Primary aberrations of a thin lens with different ... - OSA Publishing

L. N. Hazra and C. A. Delisle

Vol. 15, No. 4 / April 1998 / J. Opt. Soc. Am. A

945

Primary aberrations of a thin lens with different object and image space media L. N. Hazra* and C. A. Delisle Centre d’Optique, Photonique et Laser, De´partement de Physique, Pavillon Vachon, Universite´ Laval, Ste-Foy, Que´bec GIK 7P4, Canada Received June 26, 1997; revised manuscript received November 6, 1997; accepted November 7, 1997 Results of our studies on the primary aberrations of a thin lens that has different object and image space media are presented. It is shown that the usual formalism for treatment of the primary aberrations of a thin lens in air can be extended to this case of unequal object and image space media by suitable redefinition of the shape factor and the conjugate variable. The inequality in refractive indices and dispersion characteristics of the object and image space media gives rise to interesting properties of the primary aberrations. The latter may be utilized in the treatment of the analysis and synthesis of unconventional optical components. © 1998 Optical Society of America [S0740-3232(98)02203-0] OCIS codes: 220.0220, 220.1010, 080.0080, 110.0110.

1. INTRODUCTION Practical design of lens systems in accordance with given specifications is essentially a problem of nonlinear optimization in a constrained multivariate hyperspace. Since the advent of digital computers, optical designers have experimented with different optimization techniques that can yield an optimum solution, given a starting point in the multivariate hyperspace.1 These experiments have been mostly successful, and they have yielded the results sought: a large number of in-house and commercial lens design software packages are in routine use around the globe.2 Nevertheless, it is well-known that these solutions represent optimum solutions in the neighborhood of the given starting point.3–5 Often different starting points lead to different local optima, the designer making use of the best among them. Practical success in any venture of lens design depends to a large extent on choice of the starting point. Several interesting investigations are currently being undertaken to circumvent this problem.6 Some are also trying to adopt stochastic approaches for seeking a global optimum.7,8 However, except for the trivial cases, the curse of dimensionality associated with the large number of variables in lens design problems seems to pose a fundamental limitation in seeking a true global optimum from an arbitrary starting point in a reasonable time frame. Traditionally, optical designers have strived to tackle this problem of large dimensionality in a clever way. Taking recourse to the principles of Gaussian optics and aberration theory, they developed a plethora of techniques with varying degrees of complexity for analysis and synthesis of lens systems, e.g., thin lens design, thirdorder design, thick lens design.9,10 This has led to the development of effective strategies for tackling formidable design problems with large degrees of freedom by means of approximately similar subproblems with considerably lower degrees of freedom.11–16 The reduced number of degrees of freedom at the subproblem level has rendered possible the emergence of several analytical and semiana0740-3232/98/040945-09$15.00

lytical approaches for seeking the desired goals in an optimum fashion. These optima provide convenient launching pads for further exploration in the final design space. The nature of optimality of the ultimate solutions obtained thereby may be open to question, yet the feasibility of obtaining optimum starting points that are likely to result in viable solutions in practice, is, in itself, a vindication of the raison d’eˆtre of this approach. The observations enunciated above have prompted us to renew our investigations on the methods for thin lens predesign of lens systems. Ab initio design of lens systems usually starts with a thin lens predesign stage. Besides the ease in manipulation with the Gaussian parameters, another feature that makes this stage particularly useful is the convenient expressions for primary aberrations of thin lenses. This in turn greatly facilitates analytical treatment of the problem at hand. It is interesting to note that, though formulas for primary aberrations of a thin lens have been in existence for more than a century, they are generally given for a thin lens in air.17–26 Wynne generalized the notion of a thin lens to mean a thin component of refractive index n 0 n in a medium of index n 0 and reported the corresponding expressions for primary aberrations.27 Welford observed that ‘‘this enables us to include the case of a thin air gap with curved surfaces between glasses of equal indices, a situation which sometimes occurs.’’ 28 In general, there is no compelling reason to be limited by this restriction of equal indices. For the sake of development of unconventional components catering to the exigencies of optics technology, it will be worthwhile to have suitable expressions for primary aberrations of a thin lens with different object and image media. The same is reported in this paper.

