EISEVIER

15 July 1996

Optics Communications 128 (1996) 292-306

OPTICS COMMUNICATIONS

Full length article

Vector diffraction in paraboloidal mirrors with Seidel aberrations. I: Spherical aberration, curvature of field aberration and distortion

Rishi Kant

Department of Mechunical Engineering, San Jose Smte University. Coiiege of Engineering,

I Washington Squure, San Jose. CA 95192-0087, USA

Received 29 August 1995; accepted I8 January I996

Abstract

We study the image formation in paraboloidal reflectors with Seidel (fourth order) aberrations, particularly the spherical aberration, curvature of field aberration and the distortion aberration. The theory developed here is based on Richards’ and Wolf’s formulation of the vector diffraction theory of focusing systems. The expressions for the electric and magnetic vectors are derived to reflect the axially symmetric wave aberrations and it is established that these expressions are similar to the expressions for aberration-free parabolic reflectors except for the presence of the factor, exp(iK@), provided that these expressions are formulated in terms of the component, in the direction of tbe optical axis, of the unit normal to the wave front at the exit pupil. Here Cp is the aberration function. It is shown that if wave fronts emerging from the exit pupil suffer from the spherical aberration, the images formed at the respective diffraction foci are just as sharp as the images formed at the Gaussian focus by an aberration-free parabolic reflector, but with a slight reduction in the intensity. The similarity also extends to the state of polarization of the focused light. In the case of the curvature of field aberration, and in contrast with the scalar theory, we show that the Strehl intensity in the plane of the diffraction is less than unity. We also show that in the case of the spherical aberration, the intensity at the diffraction focus increases with increase in the numerical aperture,

whereas for the curvature of field aberrations it decreases with the increase in the numerical aperture. For the distortion aberration, the intensity is the same as the Gaussian intensity and the generated image is merely shifted perpendicular to the

optical axis.

1. Introduction

The scalar diffraction theory is deficient in pre- dicting the .quality of the image generated by paraboloidal mirrors of wide-angular apertures be- cause of the dependence of the electric and magnetic vectors on the numerical aperture. High frequency vector diffraction theories were developed by Igna- tovsky [1,2], Luneberg [3], Wolf 141, Richards and

Wolf [5], and Boivin and Wolf [6]. Based on the works of authors [4-61, many authors like Yoshida and Asakura [7,8], Hardy and Treves [9] have solved many interesting problems of aplanatic systems. Barakat [lo] used the vector diffraction theory of Richards and Wolf (51, and presented very interesting results for paraboloidal mirrors with or without ob- scuration. Barakat’s treatment is for aberration-free paraboloidal mirrors and is limited to image forma-

0030~4018/96/$12.00 Copyright 0 1996 Published by Elsevier Science B.V. All rights reserved. PII SOO30-4018(96)0008 l-8

R. Kant/Optics Communications 128 11996) 292-306 293

tion on the focal plane alone. In designing optical instruments, it is essential that the information about the image in the neighborhood of the focal plane be available for sensitivity analysis. A very comprehen- sive treatment of problems of image generation in aplanatic systems with Seidel aberrations is found in Born and Wolf [ 111, who cite the works of Nijboer [12], Nienhaus [13], and Kingslake [14]. Work of these authors is extensive but it is confined to the scalar theory. The reader is also referred to two recent texts, Stamnes [ 151, who has presented inter- esting results for aplanatic systems with aberrations and Mahajan [16], both of these authors dealt with the scalar theories of aberrations. Neither these works, however, consider imaging by paraboloidal mirrors. Concave parabolic mirrors are very useful because they focus incoming radiation, parallel to their axis, from a source located at infinity, without introducing any aberrations. Even though parabol- oidal reflectors play an important role in radio and optical telescopes, there is very little information available in the literature about the image generated by parabolic mirrors of moderate to high numerical aperture. Vector diffraction in spherical mirrors is considered by Kline and Kay [17].

In a series of recent papers, the author, Kant [l&21], developed an analytical method to solve the diffraction integrals derived by Richards and Wolf [51 and extended the theory of aplanatic systems [ 181 to include the primary Seidel aberrations of the first kind namely the spherical aberration, the cunature of field aberration and the distortion aberration [ 191, and more recently to include astigmatism and coma [21]. In the publication [20], the author has included the generalization that the amplitude of the incident wave has Gaussian distribution (see Yoshida and Asakura [8]). In Ref. [ 181, the theory has been adapted to include parabolic mirrors as well. In these publica- tions, the electric and magnetic energy-density distri- butions in the entire image space were obtained. In cases of studies with aberrations [19,21], it was shown that when the numerical aperture is small, the results are in agreement with those of the scalar theory. The author has also verified the results of Hardy and Treves [9], who have solved the problem of a stigmatic system. The foregoing discussion sug- gests that the method proposed by the author is robust and accurate, as it produces results that are in

agreement with previously published results, and in the limit of small numerical aperture, it reduces to the scalar theory.

