Geometrical Irradiance Changes in a Symmetric Optical System

Geometrical irradiance changes in a symmetric optical system

Dmitry Reshidko Jose Sasian

Dmitry Reshidko, Jose Sasian, “Geometrical irradiance changes in a symmetric optical system,” Opt. Eng. 56(1), 015104 (2017), doi: 10.1117/1.OE.56.1.015104.

Downloaded From: https://www.spiedigitallibrary.org/journals/Optical-Engineering on 11/28/2017 Terms of Use: https://www.spiedigitallibrary.org/terms-of-use Optical Engineering 56(1), 015104 (January 2017)

Geometrical irradiance changes in a symmetric optical system

Dmitry Reshidko* and Jose Sasian University of Arizona, College of Optical Sciences, 1630 East University Boulevard, Tucson 85721, United States

Abstract. The concept of the aberration function is extended to define two functions that describe the light irradiance distribution at the exit pupil plane and at the image plane of an axially symmetric optical system. Similar to the wavefront aberration function, the irradiance function is expanded as a polynomial, where individual terms represent basic irradiance distribution patterns. Conservation of flux in optical imaging systems is used to derive the specific relation between the irradiance coefficients and wavefront aberration coefficients. It is shown that the coefficients of the irradiance functions can be expressed in terms of wavefront aberration coefficients and first- order system quantities. The theoretical results—these are irradiance coefficient formulas—are in agreement with real ray tracing. © 2017 Society of Photo-Optical Instrumentation Engineers (SPIE) [DOI: 10.1117/1.OE.56.1.015104]

Keywords: aberration theory; wavefront aberrations; irradiance. Paper 161674 received Oct. 26, 2016; accepted for publication Dec. 20, 2016; published online Jan. 23, 2017.

1 Introduction the normalized field H~ and aperture ρ~ vectors. The field vec- In the development of the theory of aberrations in optical tor and the aperture vector may be defined in either the object imaging systems, emphasis has been given to the study of or the image spaces. Two vectors uniquely specify a ray image aberrations, which are described as wave, angular, propagating in the lens system. The wavefront aberration transverse, or longitudinal quantities.1 The light irradiance function is expanded into a polynomial of the rotational variation, specifically at the exit pupil plane and at the invariants as dot products of the field and aperture vectors, H~ H~ H~ ρ~ ρ~ ρ~ image plane of an optical system, is a radiometric aspect specifically ð · Þ, ð · Þ, and ð · Þ, and to the fourth H~ ρ~ of a system that is also of interest. order of approximation on and is written as follows:

EQ-TARGET;temp:intralink-;e001;326;425 X The geometrical irradiance calculation is useful for deter- ~ ~ ~ j ~ m n WðH; ρ~Þ¼ Wk;l;mðH · HÞ ðH · ρ~Þ ðρ~ · ρ~Þ mining the apodization of the wavefront at the exit pupil. The j;m;l point spread function (PSF) describes the response of an im- ~ ~ ~ aging system to a point object. An accurate diffraction cal- ¼ W000 þ W200ðH · HÞþW111ðH · ρ~Þ ’ culation of a system s PSF requires knowledge not only of W ρ~ ρ~ W ρ~ ρ~ 2 the wavefront phase but also of its amplitude. These, the þ 020ð · Þþ 040ð · Þ ~ ~ 2 phase and amplitude, are usually calculated geometrically þ W131ðH · ρ~Þðρ~ · ρ~ÞþW222ðH · ρ~Þ at the exit pupil of the system. While the wavefront phase ~ ~ is evaluated by calculating the optical path length for þ W220ðH · HÞðρ~ · ρ~Þ rays, the wavefront amplitude is estimated by the square W H~ H~ H~ ρ~ W H~ H~ 2; (1) root of the geometrical irradiance. þ 311ð · Þð · Þþ 400ð · Þ The image plane illumination or relative illumination is where each aberration coefficient Wk;l;m represents the ampli- another characteristic that may have a strong impact on tude of basic wavefront deformation forms and subindices j, m, the performance of a lens system. and n represent integers k ¼ 2j þ m and l ¼ 2n þ m.3 The light irradiance distribution at the exit pupil plane Similarly, a pupil aberration function WðH;~ ρ~Þ can be defined and/or at the image plane of an optical system can be derived to describe the aberration between the pupils. This function is from basic radiometric principles, such as conservation of constructed by interchanging the role of the field and aperture flux.2 In this paper, we provide a study of the relationship vectors, and to fourth order, it is as follows: between irradiance at these two planes and the system’s

EQ-TARGET;temp:intralink-;e002;326;217 X wavefront aberration coefficients. Our study is based on geo- ~ ~ ~ j ~ m n WðH; ρ~Þ¼ Wk;l;mðH · HÞ ðH · ρ~Þ ðρ~ · ρ~Þ metrical optics. Edge diffraction effects are not considered in j;m;l this study. An extended object that has a uniform or ~ Lambertian radiance is assumed. The geometrical irradiance ¼ W000 þ W200ðρ~ · ρ~ÞþW111ðH · ρ~Þ that we discuss describes the illumination distribution on the W H~ H~ W H~ H~ 2 image plane or on the exit pupil plane of an optical system. þ 020ð · Þþ 040ð · Þ The concept of the wavefront aberration function is well ~ ~ ~ ~ 2 þ W131ðH · HÞðH · ρ~ÞþW222ðH · ρ~Þ established. The wavefront aberration function WðH;~ ρ~Þ of ~ ~ an axially symmetric system gives the geometrical wavefront þ W220ðH · HÞðρ~ · ρ~Þ deformation (from a sphere) at the exit pupil as a function of ~ 2 þ W311ðH · ρ~Þðρ~ · ρ~ÞþW400ðρ~ · ρ~Þ ; (2)

Optical Engineering 015104-1 January 2017 • Vol. 56(1)

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where the pupil aberration coefficients are barred to distin- assumed that the irradiance is greater than zero for any guish them from the image aberration coefficients and point defined by the normalized field H~ and aperture ~ρ subindices j, m,andn represent integers k ¼ 2j þ m vectors. and l ¼ 2n þ m. The question we pose and answer is: what is the relation- In analogy to the wavefront aberration function, we define ship between the wavefront aberration functions WðH;~ ρ~Þ the irradiance function IðH;~ ρ~Þ that gives the irradiance at the and WðH;~ ρ~Þ and the irradiance functions IðH;~ ρ~Þ and image plane of an optical system. To the fourth order of I¯ðH;~ ~ρÞ? We use conservation of flux in an optical system approximation, this irradiance function is expressed as to determine the relationship between wavefront and irradi- follows: ance coefficients. We also discuss in some detail the particular case of relative illumination at the image plane. Our EQ-TARGET;temp:intralink-;e003;63;653 X ~ ~ ~ j ~ m n IðH; ρ~Þ¼ Il;k;mðH · HÞ · ðH · ρ~Þ · ðρ~ · ρ~Þ theoretical results are in agreement with real ray tracing. j;m;n Overall, the paper provides new insights into the irradiance I I H~ H~ I H~ ρ~ I ρ~ ρ~ changes in an optical system and furthers the theory of ¼ 000 þ 200ð · Þþ 111ð · Þþ 020ð · Þ aberrations. 2 ~ ~ 2 þ I040ðρ~ · ρ~Þ þ I131ðH · ρ~Þðρ~ · ρ~ÞþI222ðH · ρ~Þ ~ ~ ~ ~ ~ þ I220ðH · HÞðρ~ · ρ~ÞþI311ðH · HÞðH · ρ~Þ 2 Radiative Transfer in an Optical System ~ ~ 2 In this section, we review radiometry and show how aberra- þ I400ðH · HÞ ; (3) tions relate to conservation of optical flux Φ in an optical system. Figure 2 illustrates the basic elements of an axially where the irradiance coefficients Ik;l;m represent basic illumi- symmetric optical system and the geometry defining the nation distribution patterns at the image plane, as shown in transfer of radiant energy. Rays from a differential area Fig. 1, and subindices j, m, and n represent integers dA in the object plane pass through the optical system k ¼ 2j þ m and l ¼ 2n þ m. We also define another irradi- and converge at the conjugate area dA 0 in the image ance function I¯ðH;~ ρ~Þ that gives the irradiance of the beam at plane. The differential cross sections of the beam dS in the exit pupil plane of an optical system: the entrance pupil plane and dS 0 in the exit pupil plane

