Download Generalized Jinc functions and their application to focusing and

Qing Cao
Vol. 20, No. 4 / April 2003 / J. Opt. Soc. Am. A
661
Generalized Jinc functions and their application
to focusing and
diffraction of circular apertures
Qing Cao
Optische Nachrichtentechnik, FernUniversität Hagen, Universitätsstrasse 27/PRG, 58084 Hagen, Germany
Received October 7, 2002; revised manuscript received December 2, 2002; accepted December 2, 2002
A family of generalized Jinc functions is defined and analyzed. The zero-order one is just the traditional Jinc
function. In terms of these functions, series-form expressions are presented for the Fresnel diffraction of a
circular aperture illuminated by converging spherical waves or plane waves. The leading term is nothing but
the Airy formula for the Fraunhofer diffraction of circular apertures, and those high-order terms are directly
related to those high-order Jinc functions. The truncation error of the retained terms is also analytically investigated. We show that, for the illumination of a converging spherical wave, the first 19 terms are sufficient
for describing the three-dimensional field distribution in the whole focal region. © 2003 Optical Society of
America
OCIS codes: 050.1940, 000.3870, 220.2560, 050.1960, 050.1220.
1. INTRODUCTION
The diffraction problem of circular apertures illuminated
by converging spherical waves or plane waves is an important topic1,2 in optical science, because this kind of
wave phenomenon is frequently encountered in various
optical systems. It is well-known that the Fraunhofer
diffraction of circular apertures has a Jinc-function
distribution3,4 (i.e., the Airy pattern distribution) with the
radial coordinate. However, this kind of simple field distribution appears only at the geometrical focal plane for
the case of converging spherical-wave illumination. And
for the case of plane-wave illumination, it is also valid
only for those transverse planes that are far from the aperture plane. In all the other regions, especially in the
focal region of a converging spherical wave, the diffracted
field has a rather complicated structure. To investigate
this kind of complicated diffraction problem, people have
developed various numerical and analytical methods. At
present, it is possible to use numerical methods to implement an accurate simulation, but one cannot obtain as
much physical insight as with analytical methods.
Among the analytical methods, the classical Debye
theory1 employs the Lommel functions to describe the
three-dimensional field distribution in the focal region for
high-Fresnel-number focusing systems. However, it is
not applicable to low-Fresnel-number focusing systems.
By use of an appropriate variable transform, Li and Wolf 5
and Li6 extended the Lommel-function description to general paraxial focusing systems with arbitrary Fresnel
numbers. The case of plane-wave illumination was also
investigated in Ref. 6 as a separate treatment.
Recently, Wang et al.7 attempted to derive a simple
closed-form expression for the Fresnel diffraction of circular apertures illuminated by spherical waves or plane
waves. Unfortunately, their result was demonstrated to
be incorrect.8,9 As an alternative, Overfelt and White9
developed the exponential polynomial description for the
1084-7529/2003/040661-07$15.00
same problem. Obviously, their treatment can be applied
to focusing and diffraction of circular apertures illuminated by converging spherical waves or plane waves.
Compared with the Lommel-function description, this
new description eliminated the need to split the computational problem into two different regions.
In another context, Wang et al.10 developed a novel
analytical tool named moment expansion to investigate
the depth of focus. Their analysis is mainly based on the
Fourier transform pairs of the partial derivatives in real
space and the corresponding moments in the spatialfrequency domain. In principle, their expression [see Eq.
(7) of Ref. 10] can be used to analytically describe an arbitrary paraxial focusing system, provided that the farfield distribution at the focal plane is known and has all
the high-order partial derivatives. As a special case, the
diffraction problem in the focal region of circular apertures illuminated by converging spherical waves can also
be treated by this method. Compared with the Lommelfunction and exponential polynomial descriptions, the moment expansion method has a more explicit connection
with the far field. This property can be easily found from
Eq. (7) of Ref. 10, where the leading term is just the far
field. However, their work is not appropriate for a planewave illumination, because, in this case, there is no geometrical focal plane within a finite region at all. This
drawback comes from the inconsistency between the
asymptotic far-field behavior, which is proportional to z ⫺1
for large z in this case, and the moment expansion
method, which actually uses the variable z ⫺ f (because
the coordinate origin used in Ref. 10 is located at the focal
point), where z is the longitudinal distance from the aperture plane and f is the curvature radius of the converging spherical waves. For the illumination of a plane
wave, the variable z ⫺ f is not appropriate because f
⫽ ⬁ in this case. Considering the continuous change of
the physical behaviors with the change of the curvature
© 2003 Optical Society of America
662
J. Opt. Soc. Am. A / Vol. 20, No. 4 / April 2003
Qing Cao
f ⫺1 of the incident spherical waves, one can deduce that it
is also inconvenient to use the moment expansion
method10 to describe the focusing systems with long focal
length (i.e., f ⫺1 is small).