We pointed out in Ref. [19] that the Seidel aberra- tions, or the fourth order wave aberrations, fall into two categories. In category one the wave fronts at the exit pupil are the surfaces of revolution about the optical axis or they are spherical but their axis of symmetry is not the optical axis. In the other cate- gory the wave fronts are asymmetrical about the optical axis. The spherical aberration, the curvature of field aberration, and the distortion aberration be- long to the first category. The astigmatism and coma belong to the second category. In this paper, we study the vector diffraction problem in paraboloidal mirrors, with or without obscuration, and which suffer from the Seidel aberrations belonging to the first category. The problem of astigmatism and coma in paraboloidal mirrors is studied in a second paper.

2. Statement of the problem

We consider the vector diffraction problem of paraboloidal mirrors with Seidel aberrations of the category one, and with or without central obscura- tion, which image a point source at infinity. We further imagine that the source emits linearly polar- ized monochromatic light. Let 01 denote the half-an- gular aperture of the system and let oI be the half-central obscuration, with CY , < (Y. The obscura- tion ratio E (0 5 E < 1), is given by

tanci , EC-

tanci ’ (1)

Ideal Gaussian Sphere

Aberrated Wave Front

Fig. 1. Tbe coordinate system and the geometry of the problem.

294 R. Kant/Optics Communications 128 (1996) 292-306

which for narrow angular optical systems reduces to e=OL,/ci. Let

E = Re{ e( P) exp( - iqt)},

H=Re{h(P)exp(-io,r)} (2)

represent the electric and magnetic fields at time t at an arbitrary point P in the focal region of the image space of the optical system (Re represents the real part). The geometry of the optical configuration is shown in Fig. 1.

The space dependent electric and magnetic vec- tors, e(P) and h(P) in the image space of the optical

(4 u=O, oc=60”, o(,=O’

“0 2 4 6 8 P=o” v, optical units

03 u=4, oc=60”, o(,=O’

“0 2 4 6 V,

8 +)” optical units

system with large Fresnel number, which were de- rived by Wolf [3] and later used by Richards and Wolf [5], are:

e(P) = -g//flo(S:,Sy) exp(iw[@(s,,s,) :

+s+‘)]}ds,ds,, (3) and

h(P) = -~/_/~b(S:‘Sy) exp(k[@(s,,s,) ;

+s.r(P)])ds,ds,, (4)

(4 u=O, a=60”, a,=30”

“0 2 4 6 V,

8 (o=oo optical units

(d) u=4, a=60”, 0(,=30”

-0 2 4 6 V,

8 (p=@ optical units

Fig. 2. Time-averaged electric energy distributions for various parabolic mirrors with or without obscuration. On the focal plane, the results

are in agreement with those in Ref. [IO]. The results on the plane u = 4 optical units (in the image space) are new.

R. Kant/ Optics Communications 128 (19961292-306 295

where rip) is the radius vector connecting the point P with the origin of the coordinate system located at the Gaussian focus (see Fig. 21, s = fsX, sy , s,) is the direction vector of a typical ray in the image space, and K denotes the wave number. The integral is taken over the entire surface of the wave front leaving the exit pupil. The complex vector functions, a(~,, syl and MS,, sy), are strength vectors of the unperturbed electric and magnetic field in the exit pupil. The function Ns, , sy ), represents the wave aberration function. In the present paper, we consider the case of paraboloidal mirrors with primary Seidel aberrations of the category one, thus MS,, sY) # 0. For later convenience, we define a set of the so-called optical coordinates, u and U,

u = K zp sin2a )

u = K Xp + yp sinol) \/2 (5)

where xp, yp, and zp are Cartesian coordinates of a point in the image space referred to a right-handed coordinate system with its origin at the Gaussian focus, z-axis pointing away from the system.. and x-axis coinciding with the direction of polarization of the monochromatic light in the object space. We also introduce spherical polar coordinates r, 0, cp (1. > 0, 0 < 8 5 T, 0 < cp < 2rr) with origin at the Gaussian focus, and the azimuth plane, cp = 0, containing the electric vector in the object space. The functions, @(s,, s,), for the types of the Seidel aberrations considered in this paper are: (a> Spherical Aberration

@( Sxr “J = u,p4 (6)

(b) Curvature of Field

@( s,, “J = UfP2 (7)

Cc> Distortion

a( Sxr sy) = u,p coscp, (8)

where the zonal radius, p, is defined as:

sin0 P=- sino . (9)

According to the scalar theory, the curvahlre of field aberration, as well as the distortion aberration, do not change the intensity patterns but merely shift them either along or perpendicular to the optical axis. While in the vector diffraction theory, this remains true for the distortion aberration, as we shall

later see, the curvature of field aberration (or equiva- lently defocus, because in this case we are only considering a monochromatic source at infinity) changes the intensity pattern and causes the intensity to be maximum at some point along the optical axis other than the Gaussian focus (see Richards and Wolf [5] and Kant [19]). Since the distortion aberra- tion causes no changes in the intensity patterns, we omit this aberration in this study, and consequently it is assumed that u,, = 0.

