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Primary coefficients for axial gradient-index

Florian Bociort

Optics Research Group, Department of Applied Physics Delft University of Technology, The Netherlands*

ABSTRACT

As for homogeneous lenses, for axial gradients the analysis of the Seidel and coefficients can be very useful in design. However, at present few commercial optical design programs list the Seidel aberrations of GRIN lenses and none of them lists the chromatic aberrations. In order to facilitate the computer implementation of the chromatic aberrations of axial GRIN lenses a new mathematical derivation for the axial and lateral color coefficients is presented. Also, new qualitative insight into the properties of axial GRIN lenses is obtained by introducing the thin-lens approximation in the aberration expressions. Within the domain of validity of this approximation, the primary aberrations of an axial GRIN lens are equivalent to those of a pair of homogeneous aspherical lenses in contact, having a common plane surface and having refractive indices and Abbe numbers equal to the corresponding axial GRIN values at the two end surfaces.

Keywords: lens design, gradient-index , aberrations,

1. INTRODUCTION

The interest in gradient-index (GRIN) optics has increased recently as a result of substantial progress in the manufacturing technology of GRIN components. Major progress has been achieved especially for axial gradients (i.e. glasses for which the is a function of the coordinate along the symmetry axis) 1,2. It is well known that spherical surfaces of axial GRIN lenses have aberration correction capabilities that are nearly equivalent to those of aspheric homogeneous lenses, and therefore axial gradients are used basically as asphere replacements. However, the effect on the monochromatic and chro- matic aberrations of the propagation through the axial GRIN medium between the surfaces is more difficult to be ana- lyzed. It is convenient for this purpose to examine the Seidel and chromatic paraxial aberration coefficients (i.e. the primary aberrations) of the axial GRIN lens: Unlike propagation in homogeneous media, ray propagation through gradients gener- ates so-called transfer contributions to the primary aberrations. For instance, the values of the primary aberrations can be used to investigate the sensitivity of the aberrations when the gradient material is changed.

Analytic expressions of the primary aberrations of axial GRIN lenses have been derived some time ago by Sands3,4,5,6. How- ever, Sands’ complex mathematical derivation is quite a challenge for many interested readers. At the time of this writing only one of the major commercial optical design programs (CODE V) lists the Seidel aberrations of gradients and none of them lists the chromatic aberrations.

The present paper has two objectives. First, in order to facilitate the computer implementation of the chromatic aberrations of axial GRIN lenses a new mathematical derivation for the transfer contributions axial and lateral color coefficients will be presented. The derivation described in Section 4 is based on the method first developed by this author for the aberration coefficients of radial GRIN lenses7. The results are equivalent with those of Sands, but the derivation itself is perhaps sim- pler and certain intermediate results can also be used to check the accuracy of the computer implementation of the formulas and to avoid possible implementation errors. As a prerequisite, will be reviewed in Sec. 2. The formulas that are necessary for the computation of the primary aberrations are summarized in Sec.3.

The second objective of this paper is to gain a better qualitative understanding of the effect of the transfer contributions for all primary aberrations of axial gradients. For this purpose, in Section 5 the thin-lens approximation will be introduced in

* Address: Lorentzweg 1, 2628 CJ Delft, E-mail:[email protected], Permanent address: National Institute for Laser, Plasma and Radiation Physics, Department of Lasers, P.O. Box MG-36, 76900 Bucharest, Romania the aberration expressions. It will be shown that, within the domain of validity of this approximation, the primary aberra- tions of an axial GRIN lens are equivalent to those of a pair of homogeneous aspherical lenses in contact, having a common plane surface and having refractive indices and Abbe numbers equal to the corresponding axial GRIN values at the two end surfaces.

2. RAY TRACING IN AXIAL GRADIENTS

As for homogeneous lenses, the primary aberrations of axial gradients are computed by using paraxial ray-tracing data for the marginal and chief ray. Therefore, ray tracing in axial gradients will be briefly reviewed below.

The path of a ray in an inhomogeneous medium is described by the well - known differential equation

d ⎛ dR ⎞ ⎜n ⎟ = ∇n (1) ds ⎝ ds ⎠ where n(x,y,z) is the refractive index of the medium, R = (x,y,z) is the position vector and ds denotes the arc length between two neighboring points along the ray path, such that we have dsdxdyd2 =++2 2 z 2 . The vector n dR/ds is of length n and has as its components the three optical direction cosines ξ,η,ζ with respect to the axes x, y and z , respectively, which are related by ξ222++=η ζ n2 .