EQ-TARGET;temp:intralink-;e004;63;469 X I¯ H;~ ρ~ I¯ H~ H~ j H~ ρ~ m ρ~ ρ~ n are optically conjugated to the first order. ð Þ¼ l;k;mð · Þ ð · Þ ð · Þ In a lossless and passive optical system, the element of j;m;l radiant flux dΦ is conserved through all cross sections of ¯ ¯ ¯ ~ ¯ ~ ~ ¼ I000 þ I200ðρ~ · ρ~ÞþI111ðH · ρ~ÞþI020ðH · HÞ the beam. The equation for the conservation of flux is written ¯ ~ ~ 2 ¯ ~ ~ ~ as follows: þ I040ðH · HÞ þ I131ðH · HÞðH · ρ~Þ ¯ ~ 2 ¯ ~ ~ dAdS cos4ðθÞ dA 0dS 0 cos4ðθ 0Þ þ I222ðH · ρ~Þ þ I220ðH · HÞðρ~ · ρ~Þ dΦ L L dΦ 0; (5) EQ-TARGET;temp:intralink-;e005;326;421 ¼ o ¼ o ¼ t2 t 02 ¯ ~ ¯ 2 þ I311ðH · ρ~Þðρ~ · ρ~ÞþI400ðρ~ · ρ~Þ ; (4) where Lo is the source radiance. We assume that the source is ¯ where each irradiance coefficient Il;k;m represents basic Lambertian, and, consequently, Lo is constant. The object apodization distributions at the exit pupil, as shown in space angle θ is between the ray connecting dA and dS Fig. 1, and subindices j, m, and n represent integers and the optical axis of the lens system. Similarly, θ 0 is k ¼ 2j þ m and l ¼ 2n þ m. In these functions, it is the image space angle between the ray connecting dA 0 and dS 0 and the optical axis. t (t 0) is the axial distance between the object (image) plane and the entrance (exit) pupil plane, respectively. Equation (5) provides the radiant flux along a particular ray in an optical system. Since the normalized field H~ and aperture ~ρ vectors uniquely specify any ray propagating in the optical system, Eq. (5) gives the radiant flux as a function of the normalized field H~ and pupil coordinates ~ρ. The irradiance on the exit pupil plane is obtained by dividing the radiant flux that is incident on a surface by the unit area. It follows that

EQ-TARGET;temp:intralink-;e006;326;202 dΦ 0 dA 0 ¯ ~ 4 0 dIðH; ρ~Þ¼ ¼ Lo cos ðθ Þ dS 0 t 02

dA dS 4 dΦ ¼ Lo cos ðθÞ¼ (6) t2 dS 0 dS 0

Fig. 1 Second- and fourth-order apodization terms at the exit pupil where dI¯ðH;~ ρ~Þ is the differential of irradiance on the exit given by the coefficients of the pupil irradiance function presented pupil plane. Similarly, the irradiance on the image plane as a function of the aperture vector ~ρ or irradiance patterns at the image given by the coefficients of the irradiance function presented is obtained by dividing the radiant flux by the image surface as a function of the field vector H~ . unit area as in

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Fig. 2 Geometry defining the transfer of radiant energy from differential area dAin the object space to the conjugate area dA0 in the image space. The radiant flux through all cross sections of the beam is the same.

EQ-TARGET;temp:intralink-;e007;63;565 EQ-TARGET;temp:intralink-;e008;326;565 dΦ 0 dS 0 ~ ¯ ~ 4 0 ~ 4 0 IpinholeðH;ρ~Þ¼IpinholeðH;ρ~Þ¼I0∕pinhole · cos ðθ Þ dIðH; ρ~Þ¼ ¼ Lo cos ðθ Þ dA 0 t 02 02 ~ ~ 02 0 ¼ I0∕pinhole · ½1 − 2 · u¯ ðH · HÞ − 2 · u ðρ~ · ρ~Þ dS dA 4 dΦ ¼ Lo cos ðθÞ¼ (7) t2 dA 0 dA 0 − 4 · u 0u¯ 0ðH~ · ρ~Þ::: 4 ~ ~ 2 3 ~ and dIðH;~ ρ~Þ is now the differential of irradiance on the þ3 · u¯ 0 ðH · HÞ þ 12 · u 0 u¯ 0ðH · ρ~Þðρ~ · ρ~Þ ¯ ~ image plane. The differentials of irradiance dI H; ρ~ and 2 ð Þ 12 u 02u¯ 02 H~ ρ~ ::: dIðH;~ ρ~Þ are functions of the field and aperture vectors. þ · ð · Þ Equations (6) and (7) are general and do not involve any þ6 · u 02u¯ 02ðH~ · H~ Þðρ~ · ρ~Þ approximations. However, care must be taken in evaluating these expressions since the angles θ and θ 0 may vary due to þ 12 · u 0u¯ 03ðH~ · H~ ÞðH~ · ρ~Þþ3 · u 04ðρ~ · ρ~Þ2; dA dA 0 dS aberrations. In addition, the differential areas , , , (8) and dS 0 may also vary for different points in the aperture and in the field of the lens. where u 0 and u¯ 0 are the first-order marginal and chief ray slopes in the image space and I0∕pinhole is a constant corre- 3 Irradiance Function of a Pinhole Camera sponding to the irradiance value for the on-axis field point or In this section, we calculate coefficients of the irradiance zero-order term of the irradiance function. Without loss of function defined in Eqs. (6)and(7) for a pinhole camera. generality, we set this zero-order term of the irradiance func- The pinhole camera produces images of illuminated objects tion equal to 1. as light passes through an aperture, as shown in Fig. 3. In aberration theory, the reference sphere is used as the In this model, we define the field vector H~ on the image reference to measure the wavefront deformation. In analogy, plane and the aperture vector ~ρ on the plane of the aperture. the geometry in Fig. 3 can be used to define a model of irra- ¯ ~ ~ ~ The irradiance functions IðH; ρ~Þ and IðH;~ρÞ give the relative diance IpinholeðH; ρ~Þ, which is helpful as a reference and for irradiance along the ray specified by the field and aperture further calculations. In an actual system, image and pupil vectors. It follows from Eqs. (6) and (7) that irradiance dis- aberrations affect the irradiance distribution. tribution at a point specified by H~ on the focal plane or at a ~ρ point specified by on the aperture plane are proportional to 4 Irradiance on the Image Plane cos4 θ 0 of the particular ray. To calculate the coefficients of ð Þ ~ρ the irradiance functions to fourth order, we express cos4ðθ 0Þ The aperture vector is selected at the exit pupil plane, and, in terms of first-order system quantities as (see Appendix A): thus, it defines the ray intersection with this plane. Rays at the exit pupil pass through a uniform grid by construction, and differential areas dS 0 are constant given that we choose to define rays at the exit pupil. To calculate the image plane irradiance, we evaluate Eq. (7) in the image space as follows:

dI I cos4 θ 0 ; (9) EQ-TARGET;temp:intralink-;e009;326;190 ¼ 0 · ð Þ

where dS 0 I L ; (10) EQ-TARGET;temp:intralink-;e010;326;147 0 ¼ o t 02

where I0 ¼ 1 is a constant corresponding to the irradiance value for the on-axis field point. In contrast to a pinhole cam- Fig. 3 Geometry defining a pinhole camera. Images are formed as era, in an actual system, the ray angle in Eq. (9) is modified light passes through the aperture. by image aberrations. To calculate the irradiance distribution

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Ið2Þ Table 1 Second-order irradiance coefficients k;l;m at the image and plane of an optical system. dA I ≡ L EQ-TARGET;temp:intralink-;e013;326;741 ; (13) 0 o t2 02 I020ðρ~ · ρ~Þ¼−2u ðρ~ · ρ~Þ