In Eqs. (6), (7) the constants u, and, uf charac- terize the departure of the wave front from the ideal Gaussian sphere at the periphery of the exit pupil. In the category of aberrations considered in this paper, the aberrated wave fronts are circularily symmetric about the optical axis. The coefficient ud also char- acterizes the departure from the ideal Gaussian sphere, but the aberrated wave front is a sphere whose axis is different from the optical axis. How- ever, this measure does not uniquely define the surface of the aberrated wave front. In order to define this surface, one needs the direction as well. Strictly speaking this should be in the direction of the ray, but to a very good approximation, this could be taken in the direction of the normal to the refer- ence sphere (see Ref. [ 111). Thus the radial distance of any point to the origin is:

r(e) =f+ u,p4 + Ufp*. (10)

In Eq. (lo), f is the radius of the focal sphere. The parametric equations of the surface of the wave front are given by:

x = I( t3) sine coscp,

y = r(e) sine sincp,

~=t-(e)cose. (11) Eqs. (11) are used to evaluate the expression for

the unit normal to the surface of the aberrated wave front using the method outlined in Ref. [22]. For algebraic details, the reader is referred to a previous publication [19]. Here, we only report the final re- sult:

s=pcoscpi+psincp j+qk,

where (12)

de) = r(e)sine- rye)cOse

\Iqyqi$'

296 R. Kant/Optics Communications 128 (1996) 292-306

q(B) = r( B)cos0 + r’( B)sinfl

/r’(e)+ (13)

From Eqs. (12) and (13), it is seen that

/s/=p2+q*= 1, pp’= -qq’. (14

For later use, we also calculate,

ds,ds,=lJIdOdq,

IJI= as, as, a~, as, xa(p-a(p~ =PP’= -qq’, ( 15)

where I Jl is the Jacobian of the transformation. We further note that position vector r(P) is

f(P) = r( P)(sinB Pcoscp, i + sine,sincp, j

+cose,k). c 16)

Next, we calculate the strength vectors c~fs,, sY) and b(s,, s,). We accomplish this by ray tracing. For algebraic details of this approach the reader is referred to Richards and Wolf 151 and Kant 1191. For the sake of brevity only, we provide here the final result:

+Vy)

2

=fQ +q) (

[(1+4) +cos2tp(l-d i

2

+ sin2v(q- *) 2

( 17)

The factor 2/(1 + q> appearing in the above equation comes from the requirement that the inci- dent energy contained in the area of an infinitesimal annulus of ray tube at the entrance pupil is equal to the energy in the corresponding area of the reflected wave front. This expression is derived in Ref. 1151. In Eq. (16), the constants E and p represent electric permitivity and magnetic permeability of the medium.

pressed in terms of very rapidly converging series consisting of products of the Gegenbauer polynomi- als and the spherical Bessel functions. Both of these functions are evaluated by using very stable reoccur- rence relations. Moreover, for both of these func- tions, the first two are expressed analytically, the remainder are evaluated extremely accurately with very little numerical effort. For details, the reader is referred to Refs. [18,19]. The integrals in the series form are:

Substitution of Eqs. (5), (12), (15), (161, and (17) 1, = 2 C 0fi”Cf/‘(cos+)j,( w),

in Eqs. (3) and (4) and carrying out integration with s-0

(18)

respect to 4, we get electric and magnetic fields vectors:

e,= -i A(Io+cos2qZ2),

e,,= -i Asin2q12,

e,= -2 Aces (PI,,

h,= -ifiAsin 2qI,,

h,= -imA(ZO-cos2g12),

h,= -2\/E/CLAsincpI,. (19)

In Eqs. (18) and (19), the constant A = KZ~~,

where K is the wave number and IO is the amplitude of the incident light at the entrance pupil. The inte- grals IO, I,, and Z2 are given as follows:

1, = / qTz’)Jo( pu/sina) exp[iK#( q, us7 q)]

,

I, = / kz”& Jd w/sin~)

xexp[iKK( q,us, q)] exp ,

I, = /

gfa,) 1 -9 9(u) (4+1)J2( pv/sina)

Xexp[ilcn( q, us, uf)] exp i: ( 1 sm*ci dq, (20)

where we have defined a new function:

N(q,u,,Q =@‘(pdJs,q). (21)

The integrals appearing in Eq. (20) can be ex-

R. Kant/Optics Communications 128 (1996) 292-306

I, = 2sinQ E ~~i”Cl”(cos*)j,+,(w), S=O

Z, = 2sin** i u3i”CJ”(cos*)j,+,(o). t: 22) s=o

In (22), the variables IJJ and o are related to u and u as follows:

u u 0 COSqJ = 7

sin*o ’ and w sin@ = -

sincr ’ r: 23)

and a:, uf and uf the coefficient of superposition of the Fourier-Gegenbaurer series:

a! = /

Olexp(iKH) Cf/‘( q) dq,

aI

u’ = s / a( 1 - q) exp(iKH) C,‘/‘(q) dq,

a1

af= f

ff(1 -4)2exp(iK~t)C,5’2(q)dq. (24) aI

Once the coefficients of superposition are deter- mined, the value of the integrals !,, I, and ,I, is easily evaluated. Note that when the results on the optical axis are desired, the integral I, takes a particularly simple form; at u = 0, the integrals I, and I, are zero, and

cc u I, = 2 C ufi’j, -

s=o ( 1 sir& ’ (25)