In axial gradients, the refractive index changes only along the symmetry axis, n=n(z). Since ∂∂nx//== ∂∂ ny0 we have dξ //ds== dη ds 0 and it follows that in axial gradients the optical direction cosines with respect to x and y

dx dy n ==ξ , n η (2) ds ds are invariant along the ray path, retaining the initial values ξ and η they had at the entrance in the medium. The third optical direction cosine dz n ==ζ nz2 () −ξ 2 −η 2 (3) ds is then simply a function of z . (We assume that the ray propagates from left to right and therefore choose the plus sign for the square root in Eq.(3).)

For determining the ray path, note that Eqs. (2-3) yield

dx ξ dy η ==, (4) dz ζ dz ζ

The equations (4) can be integrated directly. Let x and y be the ray coordinates at the entrance in the medium and let x' and y' be the corresponding values after the ray travels a distance d in the medium. The solutions read

xxA′ =+ξ ()ξ,ηη, yyA′ =+ ()ξ,η (5) where we have denoted d dz A()ξ,η = ∫ (6) 0 nz2 ()−−ξ 2 η 2 ω ==+16−1 ξ 2 η2 Introducing the abbreviations nz, Cξη we can write

ω 1 1 3 (7) = =+ωωCξη +... 222 2 nz()−−ξ η 1 − Cξηω 2 Therefore we have 1 Andnd()ξ,η =+−−1 3 ()ξ 2 ++η 2 ... (8) 2 where we have denoted the average values of 1/n and 1/n3 by 1 d dz n −1 = ∫ (9) d 0 nz() and 1 d dz n −3 = ∫ (10) d 0 nz()3

In the paraxial approximation only the first term in the power series expansion (8) is retained, and Eqs. (5) become

xxnd′ =+ −−1 ξ , yynd′ =+ 1 η (11)

Note that the paraxial transfer equations for axial gradients are the same as for homogeneous media, excepting that the re- ciprocal of the homogeneous refractive index is replaced by its average value given by Eq. (9). The second term in Eq.(8) leads to a third-order term in Eqs.(5) and therefore the quantity (10) will enter in the expressions of the Seidel coefficients for transfer.

For axial gradients, the of paraxial rays at a surface is described by precisely the same equation as in the homoge- neous case. However, since the refractive index is variable within the lens, the value of the refractive index to be used is that at the surface vertex.

Consider a rotationally - symmetric optical system consisting of homogeneous and gradient - index lenses. We define an τ τ arbitrary ray through the system by its normalized field coordinates ( x, y) in the object plane P, and by the coor- σ σ dinates ( x, y) in the entrance pupil plane EP. Thus, if rEP is the radius of the entrance pupil and rP is the maximum object height, the Cartesian ray coordinates at the object plane and at the entrance pupil plane are given by ==ττ ==σσ xPPxPPryr, y , xEPryr EP x, EP EP y (12)

At each surface, the position and direction of the ray are given by the x and y coordinates of its point of intersection with the surface and by the optical direction cosines ξ and η.

A j B

m i h u w O S C P

Fig. 1 Ray parameters of the marginal ray OB and chief ray AP at the first surface of the system

In the following sections we will need the position and direction of the ray only in the paraxial approximation. In this case it ξ η σ σ τ τ can be shown that at each surface of the system x,y, , are given by linear combinations of x, y, x, y and that the coeffi- cients are the height and slope of the paraxially traced marginal and chief rays at that surface xm=+τσ h, ξ =−− nwτσ nu xx0 x0 x . (13) =+τσ η =−− τ σ ymyy h, nw0 y nu0 y

Here, the paraxial marginal and chief ray heights are denoted h and m and the corresponding marginal and chief ray slopes are denoted u and w. (See Fig.1.) The sign convention adopted for u and w is that their signs are the opposite of those of the corresponding direction cosines. (This is why in Eqs. (13) we have minus signs in the equations for ξ and η.) The refractive index at the vertex of the surface is denoted by n0.

= ′ = ′ ξ =− ξ′ =− ′′ At transfer through axial gradients, inserting in Eqs. (11) xhx, h and nu0 , n0u we obtain for the marginal ray the transfer equation ′= − −1 h hdnn0u (14) while the invariance of ξ yields ′′= nu00 nu . (15)

For the chief ray, the inclination u and height h must be replaced by w and m.