~ 0 0 ~ where I0 1 is a constant corresponding to the irradiance I111ðH · ρ~Þ¼−4u u¯ ðH · ρ~Þ ¼ value for the on-axis field point. A comparison of ~ ~ 02 ~ ~ dA 0 I200ðH · HÞ¼−2u¯ ðH · HÞ Eqs. (12) and (9) reveals an additional term dA that is given by the ratio between the elements of the area at the at the image plane of an optical system with the aperture stop object and image planes. In an optical system with aberra- 4 dA 0 at the exit pupil, we express cos ðθ 0Þ in terms of first-order tions, the ratio dA may vary over the pupil and over the system quantities and wavefront aberration coefficients (see field. Two differential areas are related by the determinant Appendix B). Table 1 provides a summary of the second- of the Jacobian JðH;~ ρ~Þ of the transformation: order image plane irradiance coefficients. We conclude dA 0 J H;~ ρ~ dA: (14) that the second-order irradiance terms are not affected by EQ-TARGET;temp:intralink-;e014;326;605 ¼ ð Þ image aberrations and that the image plane irradiance to the second order is equivalent to the ideal irradiance of a pinhole camera: ¯ð2Þ Table 3 Second-order irradiance coefficients Ik;l;m at the exit pupil I H;~ ρ~ ð2Þ I H;~ ρ~ ð2Þ EQ-TARGET;temp:intralink-;e011;63;551 ð Þ ¼ pinholeð Þ plane of an optical system. ¼ 1 − 2u 02ðρ~ · ρ~Þ − 4u 0u¯ 0ðH~ · ρ~Þ − 2u¯ 02ðH~ · H~ Þ: I¯ H~ H~ −2u¯ 02 − 4 W H~ H~ (11) 020ð · Þ¼ Ж 311 ð · Þ ¯ ~ 0 ¯ 0 4 6 ~ Table 2 summarizes the fourth-order image plane irradi- I111ðH · ρ~Þ¼ −4u u − Ж W 220 − Ж W 222 ðH · ρ~Þ ance coefficients. The fourth-order irradiance coefficients are the sum of two components. The first component is repre- ¯ 02 4 I200ðρ~ · ρ~Þ¼ −2u − W 131 ðρ~ · ρ~Þ sented by products of the first-order ray slopes u 0 and u¯ 0 Ж in the image space. The second component includes additional terms that are functions of the fourth-order image aberration coefficients and the first-order ray slopes. The constant Ж in Table 2 represents the Lagrange invariant of the system. ¯ð4Þ Table 4 Fourth-order irradiance coefficients Ik;l;m at the exit pupil plane of an optical system. 5 Irradiance on the Exit Pupil Plane 2 2 I¯ H~ H~ 3u¯ 04 − 6 W 3 W u¯ 02 3 W W H~ H~ To calculate the irradiance distribution on the exit pupil plane 040ð · Þ ¼ Ж 511 þ Ж 311 þ Ж2 311 311 ð · Þ of an optical system, we evaluate Eq. (6) in the image space ¯ ~ ~ ~ 0 03 10 8 5 02 as follows: I131ðH · HÞðH · ρ~Þ¼ 12u u¯ − W 422 − W 420 þ W 222u¯ Ж Ж Ж 0 2 ¯ 02 6 0 ¯ 0 4 10 dA þ W 220u þ W 311u u þ 2 W 220W 311 þ 2 W 222W 311 dI I cos4 θ 0 (12) Ж Ж Ж Ж EQ-TARGET;temp:intralink-;e012;63;339 ¼ 0 · · ð Þ dA ~ ~ ~ × ðH · HÞðH · ρ~Þ ð4Þ 2 Table 2 Fourth-order irradiance coefficients Ik;l;m at the image plane ¯ ~ 02 ¯ 02 12 4 2 ¯ 02 I222ðH · ρ~Þ ¼ 12u u − Ж W 333 − Ж W 331 − Ж W 131u of an optical system. 12 0 ¯ 0 8 0 ¯ 0 2 02 8 þ W 222u u þ W 220u u − W 311u þ 2 W 222W 222 Ж Ж Ж Ж 2 I ρ~ ρ~ 2 3u 04 16 W u 0u¯ 0 ρ~ ρ~ 2 16 W W − 4 W W H~ ρ~ 040ð · Þ ¼ þ Ж 040 ð · Þ þ Ж2 222 220 Ж2 311 131 ð · Þ ~ 03 0 16 02 12 0 0 ~ ¯ ~ ~ 02 02 6 5 02 2 0 0 I131ðρ~ · ρ~ÞðH · ρ~Þ¼ 12u u¯ þ W 040u¯ þ W 131u u¯ ðρ~ · ρ~ÞðH · ρ~Þ I220ðH · HÞðρ~ · ρ~Þ¼ 6u u¯ − W 331 þ W 131u¯ − W 222u u¯ Ж Ж Ж Ж Ж 4 0 ¯ 0 5 02 10 8 − W 220u u W 311u 2 W 311W 131 − 2 W 222W 220 2 2 Ж þ Ж þ Ж Ж ~ 02 02 8 02 8 0 0 ~ I222ðH · ρ~Þ ¼ 12u u¯ þ W 131u¯ þ W 222u u¯ ðH · ρ~Þ Ж Ж ~ ~ × ðH · HÞðρ~ · ρ~Þ ~ ~ 02 ¯ 02 4 ¯ 02 8 0 ¯ 0 ~ ~ I220ðH · HÞðρ~ · ρ~Þ¼ 6u u þ Ж W 131u þ Ж W 220u u ðH · HÞðρ~ · ρ~Þ ¯ ~ 03 ¯ 0 10 8 5 02 I311ðH · ρ~Þðρ~ · ρ~Þ¼ 12u u − Ж W 242 − Ж W 240 þ Ж W 222u 3 8 2 8 2 4 2 W u 02 6 W u 0u¯ 0 4 W W 10 W W I H~ H~ H~ ρ~ 12u 0u¯ 0 W u¯ 0 W u¯ 0 W u 0u¯ 0 þ Ж 220 þ Ж 131 þ 2 220 131 þ 2 222 131 311ð · Þð · Þ¼ þЖ 222 þЖ 220 þЖ 311 Ж Ж ~ × ðH · ρ~Þðρ~ · ρ~Þ ~ ~ ~ ×ðH · HÞðH · ρ~Þ 2 2 I H~ H~ 3u¯ 04 4 W u¯ 02 H~ H~ I¯ ρ~ ρ~ 2 3u 04 − 6 W 3 W u 02 3 W W ρ~ ρ~ 2 400ð · Þ ¼ þ Ж 311 ð · Þ 400ð · Þ ¼ Ж 151 þ Ж 131 þ Ж2 131 131 ð · Þ

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If the Jacobian determinant is expressed in terms of wave- We have set the field vector H~ at the object plane and the front aberration coefficients, the differential of irradiance on aperture vector ρ~ at the exit pupil plane. We assume that the the exit pupil plane is calculated by evaluating Eq. (12) and aperture stop coincides with the location of the aperture vec- keeping terms to the fourth order (see Appendix C). Tables 3 tor, which is the exit pupil plane. Our formulas would change ~ and 4 provide a summary of the exit pupil plane irradiance depending on whether the field vector H is at the object or coefficients to the fourth order of approximation. image plane and on whether the aperture vector ρ~ and the ¯ The coefficients Ik;l;m are the sum of several components. stop are at the entrance pupil, the exit pupil, or an intermedi- The first component is represented by products of the first- ate pupil. order ray slopes u 0 and u¯ 0 in the image space and is equivalent to the first component of Ik;l;m coefficients. The second 6 Coefficient Relationship component includes additional terms that are functions of the For the case in consideration of having the field vector at the fourth-order image aberration coefficients and first-order ray object plane and the aperture vector at the exit pupil plane, slopes. Note that the second-order terms of the exit pupil irra- we can write the second order relationships in Table 5. diance function are modified by image aberrations, in con- Thus, it is possible to determine aberration coefficients trast to the second-order terms of the image plane irradiance. from measurements of the irradiance at the exit pupil ¯ð4Þ 4 The third component of the coefficients Ik;l;m is proportional plane and at the image plane of an optical system. to the six-order image aberration coefficients. Finally, we However, since the field vector is defined at the object plane, the image plane irradiance refers to image points have additional terms that involve products of the fourth- ~ ~ order aberrations. H þ ΔH, and, as a consequence, measurements should be made at conjugate object-image points.

Table 5 Relationships between second-order irradiance coefficients 7 Relative Illumination Ið2Þ I¯ð2Þ k;l;m and k;l;m. A practical and important case is the relative illumination RIðH~ Þ at the image plane of an optical system, which can ¯ 4 be written to the fourth order as follows: I020 − I200 ¼ Ж W 131 2 ¯ 4 6 RI H~ 1 I H~ H~ I H~ H~ : (15) I111 − I111 ¼ Ж W 220 þ Ж W 222 EQ-TARGET;temp:intralink-;e015;326;475 ð Þ¼ þ 200ð · Þþ 400ð · Þ I − I¯ 4 W 200 020 ¼ Ж 311 This is given in the limit of a small aperture by the terms ~ ~ ~ ~ 2 I200ðH · HÞ and I400ðH · HÞ of the irradiance function

Table 6 Second and fourth-order irradiance coefficients describing relative illumination.

Case I200 I400

Pinhole camera. −2 · u¯ 02 3 · u¯ 04

Field vector at the image plane. Stop and −2 · u¯ 02 3 · u¯ 04 aperture vector at the exit pupil plane.

¯ 02 ¯ 04 4 ¯ 02 Field vector at the object plane. Stop and −2 · u 3 · u þ Ж W 311 · u aperture vector at the exit pupil plane.