3. The Strehl intensity, diffraction focus and the polarization

In the absence of aberrations, the intensity is maximum at the Gaussian focus. When the aberra- tions are present, the point of maximum intensity is called the diffraction focus. The intensity at the diffraction focus normalized with respect to the in- tensity at the Gaussian focus in the absence of aberrations is cahed the Strehl intensity. For the category of aberrations studied in this paper., the wave fronts, after reflection from the paraboloidal mirrors are symmetric with respect to the optical axis, and therefore, it is reasonable to assume that

the diffraction focus lies optical axis, the integrals integral 1, becomes,

Zo[u,(oru,),o,o,,u]

291

on optical axis. On the I, and I, are 0, and the

7;

=2 C up[U,(oruf),a,a,]i.‘j, & ( 1 (26) s=o

and the intensity along the optical axis [5] is:

w = --!-[ A*], * i,], where Jo denotes the complex conjugate of I,. The integral IO can also be expressed in terms of a power series of a function containing the aberration func- tion and the distance u from the Gaussian focus. At the diffraction focus, aw/au = 0. We find that loca- tions for the diffraction foci are approximated by the formulae:

u=uJl +q(a!)] (27)

for systems with the curvature of field aberrations,

(28)

for systems with the spherical aberrations. We need to caution that the above formulae are semi-em- pirical results and some further work is necessary to determine the locations of diffraction foci more pre- cisely.

Next, we examine the polarization in the image space. To do this, we follow the treatment given in Ref. [l I]. Let ZLi’ and I!,” (n = 1, 2, 3) denote the real and imaginary parts of the integrals given in Eq. (22) and let,

e(u,u,cp) =P(u,u,(P) +iQ<u.u,cp), (29)

where P and Q are real vectors; they are a pair of conjugate semi-diameters of the polarization ellipse of the electric vector. Then the components of the vectors P and Q are given as follows (see Ref. f5]>:

P, = -A[ Ihi’ + cos 2~41 I;‘)],

Py = -A sin 29 Ii’),

Pz= -2AcoscpZ,(‘), (30)

298 R. Kant/Optics Communications 128 (1996) 292-306

and

Q, = -A[ [A” + cos 2q @‘j,

Q, = -A sin 2tp I:“,

Q;= -2Acosq4”. (31)

The integrals I,, I,, and I, are in general com- plex, including at the focal plane and the plane perpendicular to the optical axis that contains the diffraction focus. The polarization ellipse of the elec- tric vector is characterized by the two semi-axes of length a and b such that

a2=f[p2+ez+ (P~-Q~)~+~(P.Q)~],

b2=(P2+Q2- (P2-~2)‘+YPsQ)2].

(32)

and the angle + that the major axis, a, makes with the vector P is given by

b tan* = -tarE,

a (33)

where

tan 2E = tan 2p cos y,

in which

(34)

tar@ = P/Q (35)

and y is the angle between the vectors P and Q. This formally completes the solution of the problem. Below we discuss the results.

4. Discussion of results

In this section, we apply the theory just developed to the case of diffraction in paraboloidal mirrors with primary Seidel aberrations. First for comparison pur- poses, and to show that the method proposed here produces previously known results when the aberra- tions are absent, we plot the time-averaged electric energy distribution for two cases of paraboloidal mirrors of semi-angular aperture of 60”, one with central obscuration of 30” and the other without obscuration. These distributions are shown on the focal plane (Fig. 2a and Fig. 2c) and on the plane I( = 4 (Fig. 2b and Fig. 2d). We find that the energy

distribution on the focal plane is in agreement with the result given in Ref. [lo]. We believe that for paraboloidal reflectors the intensity distribution on any plane other than the focal plane has not been reported before. We have shown in Ref. [ 181 that for aplanatic systems, the method presented here pro- duces results that are in agreement with those of Ref.

(4 u=o.o, u,=o.o, cx=75”, cx,=15O

\ 0.5

.e

= 6

r-l

“0 2 4 6 8 (o=o” v, optical units

fb) U=Udf, u,=O.48, cx=75”, a,=1 5”

u-l 8 .?d

= 6

E 0 .- 4

z

O 2 i

n 2 4 6 8 (p=oo

v, optical units

Fig. 3. Time-averaged electric-energy density for a paraboloidal

mirror of semi-angular aperture of 75” and with semi-angular

central obscuration of 15”: (a) on the focal plane of an aberration-

free mirror and (b) for the same mirror with spherical aberration

of u, = 0.4% The electric energy distribution is shown on tbe

best receiving plane located at IQ, = 4.3546. The Strehl intensity

is 0.9958. The focal length of the mirror is WOOOX.

R. Kant/ Optics Communications I28 (1996) 292-306 299

[6], who calculated the time-averaged electric energy density in the entire image space.