Finally, for an arbitrary paraxial ray we consider the quantity Λ =+ξ x nux0 h (16) formed with the marginal ray and the projection on the xz plane of the given ray. It can be verified by direct substitution of the corresponding paraxial refraction or transfer equations that at refraction at a surface separating homogeneous or gradient media or at transfer through axial gradients, the quantity (16) remains unchanged. In fact, Eq.(16) is a paraxial invariant for the entire optical system that will play an important role in the derivation of the chromatic aberration coefficients in Sec. 4. In homogeneous media, an equivalent invariant is known as the Lagrange invariant.

3. PRIMARY ABERRATION COEFFICIENTS

As has been first shown by Sands3,4,5,6 for a rotationally - symmetric optical system consisting of homogeneous and gradient Γ - index lenses, the total Seidel aberration coefficients p,P are sums over all surfaces and inhomogeneous media of ordinary surface contributions Sp,PS , inhomogeneous surface contributions Sp* and inhomogeneous transfer contributions Tp,PT , where the indices 1,2,3,4 denote , , astigmatism and , respectively, and P is the Petzval sum. Γ =++()* = pppp∑ SS∑ T, p1,2,3,4 surfaces GRIN media (17) =+ PP∑ ST∑ P surfaces GRIN media

Γ The chromatic paraxial aberration coefficients λ p are sums of surface contributions Sλp and inhomogeneous transfer con- tributions Tλp, where the indices 1 and 2 denote the axial color and lateral color coefficients

Γ =+ = (18) λλλppp∑ STp∑ , 1,2 surfaces GRIN media

All these aberration coefficients are calculated using paraxial marginal and chief ray data at the lens surfaces and several paraxial invariants. In what follows Δ ( ) denotes the difference between the values after and prior to refraction or transfer of the quantity in the parentheses. In the final expressions for the chromatic aberrations given below, δ λ () denotes the differ- ence between the values at the F and C of the quantity in the parentheses, while all other data are computed at the reference . For the ordinary contributions to the primary aberrations of a spherical surface with curvature ρ we have exactly the same expressions as in the homogeneous case

= ()2 Δ() S1 n0i h u n0 S = n in jhΔ()u n 2 0 0 0 (19) = ()2 Δ() S3 n0 j h u n0 = −ρ 2Δ() PS H 1 n0 = ()2 Δ()+ Δ() S4 n0 j m u n0 n0 jH w n0 and ⎛ δ ⎞ = Δ⎜ λ n0 ⎟ Sλ1 hn0i ⎜ ⎟ ⎝ n0 ⎠ (20) ⎛ δ ⎞ = Δ⎜ λ n0 ⎟ Sλ 2 hn0 j ⎜ ⎟ ⎝ n0 ⎠ δ = − λ n0 n0F n0C

′ ′ = + ρΔ With our notation and sign convention, the paraxial refraction equation for the marginal ray is given by n0u n0u h n0 . (Recall that n0 is the refractive index at the vertex of the surface.) In Eqs.(19) and (20) we also use the paraxial refraction invariants = ρ − = ρ − = − n0i n0h n0u , n0 j n0m n0w , H mn0u hn0w (21)

By denoting the first derivative of the refractive index at the surface vertex by

= dn (22) N z dz z=0 the inhomogeneous surface contributions to the Seidel coefficients are given by:

* = 4 ρ 2Δ S1 h N z * = 3 ρ 2Δ S2 h m N z (23) * = 2 2 ρ 2Δ S3 h m N z *= 3 ρ 2Δ S4 h m N z

Finally, by using the abbreviations (9) and (10) the transfer contributions to the primary aberrations are given by  " − h T =+nu3 3 nudn3 Δ # (24) 1 0 0  2 # ! n0 $ T T T n w 4 = 3 = 2 = 0 (25) T3 T2 T1 n0u = PT 0 (26) where we recall that n0u and n0w are transfer invariants. The transfer contributions to the chromatic coefficients are  " − δ n T =−nunud δ n 1 Δ λ 0 h# λλ100   ! n0 $ (27)  δ " −1 λ n0 Tλλ=−nwnud δ n Δ h# 200   ! n0 $ δ −1 =−−−11 λ nnFC n

Equations that are equivalent with (19-27) have been first obtained by Sands. The present author has derived Eqs. (19-27) independently and has also implemented them in a computer program. In a paper describing a four-lens system that includes two axial GRIN lenses, Pfisterer uses the results of Sands and lists the numerical values of all seven primary aberrations8. The program using Eqs. (19-27) produces for the Pfisterer system the same result. (See Appendix A.)

Remarkably enough, at present the commercial program CODE V computes the Seidel aberrations, but not the chromatic aberrations of gradients. (For various lenses with axial gradients of AXG and URN type, the Seidel aberrations obtained with our program turned out to be identical with those listed by CODE V.)

