15 July 1995
OPTICS
COMMUN~~ATI~S
Optics Communications 118( 1995) 207-2 14
Analysis of the Strehl ratio
using the Wigner distribution function
Dobryna Zalvidea a, Mario Lehman a, Sergio Granieri by*,Enrique E. Sicre b
a ~epurta~eato
de Ft’sica, ~aiversid~
~~~io~l def Sar, (8~) Bahth Blanca, Argentina
h Centro de Investigaciones Upticas (CiOp), C.C. 124, (1900) La Plats. Argentina
Received 19 September 1994; revised version received 20 December 1994
Abstract
The on-axis intensity distribution, or the Strehl ratio versus defocus, is described in a phase-space domain in terms of
the Wigner distribution function. A coordinate transformation that is defined for a general annular aperture is employed to
convert the pupil amplitude transmittance into a one-dimensional function. We show that the single display of the Wigner
distribution function associated with this modified pupil contains ail the values of the Strehl ratio along the optical axis.
The analysis is performed for two primary aberrations: coma and asti~atism.
1. Introduction
There are several criteria to analyze the performance of an optical imaging system to aberrations and/or focus
errors based on the on-axis image irmdiance as the relevant quantity [ 1.21. The Strehl ratio (SR), defined as
the ratio of the intensity values at the diffraction focus with aberrations and one without aberrations, becomes
an important figure of merit from the viewpoint of image evaluation. In order to link the SR with the design
properties of the optical system, the image intensity can be related to the amplitude transmittance of the pupil
function using Fourier optics relationships. However, some geometrical insight is useful for analyzing tolerance
criteria. In recent years phase-space representations, like the ambiguity function [ 31 and the Wigner distribution
function ( WDF) [ 4-71, have been employed in optics for several applications [ 8- 151. More s~ific~Iy,
the
WDF which can be considered as the “amplitude” of parageometrical rays involving diffraction effects was
applied to study the behavior of the SR for varying focus errors and object source locations [ 161. Furthermore,
Ojeda-Castaiieda et al. [ 171 described the SR of a rotationally symmetrical system versus defocus in terms of
the ambiguity function. They showed that the Fourier spectrum of the SR for variable spherical aberration and
defocus can be analyzed from a single, phase-space picture.
The purpose of this paper is to show that the SR for varying position along the optical axis (i.e., focus
errors) and different types of primary aberrations can be visualized in a single, phase-space representation
coordinate/spatial frequency. We derive a two-dimensional WDF (instead of a four-dimensional one), which
is associated with a modified one-dimensional pupil function. The value of the SR can be obtained by properly
’ Also: Facultad de Ciencias Exactas, Uaive~idad National de La plats, Argentina.
0030-4018/95/$09.50 @ 199.5Elsevier Science B.V. All rights reserved
SSDlOO30-4018(95)00196-4
D. Zalvia’ea et al. /Optics Communications 118 (1995) 207-214
208
projecting this WDE This rest& is used for comparing the tolerance to defocus of a transparent, finite size
circular pupii with an annular aperture for both, coma and astigmatism.
2. Basic theory
An optical imaging system can be described by the amplitude point-spread function
(1)
where zi is the axial distance between the plane of the exit pupil and the image plane located at Z = 0. The
complex amplitude transmittance t(&, q), associated with the exit pupil, can be written as
being to( 5, v) the nonnegative amplitude-transmission function of the aperture, and w( 6,~) the wave-aberration
function introduced by the optical system. The Wigner distribution function (WDF) associated with the field
~pIitude at the image plane is given by either of the two following expressions 141:
x’
x-7
I
Y-q/k
2
I
,y - $-;O
$;O
exp[-2ti(x’v
exp[2ti(xv’+yp’)]
+y’,u)]
dv’dp’,
dx’dy’
(3a)
(3b)
where (v, ,u) are the spatial frequencies and p”(v,p) denotes the Fourier transform of p(x, y). A close
relationship between this WDF and t([, T) is obtained by combining Eqs.( 1) and (3)
=(AZi)2Wt x-AZiv,Y-Azi/l;;iX;,~
~
Wi(X,y;V,lu-)
(
i
*>
(4)
being W,( x, y; v, CL)the WDF of the pupil function given by Eq. (2).
