Full Terms & Conditions of access and use can be found athttp://www.tandfonline.com/action/journalInformation?journalCode=tmop19

Download by: [National Chiao Tung University ] Date: 12 November 2015, At: 07:20

Optica Acta: International Journal of Optics

ISSN: 0030-3909 (Print) (Online) Journal homepage: http://www.tandfonline.com/loi/tmop19

A New Series Representation of the FresnelDiffraction Field of Axially Symmetrical Filters

M. Novotn

To cite this article: M. Novotn (1977) A New Series Representation of the Fresnel DiffractionField of Axially Symmetrical Filters, Optica Acta: International Journal of Optics, 24:5, 551-565,DOI: 10.1080/713819599

To link to this article: http://dx.doi.org/10.1080/713819599

Published online: 16 Nov 2010.

Submit your article to this journal

Article views: 16

View related articles

Citing articles: 3 View citing articles

OPTICA ACTA, 1977, VOL . 24, NO. 5, 551-565

A new series representation of the Fresnel diffraction field

of axially symmetrical filters

M. NOVOTN~'

Institute of Instrument Technology, Czechoslovak Academy of

Sciences, Kralovopolska 147, 612 64 Brno 12, Czechoslovakia

(Received 19 March 1976)

Abstract. A new series representation of the diffraction field G(u, v) due to

the axially symmetrical filters is derived . The coefficients of the series are

formed by certain scalar products of the pupil function and the Bernoulli poly-

nomials. Unlike the previous representations of the Fresnel diffraction field

containing the Lommel functions of two variables, this representation operates

only with special functions of one variable . Applying the theory to the focal

diffraction patterns G(4ITML, v), Linteger, due to the filters with transmissivities

periodic with the squared distance from the axis, the previous result [3] directly

follows: the focal patterns of any filter approach the Airy pattern if the number

of the periods M increases . The series coefficients for the Fresnel diffraction

fields of the ideal lens and also of the polynomial filter transmissivities are derived .

The results are documented for the diffraction fields of the Fresnel and Gabor

zone plates .

1. Introduction

It is known [1, 2] that the diffraction fields of the Soret zone plates in the focal

planes u=4irMl, 1=0, 1, 3, . . . take the form of the Airy pattern as soon as

the number of the transparent zones M is sufficiently large . This result has been

extended in [3] to all axially symmetrical filters simultaneously with the term of

focus : the diffraction field G(u, v) of a filter g(t) near its foci approaches that of an

ideal lens, providing the filter transmissivityg(t) is periodic in the variable t= p 2 la 2,

p being the distance from the axis and a the radius of the circular pupil . The role

of M is taken over by the number of periods .

In contrast to the general character of considerations in the previous paper [3]

inspired by the Arsenault-Boivin representation [4], we shall study the behaviour

of the focal diffraction patterns in more detail . The representation from [4] can be

transformed to the series expansion in which the number of periods M directly

control the rapidity of the series convergence (3) . This property of the new

series representation facilitates considerably the discussion and calculation of the

diffraction patterns due to the filter transmissivities with different number of

periods M-cf. 7 .

The series comprises newly established coefficients yk(u) and functionsf k(v) .

The coefficients yk(u) (so called diffraction coefficients) are determined by the

course of the pupil function within one period . They can be calculated either as

a sum of the Fourier coefficients of the pupil function g(t) or in the form of a

scalar product of g(t) and the Bernoulli polynomials (4) . The mathematical

flexibility of the Bernoulli polynomials permits one to obtain the analytical form of

the focal diffraction coefficients yk(4irML), L being an integer, associated with the

focal diffraction patterns due to a number of simple filter transmissivities (cf. 7) .

Dow

nloa

ded

by [N

ation

al Ch

iao T

ung U

nivers

ity ]

at 07

:20 12

Nov

embe

r 201

5

552

M. Novotny

The diffraction coefficients associated with the general case of Fresnel diffraction

y k(u) are derived in 6 for the ideal lens, the Gabor zone plate or to the poly-

nomial filter transmissivities .

The introduced functions / k(V) may be generated with the help of their

recurrence properties or may be established in a simple way using the Bessel

functions ( 5) . Their behaviour for small values of v somewhat resembles that

of the Bessel functions . However, they tend to zero more rapidly for v-->0 and

k-oo so that a rapid convergence of the series representation near the axis is

guaranteed .