With the notable exception of Ref. 8, Sands’ formulas for the transfer contributions to axial and lateral color do not seem widely used. At the time of this writing no widespread commercial optical design program lists them. In the past, some members of the GRIN community were somewhat skeptical to use Sands’ formulas for the chromatic aberrations of rota- tionally symmetric gradients, because they suspected an error. Recently a minor computation error was indeed detected and corrected by the present author7, but this error affects only radial gradients. Sands’ results for the chromatic aberrations of axial gradients are correct. (Note however that in Refs. 5 and 6 the two chromatic coefficients are defined with the opposite sign as compared to our definition.) In order to stimulate their use in computer programs, an independent derivation of the transfer contributions (27) and a reliable method to check the accuracy of their computer implementation will be given in the following section.

4. TRANSFER CONTRIBUTIONS TO AXIAL AND LATERAL COLOR

The method used below for the new derivation of the transfer contributions (27) to the chromatic aberrations of axial gradi- ents has been first developed by this author to derive the chromatic aberrations to radial gradients7. The starting point of the derivation is the paraxial invariant (16).

In a rotationally - symmetric optical system consisting of homogeneous and GRIN media, let Q be the image plane conju- λ gated to the object plane P for some reference wavelength 0. Assume that the media at the planes P and Q are homogene- ous. Because of , paraxial rays starting from the same object point (xyPP, ) but having different wavelengths in- λ Ξ =− tersect the plane Q at different points. For a ray having an arbitrary wavelength , the departures λλxQQxx and Ξ =− λλyQQyy define the components of the transverse chromatic paraxial aberration vector of the ray. For the convenience of notation, each quantity at the wavelength λ is denoted by the subscript λ at the symbol of the quantity. All quantities λ without this subscript are considered at the reference wavelength 0.

We first seek a way to write the chromatic aberration vector as a sum of contributions from all at surfaces and Λ transfers through inhomogeneous media. Consider therefore the paraxial invariant x given by Eq.(16). Noting that the Λ Λ Λ λ=λ change at refraction or transfer of x vanishes, the basic idea is to construct a quantity λx which reduces to x for 0 , Ξ Λ in such a way that the contributions to λx of any refraction or transfer are related to the corresponding changes of λx . 9 Λ Following Buchdahl , λx will be called the chromatic paraxial quasi - invariant.

P Q h=0 Λ ΛΛ== At the planes and , where we have , the simplest guess for λx is λλλλxPnux P P P, xQ nux Q Q Q . Consequently, we can write 38−=ΛΛ − . However, since Λ is an invariant and since at the object plane P we have nQux Qλλ Q x Q xQ xQ x = ΛΛΛ== ΞΛ=−=− Λ Λ Λ Λ xxλP P , it follows that xQ xPλ xP and we have nQQu λλ x xQ xQ λ xQ λ xP. Assuming that λx can be Λ properly defined at each surface of the system, by summing up the variation of λx over all refractions and transfers in the system, we obtain the desired decomposition ΞΔΛ= nQQu λλ x ∑ x (28)

Λ Λ The precise form of λx can be determined by observing that the variation of λx at transfer through homogeneous media λ λ must vanish. Let n and nλ be the refractive indices at wavelengths 0 and and d be the thickness of the homogeneous me- ξ ΔΔ==−ξ Λ dium. Since u and λ remain unchanged, we have xdnhuλλλ, d and it can then be easily seen that λx written as Λ =+n ξ (29) λλx nux h λ nλ satisfies the required condition: ΔΛ=+ Δn ξ Δ = (30) λλx nu x λ h 0 nλ

For axial GRIN lenses a straightforward generalization of Eq. (29) is

n Λ =+0 ξ (31) λλx nux0 h λ n0λ

The equations (28) (where the sum now extends only over surfaces and GRIN media) and (31) give the required decompo- Ξ ξ η sition of λx in surface and transfer contributions. Of course, by replacing in these equations x and by y and , a similar Ξ decomposition can be written for λy , but for the purpose of this study this will not be necessary.

ξ Ξ Using at the image plane Q the equations (13) for xλ and λ it follows that λx is a linear quantity in aperture and field co- ordinates

−=−=+ΞΔΛΓΓστ nuQQλλλλ x∑ x1 x 2 x (32)

Γ Γ The coefficients λ1 and λ2 are the total chromatic paraxial aberration coefficients of the system.