In order to analyze the SR employing the WDF formalism, we take advantage of the two following properties
t4,51:
w, (x, y; I/, CL) = Wo(x - Azv, y - AZ& v1/xl I
(5)
00
I(x,y;z)
Wz(x,y;v,~)dvd/~.
=
(6)
--M
IQ. (5) states that the WDF in free space propagation, within the paraxial approximation, can be found from
the original WDF by performing a coordinate shearing, and Eq. (6) gives the signal intensity as the WJJF
projection on the spatial frequency axis. Using Qs. (4) -( 6)) the normalized on-axis intensity for variable z,
in the neighborhood of the image plane, becomes
cw
Z(O,O;z) =
~
-03
Wi( -AZv, -A&U; V’rpf dv dg,
(7)
D. Z&idea et al. /Optics Communicafio~ ii’8 (1995) 207-214
209
and the SR versus defocus results as
where (1 = -~Z/Zi>V , 52 = -(Z/‘Zt)~ are redefined spatial frequencies, lo is the on-axis image intensity
in the absence of aberrations, and K = ~~z~/zA)~ being A the pupil area. Although Ekl. (8) provides a
complete description of the SR for any plane z, # 0, it involves the manipulation of a function defined in a
four-dimensional phase-space. Therefore, we follow an alternative approach in order to reduce the information
content of the SR to a two-Dimensions phase-space. We start by writing the intensity in the neighborhood of
the image plane as
(9)
Then, the on-axis intensity is
Since we restrict the analysis to radially symmetric apertures (i.e., fe(&, 77) = f&p>, being p = (s* -I- q2) f ),
EIq. (IO) becomes
(11)
For the particular case of a free-aberration system, w( p, #J) = 0, and Eq. ( 11) takes the form
Now, we employ the change of variable p = p(g), intr~uc~
in Ref. [ IS], which transfo~ any twodimensional radially symmetric aperture to(p), defined for 0 5 p 2 po, into a modified one-dimensional pupil
function qo( 5) that is different from zero only in the interval -l/2 5 5 5 l/2; that is
(p/po12=U
-e2M+;H-E2.
(131
as is shown in Fig. 1 Eq. ( 13) takes into account the possible appearance of a central obscuration of radius EPO
(0 5 E < 1) , so that the case of a complete circular aperture is recovered with E = 0. Thus, using Rq. ( 13)
we get the Fourier relationship between the on-axis intensity and the pupil function in the form
210
D. Z&idea
et al. /Optics
Communications 118 (1995) 207-214
-0.5
PO
P
0
0.5
c
Fig. 1. Coordinate transformation p = p(l).
or, alternatively, in terms of the SR
(14)
The limits of integration extend from -co to +oo, since qo( 4’) is equal to zero outside the domain ( - l/2, l/2).
By means of the WDF (see Eq. (3b)), Eq. ( 14) can be rewritten as
(15)
Therefore, the SR for a rotationally symmetric free-aberration system can be obtained for any out-of-focus
plane from a proper spatial frequency projection of the WDF associated with the transformed pupil function
qo( 5) defined in the two-dimensional phase-space
(x=z/zo, y=5>,
where zu = 2Azf/&( 1 - l
*).
Next, we analyze how this result is modified when some aberrations are present. To illustrate this approach,
two primary aberrations are considered: coma and astigmatism. In the first case, the wave-aberration function
becomes
~‘“‘(PJM
(16)
= &(~/po)~cos#,
where A, is the coma coefficient. By replacing Eq. ( 16) into Eq. ( 11)) and performing the &integral together
with the coordinate change given by Eq. ( 13)) the SR can be expressed as
(17)
being
p(J)
(2?TAc
= qo(!aJo
F((l
-e2)(l+
4) +e2)3’2
1
(18)
>
the modified one-dimensional pupil function corresponding to coma. Jo denotes a Bessel function of the first
kind and zero order. The equivalent result to that given by Eq. ( 15) is
D. Zaivideu et al. /Optics
Communica?ions 118 (1995) 201-214
211
Fig. 2. Wigner dist~bution functions. (a) ~nifo~ circular aperture, (b) circular aperture with coma (A, = 0.3). (c) circular apenure
with coma (AC = 0.6) and (d) annular aperture (E = 0.75) with coma (A, = 0.6).
m
SCC’(z)
=
,t
--oo
)
d5.