2. SHAPE FACTOR AND CONJUGATE VARIABLE Figure 1(a) shows a single lens formed by two spherical refracting interfaces of curvatures c 1 and c 2 . The lens © 1998 Optical Society of America

946

J. Opt. Soc. Am. A / Vol. 15, No. 4 / April 1998


leads to simplified expressions for the primary aberrations of thin lenses. In line with the symmetrical variables of Coddington19 and Hopkins,22 we define a normalized shape variable X as X5

~ m 2 n 1 !c 1 1 ~ m 2 n 28 !c 2 K

(7)

c1 5

K~ X 1 1 ! , 2~ m 2 n1!

(8)

c2 5

K~ X 2 1 ! . 2~ m 2 n 28 !

(9)

so that

Note that a change in shape factor DX leads to different changes in the curvatures,

Fig. 1. a) Paraxial ray paths through a single lens of refractive index m. The object and image spaces have indices n 1 and n 2 8 . b) Equivalent thin lens for the singlet assuming d → 0. Stop is on the lens.

has refractive index m, and the object and image spaces have refractive indices n 1 and n 2 8 , respectively. The stop is assumed to be on the lens. Two paraxial ray paths are shown: one, the paraxial marginal ray (PMR), is indicated by arrows, and the other, the paraxial pupil ray or the paraxial principal ray (PPR), is indicated by solid triangles. The convergence angles of the PMR in the object and the final image spaces are u 1 and u 2 8 , respectively. The heights of the PMR on the consecutive refracting surfaces are h 1 and h 2 . The axial thickness of the lens is d. As usual, the subscripts refer to quantities at that surface; addition of a prime indicates quantities in the image space of that surface. In our treatment, we follow the sign convention of Born and Wolf.24 Using paraxial refraction and transfer equations consecutively for the PMR on the two surfaces of the lens, we have

m u 18 2 n 1u 1 5 h 1~ m 2 n 1 ! c 1

(1)

u 2 5 u 18,

(2)

h 2 5 h 1 2 du 2 ,

(3)

n 28u 28 2 m u 2 5 h 2~ n 28 2 m ! c 2 .

(4)

In the thin lens approximation, d → 0, so that h 1 , h 2 → h. The equivalent thin lens corresponding to the singlet of Fig. 1(a) is shown in Fig. 1(b). For the thin lens, we thus have n 2 8 u 2 8 2 n 1 u 1 5 hK

(5)

where K, the power of the thin lens, is given by K 5 ~ m 2 n 1 !c 1 2 ~ m 2 n 28 !c 2 .

(6)

From the early days of systematic investigations on primary aberrations of thin lenses, it has been observed that introduction of composite parameters for the structure of the lens and its imaging configuration, namely, the shape factor (also known as coflexure,20 Durchbeigung,17 or cambrure25) and the conjugate variable, respectively,

Dc 1 5

K DX, 2~ m 2 n1!

(10)

Dc 2 5

K DX, 2~ m 2 n 28 !

(11)

but the power K remains unchanged. A normalized conjugate variable Y that remains unchanged with changes in the bending or shape variable X can be defined as Y5 5

n 28u 28 1 n 1u 1 n 28u 28 2 n 1u 1 n 28u 28 1 n 1u 1 . hK

(12)

It can be easily seen that the convergence angles u 1 and u 2 8 in the object and image spaces are given by u 28 5 u1 5

hK ~ Y 1 1 !, 2n 2 8

(13)

hK ~ Y 2 1 !. 2n 1

(14)

Also, the paraxial magnification M 5 n 1 u 1 /n 2 8 u 2 8 , so that Y can also be expressed as Y5

11M . 12M

(15)

3. MONOCHROMATIC PRIMARY ABERRATIONS Since the arbitrary location for the stop can be taken care of by stop shift formulas, it is sufficient to derive expressions for the central aberrations, i.e., aberrations introduced by the lens when the stop is in the lens. It is well known that the primary aberrations of the singlet are given by the sum of the corresponding primary aberrations at the two refracting surfaces. Instead of rederiving the latter, we make use of the standard expressions for primary aberrations at an individual refracting surface, as given by Hopkins29 and Welford.30 The paraxial refraction invariant for the PMR at the first surface, A 1 , is



A 1 5 n 1 ~ hc 1 2 u 1 !