Λ We now derive Eqs.(27) by examining the variation of λx at transfer through axial gradients. First, we will temporarily δ λ λ redefine the operator λ () so that in this section it will mean the difference between the values at the wavelengths and 0 of the quantities in the parentheses (instead of the values at the wavelengths F and C, as in Section 3). For instance, the δ = − change with wavelength of the refractive index at the surface vertex is λ n0 n0λ n0 .

Noting that n n −δ n δ n 0 = 0λ λ 0 =1− λ 0 (33) n0λ n0λ n0λ

Λ the variation of λx given by Eq.(29) at transfer through an axial gradient is given by

   δ  n0 λn0 (34) ΔΛΔΔλλλ=+nu x ξ  hnux =+− ΔΔΔλλξ hξ λ h x 0   0   n0λ n0λ

From Eqs. (11) and (14) it follows that ΔΔ==−ξ −1 −1 . Writing δ −1 =−−1 −1 we then have xdnhdnnλλλ , 0u λλnn n  δ   δ " −1 −−1 λ n0 1 λ n0 (35) ΔΛλλλλ=−−nuξ dnξ dn nu ξ λ Δ h =−ξ λλ n udδ n Δ h# x 0 0   0   n0λ ! n0λ $ Using for ξ the substitution (13), Eq. (35) becomes λ  " − δ n ΔΛ=−1nuστδ + nw6 nud n1 − Δ λ 0 h# . (36) λλλλλλxxx0 0 0   ! n0 $

On the other hand, by comparing Eqs. (18) and (32) we observe that at transfer through an axial gradient we have −=ΔΛ στ + λλxxTT1 λ2 x . It follows immediately that the transfer contributions to the primary chromatic aberrations are  "  " −−δ n δ n T =−nu nudδ n1 ΔΔ λ 0 hTnwnudn# , =− δ 1  λ 0 h# λλλλ1 0 0   λλλλ2 0 0   (37) ! n0 $ ! n0 $

where Tλ1 stands for axial color and Tλ2 for lateral color.

Λ Using the same technique, the change at refraction at a surface of λx given by Eq.(31) leads to

 δδ   λ n0 λ n0 (38) Sλλλ= hn i ΔΔ  , Shnjλλλ=   1 0   2 0   n0λ n0λ

Together with Eqs. (18) and (32), the equations (37-38) describe the chromatic aberrations of paraxial rays at the given wavelength λ. It must be emphasized at this point that Eqs.(37-38) are exact. However, in order to describe more conven- iently the chromatic behavior for the full wavelength range over which the system is to be used, two approximations are δ = − −1 =−−1 −1 δ = − introduced in Eqs.(37-38): i) Instead of λ n0 n0λ n0 and δ λλnn n we use λ n0 n0F n0C and δ −1 =−−−1 1 and ii) we compute all quantities that have the index λ at the reference wavelength λ . In this case λ nnFC n 0 the equations (37-38) become Eqs. (20) and (27).

The exact equations (37-38) can also be used to verify the correctness of the computer implementation of the chromatic ab- σ τ σ τ Γ errations. If we first set in Eq.(32) x =1 , x=0 and then x =0 , x=1 we obtain for the total axial color coefficient λ1 and Γ for the total lateral color λ2 the alternative expressions

Γ =− , Γ =−38 − (39) λλ1 nuhQQ Q λλ2 nuQQ m Q m Q

λ We assume that for a given optical system the image plane is at its paraxial position Q for the reference wavelength 0. For λ an arbitrary wavelength we can then compute the marginal and chief ray heights hλQ and mλQ at the image plane in two different ways: i) directly from paraxial ray tracing and ii) from Eqs. (39) by using the equations (18) and (37-38). The computer implementation of the chromatic aberrations is correct if the results are identical in both cases. An example is shown in Appendix B.

5. THIN LENS APPROXIMATION

The objective of this section is to provide qualitative insight into the influence of the transfer contributions (24-27) on the total Seidel and chromatic aberrations. As in the homogeneous case, a useful tool for qualitative analysis is the thin – lens approximation. We will see that if we neglect the change of the ray height within the lens we can find for any axial GRIN lens an equivalent homogeneous lens having exactly the same primary aberrations.

For homogeneous lenses, the thin-lens expressions are obtained by substituting d=0 into the exact paraxial relations and into the aberration expressions for the surface contributions. For lenses having a finite but not too large thickness, the approxi- mate aberrations differ to some extent in absolute magnitude from the exact aberrations, but show approximately the same variation as the latter when lens parameters are changed10. Since in the homogeneous case the paraxial transfer relation is h’=h-ud , neglecting the change of the ray height means that in the expressions of the primary aberrations the ratio between the terms neglected and those kept is of the order of magnitude of ud/h.