−2u¯ 02 4 W 3u¯ 04 6 W − 3 W u¯ 02 3 W W Field vector at the image plane. Stop and þ Ж 131 þ Ж 151 Ж 131 þ Ж2 131 131 aperture vector at the entrance pupil plane. − 1 32W W 24W W Ж2 ½ 040 220 þ 040 222

−2u¯ 02 4 W 3u¯ 04 4 W u¯ 02 6 W − 3 W u¯ 02 3 W W Field vector at the object plane. Stop and þ Ж 131 þ Ж 311 þ Ж 151 Ж 131 þ Ж2 131 131 aperture vector at the entrance pupil plane. − 1 32W W 24W W 8W W Ж2 ½ 040 220 þ 040 222 þ 131 311

−2u¯ 02 4 W 3u¯ 04 6 W − 3 W u¯ 02 3 W W Stop and aperture vector at plane between þ Ж 131B þ Ж 151B Ж 131B þ Ж2 131B 131B components of the lens. Field vector at the − 1 32W W 24W W image plane. Ж2 ½ 040B 220B þ 040B 222B

¯ 02 4 ¯ 04 4 ¯ 02 4 ¯ 02 6 3 ¯ 02 Stop and aperture vector at plane between −2u þ Ж W 131B 3u þ Ж W 311Au þ Ж W 311B u þ Ж W 151B − Ж W 131B u components of the lens. Field vector at the − 1 32W W 24W W 8W W object plane. Ж2 ½ 040B 220B þ 040B 222B þ 131B 311B 3 W W − 8 W W þ Ж2 131B 131B Ж2 131B 311A

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IðH;~ ρ~Þ. Although the calculation of the relative illumination more accurately for a lens system with a finite aperture. is typically done over the entire pupil, we have previously The relative illumination is obtained by the integration of shown that the approximation of a small diaphragm is suffi- the irradiance function over the exit pupil area as follows: 5 ZZ ciently accurate for practical purposes. The relative illumi- 1 RI H~ I H;~ ρ~ ρdρ dα; (20) nation on the image plane depends on the position of the EQ-TARGET;temp:intralink-;e020;326;719 ð Þ¼ ð Þ aperture stop, and the irradiance coefficients also depend π ρ;α on the location of the field and aperture vectors. I I where α is the angle between the field vector H~ and aperture Table 6 presents the second 200 and fourth order 400 ~ coefficients of the relative illumination function RIðH~ Þ for vector ρ~. The irradiance terms that are proportional to ðH · ρ~Þ several cases in the location of the aperture and field vectors. do not contribute to the relative illumination of a lens since When the stop is between system components, the system is the integral of cosðαÞ over the circle is zero. The field-inde- then divided into part A preceding the stop and part B fol- pendent irradiance terms give a constant integral value, while lowing the stop, and aberrations of each part are then used to other fourth-order irradiance terms modify second- and define the irradiance coefficients. Note that both image and fourth-order relative illumination coefficients. Table 7 sum- pupil aberration coefficients are used.5 marizes the integrated values of the irradiance terms over the ~ Examination of the functional form of the coefficients exit pupil. It follows that the relative illumination RIðHÞ for allows one to make several points of interest. First, to the an optical system with a finite aperture can be written as second–order, pupil coma influences the relative illumina- follows: I tion and can nullify the coefficient 200 in some cases. EQ-TARGET;temp:intralink-;e021;326;554 1 1 1 This is known as the Slyusarev effect.6 Alternatively, RI H~ 1 I I I ð Þ¼ 1 1 þ 200 þ 4 222 þ 2 220 the relationship between image and pupil aberrations is as 1 þ 2 I020 þ 3 I040 follows: ~ ~ ~ ~ 2 × ðH · HÞþI400ðH · HÞ : (21) Ж W W − Δ u¯ 2 ; (16) EQ-TARGET;temp:intralink-;e016;63;508 131 ¼ 311 2 f g As an example, we analyze the image plane illumination which can be used to determine how image distortion W311 of a mirror system in Fig. 4. This optical system is telecentric influences relative illumination.7 In this case, similar to the in the image space and is designed to satisfy the aplanatic standard cos4-law of illumination fall off, slopes in the object condition between the pupils to the sixth order. As a conse- space are used in the expression of the relative illumination.5 quence of aplanatic imaging between the pupils, the system Second, if the system is telecentric in the image space suffers from residual image aberrations. The mirror system 0 u¯ ¼ 0 and if the system is also aplanatic W131 ¼ 0, operates at f∕10 and covers a field-of-view (FoV) of 20 deg. W151 ¼ 0, and W040 ¼ 0, then the relative illumination is uniform to the fourth order. Such a system would have a f sin θ 8 Table 7 Second and fourth-order irradiance coefficients integrated ð Þ image height mapping. Other cases for uniform over the exit pupil area. 4 02 illumination are also possible, requiring Ж W131 − 2u¯ ¼ 0. Third, if the system is telecentric in the object space u¯ ¼ 0 and if the Herschel condition of the pupils is satisfied as Irradiance term Integral value I ~ρ ~ρ 1 I Ж 020ð · Þ 2 020 W Δ u¯ 2 ; (17) EQ-TARGET;temp:intralink-;e017;63;335 131 ¼ f g ~ 8 I111ðH · ρ~Þ 0

~ ~ 2 then the relative illumination would follow to the second I200ðH · HÞ I200H order cos3ðθ 0Þ rule in a system, where the aperture vector I ρ~ ρ~ 2 1 I is defined at the entrance pupil plane or at the intermediate 040ð · Þ 3 040 1 ~ pupil. This follows since I131ðH · ρ~Þðρ~ · ρ~Þ 0 ~ 2 1 2 4 I222ðH · ρ~Þ I222H RI H~ 1 −2u¯ 02 W ~ ~ 4 EQ-TARGET;temp:intralink-;e018;63;249 H H ð Þ¼ þ þ Ж 131 ð · Þ I H~ H~ ρ~ ρ~ 1 I H2 220ð · Þð · Þ 2 220 1 1 −2u¯ 02 u¯ 02 H~ H~ (18) I H~ H~ H~ ρ~ ¼ þ þ 2 ð · Þ 311ð · Þð · Þ 0 ~ ~ 2 4 I400ðH · HÞ I400H and 3 cos3 θ 0 1 − ¯ 02 : (19) EQ-TARGET;temp:intralink-;e019;63;164 u ð Þ¼ 2 þ ···

Fourth, contrary to intuition, pupil spherical aberration ¯ W040 may not influence the relative illumination when W220 ¼ W222 ¼ 0. However, pupil spherical aberration can change the angle of incidence of light on the image plane. Fig. 4 An image space telecentric mirror system. The mirror system is If all fourth-order irradiance terms are considered, it designed to satisfy the aplanatic condition between the pupils to six is possible to analytically calculate relative illumination order.

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Table 8 Second and fourth-order irradiance coefficients of the mirror L H;~ ρ~ L 1 A ρ~ ρ~ B H~ ρ~ C H~ H~ ; EQ-TARGET;temp:intralink-;e022;326;752 ð Þ¼ 0 · ½ þ ð · Þþ ð · Þþ ð · Þ system in Fig. 4. (22)

Irradiance term Value where A, B,andC are coefficients describing the emission profile. We substitute Eq. (22)intoEqs.(9)and(10)and I020ðρ~ · ρ~Þ 0.141052 write the irradiance coefficients to the second order as follows: I H~ ρ~ −0 132590 111ð · Þ . EQ-TARGET;temp:intralink-;e023;326;682 I H;~ ρ~ ð2Þ 1 −2u 02 A ρ~ ρ~ −4u 0u¯ 0 B H~ ρ~ ~ ~ ð Þ ¼ þ½ þ ð · Þþ½ þ ð · Þ I200ðH · HÞ 0.000000 −2u¯ 02 C H~ H~ : (23) 2 þ½ þ ð · Þ I040ðρ~ · ρ~Þ −0.008925 ~ I131ðH · ~ρÞð~ρ · ~ρÞ 0.026928 Thus, second-order variations in the source radiance pro- duce second-order variations in irradiance at the image plane. ~ ~ 2 I222ðH · ρÞ −0.011326 In addition, other higher-order irradiance terms, not pre- ~ ~ sented here, result from different combinations of the source I220ðH · HÞð~ρ · ~ρÞ −0.016355 radiance terms with irradiance terms, as shown in Tables 1 ~ ~ ~ I311ðH · HÞðH · ρ~Þ 0.009353 and 2. These higher order terms can be considered as extrin- ~ ~ 2 sic irradiance aberrations that result from the interaction of I400ðH · HÞ 0.000000 the incoming irradiance variations and aberration in the system. However, Eq. (23) shows that, with respect to the irradi- ~ ance of a pinhole camera IpinholeðH; ρ~Þ, second-order variations in the object space simply add to obtaining the irradiance in the image space.