F

G

X m 1 Y n hK 5 2 1 . 2 1 m 2 n1 n1 n 1~ m 2 n 1 !

a2 5 (16) a3 5

The same invariant for the PMR at the second surface, A 2 , is A 2 5 n 2 8 ~ hc 2 2 u 2 8 ! 5

F

G

X m 1 Y n 8 hK 2 2 . 2 2 m 2 n 28 n 28 n 28~ m 2 n 28 !

a4 5

m2 21

m

We represent all quantities with reference to the PPR with a bar on the symbol. Since the stop is at the thin ¯ 5 0, the paraxial refraction invariants for the lens, h PPR take particularly simple forms as

a6 5

¯ 5 n 8 ~ ¯h c 2 ¯u 8 ! 5 2n ¯u 5 H , A 1 2 2 2 1 1 h

(18)

a7 5

¯ 5 n ~ ¯h c 2 ¯u ! 5 2n 8 ¯u 8 5 H , A 2 1 1 1 2 2 h

(19)

a8 5

where H, the Smith–Helmhotz–Lagrange invariant is given by

D1 5

u 18 u1 2 , m n1

(21)

D2 5

u 28 u2 2 . n 28 m

(22)

Using Eqs. (1)–(4), (8) and (9), and (13) and (14) for algebraic manipulations, we can represent D 1 and D 2 as D1 5

D2 5

1 2

hK

1 2

hK

S F

1 n1

2

1

1 ~ n 28 !2

X

m 2

2

2

X

m2

m 2 2 n 12 m n1 2

1

2

D

Y ,

m 2 2 ~ n 28 !2 m 2~ n 28 ! 2

(23)

G

Y .

S I 5 A 1 hD 1 1 A 2 hD 2 . 2

2

(25)

m

1 4 3 h K ~ a 1 X 3 1 a 2 Y 3 1 a 3 X 2 Y 1 a 4 XY 2 1 a 5 X 2 8 1 a 6 Y 2 1 a 7 XY 1 a 8 X 1 a 9 Y 1 a 10! ,

where a1 5

1

F

n2

m2 ~ m 2 n !2

2

~ n8!2 ~ m 2 n8!2

G

F

m1n

F F

n ~m 2 n!

a 10 5 m 2

~ m 2 n8!2

3 m 1 2n 8

2

~ n8!2

n ~m 2 n!

D

G G

,

(30)

(31)

,

(32)

m 1 n8 n 8~ m 2 n 8 !

G

G

,

(34)

~ n 8 ! 2~ m 2 n 8 !

1

(33)

,

3m 1 n8

2

2

(28)

,

n 8~ m 2 n 8 ! 2

1 2

,

(29)

2m 1 n8

2

3m 1 n

n8

G

,

2 m 1 3n 8

1

n~ m 2 n !

D

m 1 2n 8

1

2

n~ m 2 n !

m 2 n8

1

2

2

m 1 3n 8

2

n

2m 1 n

G

,

1 ~ n8! ~ m 2 n8!2 2

(35)

G

.

(36)

For the sake of simplification in notation, we drop the subscripts in n 1 and n 2 8 and represent them as n and n 8 , respectively. It is significant to note that the variation of S I with X and Y is cubic in nature. In the special case of equal optical media for the object and image spaces, i.e., when n 5 n 8 , we have a 1 5 a 2 5 a 3 5 a 4 5 a 8 5 a 9 5 0, so that ~ S I ! n5n 8 5

1

h 4K 3

4

F

m 1 2n m~ m 2 n !

4~ m 1 n !

2

mn~ m 2 n !

3 m 1 2n

X2 1

2

XY 1

(37)

mn2 m2

n 2~ m 2 n ! 2

G

Y2

.

(38)

For the special case of a thin lens in air, n 5 n 8 5 1, Eq. (38) reduces to ~ S I ! n5n 8 51 5

1 4

h 4K 3

2 5

1 4

(26)

(27)

F

m

1 ,

m2n

n

~ n8!2

2

m 1 2n

24

Substituting terms for A 1 , A 2 , D 1 , and D 2 in Eqs. (16), (17), (23), and (24), we obtain SI 5

m 1 3n

1 3 m 1 2n

(24)

A. Primary Spherical Aberration The primary or Seidel spherical aberration coefficient S I of the thin lens is given by

n2

m ~m 2 n!

a9 5 2

(20)

The changes in the ratio of the convergence angle in a medium with respect to the refractive index of that medium for the two surfaces are

F F

1

m2 2 ~ n8!2

2

2 m 1 3n

1

H 5 n 1 ~ u 1 ¯h 2 ¯u 1 h ! 5 2n 1 ¯u 1 h 5 n 2 8 ~ u 2 8 ¯h 2 ¯u 2 h ! 5 2n 2 8 ¯u 2 h.

