017) 128–134
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3.3. Effect of polarization on the image quality of the optical system
The 3D PAFs have been discussed in Section 2, and we can find thatthe imaginary part of 3D PAFs have contributions to the traditionalwavefront aberration. In order to verify the influence of polarization ontraditional wavefront aberration, we will analyze the wavefront aberra-tions and point spread functions (PSFs) of the optical system respec-tively by geometric ray tracing (GRT) and polarization ray tracing(PRT) in this section.
As shown in Table 1, when the incident beam is a completely non-polarized light beam, the peak-to-valley (PV) value and the root meansquare (RMS) of the optical system for all design wavelengths arereduced when the FOV=0° and the polarization effect is considered. Forexample, when λ = 546 nm and FOV=0°, the percentage increase (PI)
of PV is −3.35% and PI of RMS is −3.93%, i.e., the imaginary part of 3DPAFs and wavefront aberration have offset each other, hence, the imagequality of the optical system on the axis filed is improved. But whenFOV equal to 5° and 9°, both of the PI of PV and RMS are increased,i.e., the image quality of the optical system turns to be serious for PRT.
Why is there a deviation when using GRT and PRT to calculate thewavefront aberration? The fundamental reason is that the phase shiftson the coatings are not considered for GRT. When we use PRT tocalculate the wavefront aberration, the real OPD is equal to thegeometric OPD plus the phase shifts on coatings. In general the phaseshifts for s- and p-polarization components are different, so the phaseshifts are closely related to the polarization state of input ray, and thiscauses the imaginary part of 3D PAFs. Therefore, the actual wavefrontaberration is influenced by the polarization effect of optical system.
Fig. 5. (a) Real part and (b) Imaginary part of the 3D PAFs on the exit pupil for the edge FOV with AR coating.
Fig. 6. Diattenuation pupils for (a) the on-axis FOV without coating, (b) the on-axis FOV with AR coating, (c) the edge FOV without coating, and (d) the edge FOV with AR coating.
Fig. 7. Retardance pupils for (a) the on-axis FOV without coating, (b) the on-axis FOV with AR coating, (c) the edge FOV without coating, and (d) the edge FOV with AR coating. Theunit of retardance is degree.
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Moreover, the effects of polarization on wavefront aberration aredifferent for different wavelengths and FOVs. Thus, selecting anappropriate FOV and wavelength to correct the PAs is very importantto reduce the effect of polarization on wavefront aberration.
PSF is an image quality evaluation method based on diffractiontheory which is widely used. And the strehl ratio is the most importantparameter when calculating PSF. As shown in Table 2, considering thepolarization effect, the PI of strehl ratio is 12.12% when the FOV=0°;but for the FOV equal to 5° and 9°, the PI of strehl ratios are −6.33%and −3.28%, and the image quality deteriorates. Therefore, when weanalyze the influence of polarization on the image quality of opticalsystem, the same conclusion is obtained whether by calculatingwavefront aberration or PSF. Correcting PA has an important guidingsignificance to optimize high performance optical system and improvethe image quality.
4. Conclusion
In this paper, we have further investigated 3D PA by introducingthe 3D PAF to a real optical system with complex multi-element lens.In order to avoid phase wrapping, we combined the 3D polarizationray-tracing matrix with the scalar OPD to acquire the 3D PAF. 3D PAFscan be viewed as a generalization of the polarization function repre-sentation to 3D global coordinate, but they can simultaneouslyrepresent traditional wavefront aberration and polarization character-istics of optical system.
The 3D PAFs of a microscope objective with NA=0.59 (U.S. patent2,604,013) are calculated. The influences of the FOV and the coating onthe distribution rules of the 3D PAFs have been discussed. Coating isthe main cause of the retardance for the refractive optical system. Thus,controlling the parameters of coating is a viable method to reduce thePAs. The common practice is to minimize the phase difference betweenp-polarization and s-polarization components of optical film on eachsurface, but it's very difficult because the coating also need to meet therequirements of the transmittance. It may be simple and effective, if wedesign the polarization properties of the coating on a particular surface,and make the PA caused by this surface to balance the PAs caused bythe other surfaces.
The diattenuation pupils, retardance pupils, and polarization-dependent intensity maps have been shown. The effects of polarizationon the image quality of the optical system are analyzed via wavefrontaberration and PSF. For an optical system with broad spectrum andlarge FOV, PAs are functions of wavelengths and FOVs. It's impossibleto correct the PAs for all wavelengths and FOVs perfectly. Thus, whenwe try to correct the PAs of optical system, the most important point isto define the reasonable evaluation function of PAs and select anappropriate FOV and wavelength.
3D polarization ray-tracing method is a powerful tool for polariza-tion analysis, and the 3D PA of a complex optical system with foldingmirror and bionic moth-eye microstructure is still under study toexplore the polarization characteristics of bionic optical system.
Acknowledgments
We thank the financial support by National Natural ScienceFoundation of China under No.11474037.
Fig. 8. Distributions of the intensity on exit pupil for (a) the on-axis FOV with x- polarized input beam, (b) the on-axis FOV with y- polarized input beam, (c) the on-axis FOV withcircularly polarized input beam, (d) the on-axis FOV with elliptical polarized input beam, (e) the edge FOV with x- polarized input beam, (f) the edge FOV with y- polarized input beam,(g) the edge FOV with circularly polarized input beam, and (h) the edge FOV with elliptical polarized input beam.
Table 1Effect of Polarization on Wavefront Aberration.
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