(19)
For the case of astigmatism, the wave-abe~ation function is
w““(P,&
= A,(p/~o)~cos~4,
(20)
and following a similar procedure to that carried out to obtain J3q. ( 19), it results
(21)
212
D. Z&idea
et al. /Optics
Communications
118 (199.5) 207-214
0.9
0.8
0.7
0.6
ET
;:- 0.5
0.6
$,
-rn
0.5
-rn
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0.0
0.0
0
1
2
3
0
4
1
2
Z/Z,
wzo
(a)
(bf
3
4
Fig. 3. Strehl ratio versus defocus for coma. (a) Ac = 0.2 and (b) AC = 0.4.
0.9
Aa=0.3
-
E=O
0.8
0.7
0.6
z
=- 0.5
-&I
0.4
$-, 0.5
0.3
0.3
0.2
0.2
0.1
0.1
G-r/l
-3-2-l
0
Uz,
(a)
12
3
0.4
-3
-2
-I
0
1
z/z,
@I
Fig. 4. Strehl ratio versus defocus for astigmatism. (a) As = 0.3 and (b) Aa = 0.5.
where
2
3
D. Z&idea
et al. /Optics
1.0
1.0
0.9
0.9
-
0.8
0.6
E=O
c=o.5
0.7
------
J
0.7
EZO.75
0.6
0.6
z
;;
0.5
-r/l
0.4
213
Communications 118 (1995) 207-214
E
0.5
3
-WI 0.4
0.3
0.3
I
0.2
0.2
0.1
0.1
0.0
0
0.0
1
2
3
4
0
1
2
A=
Aa
(a)
(b)
3
4
Fig. 5. Strehl ratio at the image plane. (a) Coma and (b) astigmatism.
the corresponding pupil function.
As can be concluded from Eqs. ( 15), ( 19) and (21), the SR of a rotationally symmetric optical system, for
any amount of defocus, can be analyzed from a single picture representation: the WDF associated with a pupil
function given by the product of a one-dimensional transmittance qo( 5) and a Bessel function JO(U), where u
depends on the aberration coefficient and the size of the annular domain.
is
3. Examples
In order to illustrate this approach we analyze the behavior of the SR for a full circular pupil (E = 0)
and two annular apertures (E = 0.5 and E = 0.75). Some work about the influence of central and noncentral
obscurations on the imaging properties of optical systems can be found in Refs. [ l&19]. Fig. 2 shows the gray
level representations of the WDF corresponding to: (a) free-aberration circular aperture, (b) circular aperture
with A, = 0.3, (c) circular aperture with A, = 0.6, and (d) annular aperture (E = 0.75) with A, = 0.6. By
simple inspection of the WDF displays, it can be derived the higher tolerance to defocus of the annular aperture
as compared with the circular one (for the same amount of aberration) and the decrease of the SR for growing
aberrations. As follows from Eqs. ( 19) and (21), the SR for variable z is obtained by adding the values of the
WDF along different slices of the spatial coordinate. Figs. 3 and 4 show the SR, S( z ), for two different values
of coma (A, = 0.2 and A, = 0.4) and astigmatism (A, = 0.3 and A, = 0.5), respectively. Finally, in Fig. 5 it is
shown the variation of the SR at the image plane (z = 0) as a function of the aberration coefficients. As can
be expected, for a larger central obscuration, the SR exhibits lower tolerance to an increase of the aberrations.
214
4.
D. Zalvidea et al. /Optics
Communications 118 (1995) 207-214
Conclusions
We have shown that tolerance criteria based on the Strehl ratio (SR) can be visualized in a single phasespace representation,
the Wigner distribution function (WDF) of a one dimensional modified pupil function.
This result allowed us to compare the performances of a circular uniform pupil and two annular apertures, for
varying focus errors and for two different aberrations: coma and astigmatism. For more general pupil functions,
it should be taked into account that the WDF projection which gives rise to the SR can be similar even if
the shapes of the different WDFs deviate from each other. Besides, this approach can be applied to analyze
comparatively different shade masks which are proposed to reduce the influence of the several aberrations.
Acknowledgments
Financial support from Consejo National
acknowledged.
de Investigaciones
Cientificas
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y T6cnicas
(CONICET)
is gratefully