2

m2

(17) a5 5

F S S

21 m 2 2 n 2

947

F

m12 m~ m 2 1 !

4~ m 1 1 !

m~ m 2 1 ! h 4K 3

F

H

X2 1

2

m~ m 2 1 !

m2 ~m 2 1!

2

2

2

m

m2

XY 1

m12

3m 1 2

~ m 2 1 !2

F

X2

m m12

Y2

Y2

G

2~ m2 2 1 !

GJ

m12 .

Y

G

2

(39)

The last expression tallies with the one given by Hopkins.31

948



The dimensionless normalized spherical aberration coefficient S I is given by SI 5

SI h 4K 3

.

(40)

The dimensionless primary aberration coefficients S i are identical to the structural aberration coefficients as proposed by Shack.32 Figures 2–4 give the variation in S I with the shape factor X for three thin lenses of refractive indices 1.5, 1.6, and 1.7 with conjugate variables Y 5 0, Y 5 21, and Y 5 2, respectively. For all the lenses, the refractive index for the object space n 5 1. In each figure we show different curves corresponding to values of the refractive index of the image space over the range 1 (0.2) 2. The curve a corresponds to the special case of a thin lens in air. For all the other curves, the object and image space indices are different and the strong cubic nature of variation in S I with X is apparent. The cubic curves demonstrate a point of inflection in the neighborhood of X 5 1. The exact location of the point of inflection for an individual curve can be obtained by the relation d2 S I dX 2

5 0.

(41)

Fig. 3. Variation of S I with X. Y 5 21, m 5 1.6, n 5 1. (a) n 8 5 1, (b) n 8 5 1.2, (c) n 8 5 1.4, (e) n 8 5 1.8, (f) n 8 5 2.

From Eqs. (26), (40), and (41), we can determine the value ˜ at the point of inflection. It is given by of shape factor X ˜ 5 2 a3 Y 2 a5 . X 3a 1 3a 1

(42)

Using the detailed expressions for a 1 , a 3 , and a 5 as given by Eqs. (27), (29), and (31) respectively, we note that

Fig. 4. Variation of S I with X. Y 5 2, m 5 1.7, n 5 1. (a) n 8 5 1, (b) n 8 5 1.2, (c) n 8 5 1.4, (d) n 8 5 1.6, (e) n 8 5 1.8, (f) n 8 5 2.

as n 8 → m ,

Fig. 2. Variation of S I with X. Y 5 0, m 5 1.5, n 5 1. (a) n 8 5 1, (b) n 8 5 1.2, (c) n 8 5 1.4, (d) n 8 5 1.6, (e) n 8 5 1.8, (f) n 8 5 2.

a3 a5 → 0 and → 23, a1 a1

(43)

˜ → 11. Indeed, the particular case of n 8 5 m so that X deserves special attention. This indicates the practical absence of any second refracting surface. From Eqs. (4) and (22) we note that when n 8 5 n 2 8 5 m , u 2 8 5 u 2 and



D 2 5 0. From Eqs. (6) and (7) it may be seen that in this case, K 5 ( m 2 1)c 1 and X 5 1. Since the total primary spherical aberration arises at the first surface, we have, from Eq. (26), ~ S I ! n 8 →m

1 5 h 4K 3~ a 18 1 a 28Y 3 1 a 38Y 1 a 48Y 2 1 a 58 8 1 a 6 8 Y 2 1 a 7 8 Y 1 a 8 8 1 a 9 8 Y 1 a 108 ! , (44)

where a 1 8 , a 2 8 , . . . , a 108 are the first terms in the expressions for a 1 , a 2 , . . . , a 10 in Eqs. (27)–(36); e.g., a 18 5

F

1

n2

m2 ~ m 2 n !2

G

.

and the expression for S II reduces to ~ S II ! n5n 8 5

1 2

(46)

B. Primary Coma The primary coma coefficient S II of the thin lens is given by ¯ A hD 1 A ¯ A hD . S II 5 A 1 1 1 2 2 2

(47)

Using Eqs. (18) and (19), we can express S II as S II 5 H ~ A 1 D 1 1 A 2 D 2 ! .