However, unlike the homogeneous case, for GRIN lenses transfer through the medium also contributes to the total aberra- tions. In order to avoid a possible loss of some specific gradient contribution, we will ensure that by introducing the thin lens approximation in the transfer relations (24-27) the ratio between the terms neglected and those kept will also be of the order of magnitude of ud/h , as in the homogeneous case.

We have seen that the paraxial transfer equation (14) and the transfer contributions (24-27) contain three functions of the −1 −3 −1 refractive index n , n and δ λ n that must be computed by quadrature. We first evaluate the order of magnitude of these three quantities. Since all these quantities can be regarded as average values over the lens thickness of given functions of gradient material properties, it is reasonable to assume that their order of magnitude is the same as that for the corre- sponding function at any of the lens surfaces. By denoting the values of n0 at the first and second end surface by n01 and n02 respectively, the orders of magnitude for the three quantities are

d δ 16 −− −− − λ nz δ n 1 1 3 3 δ 1 1 λ 01 (40) nn~ , nn~ , λ n~ −−I dz ~ 01 01 0 2 16 2 d nz n01

Appendix C shows that the assumptions (40) are verified for several commercially available axial GRIN materials.

In the paraxial transfer relation (14) the ratio between the second term −−−1 and the first one is then the same dn nu01 ~ ud as in the homogeneous case ud/h . Thus, neglecting the change of the ray height within the lens introduces in the surface = ′ contributions of axial gradients an error of the same magnitude as for homogeneous lenses. Recalling that n01u n02u we note that unlike other types of gradients such as for instance the radial ones, the transfer through an axial gradient does not contribute to the lens power.

We now consider the transfer contributions to the primary aberrations. It suffices to consider the expressions for spherical aberration and axial color, since the expressions of the remaining aberrations result then immediately. For the spherical ab- erration (24) we have

− −1 "    " − hnnud01 h 1 1 1 −−1 T =+nu3 3 n n3 ud − # =−nu3 3 h  −−ud nn1 nn3 3 # (41) 1 01 01 2 2 01  2 2   2 01 2 01  ! n02 n01 $# ! nn02 01 n02 n01 $

The second term in the square bracket of Eq.(41) has the order of magnitude     1 −−1 1 1 −−ud nn1 nn3 3  ~ −− ud  . (42)  2 01 2 01   2 2  n02 n01 nn02 01

Similarly, for axial color we have  "    " −−δδδδδn n n n n −− T =−−−nun δ n1 ud  λλλλλ02 49hnnud1 01 hnuh# =−  02 − 01  −+ud 02 nn1 nδ n1 # (43) λλ1 01 01  01  01    01 01 λ  ! n02 n01 $ ! n02 n01 n02 $ where the order of magnitude of the second term in the square bracket is     δ n −−δδn n −+ud λ 02 nn1 nδ n1  ~ −− ud λλ02 01  . (44)  01 01 λ    n02 n02 n01 As shown in Appendix C the estimations (42) and (44) turn out to be correct for the axial GRIN materials examined there. It can then be seen that both in Eq. (41) and in Eq. (43) the ratio between the second and the first term in the square bracket is of the order of magnitude of ud/h . Thus, setting d=0 in the transfer contributions of axial gradients is consistent with the traditional thin-lens approximation. The equations (41) and (43) then become

 1 1  2  δδn n   δn   n −1 T =−nuh3 3   = 161nu hΔ un 6 , T =−nuh λλ02 − 01  =−hn uΔΔ λ0  =−hn u  0  (45) 1 01  2 2  0 0 λ1 01   0   0  ν  nn02 01 n02 n01 n0 n0

νδ=− In the second equation, the Abbe number (n0 1) / λn0 has been introduced to describe dispersion. It can be observed that in the thin lens approximation only the refractive indices and Abbe numbers at the two end surfaces appear in the ex- pressions of the transfer contributions to the primary aberration coefficients. Thus, the effect of transfer through the axial GRIN medium does not depend on the specific form n=n(z) of the refractive index distribution.

Remarkably enough, Eqs. (45) and the similar equations for the other primary aberrations are precisely the same as in the special case of a medium consisting just of two homogeneous glasses which have the refractive indices n01 and n02 and the ν ν Abbe numbers 1 and 2 respectively and which are separated by a plane surface. Noting that for a plane surface we have =− according to Eqs. (21) n0inu0 , the equations (45) are identical with the surface contributions (19) and (20) of the plane surface separating the two media above.