4.1. The spherical aberration

In this section, we discuss the images formed by the paraboloidal mirrors with spherical aberrations. In Eq. (10) the only nonzero coefficient is U, and r(0) is given by:

r( 0) Pf+ u,p4. (36)

In addition to presenting results on various planes in the image space, this section establishes that the distribution of the time-averaged electric energy den- sity on the best receiving plane in the presence of spherical aberrations is similar to the distribution of the same quantity on the focal plane of an aberra- tion-free mirror. Thus, if the wave in the object space is linearly polarized, the solution presented here stipulates that a detector of electric energy (such as a photographic plate) placed on the best receiving plane would record a distant source’s image that is just as resolved as if the detector were placed on the focal plane of the aberration-free paraboloidal mirror with similar attributes. Furthermore, the larger the numerical aperture of the paraboloidal mirror, the greater the similarity in the resolution of the images formed at the focal plane of an aberration-free mirror and the images formed at the best receiving plane of a mirror with spherical aberration. Thus if a paraboloidal mirror is found to have spherical aberra- tion, it still can be used for quality imaging, provided the detector is placed on the best receiving plane. As we shall see later, the forgoing assertion is also true for paraboloidal mirrors with curvature of field aber- rations.

In Fig. 3a, we show the distribution of the time- averaged electric energy density on the focal plane

Fig. 4. Time-averaged electric-energy density for a paraboloidal

mirror of focal length of 10000X and semi-angular apemIre of

60”. and with semi-angular central obscuration of 15”: (a) on the

focal plane of an aberration-free mirror, (b) for the same mirror

but with spherical aberration of U, = IA. The Strehl intensity is

0.9430 and the best receiving plane is located at ud, = 10.5781

optical units, and (c) for the same mirror but with spherical

aberration of U, = 2X. The Strehl intensity is 0.8634 and the best

receiving plane is located at udf = 22.545 optical units.

(4 u=o.o, u,=o.o, a=60”, a,=1 5”

2 4 6 V,

8 p=oo optical units

(b) u=u&, lJ,=l.O, 0(=60”, CQ=15”

-0 2 4 6 8 cp& v, optical units

6) U=&,f, C&=2.0, (x=60”, (x,= 15”

-0 2 4 6 8 PO=00 v, optical units

300 R. Kant/ Optics Communications 128 f 1996) 292-306

of an aberration-free paraboloidal mirror of semi-an- gular aperture of 75” and with semi-angular obscura- tion of 15”. In Fig. 3b, we consider the above mirror but with spherical aberration, u, = 0.48X, and we show the distribution of the time-averaged electric energy density on the best receiving plane located at Key= 4.3546 optical units. The Strehl intensity in this case is 0.9958. From these figures, it is noted that in these two cases, the distribution of the time- averaged electric energy density is almost identical.

This similarity also extends to the cases when the amount of spherical aberration is large as can be seen from Fig. 4b and Fig. 4e. In these figures, we show the distribution of the time-averaged electric energy density for two paraboloidal mirrors of semi- angular aperture of 60” and semi-angular obscuration of 15”, but with spherical aberrations of 1X and 2X respectively. For the case when the spherical aberra- tion is 1 A, the Strehl intensity is 0.9430 and the best receiving plane is located at u = 10.5781 optical units, whereas for the second case the Strehl inten- sity is 0.8634 and the best receiving plane is located at u = 22.545 optical units. From these figures, it is seen that the distribution of the time-averaged elec- tric energy density is similar to the distribution on the focal plane of an aberration-free paraboloidal mirror of the same attributes (see Fig. 4a). The resolution of the images formed by these paraboloidal mirrors with spherical aberrations is just as sharp as

0.1 0 2 4 6 a 10

v, optical units

Fig. 5. Time-averaged electric energy distribution on the focal

planes of paraboloidal mirrors with various amounts of spherical

aberrations. The intensities are normalized to 100 at the focus of

an aberration-free mirror. It is seen that the image at the focal plane formed by the mirrors with aberration is of considerably

poor quality.

0 2 4 6 8 v, optical units

0 2 4 6 10

v,opticd units

Fig. 6. Time-averaged energy distributions along the two azimuth

directions, cp = 0” and cp = 90”. on the best receiving planes for

paraboloidal mirrors with various amount of spherical aberrations.

It can be seen that on the best receiving planes, the images formed

by the mirrors with spherical aberrations have almost the same

resolution as the image formed by the aberration-free mirror on

the focal plane. Furthermore, the resolution in the up = 90” direc-

tion is better than the resolution in the cp = 0’ direction (see Ref.

151).

the image formed by the aberration-free mirror, but the peak intensity is some what reduced.

In Fig. 5, we show the variations of the time-aver- aged electric energy density along the azimuth cp = 0” on the focal planes of the above paraboloidal mirrors in presence of spherical aberrations of IA and 2A and compare these with the variation for the case of an aberration-free mirror. We see that when the aberrations are present, the quality of the image on the focal plane is very poor.