In comparison with the former representations of the axially symmetrical

fields, two points should be emphasized . The theory presented is based on the

Fourier series expansion of the pupil function . It can then be applied in a proper

way, not only to the filters with continuous transmissivities but even to the filters

characterized by discontinuous pupil functions, such as the Fresnel plate . In

the presence of discontinuities most of the known methods fail because they use

the Taylor or Maclaurin expansions of the pupil functions ([l], p. 345) . More-

over, the periodicity of the Fourier series enables us to facilitate the calculations of

the diffraction patterns due to the periodic filter transmissivities while in appli-

cations of the former approaches the great number of periods of the pupil function

causes numerical difficulties . This was also the reason why a special theory of

the Fresnel plate was elaborated ([1], p. 407) .

Unlike the previous representations of the Fresnel diffraction fields containing

functions of two variables ([1], p . 354), our representation always operates with

the functions/k(V) of one variable . The other variable u is involved just in the

diffraction coefficients y k(u) . Such separation of variables is not only convenient

for numerical calculations but it enables us to solve in principle inverse diffraction

problems in terms of the diffraction coefficients . Thus, the pupil function may

be established from the set of the diffraction ceofficients {yk(u)}, k=0, 1, . . .

associated with any Fresnel diffraction pattern u = const-cf . [8] .

After reviewing the necessary results from the scalar diffraction theory in the

next section we shall derive in 3 the new series representation.

2. The representations of the diffraction integral

In the frame of the scalar theory of the optical diffraction, the diffraction field

G(u, v) of an axially symmetrical filter g(t) may be expressed in the form [1, 3]

2

i

ZtG(u, v) =2 f g(t)exp ()Jo(vth12)dt

2(1)

0

with the symbols u, v, t, S2 introduced in [3], p. 219 . Further, we set iQ/2=1

for each studied plane u = const because it represents here an unsubstantial

proportionality factor .

We shall investigate the filters that display the periodic pupil functions g(t)

having the period 1/M, M natural . The case of non-periodic pupil functions is

also included by setting M= 1 when the period contains the whole pupil area .

Such periodic functions may be expanded in the orthogonal set {exp (i27rMmt)}

m=0, 1, 2, . . . having the same basic period 1/M-cf . [3] . Hence,

g(t) = Ig, exp (i27rMmt),

(2)

M

Dow

nloa

ded

by [N

ation

al Ch

iao T

ung U

nivers

ity ]

at 07

:20 12

Nov

embe

r 201

5

Axially symmetrical Fresnel diffraction

553

1/M

1

gm= M f g(t) exp (- i27rMmt) dt=f g(t) exp (- i2rrMmt) dt .

0

0

(3)

The last relation can be rewritten with the help of substitution t'= Mt as follows

1

gm= f g(t'/M)exp(-i2rmt')dt' .

(4)

0

Therefore, the Fourier coefficients g n are fully determined by the form of the

pupil function within one period interval and do not depend on the number of

periods M. According to (1) and (3), they represent the values of the diffraction

field in the axial points [-4irMm, 0], i .e .

gm =G(-4rrMm, 0) . (5)

These points, wheregm

:A 0, may be considered as the foci of the filter [3] . The

total energy transmitted through the filter g(t),

1

Eg=

I

Ig(t)I2dt

(6)

0

is entirely distributed among them, in agreement with the completeness relation

(m=0, 1, 2, . . .)

Eg = ,I

Igml2=~ IG(-4irMm, 0)1 2 .

m

m

(7 )

By means of the Fourier series (2) the diffraction integral (1) may be expressed

as the sum [3, 4]

G(u, v)= I

g.0(u.,v),

(8 )

n

where un

1

O(u, v) = f exp (iut/2)J0(vt1/2 ) dt .

0

(9)

The sum (8) may be then interpreted as the superposition of the diffraction fields

due to the ideal lenses exp (-i27rMnt) submitted to a certain uniform filtration

given by the Fourier coefficients gn . The foci of these lenses [4IrMn, 0] coincide

with those of the filter .

We shall need the Lommel representation of the diffraction field O(u, v) .

Considering the relation (9) to be the integral representation of the Lommel

functions of the first kind ([5], p . 309), we find

O(u, v)= uexp (2)[U2 (u, v)+iU 1 (u, v)] .

(10)

Dow

nloa

ded

by [N

ation

al Ch

iao T

ung U

nivers

ity ]

at 07

:20 12

Nov

embe

r 201

5

554

M. Novotny

Expressing the Lommel functions of the first kind in terms of those of the second

kind ([5], p. 310), we arrive at another relation :

2

2

O(u,v) -u exp

(2)1

V0 (u,v)+iV 1(u,v)- exp

(12u v )] .