In this paper, we shall modify the moment expansion
method for focusing and diffraction of circular apertures
illuminated by converging spherical waves or plane
waves. This modified treatment employs the suitable
variables z ⫺1 and z ⫺1 ⫺ f ⫺1 (in the concrete employment,
we shall use the corresponding Fresnel number) to replace z ⫺ f. We shall also define and analyze a family of
generalized Jinc functions. In terms of them, our treatment can lead to an elegant series-form expression, which
is valid for arbitrary spherical-wave (including planewave) illumination. The paper is organized as follows.
In Section 2, we define the family of generalized Jinc
functions and outline their main properties. In Section
3, the series-form expression for the diffraction problem of
circular apertures is presented in terms of this family of
functions, and we analyze the truncation error of the retained terms and check the validity. And, in Section 4,
we conclude this paper and discuss some related problems. As we shall show below, compared with other analytical methods, our treatment has the following four advantages: (1) It has a more explicit connection with the
far field and therefore provides an explicit insight showing how the defocused field distributions gradually deviate from the far-field distribution. (2) The case of planewave illumination is automatically included as a special
case. Therefore a separate treatment6 for the case of
plane-wave illumination is no longer needed. (3) Our
treatment allows a simple analytical estimate for the
truncation error of the retained terms. (4) The generalized Jinc functions used in our treatment are one-variable
functions, which are simpler than the two-variable Lommel function1,5,6 and the two-variable exponential
polynomials.9 Because of these advantages, this approach is more suitable for describing the focusing and
diffraction problem of circular apertures.
It is well-known that the Fraunhofer diffraction of circular apertures has the simple Jinc-function distribution3,4
(i.e., the Airy pattern distribution). The Jinc function is
given by
J 1共 u 兲
u
,
(1)
where J 1 (v) is the first-order Bessel function of the first
kind and u is the variable. From the properties of Bessel
functions,11 one knows that the Jinc function can also be
written as the integral form
Jinc共 u 兲 ⫽
1
u2
冕
1
Jincn 共 u 兲 ⫽
u 2n⫹2
(2)
0
where J 0 (u) is the zero-order Bessel function of the first
kind and v is the integral variable. In fact, in the
derivation1 of the Jinc-function distribution of the Fraunhofer diffraction of circular apertures, the form of Eq. (2)
is encountered before that of Eq. (1).
u
v 2n⫹1 J 0 共 v 兲 dv,
(3)
0
n
Jincn 共 u 兲 ⫽
兺
共 ⫺2 兲 m
m⫽0
n!
J m⫹1 共 u 兲
共 n ⫺ m 兲!
u m⫹1
,
(4)
where J m⫹1 (u) is the (m ⫹ 1)th-order Bessel function of
the first kind and 0! ⫽ 1. In particular, the first several
functions are given by [besides Jinc0 (u), given by Eq. (1)]
Jinc1 共 u 兲 ⫽
Jinc2 共 u 兲 ⫽
Jinc3 共 u 兲 ⫽
J 1共 u 兲
u
J 1共 u 兲
u
J 1共 u 兲
u
⫺ 48
⫺2
⫺4
⫺6
J 2共 u 兲
u2
u4
,
J 2共 u 兲
u
u
(5)
⫹8
2
J 2共 u 兲
J 4共 u 兲
J 3共 u 兲
⫹ 24
2
u3
,
(6)
J 3共 u 兲
u3
.
(7)
Let us now investigate some important properties of
this family of functions: (1) The nth-order Jinc function
has n ⫹ 1 subterms, and each subterm has the factor
J m⫹1 (u)/u m⫹1 , where the order m ⫹ 1 of the Bessel function in the numerator is just the power of u in the denominator. (2) The first subterm of each Jinc function is always J 1 (u)/u. This property is more explicitly shown in
Eqs. (5)–(7). (3) From the asymptotic behavior11
冑
2
␲u
冉
cos u ⫺
n␲
2
⫺
␲
4
冊
for large u, one can deduce that, for each high-order Jinc
function, all the other subterms decrease faster than the
first subterm when u becomes large. As a consequence,
we obtain the important property that all the Jinc functions approach Jinc0 (u) for large u, i.e.,
lim Jincn 共 u 兲 ⫽ Jinc0 共 u 兲 ⫽
u→⬁
J 1共 u 兲
u
.
