Zone plate for
arbitrarily
high focal depth
Jorge OjedaCastaneda and L. R. BerrielValdos
We show that a zone plate with a prespecified number of foci, which are separated axially by Rayleigh's limit ofresolution, can create an arbitrarily high focal depth when used as an apodizer over an otherwise clear
aperture. We discuss the resolution and light gathering power of this method.
1. Introduction
Various efforts are going on involving the use of
apodizers for designing modern optical instruments
with novel imaging characteristics.1
7
Imaging systems used in microelectronics, medicaldiagnosis, and machine vision require high focal depth.In a previous publication we indicated that the conceptof superresolution can be used to achieve an arbitrarilyhigh focal depth.
6
Furthermore, in a recent Letter,
7
we reported that an optical system with a double focuscan be used to increase focal depth, provided that the
two axial amplitudes from each focus are axially sepa
rated by a distance equal to or less than Rayleigh'slimit of resolution.Our aim here is to discuss a method for achieving anarbitrarily high focal depth by the use of a zone platethat generates a prespecified number of identical multiple foci. The amplitude distribution along the opti
cal axis of any two adjacent foci of the zone plate are at
Rayleigh's limit of resolution. Our approach can berephrased heuristically with the help of Fig. 1 as follows: We propose using a zone plate that creates aprespecified number of multiple foci that have identical amplitude responses along the optical axis. Ifthese axial amplitude responses are separated by Rayleigh's limit of resolution, their superimposition generates within a prespecified interval a practically continuous axial distribution.In Sec. II we describe the design of the multiple focizone plate. In Sec. III, we discuss the light gatheringpower and the resolution of our method.
The authors are with National Institute of Astrophysics, Optics, &Electronics, Apdo.Postal 216, Puebla 72000 Pue, Mexico.
Received 23 March 1989.00036935/90/07099404$02.00/0.C 1990 Optical Society of America.
II.
Zone Plate Design
Let us consider the complex amplitude p(r;W
20
) atthe image of a point source as produced by a 2Doptical system with radial symmetry and sufferingfrom focus errors:
p(r;W
20
)
=
27r
J p)JO 2rrp)
exp[i2r(p/U)
2
W
2
0
]pdp. (1)
In Eq. (1) r is the radial coordinate in the image plane;W
2
0
is the defocus coefficient measured in wavelengthunits; p(p) is the complex amplitude at the exit pupil;Jo denotes the Bessel function of zero order and of thefirst kind; and p is the radial spatial frequency at theexit pupil whose maximum value is Q.It is convenient to rewrite Eq. (1) for
r
= 0 as follows:
p(r
= ,W
20
)
=
7rU2
exp(irW
20
)q(W
20
), (2)
where
q(W
20
)
= J5(p)
X
expti2rW
20
[(p/Q)
2

0.5]ld[(p/)
2
]
=
J
4(t) rect(r) exp(i2rW
20
t)dD. (3)
In Eq. (3) we define
=
(p/Q)
2

0.5, q(t)
rect()
=
p(p).
(4)
For a clear circular aperture the axial amplitude
response q(W
20
) is
q(W
20
)
= sinc(W
20
), (5)
where sinc(O) = sin(7r0)/(7rO). We propose using a zoneplate over the exit pupil to replicate several times theaxial amplitude response in Eq. (5). The separationbetween any two adjacent axial amplitude responses isRayleigh's limit of resolution, in this case along theoptical axis.Consequently, we aim to obtain with a zone plate thefollowing axial amplitude response:
994
APPLIED OPTICS / Vol. 29, No. 7
/
1 March
1990
= /Xf
N\
EXIT PUPIL
IMAGE PLANE
W
20
Fig. 1. Schematic diagram that shows he superposition of multiple
axial amplitude responses.
I
(n
D
0.8
00
n 0.6
0
cr 0.4
a:J
0.20.0
0.1
Fig. 2. Streb
0
1.0
~
0.9
,
0.8
U
Z 0.7 
t
0.6
U
0.5

z
<
0.4
W 0.3
s
0.2
0.1

2 0.0
0.1

0.2

0.3

0.00.1 0.2 0.3 0.4 0.5
DIMENSIONLESS VARIABLE
Fig. 3. Amplitude transmittance of the equivalent 1D pupil function.
1.0
0.8
w
z
U
z
0I
Q.
1.0 2.0 3.0 4.0 5.0 6.0
DEFOCUS COEFFICIENT W
20
ratio vs defocus of the proposed apodizer for anincreased number of terms.
Mq(W
20
)
=
E
6(W20 m)*sinc(W
2
0), (6)m=M
where the asterisk denotes convolution. The sum in
Eq. (6) reduces to Eq. (5), q(W
20
) = sinc(W
20
) for
M
=
0, while if
M