(48)

On algebraic manipulations after substituting for A 1 , A 2 , D 1 , and D 2 in Eqs. (16), (17), (23) and (24), respectively, we obtain 1 2 2 h K H ~ p 1 X 2 1 p 2 Y 2 1 p 3 XY 1 p 4 X 1 p 5 Y 4 1 p6!,

p1 5

(49)

p2 5 p3 5 p4 5 p5 5

1

m

2

S F F

n

m2n

2

m2 2 n2

1

m2

n2

21 m 1 2n

m

2

1

F

n

m1n

m n~ m 2 n !

F

21 2 m 1 n

m

p6 5 m

F

n

2

1 n ~m 2 n! 2

n8

m 2 n8

2 2 1 1 2

D

,

m 1 2n 8 n8

G

G

,

2m 1 n

mn2

G

Y . (57)

~ S II ! n5n 8 51 5

F

G

m11 1 2 2 2m 1 1 h K H X2 Y . 2 m~ m 2 1 ! m

(58) The dimensionless normalized primary coma S II is given by S II 5

S II 2

h K 2H

.

(59)

Figures 5–7 show the variation in S II with X for three thin lenses of refractive index 1.5, 1.6, and 1.7 with conjugate variables Y 5 0, Y 5 21, and Y 5 2, respectively. The refraction index of the object space n 5 1 for all lenses. In each figure we show different curves corresponding to values of the refractive index of the image space over the range 1 (0.2) 2. The curve a in each figure corresponds to the special case of a thin lens in air. In the limit as n 8 → m , D 2 → 0 and X 5 1. Since the total primary aberration is contributed by the first surface, we have from Eq. (49) ~ S II ! n 8 →m 5

1 2 2 h K H~ p 18 1 p 28Y 2 1 p 38Y 1 p 48 4 1 p 58Y 1 p 68 !,

(60)

where p 1 8 , p 2 8 , . . . , p 6 8 are the first terms in expressions for p 1 , p 2 , . . . , p 6 in Eqs. (50)–(55); e.g.,

n 8~ m 2 n 8 ! 2m 1 n8

(51)

,

m 1 n8

~ n8!2

mn~ m 2 n !

X2

(50)

m2 2 ~ n8!2 ~ n8!2

m1n

(45)

S I ~ X, Y, n, n 8 , m ! 5 S I ~ 2X,2Y, n 8 , n, m ! .

where

F

Further, in the special case of a thin lens in air, i.e., n 5 n 8 5 1, we have the well-known relation

From Eqs. (26)–(36) it may be noted that, in general,

S II 5

h 2K 2H

949

G

(52)

G

,

,

1 ~n8! ~m 2 n8! 2

(53)

(54)

G

.

(55)

As before, the subscripts of n 1 and n 2 8 are dropped in the above relations for the sake of simplicity. Equation (49) shows that, in general, the variation of S II with X and Y is quadratic. In the special case of equal optical media for the object and image spaces, i.e., when n 5 n 8 , we have p 1 5 p 2 5 p 3 5 p 6 5 0,

(56)

Fig. 5. Variation of S II with X. Y 5 0, m 5 1.5, n 5 1. (a) n 8 5 1, (b) n 8 5 1.2, (c) n 8 5 1.4, (d) n 8 5 1.6, (e) n 8 5 1.8, (f) n 8 5 2.

950



C. Primary Astigmatism The primary astigmatism coefficient S III of the thin lens is given by ¯ 2 hD 1 A ¯ 2 hD . S III 5 A 1 1 2 2

(63)

Using Eqs. (18), (19), (23), and (24), we obtain S III 5

1 2

H 2K

HF

1 n

2

1

1 ~n8!

2

G F 2

1 n

2

2

1 ~ n8!2

GJ

Y . (64)

In the special case n 5 n 8 , we have ~ S III ! n5n 8 5

H 2K n2

.

(65)

For a thin lens in air, we have ~ S III ! n5n 8 51 5 H 2 K.

Fig. 6. Variation of S II with X. Y 5 21, m 5 1.6, n 5 1. (a) n 8 5 1, (b) n 8 5 1.2, (c) n 8 5 1.4, (e) n 8 5 1.8, (f) n 8 5 2.