(8)
(4) Putting the relation limu→0 J 0 (v) ⫽ 1 into Eq. (3) and
integrating, one can easily prove that
u
vJ 0 共 v 兲 dv,
冕
where the order n is a nonnegative integer, namely, n
⫽ 0, 1, 2,... . From Eq. (3), one can see that the zeroorder one is just the traditional Jinc function. By use of
the properties of Bessel functions and the mathematical
inductive method, one can prove the following closed-form
expression (see Appendix A):
J n共 u 兲 ⫽
2. GENERALIZED JINC FUNCTIONS
Jinc共 u 兲 ⫽
We now define a family of generalized Jinc functions as
the following forms:
Jincn 共 0 兲 ⫽
1
2共 n ⫹ 1 兲
(9)
when u ⫽ 0. (5) Through a great number of observations, it seems that the principal maximum of each Jinc
function always appears at the point u ⫽ 0. These maximum values, just as shown in Eq. (9), decrease with the
increase of the order n. We also observe that, just as
Qing Cao
Vol. 20, No. 4 / April 2003 / J. Opt. Soc. Am. A
663
where U(R, z) is the diffracted field at the z ⫽ z plane,
U 0 (r) is the incident field at the aperture plane, ␭ is the
wavelength in free space, k ⫽ 2 ␲ /␭ is the wave number
in free space, and j ⫽ 冑⫺1 is the imaginary unit. Note
that in Eq. (11) we have ignored the factor
⫺2j exp( jkz)exp关 jkR2/(2z)兴. Substituting the complex
amplitude distribution U 0 (r) ⫽ exp关⫺jkr2/(2 f )兴 of the
converging spherical wave into Eq. (11) and employing
the normalized coordinates ␳ 1 ⫽ r/a and ␳ ⫽ R/a, one
can obtain
Fig. 1.
Functional curves of Jinc0 (u), Jinc1 (u), and Jinc2 (u).
U共 ␳, z 兲 ⫽ ␲N1
冕
1
0
exp共 j ␲ N 2 ␳ 12 兲 J 0 共 2 ␲ N 1 ␳␳ 1 兲 ␳ 1 d␳ 1 ,
(12)
where N 1 ⫽ a /(␭z), N 2 ⫽ N 1 ⫺ N, and N ⫽ a /(␭f ),
which is the Fresnel number of the aperture illuminated
by a converging spherical wave with a curvature radius f.
We refer to N 1 as the Fresnel number of the aperture itself, because it corresponds to the case of plane-wave illumination. As we show below, the difference N 2 between
N 1 and N plays an important role for describing the defocused field distributions. For clarity, we write N 2 as the
form
2
Fig. 2. Asymptotic behavior of Jincn (u) for large n. For comparison, Jinc10(u) and Jinc30(u) have been amplified 22 and 62
times, respectively.
partly shown in Fig. 1, all the Jinc functions have similar
curves and these curves gradually change with the
change of the order n. In fact, it is this similarity that
stimulates us to call them the generalized Jinc functions.
(6) We prove that, when the order n approaches ⬁, the
normalized Jincn (u) functions approach J 0 (u); concretely
(see Appendix B),
lim Jincn 共 u 兲 ⫽
n→⬁
2共 n ⫹ 1 兲
J 0共 u 兲 .
␭z
冕
0
冉 冊 冉 冊
U 0 共 r 兲 exp jk
␭
z
r2
2z
exp共
j ␲ N 2 ␳ 12 兲
⫽
兺
n⫽0
⫺
1
f
冊
.
jn
n!
共 ␲ N 2 兲 n ␳ 12n .
z
(14)
⬁
兺
U n共 ␳ , z 兲 ,
(15)
共 ␲ N 2 兲 n N 1 Jincn 共 2 ␲ N 1 ␳ 兲 .
(16)
n⫽0
U n共 ␳ , z 兲 ⫽
␲jn
n!
kRr
J0
(13)
Unlike the treatment of Ref. 10, our expansion is not
about the variable z ⫺ f but about ␲ N 2 , which is proportional to the variable z ⫺1 ⫺ f ⫺1 . Substituting this expansion into Eq. (12), one can expand the field distribution U( ␳ , z) as the series form of ␲ N 2 :
U共 ␳, z 兲 ⫽
Consider a circular aperture of radius a, as shown in Fig.