(11)

The Neumann expansion of the Lommel functions ([5], p . 309 or [6], p. 537)

2k

Uj(u, v)-

kG

(__ 1)k w)+jJ2k+j(v),

(12)

m (-1),(-V)

J 21+j( v )

1=0

u

yield for special values of j, u, v some simple results, in particular, the Airy diffrac-

tion pattern in the focal plane O(0, v) = 2J1(v)/v, and the axial distribution

O(u, 0) = exp (iu/4) sin (u/4)/(u/4) = exp (iu/4) sine (u/4) .

3. The representation of the diffraction field in the focal planes

We shall adapt the expression (8) for the focal planes defined by u=4irML,

L = 0, 1, 2, . . . and g_1=A 0

. The general case of the Fresnel diffraction will

be dealt with at the end of the next section . Let us rewrite the relation (8) in

the form (n = 0, 1, 2, . . .)

G(4irML, v)=g-LO(0, v)+ I gn O(4TrM(L+n), v) .

(14)

no-L

The sum gives a deviation of the total diffraction field from the Airy pattern repre-

sented by the separated term g_LO(0, v) . It has been found in [3] that the dif-

fraction field due to a periodic transmissivity approaches the Airy pattern if the

number of periods M increase . Thus for M->co the series in (14) brings no

contribution to the total diffraction field. It will then be reasonable to rearrange

this series with respect to the increasing powers of 11M.

First, let us replace the Lommel functions of the second kind in (11)

by their Neumann expansions (13). Inserting the expression obtained instead of

O(47rM(L+n), v), the series at the right-hand side of (14) takes the form

_ 2

ao

k

i2,rMn~L

n+L~-

exp[87rM(L+n)]+k=0

(-Z)k[47rM(L+n)] J

k (v)} .

Further, the exponential function will be expanded into its Taylor series, i .e .

G~~`

k 1

gn

v k

1 nL k =O

[27r(n+L)]k+1(C2)

kl

(V)k +

2

Jk(v)] .

(15)

Let us now introduce the notations (k=0, 1, 2, . . .)

YO(4rrML) =

g-L ,

Yk+1(47rML) = -ik+

1 gn

n-L

[2ir(n+L)]k+1

Dow

nloa

ded

by [N

ation

al Ch

iao T

ung U

nivers

ity ]

at 07

:20 12

Nov

embe

r 201

5

and

Axially symmetrical Fresnel diffraction

555

/o(v) = A1(v) = 2J1(v)/v,

d k+1(v)=(2)kCk~ (V)

2 k-J

k(v)1 .

Exchanging the order of summing in the double sum (15) and involving the

term g_LO(0, v) in the series we finally get from the original relation (14)

Go

G(47TML, v)= I (-M)-k yk(4rrML)fk(v) .

(18)

k=0

We obtained the representation of the focal diffraction field in the form of the

series of terms arranged according to the increasing powers of 1IM. The period

of the pupil function 11M actually enters the expression just as (-M)-k; the

coefficients y k (u) do not depend on M, in agreement with their definition (16)

and the relation (4). Consequently, the diffraction fields belonging to the filters

mutually differing just by the number of periods M can be evaluated by means

of the representation (18) always with the same coefficients yk(u) . This fact

facilitates considerably the analysis of the focal diffraction patterns of such filters .

Since the coefficientsYk

determine the form of the diffraction pattern (together

with the number of periods M), we shall call them the diffraction coefficients .

Before employing and discussing the revealed representation (18), we shall

briefly deal with the diffraction coefficients y k(u) and the functions /k(v) alone .

4. The diffraction coefficients yk(u)

We shall derive an integral representation of the diffraction coefficients y k (0) .

This representation can be readily extended to the case of the Fresnel diffraction

U 0 .

The definition (16) may be rewritten in the form of the completeness relation

([7], p. 52)

1

Yk(o)-

gn[-(-i2lrn)-k]-

gn qn(k) fg(t)Qk(t)dt=(g,Qk)I (19)

n#0

n

0

where the introduced functio ns Qk(t), k=1, 2, . . ., are defined by their complex

conjugate Fourier coefficients qn(k) . Hence, (m=0, 1, 2, . . .)

Note that the same definition of the scalar product (19) is appropriate for periodic

pupil functions, too, if we transform them in such a way as in (4) . The form of

Qk(t)= I qm(k)exp (i2amt),

M

qm(k) = - (i2am)-k, m 0 0

(20)

qo(k)=0.