(66)

It is interesting to note that when object and image space optical media do not have the same values for the refractive index, i.e., n Þ n 8 , the central primary astigmatism of a thin lens is a linear function of the conjugate variable Y, but it is independent of lens shape factor X. Also, it is independent of the refractive index m of the thin lens. The dimensionless normalized primary astigmatism S III is given by S III 5

S III H 2K

.

(67)

Figure 8 shows the variation of S III with changes in Y over (23, 3). While the object space index n 5 1 for any of the lines, six different lines correspond to six different values of n 8 over 1 (0.2) 2. It is significant to note that S III is independent of n 8 when Y 5 21. In general,

Fig. 7. Variation of S II with X. Y 5 2, m 5 1.7, n 5 1. (a) n 8 5 1, (b) n 8 5 1.2, (c) n 8 5 1.4, (d) n 8 5 1.6, (e) n 8 5 1.8, (f) n 8 5 2.

p 18 5

1

m

2

S

n

m2n

D

.

(61)

In general, the normalized primary coma coefficient S II follows the relation S II ~ X, Y, n, n 8 , m ! 5 2S II ~ 2X, 2Y, n 8 , n, m ! .

(62)

Fig. 8. Variation of S III with Y. m 5 1.5, n 5 1. (a) n 8 5 1, (b) n 8 5 1.2, (c) n 8 5 1.4, (d) n 8 5 1.6, (e) n 8 5 1.8, (f) n 8 5 2.



S III ~ Y, n, n 8 ! 5 S III ~ 2Y, n 8 , n ! .

951

(68)

Thus the case of changes in refractive index of the object space when the image space index is air can be studied from Fig. 8, if only the signs on the scale for Y are reversed and the symbols n and n 8 are interchanged. D. Primary Curvature The primary curvature coefficient S IV of the thin lens is given by S IV 5 H 2 ~ P 1 1 P 2 ! ,

(69)

where

S S

D D

P 1 5 2c 1

1 1 2 , m n1

(70)

P 2 5 2c 2

1 1 2 . n 28 m

(71)

Substituting for c 1 and c 2 in Eqs. (8) and (9) respectively, we obtain S IV 5

H 2K 2m

FS

D S

DG

1 1 1 1 1 2 1 X . n n8 n n8

(72) Fig. 9. Variation of S IV with X. m 5 1.5, n 5 1. (a) n 8 5 1, (b) n 8 5 1.2, (c) n 8 5 1.4, (d) n 8 5 1.6, (e) n 8 5 1.8, (f ) n 8 5 2.

In the special case n 5 n 8 , we have ~ S IV ! n5n 8 5

H 2K . mn

(73)

For a thin lens in air, ~ S IV ! n5n 8 51 5

H 2K . m

(74)

It is significant to note that when n Þ n 8 , the primary curvature is a linear function of X. The normalized primary curvature can be written as S IV 5

S IV H 2K

.

SV 5

H3 h

2

F

1 n

2

2

1 ~ n8!2

G

,

(78)

where, as before, the subscripts for n 1 and n 2 8 are dropped in expression (78) for the sake of simplicity. For the special case n 5 n 8 , S V 5 0. In general, S V is independent of both X and Y.

(75)

Figure 9 presents a set of lines showing the variation of S IV with X for a thin lens of refractive index m with object space refractive index n 5 1. The six lines correspond to different values of refractive index of the image space, n 8 over 1 (0.2) 2. Note that S IV is independent of n 8 when X 5 1 for any value of n. If we consider the symmetry of the expression, then S IV ~ X, n, n 8 ! 5 S IV ~ 2X, n 8 , n ! ,

where P 1 and P 2 are given by Eqs. (70) and (71), respec¯ , A ¯ , D , tively. Using the expressions for A 1 , A 2 , A 1 2 1 and D 2 in Eqs. (16)–(19), (23), and (24), by algebraic manipulations we obtain

(76)

so that the same figure can be used to represent the case of changes in refractive index of the object space when the image space index n 8 5 1, if only the signs on the scale for X are reversed and the symbols n and n 8 are interchanged. E. Primary Distortion The primary distortion coefficient S V of the thin lens is given by ¯ ¯ A 1 ¯ 2 hD ! 1 A 2 ~ H 2 P 1 A ¯ 2 hD ! , SV 5 ~ H 2P 1 1 A 1 1 2 2 2 A1 A2 (77)