In Fig. 6a and Fig. 6b, we show the variation of the time-averaged electric energy density along the azimuth lines, cp = 0” and cp = 90”, on the best re- ceiving plane of the paraboloidal mirrors with (Y = 60” and cx , = 15”, and with spherical aberrations of

R. Kant/Optics Communications 128 (1996) 292-306 301

1X and 2X. In the same figures, we compare these with the time-averaged electric energy distribution for the case of an aberration-free paraboloidal mirror of the same attributes. These figures also confirm the assertion made earlier that on the best receiving planes, the paraboloidal mirrors with aberrations lbrm images of high resolution that are comparable to those formed on the focal planes of aberration-free mirrors. These figures also confirm that if in the object space, the wave is linearly polarized in the cp = 0” direction, the resolution of the image is higher in the q = 90” direction. This has also been noted by Richards and Wolf [51 in their study of aplanatic systems.

4.2. The curvature of field aberration

In this section, we discuss the images formed by paraboloidal mirrors with curvature of field aberra- tions. In this case, the only coefficient uf # 0, and Eq. ( 10) becomes

r( 0) Zf+ ufp2. 1: 37)

In Fig. 7, we show the distribution of the time-

U=&,f, u,=2.0, a=60°. crl= 15”

2 4 6 8 Po=oo V, optical units

Fig. 7. Time-averaged electric energy density distribution on the

best receiving plane of paraboloidal mirrors of semi-angular aper-

ture of 60” and semi-angular aperture of 15” and with the ctrrva-

hrre of field aberration of U, = 2X. The intensities are normalized

to 100 at the focus of an aberration-free mirror. The Strehl

intensity on the best receiving plane is 0.9279 and the best

receiving plane is located at u

0.0’ . ’ ’ ’ ’ ’ ’ 0 2 4 6 6 IO

v, optical units

Fig. 8. Time-averaged electric energy distribution on the focal

plane of paraboloidal mirrors with various amounts of spherical

aberrations. The intensities are normalized to IO0 at the focus of

an aberration-free mirror. It is seen that the image at the focal

plane formed by the mirrors with aberration is of considerably

poor quality.

averaged electric energy on the best receiving plane of a paraboloidal mirror of semi-angular aperture of 60” and semi-angular obscuration of 15”, and with the curvature of field aberration of 2h. The best receiving plane is located at udf = 18.445 optical units and the Strehl intensity on this plane is 0.9279. We find that the distribution of the time-averaged electric energy density is similar to the distribution on the focal plane of an aberration-free mirror of the same attributes (Fig. 4c). In Fig. 8, we show the distribution of the time-averaged electric energy on the focal plane of the aberration-free paraboloidal mirror and those with curvature of field aberrations. We see that in the presence of the curvature of field aberrations, the quality of the image formed at the Gaussian focus is very poor. In Fig. 9a and Fig. 9b, we show the distribution of the time-averaged elec- tric energy density along the azimuth lines cp = 0” and cp = 90” in the best receiving planes of paraboloidal mirrors with curvature of field aberra- tions of 1 A and 2X and on the focal plane of an aberration-free mirror. We see that the image formed on the best receiving planes is just as resolved as the image formed on the focal plane of an aberration-free mirror. Here, however, in contrast with the scalar theory, the vector theory not only predicts a shift in the intensity patterns but the changes in the image

XL? R. Kant/Optics Communications 128 (1996) 292-306

Cuvature of Field itmnaion

+ aberration-free

-m- ‘A

--e- 2x

4 6 v, optical units

0 2 4 6 8 10 v, optical units

Fig. 9. Time-averaged energy distributions along the two azimuth

directions, cp = 0” and cp = 90”. on the best receiving planes for

paraboloidal mirrors with various amount of curvature of field

aberrations. It can be seen that on the best receiving planes, the

images formed by the mirrors with spherical aberrations have

almost the same resolution as the image formed by the

aberration-free mirror on the focal plane. Furthermore, the resolu-

tion in the cp = 90” direction is better than the resolution in the

cp = 0” direction (see Ref. [5]).

quality as well. It is readily seen, however, that when the numerical aperture is small, the reduction in the peak intensity is also very small (see Fig. lob).

This is consistent with Richards and Wolf’s ob- servation that their theory reduces to Airy’s diffrac- tion theory when the numerical aperture is small (o + 0). 3ased on their derivation, Kant 1193 has shown that for large numerical apertures, the curva- ture of filed aberration changes the image intensity distribution as well. Thus the assertion that the cur- vature of field aberration is included in the analysis of Barakat [lo] appears to be in error.

4.3. The Strehl intensity and the location of the diffraction foci

Because the aberrated wave fronts in presence of the spherical and curvature of field aberrations are rotationally symmetric about the optical axis, the diffraction focus lies on the optical axis. In this section, we first calculate the variation of the time- averaged total energy density (intensity) along the optical axis to locate the diffraction focus for various

(4 100

0 15 30 45 60 75 90

a. 0

(W

100

.& = 3

80 CXrvatm of Field Akmtiin

-c E

z 60

I. I. I.. I .I. 1

0 15 30 45 80 75 90 0

a,

Fig. 10. Variation of the Strehl intensity at the diffraction foci

with numerical aperture of various paraboloidal mirrors: (a) for

mirrors with spherical aberration and (b) for mirrors with curva-

ture of field aberration.