3, illuminated by a converging sperical wave with a curvature radius f. We denote by r, R, and z the radial coordinate at the aperture plane, the radial coordinate at
the observation plane, and the distance between these
two transverse planes, respectively. It is known that,
when the paraxial condition is satisfied, one can use the
Fresnel approximation to describe the diffracted field
(both near field and far field) of a circular aperture.12 In
terms of the above-mentioned coordinate parameters, one
can express the Fresnel diffraction formula for rotationally symmetric fields truncated by a circular aperture as
a
冉
a2 1
From Eq. (13), one can see that N 2 ⫽ 0 when z ⫽ f and
that its absolute value increases when the observation
plane gradually deviates from the focal plane. As a consequence, it can be expected that the field distribution
U( ␳ , z) gradually deviates from the far-field Airy pattern
distribution with the increase of 兩 N 2 兩 . Similarly to the
approach taken in the moment expansion method,10 we
now expand the factor exp( j␲N2␳12) in Eq. (13) as
(10)
3. FOCUSING AND DIFFRACTION OF
CIRCULAR APERTURES
␲
N2 ⫽
⬁
1
This trend is clearly shown in Fig. 2, where the normalized Jinc30(u) is more similar to J 0 (u) than the normalized Jinc10(u).
U 共 R, z 兲 ⫽
2
rdr,
(11)
Fig. 3.
Schematic view of the system configuration.
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J. Opt. Soc. Am. A / Vol. 20, No. 4 / April 2003
Qing Cao
As shown in Eq. (16), the leading term is just the far-field
Fraunhofer diffraction of a circular aperture, and those
high-order terms are directly related to those high-order
Jinc functions. These properties provide an explicit
physical picture showing how the defocused field distribution at the observation plane gradually deviates from the
far-field distribution with the deviation of N 2 from 0. It
is interesting that all the even-order terms are pure real
and all the odd-order terms are pure imaginary. It is
more interesting that the leading term is independent of
N 2 in the expressive form. In other words, this term is
independent of the curvature radius f (or N ⫺1 ) in the expressive form. Perhaps one guesses that this is a mistake, because it is well-known that the far-field distribution of a circular lens appears at the geometrical focal
plane and this far-field distribution has a scale factor f (or
N ⫺1 ). In fact, this is not a mistake. The reason is that
all the high-order terms disappear at the focal plane (i.e.,
N 2 ⫽ 0), and the leading term automatically has the
scale factor f (or N ⫺1 ) at the focal plane because N 1
⫽ N in this case. From Eq. (16), one can find that, as an
important advantage of our analytical treatment, the case
of plane-wave illumination has been automatically included as the special case of N 2 ⫽ N 1 . Obviously, this
advantage partly results from the appropriate choice of
the variables z ⫺1 ⫺ f ⫺1 and z ⫺1 (the corresponding
Fresnel numbers are N 2 and N 1 , respectively). In addition, from Eq. (16), one can see that, just as is known, the
diffracted field has an asymmetric distribution about the
focal plane. This asymmetric distribution leads to the
well-known focal shift phenomenon.13–16 It is worth
mentioning that the first two terms of Eq. (16) for the special case of plane-wave illumination (i.e., N ⫽ 0 and N 2
⫽ N 1 ) have been used for the focusing analysis17 of a
pinhole photon sieve,18 which is a new class of diffractive
optical element for focusing and imaging of soft x rays.
In Ref. 17, the first two terms were called the far-field
term and the quasi-far-field correction term. In fact, it is
the successful employment of the first two terms for the
special case of plane-wave illumination in Ref. 17 that
stimulates us to derive Eq. (16), which consists of infinite
terms.
It is well-known that the on-axis field distribution
U(0, z) can be presented in a closed form. Putting the
relation J 0 (0) ⫽ 1 into Eq. (12), one can derive that
U 共 0, z 兲 ⫽
N1
j2N 2
SM ⫽
冕
⬁
兩 U 共 ␳ , z 兲 ⫺ V M 共 ␳ , z 兲 兩 2 ␳ d␳
0
关 exp共 j ␲ N 2 兲 ⫺ 1 兴 .
共 N2 ⫹ N 兲2
N 22
sin2
␲N2
2
(20)
when the series has gone into the fast convergent region.