4. PRIMARY CHROMATIC ABERRATIONS Following the prevalent practice in thin lens primary aberration treatments, we consider the longitudinal chromatic variation in focus and transverse chromatic variation in focus as longitudinal chromatic aberration and transverse chromatic aberration, respectively. A. Longitudinal Chromatic Aberration The longitudinal chromatic aberration C L of a thin lens is given by C L 5 A 1 hD 1 where D1 D2

S D dn n

S D S S D S dn n

dn n

1 A 2 hD 2

S D

D

dn n

,

(79)

5

dm dn1 2 , m n1

(80)

5

d n 28 dm 2 . n 28 m

(81)

D

In the above, m, n 1 , and n 2 8 are the refractive indices of the lens, the object space, and the image space, respec-

952



tively, at the working wavelength, and dm, d n 1 , and d n 2 8 are the absolute changes in refractive indices of the lens, the object space, and the image space over the chosen wavelength range at which the chromatic effects are considered. By using expressions (16) and (17) for A 1 and A 2 , respectively, we obtain CL 5

1 2 h K 2

G F D G F S DG F GJ

1 dn 1 1 dm 2 m2n m 2 n8 n~ m 2 n !

dn8 dm n n8 2 m1 n 8~ m 2 n 8 ! m m2n m 2 n8

1 2

HF

S

dn dn8 2 m2n m 2 n8

dn

X1

n

2

dn8 Y , n8

(82) where, as before, the subscripts for n 1 and n 2 8 are dropped for the sake of simplicity. In the special case of equal refractive index for the object and image spaces, i.e., n 5 n 8 , Eq. (82) reduces to ~ C L ! n5n 8 5 h 2 K

F

G

ndm 2 dnm . n~ m 2 n !

able redefinition of the shape factor X and the conjugate variable Y. Note that the primary spherical aberration is a cubic function of X and Y and the dependence of primary coma on X and Y is quadratic in nature. Primary astigmatism is a linear function of Y, and primary curvature is a linear function of X. Longitudinal chromatic aberration is seen to be a linear function of both X and Y. Primary distortion and transverse chromatic aberration are independent of both X and Y and depend on the optical properties of the object and image space media. The expressions presented here can be conveniently utilized in studying the effects of changes in refractive indices of surrounding media during the analysis and synthesis of thin optical components.

ACKNOWLEDGMENTS This work was supported by Formation de Chercheurs et Aide a` la Recherche of the Quebec Government and by the Natural Sciences and Engineering Research Council of Canada.

(83)

*Permanent address, Department of Applied Physics,

For a thin lens in air, n 5 n 8 5 1 and d n 5 d n 8 5 0, so that

University of Calcutta, 92 Acharyya Prafulla Chandra Road, Calcutta 700 009, India; e-mail: hazra@cubmb. ernet.in.

~ C L ! n5n 8 51 5 h 2 K

S

dm m21

D

5

h 2K , V

(84)

where V 5 ( m 2 1)/ d m is the Abbe number for the optical material of the lens. Defining the normalized form for longitudinal chromatic aberration as SL 5

CL h 2K

,

(85)

REFERENCES AND NOTES 1. 2. 3. 4. 5.

we note that S L ~ X, Y, n, n 8 , m ! 5 S L ~ 2X, 2Y, n 8 , n, m ! .

(86)

6.

B. Transverse Chromatic Aberration The transverse chromatic aberration C T of a thin lens is given by

7.

¯ hD d n 1 A ¯ hD d n . CT 5 A 1 1 2 2 n n

8.

S D

S D

(87)

¯ ,A ¯ , D ( d n/n), Using Eqs. (18), (19), (80), and (81) for A 1 2 1 and D 2 ( d n/n), we get CT 5 H

S

D

dn8 dn 2 . n8 n

(88)

Thus the transverse chromatic aberration is directly proportional to the change ( d n 8 /n 8 2 d n/n) and is independent of m, X, or Y. When the object and image spaces are same, we have C T 5 0.

9. 10. 11. 12. 13.

5. CONCLUDING REMARKS It is shown that the usual formalism for the treatment of thin lens primary aberration theory can be extended to the case of unequal object and image space media by suit-

14.