9 Irradiance Coefficients and Choice of Coordinates We have pointed out that the irradiance coefficients depend on the location of the field and aperture vectors. Tables 1–4 give the coefficients for the case of having the field vector at the object plane and the aperture vector at the exit pupil plane. This case is an important one, for a diffraction calculation knowledge of the amplitude of the field at the exit pupil is necessary; this amplitude is taken to be equal to the square root of the irradiance function I¯ H;~ ~ρ . For com- Fig. 5 Relative illumination curves calculated with real ray tracing and ð Þ with the fourth-order analytical approximation for the mirror system in pleteness, we present in Appendices D and E the correspond- Fig. 4 show excellent agreement over the entire FoV. The image is ing formulas for other cases on the location of the field and equally illuminated over the entire area. aperture vectors. The derivation of irradiance coefficients requires several steps: The coefficients of the irradiance function for this mirror system are calculated and summarized in Table 8. 1. Choose the location of the field and aperture vectors The irradiance terms I200 and I400 are zero since u¯ ¼ W131 ¼ and determine the relationships according to the flow W151 ¼ W040 ¼ 0 by design. The relative illumination chart in Fig. 6. curves calculated with real ray tracing in Zemax 2. With those relationships, determine cos4½θ 0ðH~ 0; ρ~ 0Þ. OpticsStudio lens design software and with the fourth- This calculation is similar to Appendix B. order analytical approximation in Eq. (21) show excellent 3. If not a constant, determine dA 0 or dS 0 by evaluating agreement over the entire FoV, as presented in Fig. 5. The the Jacobian determinant. This calculation is similar to image is equally illuminated over the entire area. More Appendix C. analysis examples of lenses that show improved image plane illumination can be found in our previous 4. Calculate the irradiance coefficients with Eqs. (6) or (7). publications.5

10 Coefficients Verification 8 Combination of Irradiance Coefficients To support the analytical derivation, the magnitude of the In the derivation of the irradiance coefficients, we assume that irradiance coefficients was determined both through the for- the object has a constant or Lambertian radiance. However, it mulas derived and numerically. A macro program was writ- may be desirable to determine the irradiance coefficients when ten to calculate the irradiance coefficients by making an the object has a different emission profile. In this case, the iterative fit to a selected set of irradiance values across the source radiance is expanded as a polynomial series of dot aperture and field of an optical system. The iterative algo- products of the field and aperture vectors and to the second rithm is similar to one used by Sasian to fit aberration order, for example, can be written as follows: coefficients.9

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Fig. 6 Flow chart that defines the relationships for different selections of the coordinates.

For example, for an optical system with the stop aperture at the entrance pupil, the routine in Table 9 was executed to find the magnitude of the irradiance coefficients I020ð~ρ · ~ρÞ 2 and I040ðρ~ · ρ~Þ . According to Eq. (7), the normalized irradiance at the specified field and aperture points is given by dS 0 4 0 ~ dS 0 4 0 ~ dS · cos ½θ ðH; ρ~Þ. The quantity dS · cos ½θ ðH; ρ~Þ was computed in a lens design program by defining a small circle at the entrance pupil and tracing real rays to calculate the area of the corresponding ellipse at the exit pupil.

Table 9 Iterative algorithm that was used to fit irradiance coefficients.

Fig. 7 Layout of the Landscape lens used to compare irradiance coef- FOR i ¼ 1 to 100 ficients computed analytically and numerically. To minimize fitting errors, the FoV of the lens is limited to 30 deg. ρ ¼ 0.2 I dS 4 θ 0; ρ real ¼ dS0 cos ð Þf g I I − 1 − I ρ4 − I ρ6 − I ρ8 − I ρ10 ρ−2 200 ¼ðreal 400 600 800 1000 Þ ·

ρ ¼ 0.4 Table 10 Comparison of irradiance coefficients computed analytically and numerically. The agreement to eighth digits supports the cor- Ri dS 4 θ 0; ρ ¼ dS0 cos ð Þf g rectness of the formulas. I I − 1 − I ρ2 − I ρ6 − I ρ8 − I ρ10 ρ−4 400 ¼ðreal 200 600 800 1000 Þ · Irradiance Analytical Numerical ρ 0 6 ¼ . coefficient Ik;l;m formula calculation Ri dS 4 θ 0; ρ I ρ~ ρ~ −0 0051310 −0 0051310 ¼ dS0 cos ð Þf g 020ð · Þ . 721 . 695 I I − 1 − I ρ2 − I ρ4 − I ρ8 − I ρ10 ρ−6 I H~ ~ρ 600 ¼ðreal 200 400 800 1000 Þ · 111ð · Þ 0.0051413131 0.0051413122 ~ ~ ρ ¼ 0.8 I200ðH · HÞ −0.0563954333 −0.0563954267

Ri dS 4 θ 0; ρ I ρ~ ρ~ 2 −0 000276 −0 000276 ¼ dS0 cos ð Þf g 040ð · Þ . 7939 . 8392 I I − 1 − I ρ2 − I ρ4 − I ρ6 − I ρ10 ρ−8 I H~ ρ~ ρ~ ρ~ 800 ¼ðreal 200 400 600 1000 Þ · 131ð · Þð · Þ 0.0003447530 0.0003447656 ~ 2 ρ ¼ 1 I222ðH · ρ~Þ 0.0003958170 0.0003958224

Ri dS 4 θ 0; ρ I H~ H~ ~ρ ~ρ ¼ dS0 cos ð Þf g 220ð · Þð · Þ 0.0004320213 0.0004318147 I I − 1 − I ρ2 − I ρ4 − I ρ6 − I ρ8 ρ−10 I H~ H~ H~ ~ρ −0 0001299 −0 0001299 1000 ¼ðreal 200 400 600 800 Þ · 311ð · Þð · Þ . 094 . 152 ~ ~ 2 NEXT I400ðH · HÞ 0.0028603412 0.0028602385

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After a few iterations of the loop in Table 9, the coeffi- 2 cients I020ðρ~ · ρ~Þ and I040ðρ~ · ρ~Þ converged to the theoretical values with insignificant error. A similar approach was applied to validate the remaining irradiance coefficients. The iterative fit methodology was used to test the coefficients’ values at several conjugate distances and aperture stop positions for both single surface and a system of several surfaces. The obvious agreement of the formulas with the coefficients found with the iterative fit supports the validity of the theory. As an example, the irradiance coefficients for a Landscape lens10 shown in Fig. 7 were calculated both with the analytical formulas and with real ray tracing. The lens operates at f∕8 and the FoV is limited to 30 deg. Fig. 8 Geometrical variables involved in computing irradiance of a Table 10 presents a comparison of coefficients where the pinhole camera.

differences in the eighth decimal place are due to numerical EQ-TARGET;temp:intralink-;e024;326;578 t 0 4 computation errors. cos4 θ 0 ð Þ¼ 0 0 2 0 0 2 02 1∕2 ½ðxs − xi Þ þðys − yi Þ þ e 8 9 1 4 < = 11 Conclusion 1∕2 x 0−x 0 2 y 0−y 0 2 ¼ 1 ð s i Þ ð s i Þ In this paper, we present a second- and a fourth-order theory : þ t 02 þ t 02 ; of irradiance changes in axially symmetric optical systems. The concept of the irradiance function is presented and an 1 ; (24) interpretation of the irradiance aberrations is discussed. ¼ 2 x 02 y 02 x 02 y 02 2 x 0x 0 y 0y 0 1 ð s þ s Þ ð i þ i Þ − ·ð s i þ s i Þ The irradiance function terms represent basic distribution þ t 02 þ t 02 t 02 patterns in the irradiance of a beam at the exit pupil plane or at the image plane of an imaging system. where t 0 is the axial distance between the image plane and The irradiance coefficients are found via basic radiometric the aperture plane. principles, such as conservation of flux, and are derived from 0 0 0 0 ~ Points ðxs;ysÞ and ðxi ;yi Þ are specified by the field H purely geometric considerations. This approach gives us the aperture ~ρ vectors, as in Fig. 8, and are expressed in specific relationship between the irradiance distribution and terms of the first-order system quantities as follows: wavefront aberration coefficients. The edge diffraction effects are not considered in this study, and unclipped and x2 y2 1∕2 y¯ 0 H~ x2 y2 1∕2 y 0 ρ~ (25) EQ-TARGET;temp:intralink-;e025;326;402ð i þ i Þ ¼ · ð s þ s Þ ¼ · unfolded beams are assumed. – Tables 1 4 give the irradiance coefficients in terms of and wavefront aberration coefficients and first-order system y¯ 0 y 0 quantities to the fourth order of approximation. Specific ¯ 0 − u 0 : (26) EQ-TARGET;temp:intralink-;e026;326;358u ¼ ¼ formulas are provided for irradiance at the image and at t 0 t 0 the exit pupil of an optical system. The irradiance coeffi- y¯ 0 u¯ 0 cients depend on the selection of coordinates. In this manu- Here and are the chief ray height and slope at the y 0 u 0 script, the field vector is defined at the object plane, while image plane, respectively; and are the marginal ray the aperture vector is defined at the exit pupil plane. The height and slope at the aperture plane, respectively. formulas for the irradiance coefficients in Tables 1–4 We substitute Eqs. (25) and (26) into Eq. (24) and write show excellent agreement with the results from real ray the first two terms of a Taylor series expression to obtain the tracing. We also have discussed relative illumination and fourth order approximation to the cosine-to-the-fourth-power several cases of interest where the coordinate position of the ray angle as follows:

EQ-TARGET;temp:intralink-;e027;326;244 changes. cos4½θ 0ðH;~ ρ~Þ The theory of irradiance aberrations enhances our 1 knowledge about the behavior of light as it propagates in ¼ 2 optical systems and provides insights into how individual ½1 þ u¯ 02ðH~ · H~ Þþu 02ðρ~ · ρ~Þþ2 · u 0u¯ 0ðH~ · ρ~Þ wavefront aberration terms affect the light irradiance pro- 1 − 2 u¯ 02 H~ H~ u 02 ρ~ ρ~ 2 u 0u¯ 0 H~ ρ~ ::: duced by a lens system at its image plane or at the exit ¼ · ½ ð · Þþ ð · Þþ · ð · Þ 2 pupil plane. þ3 · ½u¯ 02ðH~ · H~ Þþu 02ðρ~ · ρ~Þþ2 · u 0u¯ 0ðH~ · ρ~Þ ¼ 1 − 2 · u¯ 02ðH~ · H~ Þ − 2 · u 02ðρ~ · ρ~Þ − 4 · u 0u¯ 0ðH~ · ρ~Þ::: 2 Appendix A: Series Expansion of the þ3 · u¯ 04ðH~ · H~ Þ þ 12 · u 03u¯ 0ðH~ · ρ~Þðρ~ · ρ~Þ Cosine-to-the-Fourth-Power of the Ray Angle 2 þ 12 · u 02u¯ 02ðH~ · ρ~Þ ::: By definition, the cosine-to-the-fourth-power of the angle 0 0 2 2 ~ ~ between the ray connecting the point ðxs;ysÞ on the aperture þ6 · u 0 u¯ 0 ðH · HÞðρ~ · ρ~Þ plane and point ðx 0;y0Þ on the image plane and the optical i i 12 u 0u¯ 03 H~ H~ H~ ρ~ 3 u 04 ρ~ ρ~ 2; (27) axis is as follows: þ · ð · Þð · Þþ · ð · Þ

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0 ~ 4 0 ~ ~ where θ ðH; ρ~Þ in Eq. (27) indicates that the function is Finally, the terms ∇H cos ½θ ðH; ρ~Þ · ΔH are summa- evaluated for a ray connecting a point defined by ~ρ on the rized in Table 11. aperture plane and a point defined by H~ on the image plane. Appendix C: Derivation of the Irradiance Appendix B: Derivation of the Irradiance Function at the Exit Pupil of an Optical System Function at the Image Plane of an Optical with the Aperture Vector at the Exit Pupil System with the Aperture Vector at the To obtain the Jacobian determinant, we express the trans- Exit Pupil verse ray aberration vector ΔH~ in orthogonal components ρ^ k^ In an actual system, rays may not pass through the ideal along unit vectors and as follows: image point due to aberrations, as shown in Fig. 9. The ΔH~ ΔH ^ ^ EQ-TARGET;temp:intralink-;e030;326;641 ρ ΔH k; (30) ray intersection with the image plane is determined by con- ¼ ρ þ k ΔH~ sidering the transverse ray errors , and it is defined by the ρ^ ~ρ k^ H~ ΔH~ where is a unit vector along the aperture vector and is a vector þ . ρ~ In the presence of image aberrations, the cosine-to-the- unit vector perpendicular to the aperture vector , as shown in 4 Table 12. fourth-power of the ray angle cos ½θ 0ðH~ þ ΔH;~ ρ~Þ is evaluated by writing the differential as Table 12 Third-order transverse ray aberrations in the orthogonal

EQ-TARGET;temp:intralink-;e028;63;560 coordinates. 4 0 ~ ~ 4 0 ~ 4 0 ~ cos ½θ ðH þ ΔH; ρ~Þ ≃ cos ½θ ðH;ρ~Þ þ ∇H cos ½θ ðH; ρ~Þ ΔH;~ (28) · Ray aberration ΔH~ ρ^ k^

~ 3 4 0 ~ ΔH040 4ρ where ∇H cos ½θ ðH; ρ~Þ is the gradient of the function in ~ H ~ 2 2 Eq. (27) with the respect to the field vector . With no sec- ΔH131 3ρ Hρ ρ Hk ond-order terms in the aberrations function, the terms 4 0 ~ ~ ΔH~ 2ρH2 2ρH H ∇H cos ½θ ðH; ρ~Þ · ΔH result in irradiance terms that are 222 ρ ρ k at least of the fourth order. Thompson has shown that the ΔH~ 2ρ H2 H2 gradient operator is given by the derivative of the function 220 ð ρ þ k Þ 3 ~ 2 2 2 2 with respect to the designated vector. It follows that to ΔH311 ðHρ þ Hk ÞHρ ðHρ þ Hk ÞHk the first order: ~ ΔH400 ∇ cos4 θ 0 H;~ ρ~ −4 u¯ 02 H − 4 u 0u 0 ρ~ : (29) EQ-TARGET;temp:intralink-;e029;63;419 H ½ ð Þ ¼ ½ · · · · Þ Table 13 Third-order transverse ray aberrations derivatives in orthogonal coordinates.

ΔH~ ∂ ΔH~ ∂ ΔH~ ∂ ΔH~ ∂ ΔH~ Ray aberration Hρ ρ Hk k Hk ρ Hρ k ~ ΔH040

~ 2 2 ΔH131 3ρ ρ ~ ΔH222 4ρHρ 2ρHρ 2ρHk

ΔH~ 4ρH 4ρH Fig. 9 Geometrical variables involved in computing irradiance on the 220 ρ k focal plane of an optical system with the aperture stop at the exit pupil. ΔH~ 3H2 H2 H2 3H2 2H H 2H H Real rays (solid lines) and first-order rays (dashed lines) coincide at 311 ð ρ þ k Þðρ þ k Þ ρ k ρ k the exit pupil. Real rays may not pass through the ideal image point ΔH~ due to aberrations. 400

Table 11 Summary of contributions to irradiance from the transverse ray errors ΔH~ .

~ 2 ~ ~ ~ Ray aberration ΔH Correction term −4u¯ 0 ðH · ΔHÞ Correction term −4u 0u¯ 0ðΔH · ρ~Þ

4 16 ¯ 02 ~ 16 0 ¯ 0 2 − Ж W 040ðρ~ · ρ~Þρ~ Ж W 040u ðH · ρ~Þðρ~ · ρ~Þ Ж W 040u u ðρ~ · ρ~Þ 2 2 ~ 8 ¯ 02 ~ 8 0 ¯ 0 ~ − Ж W 131ðH · ρ~Þρ~ Ж W 131u ðH · ρ~Þ Ж W 131u u ðH · ρ~Þðρ~ · ρ~Þ

1 ~ 4 ¯ 02 ~ ~ 4 0 ¯ 0 ~ − Ж W 131ðρ~ · ρ~ÞH Ж W 131u ðH · HÞðρ~ · ρ~Þ Ж W 131u u ðH · ρ~Þðρ~ · ρ~Þ 2 ~ ~ 8 ¯ 2 ~ ~ ~ 8 ¯ ~ 2 − Ж W 222ðH · ρ~ÞH Ж W 222u 0 ðH · HÞðH · ρ~Þ Ж W 222u 0u 0ðH · ρ~Þ

2 ~ ~ 8 ¯ 02 ~ ~ ~ 8 0 ¯ 0 ~ ~ − Ж W 220ðH · HÞρ~ Ж W 220u ðH · HÞðH · ρ~Þ Ж W 220u u ðH · HÞðρ~ · ρ~Þ 2 1 ~ ~ ~ 4 ¯ 02 ~ ~ 4 0 ¯ 0 ~ ~ ~ − Ж W 311ðH · HÞH Ж W 311u ðH · HÞ Ж W 311u u ðH · HÞðH · ρ~Þ

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Table 14 Fifth-order transverse ray aberrations for an optical system with the stop aperture at the exit pupil.

~ 1 ~ 6 ~ ð5Þ Ray aberration ΔH − Ж ∇ρW ðH;ρ~Þ Terms O

~ 2 6 6 02 ΔH060ρð~ρ · ~ρÞ ~ρ − W 060 − W 040u Ж Ж ~ 2 ~ 1 1 1 02 0 ¯ 0 ΔH151H ðρ~ · ρ~Þ H − Ж W 151 − Ж 2 · W 131u þ 4 · W 040u u

~ ~ 4 1 02 0 ¯ 0 ΔH151ρðH · ρ~Þðρ~ · ρ~Þρ~ − Ж W 151 − Ж ½4 · W 131u þ 8 · W 040u u 2 ~ ~ ~ 3 1 02 0 0 ΔH333H ðH · ρ~Þ H − W 333 − ½2 · W 131u¯ þ 4 · W 222u u¯ Ж Ж ~ ~ ~ ~ 1 1 0 ¯ 0 1 02 3 ¯ 02 ΔH331H ðH · HÞðρ~ · ρ~ÞH − Ж W 331 − Ж 2 · W 220u u þ 2 · W 311u þ 2 · W 131u