At both surfaces of an axial GRIN lens the inhomogeneous surface contributions (23) are equivalent to those of a homoge- neous aspherical surface3,4. We arrive then at the following conclusion: Within the domain of validity of the thin lens ap- proximation, the primary aberrations of an axial GRIN lens are equivalent to those of a pair of homogeneous aspherical lenses in contact, having a common plane surface and having refractive indices and Abbe numbers equal to the correspond- ing axial GRIN values at the two end surfaces.

6. CONCLUSIONS

A computer program has been written that computes the Seidel and chromatic paraxial aberration coefficients for rotation- ally symmetric systems containing axial gradients (using Eqs. (9-10),(14-15) and (17-27)) or radial gradients (using formu- las first developed in Ref. 7). In order to facilitate the computer implementation of the chromatic aberrations of gradients an independent derivation of the expressions of the transfer contributions of axial gradients (Eqs. (27)) has been presented. Using Eqs. (37-39) the correctness of any computer implementation of the chromatic aberrations can be reliably verified. If we neglect the change of the ray heights within the axial GRIN lens, an equivalent homogeneous lens can be found that has exactly the same Seidel and chromatic paraxial aberration coefficients as the GRIN lens.

7. REFERENCES

1. B. V. Hunter, V. Tyagi, D. A. Tinch, P. Fournier, “Current developments in GRADIUM glass technology”, Proc. SPIE 3482, pp. 789-800, 1998 2. B. V. Hunter, B. Walters, “How to design and tolerance with GRADIUM glass” , Proc. SPIE 3482, pp. 801-812, 1998 3. P. J. Sands, “Third-order aberrations of inhomogeneous lenses”, J. Opt. Soc. Am. 60, pp. 1436-1443, 1970 4. P. J. Sands, “Aberrations of lenses with axial index distributions”, J. Opt. Soc. Am. 61, pp. 1086-1091,1971 5. P. J. Sands, “Inhomogeneous Lenses, II. Chromatic Paraxial Aberrations”, J. Opt. Soc. Am. 61 , pp. 777-783,1971 6. P. J. Sands, “Inhomogeneous Lenses, V. Chromatic Paraxial Aberrations of Lenses with Axial or Cylindrical Index Distributions”, J. Opt. Soc. Am. 61 , pp. 1495-1500, 1971 7. F. Bociort, “Chromatic paraxial aberration coefficients for radial gradient-index lenses”, J. Opt. Soc. Am. A13, pp.1277-1284, 1996 8. R. N. Pfisterer, “Design of a 35-mm photographic objective using axial GRIN materials” Proc SPIE 2000, pp. 359-368, 1993 9. H. A. Buchdahl, Optical Aberration Coefficients, Dover Publications, New York, 1968 10. H. H. Hopkins, Wave Theory of Aberrations, Clarendon Press, Oxford, 1950 8. APPENDIX A

A C program has been written that computes the primary aberrations of rotationally symmetric systems containing axial gradients (using Eqs. (17-27)) or radial gradients (using formulas developed in Ref. 7). The necessary marginal and chief ray data as well as the refractive index data are read from a file written by an external program. For generating these data a macro has been written for CODE V 8.30.

Table 1 lists the primary aberrations generated by our program for a system that includes two axial GRIN lenses with linear refractive index distributions designed by Pfisterer. The values are very close to those listed in Ref. 8, the minor differences being probably caused by the numerical techniques. However, Pfisterer lists the output in a different, but fully equivalent −Γ Γ way. For instance, he lists the chromatic aberrations as λ1/nQuQ and - λ2/nQuQ . These two quantities are also listed in the last two columns of Table 1 in order to facilitate the comparison. For this sytem we have nQ=1 and uQ = 0.25. The total Seidel aberrations ( i.e. the last line in the first five columns) are in complete agreement with those listed by CODE V.