(21)

Dow

nloa

ded

by [N

ation

al Ch

iao T

ung U

nivers

ity ]

at 07

:20 12

Nov

embe

r 201

5

556 M. Novotny

the first function Qo(t) follows directly from the relation (16) .

from (19) we have to set Qo(t)=1 and therefore

qo(0)= 1 ; q,,,(0)=0, m=1, 2, . . . .

To get yo(

0) =go

(22)

A recurrent process can be used to derive the other functions Q k(t), k=1, 2, . . . .

Integrating the relation (20) we get

fQk(t) dt=

I

qm(k)

[exp (i27rmt) -1 ] .

0

m#o i27rm

Further integration yields

I dx f Qk(t)dt=

qm(k)

0

0

m#0-i2irm

(23)

Inserting this expression into the previous one and regarding the recurrent property

of the Fourier coefficinets q,,,(k+1) .=qm(k)/(i2irm)-cf. (21), we find

t

1

v

Qk+l(t)= f Qk(x) dx - f dy f Q k(x) dx.

(24)

0

0

0

Thus, each function Qk+l(t) can be derived from the previous one, Q k (t) . We

have found this recurrence for k = 1, 2, . . . and we can so generate all functions

Qk(t) by means of (24) starting with Qo(t)=1 . The Fourier coefficients of the

functions Q k (t) formed in this way are those prescribed by (20), as we can check

additionally. The first functions Qk (t) are of the form (figure 1)

Evidently, the function Qk(t) is a polynomial of the kth degree . These poly-

nomials may be easily distinguished to be exactly the Bernoulli ones except for a

different way of normalization ([5], p. 20) .

The polynomials Qk (t) have a simple asymptotic behaviour as follows from

their Fourier series (20) . From (20) and (21) we get subsequently for k even

(k=2r, r=1, 2, . . .)

cc

and odd (k=2r+1, r=0, 1, 2, . . .)

cc

(2-) 2r+1

Q2r+1(t) _ ( -

1)r+12I m-2r-1

sin (2irmt) .

Q0(t) =1 ,

Q1(t) = t -1 /2,

Q2(t) = t 2/2 - t/2 + 1/12,

Q3(t) = t3/6 - t

2

/4 + t/12 .

(2-)2rQ2,#) _ (-

1)r+

1

2 I m-2'' COS (2lrmt)

(26)

M=1

m=1

(25)

(27)

Dow

nloa

ded

by [N

ation

al Ch

iao T

ung U

nivers

ity ]

at 07

:20 12

Nov

embe

r 201

5

(21r)r Q r(t)

Axially symmetrical Fresnel diffraction

M

V,rr-, -~W, ME0WNIAIL

MME FIAA LM10VA&, I

N4

PAIN. WIN 7~~ MIAL

10

557

0

0.25

0.5

0.75

t --- 1

Figure 1 . The polynomials Qr(t) normed by the factor (2 a)r. With increasing r they

approach the harmonic functions (28) . They are connected with the Bernoulli

polynomials Br (x) by the relation B r(2t-1)=r! Q

These series expansions yield for r large enough

(27r)rQ r(t)-~ - 2 cos 27r(t - r/4).

(28)

From figure 1 we can appreciate how fast the polynomials Qk (t) converge to their

asymptotic expansions. Practically, for k,> 4 the polynomials Qk (t) coincide with

the function (28) .

It remains to extend the derived integral formula (19) to the case of the Fresnel

diffraction, i .e. U00 . Since the diffraction coefficients are defined as complex

quantities, nothing prevents us in connecting the phase factor exp (iut/2) appearing

in the diffraction integral (1) with the pupil function g(t) . For M= 1, we thus

have

1

yk(u) =fg(t) exp (iut/2)Qk(t) dt.

(29)

0

Care should be taken if the pupil function g(t) is periodic with the period 1/M,

M > 1 . Then transforming g(t) in the same way as in (4), we arrive at a compre-

hensive form (M= 1, 2. . . . )

i

lut

Yk(u)

=fg

CM)

exp(2M) Qk(t) dt

.

(30)

Hence, in the general case, the Fresnel diffraction field has the series representation

M

G(u, v) = I (-

M)-k Yk(u)/k(v)

(31)

k=0

Dow

nloa

ded

by [N

ation

al Ch

iao T

ung U

nivers

ity ]

at 07

:20 12

Nov

embe

r 201

5

558

M. Novotny

which enables us to calculate each Fresnel diffraction pattern u = const with the

fixed coefficients yk(u) .