We now re-express Eq. (12) as the Fourier–Bessel transform form3,19
冕
⬁
W 共 ␳ 1 兲 J 0 共 2 ␲ N 1 ␳␳ 1 兲 ␳ 1 d␳ 1 , (21)
0
where W( ␳ 1 ) ⫽ exp( j␲N2␳12)/2 for ␳ 1 ⭐ 1 and W( ␳ 1 )
⫽ 0 elsewhere. This means that U( ␳ , z) and W( ␳ 1 ) are
a pair of Fourier–Bessel transforms. Similarly, one can
find that U M ( ␳ , z) and W M ( ␳ 1 ) are another pair of
Fourier–Bessel
transforms,
where
W M( ␳ 1)
⫽ j M ( ␲ N 2 ) M ␳ 12M /(2M!) for ␳ 1 ⭐ 1 and W M ( ␳ 1 ) ⫽ 0 elsewhere. By use of the Parseval theorem,3,19 one can derive the following result:
(18)
where we have written N 1 as the form N 2 ⫹ N. As a
limit case, the on-axis principal maximum appears at
(19)
兩 U 共 ␳ , z 兲 兩 ␳ d␳
U 共 ␳ , z 兲 ⫺ V M共 ␳ , z 兲 ⬇ U M共 ␳ , z 兲
冕
冕
⬁
,
,
2
where M is the number of retained terms and V M ( ␳ , z) is
the sum of the first M terms, i.e., V M ( ␳ , z)
M⫺1
⫽ 兺 n⫽0
U n ( ␳ , z). When the series goes into the fast
convergent region, those higher terms can be ignored
compared with the term U M ( ␳ , z), which is the first of
those discarded terms. Based on this approximation, one
can obtain
(17)
冉 冊
冕
⬁
0
U共 ␳, z 兲 ⫽ 2␲N1
⬁
By use of the expansion exp( j␲N2) ⫽ 兺n⫽0
jn(␲N2)n/n!,
one
can
re-express
U(0, z)
as
U(0, z)
⬁
⫽ 兺 n⫽0
j n ␲ n⫹1 N 1 N 2n / 关 2(n ⫹ 1)! 兴 . As a check of our analytical treatment, one can directly derive the same seriesform expression for U(0, z) by inserting the relations
Jincn (0) ⫽ 1/关 2(n ⫹ 1) 兴 into Eq. (16). From Eq. (17),
one can easily obtain the on-axis intensity distribution
I(0, z):
I 共 0, z 兲 ⫽ 兩 U 共 0, z 兲 兩 2 ⫽
N 2 ⫽ 0 for very large N, because in this case the factor
sin2(␲N2/2)/N 22 changes much faster than the factor (N 2
⫹ N) 2 . It is well-known that this result can be correctly
predicted by the classical Debye theory.1 As another
limit case, the on-axis principal maximum appears at
N 2 ⫽ 1 when N ⫽ 0 (i.e., plane-wave illumination), because I(0, z) ⫽ sin2(␲N2/2) in this case. In all other
cases, the on-axis principal maximum appears in the interval 0 ⬍ N 2 ⬍ 1. In addition, from Eq. (18), one can
deduce that the on-axis intensity decreases from the principal maximum to zero when N 2 ⫽ ⫾2 provided that N
⭓ 2. For this reason, we reasonably refer to the region
corresponding to ⫺2 ⬍ N 2 ⬍ 2 as the focal region.
When N ⭓ 2, this definition has direct meaning. However, when N ⬍ 2, this definition should be modified as
⫺N ⬍ N 2 ⬍ 2, because the on-axis intensity I(0, z) already decreases to zero when N 2 ⫽ ⫺N (note that this
corresponds to z → ⬁).
In the practical employment of Eqs. (15) and (16), one
needs to truncate the series. Therefore it is desirable to
analytically provide the truncation error of the retained
terms. It is appropriate to define the normalized square
truncated error S M of the retained terms as
SM ⬇
兩 W M 共 ␳ 1 兲 兩 2 ␳ 1 d␳ 1
0
⬁
0
⫽
兩 W 共 ␳ 1 兲 兩 ␳ 1 d␳ 1
2
共 ␲ N 2 兲 2M
共 2M ⫹ 1 兲共 M! 兲 2
. (22)
Qing Cao
Fig. 4. Relation between the number M 0 of the terms needed
and N 2 .
Vol. 20, No. 4 / April 2003 / J. Opt. Soc. Am. A
665
essary when 兩 N 2 兩 ⭐ 2. This actually covers the whole focal region, where the on-axis intensity decreases from the
principal maximum to zero. Therefore the first 19 terms
are sufficient for describing the three-dimensional field
distribution in the whole focal region for the illumination
of a converging spherical wave. To understand this
statement better, we draw the transverse field distribution U( ␳ , z) at the plane corresponding to N 2 ⫽ 2 (i.e.,
the boundary of the focal region) for N ⫽ 5. In the concrete computation, N 1 has been expressed as N 2 ⫹ N.
Therefore, for a given N value, U( ␳ , z) depends only on ␳
and N 2 in the expressive form. From Fig. 5, one can see
that the field distribution determined by the first 19
terms are completely indistinguishable from that calculated from the exact numerical integration in Eq. (12).