D. P. Feder, ‘‘Automatic optical design,’’ Appl. Opt. 2, 1209– 1226 (1963). M. Laikin, Lens Design (Marcel Dekker, New York, 1955). B. Brixner, ‘‘Lens design and local minima,’’ Appl. Opt. 20, 384–387 (1981). D. Sturlesi and D. C. O’Shea, ‘‘A global view of optical design space,’’ Opt. Eng. (Bellingham) 30, 207–218 (1991). M. Kidger and P. Leary, ‘‘The existence of local minima in lens design,’’ in 1990 International Lens Design Conference, G. N. Lawrence, ed., Proc. SPIE 1354, 69–76 (1990). T. G. Kuper, T. J. Harris, and R. S. Hilbert, ‘‘Practical lens design using a global method,’’ in International Optical Design Conference, G. W. Forbes, ed., Vol. 22 of OSA Proceedings Series (Optical Society of America, Washington D.C. 1994), pp. 46–51. G. Hearn, ‘‘Design optimization using generalized simulated annealing,’’ in Current Developments in Optical Engineering II, Robert E. Fisher and Warren J. Smith, eds., Proc. SPIE 818, 258–264 (1987). A. E. W. Jones and G. W. Forbes, ‘‘Application of adaptive simulated annealing to lens design,’’ in International Optical Design Conference, G. W. Forbes, ed., Vol. 22 of OSA Proceedings Series (Optical Society of America, Washington D.C. 1994), pp. 42–45. W. J. Smith, Modern Optical Engineering (McGraw-Hill, New York, 1966). R. Kingslake, Lens Design Fundamentals (Academic, New York, 1978). H. H. Hopkins and V. V. Rao, ‘‘The systematic design of two component objectives,’’ Opt. Acta 17, 497–514 (1970). L. N. Hazra, ‘‘Structural design of multicomponent lens systems,’’ Appl. Opt. 23, 4440–4444 (1984). K. Yamaji, ‘‘Design of zoom lenses,’’ in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1967), Vol. VI, pp. 105–170. I. C. Gardner, Application of the Algebraic Aberration Equations to Optical Design, Scientific Papers of the Bureau of Standards, Vol. 22, No. 550 (U.S. Department of Commerce, Washington, D.C., 1927).

L. N. Hazra and C. A. Delisle 15. 16. 17. 18. 19. 20. 21. 22. 23. 24.

A. E. Conrady, Applied Optics and Optical Design (Dover, New York, 1957), Vol. 2. A. Walther, ‘‘Eikonal theory and computer algebra,’’ J. Opt. Soc. Am. A 13, 523–531 (1996). M. Von Rohr, Geometrical Investigation of the Formation of Images in Optical Instruments (His Majesty’s Stationary Office, London, 1920). P. Turrie`re, Optique Industrielle (Delagrave, Paris, 1920). H. Coddington, A Treatise on the Reflexion and Refraction of Light (Marshall, London, 1929). H. D. Taylor, A System of Applied Optics (Macmillan, London, 1906). D. Argentieri, Ottica Industriale (U. Hoepli, Milan, 1954). H. H. Hopkins, Wave Theory of Aberrations (Oxford U. Press, London, 1950). H. H. Emsley, Aberrations of Thin Lenses (Constable, London, 1956). M. Born and E. Wolf, Principles of Optics (Pergamon, Oxford, 1980).

Vol. 15, No. 4 / April 1998 / J. Opt. Soc. Am. A 25. 26. 27. 28. 29. 30.

31. 32.

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H. Chre´tien, Calcul des Combinaisons Optiques (Masson, Paris, 1980). G. G. Slyussarev, Aberrations and Optical Design Theory (Hilger, Bristol, 1984). C. G. Wynne, ‘‘Thin lens aberration theory,’’ Opt. Acta 8, 255–265 (1961). W. T. Welford, Aberrations of Optical Systems (Hilger, Bristol, 1986), p. 227. See Ref. 22, p. 87. W. T. Welford, Aberrations of Optical Systems (Hilger, Bristol, 1986), p. 139. Note that the sign convention for paraxial angle u in this treatise is the opposite of the paraxial angle u used here. See Ref. 22, p. 123. R. V. Shack, ‘‘The use of normalization in the application of simple optical systems,’’ in Effective Systems Integration and Optical Design, G. W. Wilkerson, ed., Proc. SPIE 54, 155–162 (1974).