According to (30), only the values of the pupil function g(t) from its first

period interval (0, 1/M> are needed for evaluation of yk(u) . Thus, no numerical

difficulties arise for M large . On the contrary, the convergence of the series (31)

rapidly improves with M increasing . For a number of cases of the pupil functions

the integration (30) is possible to perform using the following properties of the

Bernoulli polynomials .

(i) They fulfil the relation following from (24)

ddt(t

) =Qk_1(t); k=1, 2, . . . .

(32)

(ii) The polynomials Q k(t) always conserve the symmetry of their asymptotic

expansion (28) which may be seen from their generation formula (24) .

(iii) Practically, fork >, 4, we may use the asymptotic expansion (28) instead of

Qk (t) in (30) . This approximation leads to the asymptotic expression for the

focal diffraction coefficients

-Zk

yk(4ML)

-

(27r)k [g-L 1 (-1)kg-L-1],

k>,4,

following as well directly from (16) .

5 . The functions/ k(v)

The definition (17) prescribes for v = 0 the zero values of all functions/ k(v)

except for the first one/0 (v) . In this regard the functions fk (v) resemble the

Bessel functions Jk(v) . There are, however, some important differences .

In agreement with (17) the functions A 2 (v), /3(v), . . . are not limited for

v--oo in contrast to the Bessel functions-cf . figure 2 . Thus, we cannot

evaluate the diffraction field G(u, v) for v-* oo by means of the series representation

(18) because an infinite number of terms must then be summed . On the

contrary, in the vicinity of the optical axis the behaviour of the functions fk(V)

guarantees a rapid convergence of the series representation (18) . With v-*0

the functions/ k(v) tend to zero in an analogous way as the Bessel functions : they

tend to zero the sooner the larger becomes index k . However, the functions/ k(v)

converge to zero with v--->0 and k-*oo much faster than the Bessel functions

do . The first term of the oscillating power series

Jk(v)=

1o(-l)-m! (m

k)

(V\ 2m k

/

determines the behaviour of the Bessel functions for v small ([6], p. 17), i .e .

k

Jk(v)

1ir)r r

(v)

(2

1.5

0.5

0

Axially symmetrical Fresnel diffraction

0 5 10 V --- 15

559

Figure 2 . The functions/r (v) . They are nonmed by the factor (21r)_r which is reciprocal

to that used for the polynomials Qr (t) in figure 1 .

with v-->O has to be characterized by a higher power of v . In fact, with the help

of (17) and (33) we can obtain the power series for the functions / k(v)

/k(v) = mo(-1)"t (m

1) (m k)1

(2)2(m k)

(35)

The term with the lowest power of v/2 is again dominant for v small enough

1 v 2k

fk(v)O by the 2kth power of v/2 . Note that the power series obtained for

k=1, 2. . . . are also valid for k=0 when/0(v)=2J1(v)/v .

Some interesting properties of the functions f k (V) follow from the power

series (35). Multiplying (35) by v/2 and then integrating, we obtain (k=0,

1,

.)

or, eventually,

vf

/k1(v) =

J2/k(v) dv

0

dfkd 1(v)

_ 2

fk(v)

(36)

Thus, the family of the functions /0 (v), / 1 (v), . . . is related by the recurrent

property (36) which gives the possibility of generating subsequently all functions

starting from the Airy pattern/0(v)=2J1(v)/v .

10

Dow

nloa

ded

by [N

ation

al Ch

iao T

ung U

nivers

ity ]

at 07

:20 12

Nov

embe

r 201

5

560

M. Novotny

Finally, we transfer the recurrent property of the Bessel functions (k = 2, 3 . . . . )

Jk(v) = 2(k -1)Jk-l(u)lu

-Jk-2(v)

(37)

to our functions fk(V) . Solving (17) with regard to the Bessel functions Jk(v)

and substituting them into (37), we come to the recurrence formula for fk(V)

in the form

d k 1(v )=

k1 (v)2k

(k-1)/k(v)-`2)2d" k-1(v) (38)

This relation was found for k = 2, 3, . . . but it holds even for k = 1 because of a

common power series (35) valid for all functions / k (v), k=0, 1, . . . . We can

then use the recurrence formula (38) for subsequent calculation of all the functions

/ k(V) from its two neighbours .