R. Kant/Optics Communications 128 (1996) 292-306 303

paraboloidal mirrors. In Fig. lOa, we show the varia- tion in the Strehl intensity for paraboloidal mirrors of various numerical aperture and with different amounts of the spherical aberration, and in Fig. lob for the mirrors with the curvature of field aberration. From Fig. lOa, we note that for low angular aperture systems, the Strehl intensity decreases with the in- crease in the amount of the spherical aberration. However, for a given amount of the spherical aberra- tion, the Strehl intensity increases with the increase in the numerical aperture of the mirror to a maxi- mum value when the numerical aperture is slightly less than unity after which it falls off slightly. While no explanation for this extremum was found, we trust our results as we have verified them by a second independent method. The same observation was found in the case of aplanatic systems [19]. From this figure, we further note that for low angular aperture mirrors, for a given amount of the spherical aberration, the Strehl intensity has the same value as for the aplanatic systems. In Fig. 11, we show the dependence of the location of the diffraction focus on the numerical aperture for the spherical aberration and for the curvature of field aberration. These loca- tions were calculated by numerically evaluating MO, d/au=O.

/ Richards and Wolf ’ a,= 0

pherieal Abenatii

Curvatum of Fiakl Aberration

- Approximation

1.0 0 15 30 45 60 75 90

0 Q,

Fig. 1 I. Variation in the location of the diffraction foci with

numerical aperture for various paraboloidal mirrors with spherical

aberration and the curvature of field aberration. The approximate

value for the location of the diffraction foci are calculated by

using Eq. (27) for the curvature of field aberration and by Eq (28)

for the spherical aberration.

u=o, u,=o.o, 0(=60”, al= 15”

v, optical units ((o-0’)

Fig. 12. Contours of p(u, cp) on the focal plane of an aberration-free

paraboloidal mirror of semi-angular-aperture of 60” and semi-at-

gular obscuration of 15”. The ratio p = a/b, where a is semi-

major axis. In the case of an aberration-free mirror, the plane

containing the semi-axis of the ellipse of polarization is perpendic-

ular to the focal plane.

This calculation is particularly simple since along u = 0, the integrals I, and Z2 are zero. The solid lines in this figure correspond to the formulae (27) and (28). We see that these formulae approximate the location of the diffraction foci well. We also note that the location of the diffraction focus moves closer the Gaussian focus from its farthest position when the numerical aperture is small (a + 0).

4.4. The state of polarization

In this section, we study the state of polarization in the image space. When the system is aberration- free, the integrals I,, I, and I2 on the focal plane are real and the vectors P and Q are perpendicular to each other. The polarization ellipse is, therefore, perpendicular to the focal plane and IPl and IQ1 represent the semi-axis of the ellipse (see Ref. [51). When the aberrations are present, the vectors P and Q are no longer perpendicular to each other and the state of polarization is determined by Eqs. (30)-(35). However, just as in the case of the aberration-free mirror, the state of polarization along the optical axis and along the y-axis is linear and is in the same direction as the polarization in the object space. In Fig. 12, we show the ratio of the semi-axes of the ellipse, p = a/b, for an aberration-free mirror, where a is the semi-major axis of the ellipse of polariza-

R. Kant/Optics Communications 128 (19961292-306

u=ud,. u,=z.o. a=60”, a,= 15”

“0 2 4 6 8 v, optical units (p=O”)

Fig. 13. Contours of p(u, cp) on the plane containing the diffrac-

tion focus for a paraboloidal mirror of semi-angular-aperture of

60” and semi-angular obscuration of 15”. and with spherical

aberration of 1X. The ratio p = a/b, where n is semi-major axis

of the ellipse of polarization. The focal length of the mirror is

1oOOOA.

tion. Since along the y-axis (cp = IT/~ and cp = 31r/2), only e., component of the vector e is differ- ent from zero, the ratio p is also zero along this axis. Therefore, nonzero contours of p must not reach the y-axis. The contours of p = 0 are crescent shaped and span between plus y-axis (cp = n/2) and minus y-axis (cp = 3rr/2). These crescents are separated from each other by a circular boundary characterized by the roots of the equation IJO, u) = 0, and that all non-zero contours are clustered in these sectors. Furthermore, contours of p = 1, imply that ellipse of

u=o.o. u,=1.0,cu=60°, a,=1 5’

"0 2 4 6 8 v. optical units (rp=O’)

Fig. 15. Contours of p(o, cp) on the plane containing the diffrac-

tion focus for a paraboloidal mirror of semi-angular-aperture of

60” and semi-angular obscuration of 15”, and with the curvature of

field aberration of 2X. The ratio p = u/b, where a is semi-major

axis of the ellipse of polarization. The focal length of the mirror is

10000X.