In particular, the zero field value at the point ␳ ⫽ 0 is accurately described by the first 19 terms of our analytical
expression.
4. CONCLUSIONS AND DISCUSSION
Fig. 5. Transverse field distribution at the plane corresponding
to N 2 ⫽ 2: (a) real part, (b) imaginary part. The Fresnel number is chosen such that N ⫽ 5. The solid lines are the exact results, and the stars are the analytical results of the first 19
terms.
Equation (22) explicitly shows how the truncation error
S M decreases with the increase of the number of retained
terms for a given 兩 N 2 兩 value (note that S M is an even function of N 2 ). It is interesting that S M is independent of
N 1 . This is due to the fact that we expand only the factor exp( j␲N2 ␳12) related to N 2 in Eq. (12) and keep the factor J 0 (2 ␲ N 1 ␳␳ 1 ) related to N 1 in Eq. (12) unchanged.
As is pointed out above, this treatment can automatically
include the case of plane-wave illumination. We have observed that the approximate field distribution V M ( ␳ , z) of
the first M terms and the exact field distribution calculated by direct numerical integration in Eq. (12) are completely indistinguishable when S M ⭐ 1 ⫻ 10⫺5 . If one
chooses S M ⫽ 1 ⫻ 10⫺5 as the critical value, then, from
Eq. (22), one can calculate the number M 0 of the terms
that should be retained, where M 0 corresponds to the case
S M 0 ⭐ 1 ⫻ 10⫺5 for the first time. Exactly speaking,
S M 0 ⭐ 1 ⫻ 10⫺5 but S M 0 ⫺1 ⬎ 1 ⫻ 10⫺5 . The relation of
M 0 to 兩 N 2 兩 is shown in Fig. 4 for the range 兩 N 2 兩 ⭐ 2.
From Fig. 4, one can see that fewer than 20 terms are nec-
We have defined a family of generalized Jinc functions
and analyzed their main properties. In terms of these
functions and the appropriate variables N 2 and N 1 , an elegant series-form expression has been presented for focusing and diffraction of a circular aperture illuminated
by converging spherical waves or plane waves. The truncation error of the retained terms has also been analytically investigated. In particular, we have shown that the
first 19 terms are sufficient for describing the threedimensional field distribution in the whole focal region
(we refer to the region corresponding to ⫺2 ⬍ N 2 ⬍ 2)
for the illumination of a converging spherical wave.
Compared with the Lommel-function1,5,6 and exponential polynomial9 descriptions, our method has the following advantages: (1) Our treatment has a more explicit
connection with the far field. As shown in Eq. (16), the
zero-order term is just the far-field Fraunhofer diffraction
of a circular aperture, and those high-order terms are directly related to those high-order Jinc functions. These
properties provide an explicit physical insight showing
how the defocused field distributions gradually deviate
from the far-field distribution. This advantage partly results from using a treatment similar to that in the moment expansion method.10 (2) The case of plane-wave illumination is automatically included as the special case
of N ⫽ 0. As a consequence, one does not need to change
the coordinate variable for the case of plane-wave illumination any longer. This advantage partly results from
the suitable employment of the variables z ⫺1 ⫺ f ⫺1 and
z ⫺1 instead of z ⫺ f. (3) As we showed above, our treatment can lead to an analytical expression for the truncation error of the retained terms. This property provides
a great convenience in practical employment. (4) The
generalized Jinc functions used in our treatment are onevariable functions, but the Lommel function and the exponential polynomials are two-variable functions. In addition, it is worth mentioning that, like the exponential
polynomial description,9 our analytical treatment does
not need to split the computation problem into two different regions.
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J. Opt. Soc. Am. A / Vol. 20, No. 4 / April 2003
Qing Cao
The drawback of our treatment is that the series converges slowly when the absolute value of N 2 becomes
large. For this case, the asymptotic solution presented
by Southwell20 is suggested. This asymptotic solution
and our treatment are complementary with each other.
The former is suitable for large 兩 N 2 兩 values, and the latter
is suitable for small 兩 N 2 兩 values.
It should be mentioned that, very recently, Janssen21
and Braat et al.22 developed the extended Nijboer–
Zernike approach for the computation of optical pointspread functions. We note that some similar mathematical problems on those integrals that are related to Bessel
functions appeared in their papers, too. By comparing
Eq. (3) here with Eq. (13) of Ref. 21, one can find that the
generalized Jinc functions investigated here are related
to the functions T n0 with f ⫽ 0 and even integers n in
Ref. 21. However, the explicit closed-form expressions
are given in different forms. In other words, Eq. (4) here
is different from the expression T n0 given by Eq. (14) of
Ref. 21 for the case of f ⫽ 0 and even integers n. The
former is much more compact than the latter, because, for
this case, we use relatively simpler mathematics. Of
course, the use of more complicated mathematics in Ref.