~ ~ ~ ~ 2 1 0 ¯ 0 ¯ 02 0 ¯ 0 02 ΔH331ρðH · HÞðH · ρ~Þρ~ − Ж W 331 − Ж ½2 · W 222u u þ W 131u þ 4 · W 220u u þ W 311u ; 2 ~ ~ 2 1 02 0 ¯ 0 ΔH242ρðH · ρ~Þ ρ~ − Ж W 242 − Ж ½2 · W 222u þ 4 · W 131u u

~ ~ ~ 2 1 ¯ 02 0 ¯ 02 02 ΔH242H ðH · ρ~Þðρ~ · ρ~ÞH − Ж W 242 − Ж ½4 · W 040u þ 4 · W 131u u þ W 222u

~ ~ ~ 4 1 02 0 ¯ 0 ¯ 02 ΔH240H ðH · HÞðρ~ · ρ~Þρ~ − Ж W 240 − Ж ½3 · W 220u þ W 131u u þ 2 · W 040u

~ ~ ~ ~ ~ 2 1 ¯ 02 ¯ 02 0 ¯ 0 ΔH422H ðH · HÞðH · ρ~ÞH − Ж W 422 − Ж ½3 · W 222u þ 2 · W 220u þ 2 · W 311u u 2 ~ ~ ~ 2 1 ¯ 02 0 ¯ 0 ΔH420ρðH · HÞ ρ~ − Ж W 420 − Ж ½W 220u þ W 311u u 2 ~ ~ ~ ~ 1 1 3 ¯ 02 ΔH511H ðH · HÞ H − Ж W 511 − Ж 2 · W 311u

Table 15 Terms corresponding to the determinant of the Jacobian transformation between the object and image planes for an optical system with the stop aperture at the exit pupil.

Fourth-order aberration Six-order aberration Orthogonal derivative Irradiance coefficient Jk;l;m contributions contributions contributions

4 J020ðρ~ · ρ~Þ − Ж W 131

~ 4 J111ðH · ~ρÞ − Ж W 220

6 − Ж W 222

~ ~ 4 J200ðH · HÞ − Ж W 311

J ρ~ ρ~ 2 − 1 5 W u 02 16 W u 0u¯ 0 − 6 W 3 W W 040ð · Þ Ж ½ · 131 þ · 040 Ж 151 Ж2 131 131

J H~ ρ~ ρ~ ρ~ − 1 7 W u 02 6 W u 02 − 10 W 4 W W 131ð · Þð · Þ Ж ½ · 222 þ · 220 Ж 242 Ж2 220 131

− 1 22 W u 0u¯ 0 − 16 W u¯ 02 − 8 W 10 W W Ж ½ · 131 · 040 Ж 240 Ж2 222 131 2 J H~ ρ~ − 1 10 W u¯ 02 20 W u 0u¯ 0 − 12 W 8 W W 222ð · Þ Ж ½ · 131 þ · 222 Ж 333 Ж2 222 222

− 1 8 W u 0u¯ 0 2 W u 02 − 4 W 16 W W Ж ½ · 220 þ · 311 Ж 331 Ж2 222 220 − 4 W W Ж2 311 131

J H~ H~ ~ρ ~ρ − 1 7 W u¯ 02 2 W u 0u¯ 0 − 6 W 10 W W 220ð · Þð · Þ Ж ½ · 131 þ · 222 Ж 331 Ж2 311 131

− 1 12 W u 0u¯ 0 3 W u 02 − 8 W W Ж ½ · 220 þ · 311 Ж2 222 220

J H~ H~ H~ ~ρ − 1 15 W u¯ 02 14 W u¯ 02 − 10 W 4 W W 311ð · Þð · Þ Ж ½ · 222 þ · 220 Ж 422 Ж2 220 311

− 14 W u 0u¯ 0 − 8 W 10 W W Ж 311 Ж 420 Ж2 222 311 2 J H~ H~ − 9 W u¯ 02 − 6 W 3 W W 400ð · Þ Ж 311 Ж 511 Ж2 311 311

EQ-TARGET;temp:intralink-;e031;326;110 0 2 ~ ~ ~ ~ Then, we consider the transformations Hρ Hρ ΔHρ y¯ 0 ∂ΔHρ ∂ΔH ∂ΔHρ ∂ΔH ¼ þ J H;~ ρ~ 1 ∇ ΔH~ k − k ; H 0 H ΔH ð Þ¼ 2 þ H þ and k ¼ k þ k, which gives the position of the y¯ ∂H~ ∂H~ ∂H~ ∂H~ given ray at the image pupil, so we find the Jacobian deter- ρ k k ρ minant: (31)

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4 ~ Table 17 Fourth-order irradiance coefficients Ið Þ at the image where ∇HΔH is the divergence of the transverse ray error k;l;m ~ ~ ~ ~ plane of an optical system with the aperture vector at the entrance ∂ΔHρ ∂ΔHk ∂ΔHρ ∂ΔHk vector, and ~ , ~ , ~ and ~ are derivatives of the pupil. ∂Hρ ∂Hk ∂Hk ∂Hρ 4,9 transverse ray error vector in the orthogonal coordinates. h 2 04 16 0 ¯ 0 6 3 02 With no second order terms in the aberration function, the I040ðρ~ · ρ~Þ ¼ 3u þ Ж W 040u u þ Ж W 511 − Ж W 311 · u ~ ~ i ∂ΔHρ ∂ΔH~ ∂ΔHρ ∂ΔH~ k − k 48 48 3 2 terms ~ ~ and ~ ~ result in irradiance terms that − 2 W 222W 040 − 2 W 220W 040 2 W 311W 311 ρ~ ρ~ ∂Hρ ∂Hk ∂Hk ∂Hρ Ж · Ж · þ Ж ð · Þ are at least of fourth order. The derivatives of the third-order h ~ 03 ¯ 0 16 ¯ 02 12 0 ¯ 0 transverse ray aberrations in orthogonal coordinates are sum- I131ðH · ρ~Þðρ~ · ρ~Þ¼ 12u u þ Ж W 040u þ Ж W 131u u marized in Table 13. 10 W 8 W − 5 W u 02 − 2 W u 2 The relationships between wavefront deformations þ Ж 422 þ Ж 420 Ж 222 · Ж 220 · 0 and transverse ray aberrations to fifth order are given in − 6 W u u¯ − 112 W W − 28 W W Ж 311 · 0 0 Ж2 131 040 Ж 220 131 Table 14. The fifth-order transverse ray errors are related i − 30 W¯ W 4 W W 10 W W H~ ρ~ ρ~ ρ~ to the wavefront deformation by the gradient of the aberra- Ж2 222 131 þ Ж2 220 311 þ Ж2 222 311 ð · Þð · Þ tion function and include additional terms that are products h 2 of the fourth-order pupil aberration coefficients and paraxial ~ 02 02 8 02 8 0 0 I222ðH · ρ~Þ ¼ 12u u¯ þ W 131u¯ þ W 222u u¯ ray slopes u¯ 0 and u 0 in the image space. The subscripts H and Ж Ж 12 4 2 2 12 ¯ ρ refer to the components along the field and aperture vec- þ Ж W 333 þ Ж W 331 þ Ж W 131 · u − Ж W 222 · uu 1 tors, respectively. 8 ¯ 2 ¯ 2 48 ~ − W 220 · uu þ W 311 · u − 2 W 131W 131 The divergence of the transverse ray errors ∇HΔH results Ж Ж Ж in terms that are at least of the second order. Sasian has − 16 W W − 8 W W − 64 W W Ж2 222 222 Ж2 220 222 Ж2 040 040 i shown the procedure to calculate the divergence operator 2 9,11 8 W W 16 W W − 4 W W H~ ρ~ of the function with respect to the designated vector. þ Ж2 222 222 þ Ж2 222 220 Ж2 311 131 ð · Þ Following this procedure and taking the required derivatives, h ~ ~ 2 ¯ 02 4 ¯ 02 8 0 ¯ 0 we find the terms in Table 15. I220ðH · HÞðρ~ · ρ~Þ¼ 6u 0 u þ Ж W 131u þ Ж W 220u u The Jacobian coefficients in Table 15 are the sum of all 6 2 02 4 0 0 5 02 ~ ~ 2 þ W 331 þ W 222 · u¯ þ W 220 · u u¯ − W 311 · u¯ components. Thus, for example, the term J400ðH · HÞ is Ж Ж Ж Ж given by − 5 W u 02 − 96 W W − 16 W W Ж 131 · Ж2 040 040 Ж2 131 131 − 16 W W − 16 W W − 4 W W Ж2 222 220 Ж2 220 220 Ж2 220 222 i 6 9 3 2 10 W W − 8 W W H~ H~ ρ~ ρ~ J − W − W ¯ 02 W W ~ ~ : þ Ж2 131 311 Ж2 220 222 ð · Þð · Þ EQ-TARGET;temp:intralink-;e032;63;441 u H H 400 ¼ Ж 511 Ж 311 þ Ж2 311 311 ð · Þ h ~ ~ ~ 0 ¯ 03 8 ¯ 02 8 ¯ 02 (32) I311ðH · HÞðH · ρ~Þ¼ 12u u þ Ж W 222u þ Ж W 220u