polarization degenerates into a circle and therefore,

the electric vector is circularly polarized. In Figs. 13 and 14, we show the same quantity on the best receiving plane and the focal plane respectively, for a paraboloidal mirror with the same attributes but with the spherical aberration of 1A. It is seen from the Figs. 12 and 13 that for an aberration-free paraboloidal mirror and for the mirror with the spherical aberration the distribution of the ratio, p, is nearly similar. This observation is true for the mirror with the curvature of field aberration as well (see Fig. 15). From Figs. 14 and 16, it is seen that on the

u=O, u,=2.0, or=60”, a,=15"

v, optical units (cp=O”)

Fig. 14. Contours of p(u, qP) on the focal plane of a paraboloidal Fig. 16. Contours of p(u, cp) on the focal plane of a paraboloidal mirror of semi-angular-apertur of 60’ and semi-angular obscura- mirror of semi-angular-aperture of 60” and semi-angular obscura- tion of 15”. and with spherical aberration of IA. The ratio tion of 15”. and with the curvature of field aberration of 2X. The p = u/b, where u is semi-major axis of the ellipse of polariza- ratio p = c1/ b, where (1 is semi-major axis of the ellipse of tion. The focal length of the mirror is IGWCIA. polarization. The focal length of the mirror is IOCOOA.

v, optical units (p=O”)

R. Kant /Opics Communicarions I28 (19961292-306 305

focal planes of the paraboloidal mirrors with the spherical and curvature of field aberrations, the dis- tribution of the ratio p is very different from for the

case when the mirror is aberration-free.

5. Concluding remarks

In this paper, we have used a previously devel-

oped method to solve the problem of electromagnetic

diffraction in paraboloidal mirrors with primary Sei- del aberrations of the first kind, namely the spherical

aberration, the curvature of field aberration, and the

distortion aberration. We have shown that our re- sults, in the limit when the numerical aperture is

small, are in agreement with the results of the scalar

theory. Theory developed here is based on Rich.srds’ and Wolf’s theory of vector diffraction for aplanatic

systems and it has been adopted to include the spherical, curvature of field, and the distortion aber- ration. It is shown that for the class of aberrations mentioned above. the diffraction integrals, if formu-

lated in terms of the zth component of the unit normal to the aberrated wave front, are similar tso the

integrals if aberrations were absent. These integrals are evaluated in terms of the products of Gegenbauer polynomials and the spherical Bessel functions. Since both of these functions are given by a finite series of trigonometric functions, their evaluation is easy and is accomplished by using stable recurrence relations.

Since the spherical Bessel functions tend to 0 very fast, only a few terms are needed for the series to attain the desired convergence. The results presented in this paper are based on computation performed on a personal computer.

In summary, we would like to make the following observations:

1. For paraboloidal mirrors with curvature of field aberrations, the Strehl intensity at the diffraction focus is less than unity. Thus according to the vector diffraction theory, the image changes in addition to being shifted laterally along the optical axis. This is in contrast with the scalar theory which predicts that when the curvature of field aberrations are present the image patterns remain unchanged but merely shift along the optical axis. For the same amount of curvature of field aberration, the diffraction focus

moves closer to the Gaussian focus from its farthest location when the numerical aperture is small (a -+

0). 2. In the presence of spherical aberrations, the

Strehl intensity increases with increasing numerical

aperture. It reaches the maximum value when the

numerical aperture is about unity and then falls off

again. No explanation of this extremum was found.

Further, for the same amount of the spherical aberra-

tion, the location of the diffraction focus moves

closer to the Gaussian focus from its farthest position

when the numerical aperture is small (a + 0). 3. In view of the above, it is necessary to revise

the Marechal’s condition for high-aperture systems.

According to Marechal’s condition, the system is considered well corrected if the Strehl intensity at

the diffraction focus is 80% or better, and the amount

of aberrations that produce lower Strehl intensity are considered excessive (see Ref. [l 11). Because of the

fact that, for the same amount of spherical aberra-

tion, the Strehl intensity increases with increasing NA, for high numerical aperture paraboloidal mirror,

greater amount of spherical aberration may be ac- ceptable. For high-aperture mirrors with curvature of field aberrations, on the other hand, it may be neces-

sary to impose a condition limiting the acceptable amount of the said aberration. This is due to the fact that for the same amount of the said aberration, the Strehl intensity at the diffraction focus decreases

with increasing numerical aperture. 4. In the presence of either spherical aberrations

or the curvature of field aberrations, the resolution of

the image formed at the diffraction focus is just as good as the image formed at the Gaussian focus of an aberration-free mirror. This observation is particu- larly important for paraboloidal mirrors with spheri- cal aberrations for it implies that such mirrors can be used for high-resolution imaging provided that the plane of the diffraction focus is used as the detector plane. These observations are based on the results

presented in the Figs. 4, 6, 7, and 9. 5. The similarity also exists, but to a lesser extent,

between the states of polarization of the electric (and magnetic) vectors at the focal plane of the aberra- tion-free system and at the plane of the diffraction focus of systems with aberration, in that the ratios of the semi-minor to semi-major axes of the ellipse of polarization are similar. However, the polarization

306 R. Kant/Optics Communications I28 (1996) 292-306

ellipse is not perpendicular to the plane of the diffraction focus as it is on the focal plane of the aberration-free system.

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