21 is mainly due to the more general subject. As we
showed in Section 2, the expression given by Eq. (4) here
is particularly suitable for investigating the important
properties of the generalized Jinc functions, such as the
asymptotic behaviors for large variable u and the similar
curves among this family of functions. One may also
note that Eq. (16) here is similar to Eq. (B4) of Ref. 22.
They become more similar if one substitutes the relation
⬁
exp( jf ) ⫽ 兺n⫽0
( jf )n/n! into Eq. (B4) of Ref. 22 and reorganizes those terms according to the power of f, where f is a
parameter used in Refs. 21 and 22 (note that this parameter is different from the focal length f used in the present
paper). In fact, after this procedure, one can find the
generalized Jinc functions in the expression. This consistency further checks the results of Eq. (16) in this paper and Eq. (B4) in Ref. 22 against each other. Obviously, this check is helpful for both equations because
they are both derived from complicated mathematics (in
particular, the latter is derived from a more complicated
mathematical background). However, it should be emphasized that the contents of the present paper are completely different from those of Refs. 21 and 22 except for
the two similar mathematical problems mentioned above.
冕
u
v 2n⫹1 J 0 共 v 兲 dv ⫽ u 2n⫹1 J 1 共 u 兲 ⫺ 2n
0
For readability, we first derive the equality
(A2)
Further employing the relation J 1 (v)dv ⫽ ⫺d关 J 0 (v) 兴
and integration by parts in Eq. (A2), one can obtain Eq.
(A1).
We use the mathematical inductive method to prove
Eq. (4):
11
1. In the first stage, one can easily prove that
Jinc0 (u) ⫽ J 1 (u)/u by use of the relation11 兰 0u vJ 0 (v)dv
⫽ uJ 1 (u). Obviously, as the special case of n ⫽ 0,
Jinc0 (u) satisfies Eq. (4).
2. In the second stage, we assume that Eq. (4) holds
for the (n ⫺ 1)th-order Jinc function. That is, we assume that
n⫺1
Jincn⫺1 共 u 兲 ⫽
兺
共 ⫺2 兲 m
m⫽0
共 n ⫺ 1 兲!
J m⫹1 共 u 兲
共 n ⫺ 1 ⫺ m 兲!
u m⫹1
where n ⫺ 1 ⭓ 0.
3. In the third stage, we prove that Eq. (4) also holds
for the nth-order Jinc function if it holds for the (n
⫺ 1)th-order Jinc function. Substituting Eq. (A1) into
Eq. (3), one can obtain
Jincn 共 u 兲 ⫽
J 1共 u 兲
u
⫹
2n
u2
J 0共 u 兲 ⫹ L 共 u 兲 ,
4n 2
L 共 u 兲 ⫽ ⫺ 2 Jincn⫺1 共 u 兲 .
u
(A4)
(A5)
Putting the assumption of Eq. (A3) into Eq. (A5), one can
obtain
n⫺1
L共u兲 ⫽
2n共⫺2兲m⫹1n! Jm⫹1共u兲
兺 共n ⫺ m ⫺ 1兲!
um⫹3
m⫽0
.
(A6)
Substituting
the
relation11
2nJ n (u)/u ⫽ J n⫹1 (u)
⫹ J n⫺1 (u) into the last subterm (i.e., the subterm corresponding to m ⫽ n ⫺ 1) of Eq. (A6), one can obtain
Jn⫹1共u兲
Jn⫺1共u兲
L共u兲 ⫽ 共⫺2兲nn! n⫹1 ⫹ 共⫺2兲nn! n⫹1
u
u
n⫺2
⫹
n
v 2n⫹1 J 0 共 v 兲 dv ⫽ u 2n⫹1 J 1 共 u 兲 ⫹ 2nu 2n J 0 共 u 兲
L共u兲 ⫽
0
兺
m⫽n⫺1
u
v 2n⫺1 J 0 共 v 兲 dv,
,
(A3)
u
冕
v 2n J 1 共 v 兲 dv.
2n共⫺2兲m⫹1n! Jm⫹1共u兲
兺 共n ⫺ m ⫺ 1兲!
um⫹3
.