,
according to the sampling theorem,
8
the sum approaches a uniform background. In Fig. 2,we display the Strehl ratio,
S(
W
20
)
=
(
W
20
)
l
24
q(0)l
2,
when
M
=0,1,2,3,4,5, and 10. As can be appreciatedfrom Fig. 2, the focal depth can be extended arbitrarilyin this fashion. We next determine the apodizer'samplitude profile.From Eqs. (2), (3), and (6) we find that
)
=
2M
+ 1Y~' 1 +
E
cos(2rmt)] rect(r), (7)
and from Eqs. (3) and (7) we obtain
p(p)
=
(2M
+
1)'
r
M
X
1
+
l)m
cos(2IrmIP2/2) circ(p/I2). (8)
m=l
In Figs. 3 and 4 we display the curves of the functions inEqs. (7) and (8), respectively. Note that when M = 10,
the curve in Fig. 3 tends to be a sinc function. One
0.60.40.20.00.20.40.0 0.2 0.4 0.6 08 1.0
RADIAL SPATIAL FREQUENCY:
/Q
Fig. 4. True amplitude transmittance of the 2D circular symmet
ric apodizer.
expects then that if
M
,
then (r)
6 ).
In otherwords, our method is another way of implementing athin annular aperture by properly weighing the amplitude of the pupil aperture trying to keep lateral resolution, as shown in Sec. III.
In Figs. 5(a), (b), and (c), we show as a grey level
picture the irradiance point spread functions for theinfocus plane W
20
= 0, and for the outoffocus planes
W
2
0
=
1.0 and
W
20
=
2.0, obtained when using theproposed zone plate, for an increasing number of foci
forM=
0,M= 1,M= 2,M= 3,M= 4,M=
5, andM=
10.
From Fig. 5, it is clear that when using our proposedzone plate, as the number of foci increase, so does the
depth of focus. However, as we show next, the incre
ment in focal depth is achieved by sacrificing lightgathering power.
Ill. Light Throughput and Resolution
Using the change of variable in Eq. (4), the normalized light throughput can be written as any of the twoexpressions:
1 March 1990 / Vol. 29, No. 7
/
APPLIED OPTICS 995
M=2t
M
=It
~~Yl
1v\
AV
V", Z
'
M= 10
'M=5
I 'M=4
11 M=3
,M= 2
,
'r"
= I
M=O
a)
W
2
0=O
b)W20=
IC
)
W
20
= 2
M=O
M=I
M=2M=3
M=4
M=5
M=10
Fig. 5. Grey level picture of the point spread functions at various
focal planes (columns) and various foci (lines) of the zone plate.
T
=
(2r2)2r I
Lb p)I
pdp
=f
qt)2¢ (9)
By substituting Eq.(7) in Eq. (9) we obtain
T
=
(2M
+
1)2
1+
4 1 cos(2rmt)dt
+4>3
>3
1/2
+4 Y Y
1/2
os(2rmt) cos(2rnt)d]
= (2M + 1)1.
T
I
.0
08
I
50.6
0
0.4
I
0 2
00
M=O
M=IM=2
2 4 6
Fig. 6. Light throughput vs number of foci.
8 M
a)W
2
0=0.0
b)W
2
o=i.o
C)W20
=
2.0
7

M=3
M=4
(10)
This formula is plotted in Fig. 6. As
M
increases, thereduction of light throughput, shown in Fig. 6, is equalto the loss of light throughput in the method reported
in Ref. 6. Note, however, that, as pointed out in Ref. 6,
the resolution for our present method is higher thanthat obtained with a clear aperture of equal lightthroughput. To support this statement, we show inFig. 7 a gray level picture of the optical transfer functions (OTFs) for a variable defocus, associated with
M=5
M=10
Fig. 7. Groy
1vol
picturo of tho OTFa aogociated with tho point
spread functions in Fig. 5.
996 APPLIED OPTICS / Vol. 29, No. 7 / 1 March 1990
I
 
use of the proposed zone plate, for various foci: M = 0,
1, 2, 3, 4, 5, and 10.
From the above results and discussion, we claim thatour method increases focal depth without substantially reducing lateral resolution for a given light throughput.
IV. Conclusions
We describe a zone plate that produces arbitrarilyhigh focal depth when used as an apodizer over anotherwise clear optical exit pupil. The proposed zoneplate creates a prespecified number of foci, all havingidentical axial amplitude responses. Any two adjacent axial amplitude responses are separated axially byRayleigh's limit of resolution. Computer calculatedcurves (Strehl ratio vs defocus, point spread functions,and OTFs) show that as the number of foci increases,so does the depth of focus. This property is achievedby sacrificing light throughput but without substantially decreasing lateral resolution.We are indebted to Alberto Gomez for numericallyevaluating the grey level pictures associated with thepoint spread functions and OTFs for various focuserrors and foci. The financial support of COSNET
704.87 is gratefully acknowledged.
References
1. P. Jaquinot and B. RoizenDossier, "Apodization," in
Progress in
Optics, Vol. 3,
E. Wolf, Ed. (NorthHolland, Amsterdam, 1964),p. 29.2. W. B. Wetherell, "The Calculation of Image Quality," in
Applied
Optics and Optical Engineering, Vol. 3,
R. R. Shannon and J. C.
Wyant, Eds. (Academic, New York, 1980).3. G. Indebetouw and H. X. Bai, "Imaging with Fresnel Zone PupilMasks: Extended Depth of Field," Appl. Opt. 23, 42994302(1984).4. J. OjedaCastaneda, L. R. BerrielValdos, and E. Montes, "Line
Spread Function Relatively Insensitive to Defocus," Opt. Lett. 8,
458460 (1983).5. J. OjedaCastaneda, L. R. BerrielValdos, and E. Montes, "Spatial Filter for Increasing the Depth of Focus," Opt. Lett. 10, 520522 (1985).6. J. OjedaCastaneda and L. R. BerrielValdos, "Arbitrarily High
Focal Depth with Finite Apertures," Opt. Lett. 13, 183185
(1988).7. J. OjedaCastaneda and A. Diaz, "High Focal Depth by Quasibi focus," Appl. Opt. 27, 41634165 (1988).
8. R. Bracewell,
The Fourier Transform and its Applications
(McGrawHill, New York, 1978), p. 189.
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