Γ Γ Γ Γ Γ Γ −Γ Γ 1 2 3 P 4 λ1 λ2 λ1/uQ - λ2/uQ so1 0.63293 0.16118 0.04105 0.54960 0.15041 -0.08102 -0.02063 0.324083 0.082531 si1 -1.22655 0.52188 -0.22205 0.09448 tr1 0.07358 -0.08836 0.10612 0.00000 -0.12744 -0.01464 0.01758 0.058547 -0.070311 so2 0.61023 -0.84493 1.16990 -0.04960 -1.55118 -0.03968 0.05494 0.158722 -0.219768 si2 0.00668 -0.00202 0.00061 -0.00018 so3 -0.17950 0.39029 -0.84861 0.23533 1.33349 0.04560 -0.09916 -0.182416 0.396634 so4 -0.11375 -0.08913 -0.06984 -0.69915 -0.60257 0.08252 0.06466 -0.330093 -0.258658 so5 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.000000 0.000000 so6 -0.55745 0.43691 -0.34243 -0.45365 0.62393 0.05378 -0.04215 -0.215136 0.168615 so7 0.01936 0.02947 0.04487 0.01950 0.09800 0.01010 0.01538 -0.040404 -0.061512 si7 0.12093 0.03605 0.01074 0.00320 tr7 0.00030 0.00219 0.01600 0.00000 0.11684 -0.00228 -0.01667 0.009131 0.066691 so8 0.94767 -0.36349 0.13942 0.47611 -0.23609 -0.06462 0.02478 0.258469 -0.099138 si8 -0.35141 -0.17686 -0.08901 -0.04480 TOT -0.01699 0.01318 -0.04325 0.07814 -0.14191 -0.01023 -0.00127 0.040904 0.005083

Table 1. Values of the primary aberrations for the system described in Ref. 8 that has F/2 and HFOV 21.8 degrees. Two axial GRIN elements are used after surfaces 1 and 7. At a surface X, soX siX and trX denote the ordinary contributions, inhomogeneous surface contributions and transfer contributions of the axial GRIN medium after the surface, respectively.

9. APPENDIX B

− λ =656.27 nm hλQ mλQQm paraxial ray tracing 0.019039 0.000603 with Eqs. (39) ,(18) and (37-38) 0.019039 0.000603

λ Table 2. In the system described in Ref. 8 the image plane Q is at its paraxial position for the reference wavelength 0 λ − =587.56 nm. For =656.27 nm the quantities hλQ and mλQQm are computed in two ways: i) with ray tracing and ii) from Eqs. (39) by using the equations (18) and (37-38). The results are identical. (See Sec. 4.)

10. APPENDIX C

For several axial GRIN materials manufactured by LightPath Technologies, Table 3 lists the values of n01, n02, δ δ λ nn01 01 and λ nn02 02 which appear in the thin-lens expressions of the transfer contributions (45), as well as the quantities − −  δ  = 1 , = 3 3 , −1 λn01 , Knn1 01 Knn2 01 Kn=−δ λ   3  2  n01

 1 −−1   1 1   δ n −−  δδn n  C =− nn1 nn3 3   −  , C =+ λ 02 nn1 nδ n1   λλ02 − 01  . 1  2 01 2 01   2 2  2  01 01 λ    n02 n01 nn02 01 n02 n02 n01

The fact that in all cases K1, K2 and K3 are close to unity shows that the assumptions (40) are correct for these glasses. The fact that in almost all cases C1 and C2 are smaller that unity shows that setting d=0 in the transfer contributions introduces a relative error smaller than ud/h .

Type n n δ δ K K K C C 01 02 λ nn01 01 λ nn02 02 1 2 3 1 2 G1SFN 1.732 1.641 0.0147 0.0112 1.03 1.11 0.87 0.36 0.34 G1SFP 1.724 1.782 0.0144 0.0166 0.97 0.93 1.06 0.35 0.41 G2SFN 1.750 1.685 0.0154 0.0129 1.01 1.05 0.93 0.53 0.49 G2SFP 1.677 1.742 0.0126 0.0151 0.98 0.94 1.06 0.50 0.54 G3SFN 1.752 1.722 0.0155 0.0143 1.00 1.02 0.96 0.54 0.46 G3SFP 1.682 1.715 0.0128 0.0141 0.99 0.97 1.02 0.52 0.58 G4SFN 1.725 1.698 0.0144 0.0134 1.00 1.02 0.96 0.46 0.38 G4SFP 1.693 1.698 0.0132 0.0134 0.99 0.99 0.99 0.71 1.13 G5SFN 1.735 1.719 0.0148 0.0142 1.00 1.01 0.97 0.52 0.37 G5SFP 1.695 1.704 0.0133 0.0137 0.99 0.99 1.00 0.61 0.83 G4LAKN 1.728 1.711 0.0146 0.0139 1.00 1.01 0.97 0.52 0.39 G4LAKP 1.684 1.700 0.0129 0.0135 0.99 0.98 1.01 0.50 0.62

Table 3. Refractive index data of several GRADIUM glasses computed with CODE V 8.30. In all cases the thickness d and offset Δz are chosen to be 3 mm. The wavelengths used are F, d, C.