(A7)
After combining the second term 关 (⫺2) n n!J n⫺1 (u)/u n⫹1 兴
and the last subterm (corresponding now to m ⫽ n ⫺ 2)
of the summed expression on the right-hand side of Eq.
(A7) and using the relation11 2(n ⫺ 1)J n⫺1 (u)/u
⫽ J n (u) ⫹ J n⫺2 (u), one can further obtain
APPENDIX A
⫺ 4n 2
u
0
m⫽0
冕
冕
(A1)
0
共⫺2兲mn! Jm⫹1共u兲
共n ⫺ m兲! u
n⫺3
⫹
共⫺2兲n⫺1n!
Jn⫺2共u兲
关n ⫺ 共n ⫺ 1兲兴!
un
2n共⫺2兲m⫹1n! Jm⫹1共u兲
兺 共n ⫺ m ⫺ 1兲!
m⫽0
though it can be found elsewhere. By use of the
relation11 vJ 0 (v)dv ⫽ d关 vJ 1 (v) 兴 and integration by
parts, one can obtain
m⫹1
⫹
共u兲m⫹3
,
(A8)
which includes two summed expressions and one isolated
term. We repeat this process again and again. At each
step, we first combine the isolated term, which can be
Qing Cao
Vol. 20, No. 4 / April 2003 / J. Opt. Soc. Am. A
written
in
the
form
(⫺2) i⫹1 n!J i (u)/ 兵 关 n ⫺ (i
⫹ 1) 兴 !u i⫹2 其 , and the last subterm of the latter summed
expression, which can be written in the form
2n(⫺2) i n!J i (u)/ 关 (n ⫺ i)!u i⫹2 兴 , where i is an integer that
starts from i ⫽ n ⫺ 2 for the first step and ends at
i ⫽ 1 for the last step. By use of the relation11
2iJ i (u)/u ⫽ J i⫹1 (u) ⫹ J i⫺1 (u), one can prove that the
sum of this combination is
共 ⫺2 兲 i n! J i⫹1 共 u 兲
共 n ⫺ i 兲!
⫹
u i⫹1
共 ⫺2 兲 i n! J i⫺1 共 u 兲
共 n ⫺ i 兲!
u i⫹1
.
n
L共 u 兲 ⫽
兺
m⫽1
共 ⫺2 兲 m n! J m⫹1 共 u 兲
共 n ⫺ m 兲!
u m⫹1
⫺
2n
u2
The author may be reached by e-mail at qing.cao
@fernuni-hagen.de.
REFERENCES AND NOTES
1.
2.
3.
We then put the first part into the former summed expression and let the second part alone (i.e., a new isolated
term). After each step, the former summed expression
increases by one subterm, and the latter summed expression decreases by one subterm. After n ⫺ 2 steps [starting from Eq. (A8)], one can finally obtain
4.
5.
6.
7.
J 0共 u 兲 .
(A9)
8.
Note that the second summed expression has disappeared
completely. It is worth mentioning that Eqs. (A7) and
(A8) already reach Eq. (A9), respectively, when n
⫽ 1 and n ⫽ 2. Substituting Eq. (A9) into Eq. (A4), one
can immediately obtain Eq. (4). This finishes the demonstration process based on the mathematical inductive
method.
9.
10.
11.
12.
APPENDIX B
13.
Changing the integral variable from v to v 1 ⫽ v/u, one
can rewrite Eq. (3) as
Jincn 共 u 兲 ⫽
冕
1
0
v 12n⫹1 J 0 共 uv 1 兲 dv 1 .
(B1)
Jincn 共 u 兲 ⫽
1
2共 n ⫹ 1 兲
冕
⬁
⫺⬁
g 共 v 1 兲 J 0 共 uv 1 兲 dv 1 ,
17.
for 0 ⭐ v 1 ⭐ 1 and g(v 1 )
where g(v 1 ) ⫽ 2(n ⫹
⫽ 0 elsewhere. Obviously, g(v 1 ) → ␦ (v 1 ⫺ 1) when n
→ ⬁, where ␦ (•) expresses the Dirac ␦ function. Substituting this relation into Eq. (B2), one can obtain
n→⬁
1
2共 n ⫹ 1 兲
冕
⬁
⫺⬁
15.
(B2)
1)v 12n⫹1
lim Jincn 共 u 兲 ⫽
14.
16.
Equation (B1) can be further expressed as
␦ 共 v 1 ⫺ 1 兲 J 0 共 uv 1 兲 dv 1 ,
18.
19.
20.
(B3)
which can directly lead to Eq. (10).
21.
ACKNOWLEDGMENT
22.
The author is indebted to Jürgen Jahns for many fruitful
discussions.
667
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