4 0 ¯ 0 10 8 þ Ж W 311u u þ Ж W 242 þ Ж W 240

5 ¯ 02 2 ¯ 02 6 0 ¯ 0 Finally, the coefficients in Tables 3 and 4 are obtained by − Ж W 222 · u − Ж W 220 · u − Ж W 131 · u u combining equations in Tables 1, 2, and 15, and keeping only − 80 W W − 28 W W − 32 W W second- and fourth-order terms. Ж2 040 131 Ж2 131 222 Ж2 131 220 − 6 W W − 4 W W 4 W W Ж2 222 311 Ж2 220 311 þ Ж2 220 131 i I 10 W W H~ H~ H~ ρ~ Appendix D: Irradiance Coefficients k;l;m þ Ж2 222 131 ð · Þð · Þ at the Image Plane of an Optical System h 2 with the Field Vector at the Object Plane ~ ~ ¯ 04 4 ¯ 02 6 3 ¯ 02 I400ðH · HÞ ¼ 3u þ Ж W 311u þ Ж W 151 − Ж W 131 · u ::: and the Aperture Vector at the Entrance − 32 W W − 24 W W − 8 W W Ж2 040 220 Ж2 040 222 Ж2 131 311 Pupil Plane i 3 W W H~ H~ 2 Tables 16 and 17 give the image plane irradiance coefficients þ Ж2 131 131 ð · Þ . for the case of having the field vector at the object plane and the aperture vector at the entrance pupil plane.

ð2Þ I Table 16 Second-order irradiance coefficients Ik;l;m at the image plane Appendix E: Irradiance Coefficients k;l;m at of an optical system with the aperture vector at the entrance pupil. the Exit Pupil Plane of an Optical System h i with the Field Vector at the Object Plane I ρ~ ρ~ −2u 02 4 W ρ~ ρ~ 020ð · Þ¼ þ Ж 311 ð · Þ and the Aperture Vector at the Entrance h i ~ 0 ¯ 0 4 6 ~ Pupil Plane I111ðH · ρ~Þ¼ −4u u þ Ж W 220 þ Ж W 222 ðH · ρ~Þ h i Tables 18 and 19 give the exit pupil plane irradiance coef- I H~ H~ −2u¯ 02 4 W H~ H~ 200ð · Þ¼ þ Ж 131 ð · Þ ficients for the case of having the field vector at the object plane and the aperture vector at the entrance pupil plane.

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I¯ð2Þ Table 18 Second-order irradiance coefficients k;l;m at the exit pupil References plane of an optical system with the aperture vector at the entrance pupil. 1. J. Sasian, Introduction to Aberrations in Optical Imaging Systems, Cambridge University Press, New York (2013). h i 2. J. M. Palmer and B. G. Grant, The Art of Radiometry, SPIE Press, I¯ H~ H~ −2u¯ 02 − 4 W H~ H~ 020ð · Þ¼ Ж 311 ð · Þ Bellingham, Washington (2010) 3. K. P. Thompson, “Description of the third-order optical aberrations of h i near-circular pupil optical systems without symmetry,” J. Opt. Soc. Am. ¯ ~ 0 ¯ 0 4 6 ~ – I111ðH · ρ~Þ¼ −4u u − Ж W 220 − Ж W 222 ðH · ρ~Þ A 22, 1389 1401 (2005). 4. C. Roddier and F. Roddier, “Wave-front reconstruction from defocused h i images and the testing of ground-based optical telescopes,” J. Opt. Soc. I¯ ρ~ ρ~ −2u 02 − 4 W ρ~ ρ~ Am. A 10(11), 2277 (1993). 200ð · Þ¼ Ж 131 ð · Þ 5. D. Reshidko and J. Sasian, “The role of aberrations in the relative illumination of a lens system,” Opt. Eng. 55(11), 115105 (2016). 6. C. Wynne, “Primary aberrations and conjugate change,” Proc. Phys. Soc. B 65(6) (1951). 7. J. Sasian, “Interpretation of pupil aberrations in imaging systems,” SPIE 4 Table 19 Fourth-order irradiance coefficients I¯ð Þ at the exit pupil Proc. 6342, 634208 (2006). k;l;m 8. R. Kingslake, Illumination in Optical System, Applied Optics and plane of an optical system with the aperture vector at the entrance pupil. Optical Engineering, Vol. II, Academic Press, New York (1965). 9. J. Sasian, “Theory of six-order wave aberrations,” Appl. Opt. 49(16), h D69–D95 (2010). 2 ¯ ~ ~ ¯ 4 6 3 ¯ 02 16 0 ¯ 0 10. R. Kingslake, Lens Design Fundamentals, 2nd ed., Academic Press, I040ðH · HÞ ¼ 3u − Ж W 511 þ Ж W 311u − Ж W 040u u i Burlington, Massachusetts (2010) 48 48 3 ~ ~ 2 11. J. Sasian, “Theory of six-order wave aberrations: errata,” Appl. Opt. − 2 · W 222W 040 − 2 · W 220W 040 2 W 311W 311 H · H Ж Ж þ Ж ð Þ 49(33), 6502 (2010). h ¯ ~ ~ ~ 0 ¯ 03 10 8 Dmitry Reshidko is a PhD candidate whose research involves the I131ðH · HÞðH · ρ~Þ¼ 12u u − Ж W 422 − Ж W 420 development of novel imaging techniques and methods for image 5 ¯ 02 2 ¯ 02 6 0 ¯ 0 16 02 aberration correction, innovative optical design, fabrication, and test- þ Ж W 222u þ Ж W 220u þ Ж W 311u u − Ж W 040u ing of state-of-the art optical systems. − 12 W u 0u¯ 0 − 112 W W − 28 W W − 30 W W Ж 131 Ж2 131 040 Ж2 220 131 Ж2 222 131 i Jose Sasian is a professor at the University of Arizona. His interests 4 W W 10 W W H~ H~ H~ ρ~ are in teaching optical sciences, optical design, illumination optics, þ Ж2 220 311 þ Ж2 222 311 ð · Þð · Þ optical fabrication and testing, telescope technology, optomechanics, h lens design, light in gemstones, optics in art and art in optics, and light 2 ¯ ~ 02 ¯ 02 12 4 2 ¯ 02 propagation. He is a fellow of SPIE and the Optical Society of America I222ðH · ρ~Þ ¼ 12u u − Ж W 333 − Ж W 331 − Ж W 131u and is a lifetime member of the Optical Society of India. 12 0 ¯ 0 8 0 ¯ 0 2 02 8 02 þ Ж W 222u u þ Ж W 220u u − Ж W 311u − Ж W 131u

− 8 W u 0u¯ 0 − 48 W W − 16 W W Ж 222 Ж2 131 131 Ж2 222 222 − 8 W W − 64 W W 8 W W Ж2 220 222 Ж2 040 040 þ Ж2 222 222 i 16 W W − 4 W W H~ ρ~ 2 þ Ж2 222 220 Ж2 311 131 ð · Þ h ¯ ~ ~ 02 ¯ 02 6 5 ¯ 02 I220ðH · HÞðρ~ · ρ~Þ¼ 6u u − Ж W 331 þ Ж W 131u

2 0 ¯ 0 4 0 ¯ 0 5 02 4 02 − Ж W 222u u − Ж W 220u u þ Ж W 311u − Ж W 131u

− 8 W u 0u¯ 0 − 96 W W − 16 W W Ж 220 Ж2 040 040 Ж2 131 131 − 16 W W − 16 W W − 4 W W Ж2 222 220 Ж2 220 220 Ж2 220 222 i 10 W W − 8 W W H~ H~ ρ~ ρ~ þ Ж2 311 131 Ж2 222 220 ð · Þð · Þ h ¯ ~ 03 ¯ 0 10 I311ðH · ρ~Þðρ~ · ρ~Þ¼ 12u u − Ж W 242

8 5 02 2 02 6 0 ¯ 0 − Ж W 240 þ Ж W 222u þ Ж W 220u þ Ж W 131u u

8 02 8 02 4 0 ¯ 0 − Ж W 222u − Ж W 220u − Ж W 311u u − 80 W W − 28 W W − 32 W W Ж2 040 131 Ж2 131 222 Ж2 131 220 − 6 W W − 4 W W Ж2 222 311 Ж2 220 311 i 4 W W 10 W W H~ ρ~ ρ~ ρ~ þ Ж2 220 131 þ Ж2 222 131 ð · Þð · Þ h ¯ 2 04 6 3 02 I400ðρ~ · ρ~Þ ¼ 3u − Ж W 151 þ Ж W 131u

− 4 W u 02 − 32 W W − 24 W W Ж 311 Ж2 040 220 Ж2 040 222 i − 8 W W 3 W W ρ~ ρ~ 2 Ж2 131 311 þ Ж2 131 131 ð · Þ

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