6. The Fresnel diffraction field of the free opening

We shall apply the derived representation to the uniform transmissivity

o(t)=1 . The diffraction field O(u, v) given by the diffraction integral (9) is

expressed in terms of the diffraction coefficients cu k (u)

0

In the planes u=4TrL, L integer, the diffraction coefficients cuk(u) represent evi-

dently the Fourier coefficients (21) and (22) of the polynomials Qk(t) . Thus, we

have for the focal plane u = 0

cu o(0)=1 ; cu k (0)=0, k=1, 2, . . .

(41)

and for the other planes u = 47rL, L 0 0

w0(4TTL)=0 ; cu k (4aL)=-(i2rL)-k, k=1, 2, . . . .

(42)

The properties of the Bernoulli polynomials discussed in 4 enable us to integrate

(40) even for a general position of the observatien plane . Assuming u ;~6 0 the

repeated integration by parts can be performed . Using (32) and the following

properties of the Bernoulli polynomials Q0(t)=1, Q 1(1)=1/2, Q1(0)=-1/2,

Q2k( 1 )= Q2k(0)# 0 Q2k 1(1)=Q2k1(0)=0 for k=0, 1, . . . we may write the

result of integration in the form

CO O(U) = exp

(4/

sine

(4)

,wk(u) = (

u2) k

wO(u) [1k

(4

U) - (

4

U) Cot

(43)

00

O(u, v)= I(-1)kwk(u)/k(v), (39)

k=0

1

wk(u)= f

exp (iut/2)Q k(t) dt. (40)

Dow

nloa

ded

by [N

ation

al Ch

iao T

ung U

nivers

ity ]

at 07

:20 12

Nov

embe

r 201

5

where

Axially symmetrical Fresnel diffraction

ul 2n5k

ul 11 .

fk

C41

= IO (-1)"Q2n(0)

C2l

The values Q2k(O) are easy to calculate in terms of the coefficients An

2kof the

polynomials Q2k (t) . If we denote

m

Qm(t) = I Anmtn ;

M=O, 1,. . . ,

n=O

then Q2k(O)=An2k. All coefficients An- are determined by the recurrent pro-

perty (24) of the polynomials Qm (t) . Consequently, we can get (m=0, 1, . . .)

Anm = A

O

--n/n!

(45)

and

m

AOm 1=

-

Aom -n/(n

2) ! .

n=o

561

(44)

(46)

These relations enable us to derive all coefficients An- if starting with the initial

one Aa = 1 . Hence, A 0 1 = - 1/2, A 0 2 =1/12, . . . .

The coefficients A02k

are also expressible in terms of the Bernoulli numbers

B2k defined to be ([5], p. 20)

B

2(-1)k

1(2k)!

-

n-2k.

2k=

(27T) 2k

n=1

From (26) follows A 02k=B

2k/(2k)! and therefore

Q2k(0 ) = B2kl(2k) ! .

(47 )

This form leads to the discovery of a remarkable property of the expression (43) .

It holds that ([5], p. 23)

u

u -

j

( - 1)nB2n (U) In

4 cot

(4/ n=0 (2n) !

(2}

(48)

We see that the series (44) represents a truncated expansion of the above function .

Thus, it holds that Wk(u)->O for k-* co and all values of u as the expression in the

square brackets turns into zero independently on u .

The series expansion (39) found for the diffraction field of the free opening

naturally gives the known distributions in the focal plane 0(0, v)=cu 0(0)f0 (v)

=2J1(v)/v and along the axis O(u, 0)=y0(u)=exp (iu/4) sinc (u/4) . At the same

time, the diffraction field of the free poening O(u, v) is that of the ideal lens

([3], p. 221). In this case the variable u stands for u - u , where u =ka2/f,

f

being the focal length, k = 2.7r/A, and a the radius of the aperture . From this may

be found the connection among our variables u - u , v and the variables used in

Dow

nloa

ded

by [N

ation

al Ch

iao T

ung U

nivers

ity ]

at 07

:20 12

Nov

embe

r 201

5

562

M. Novotn

the former Lommel representations of the diffraction field near focus of the lens-

cf. [9, 10] . Using the relations (3), (4) from [3] it is not difficult to see that it

holds for the plane of observation z =zo not too far from the focus u - u - ka2zo/f2 .

Then, our first variable differs only in the sign from that used, e .g ., in [9] . The

definitions of the second variable v coincide and so the expression O(u - u , v)

given by the relation (9) is formally identical with that in [9] . However, a

difference appears in the representations of the total diffraction field in the form

of the different geometrical factors-cf. [3], p . 221 .

In comparison with the Lommel representations (10), (11) the deduced expres-

sion (39) contains only the function of one variable . Further, note that the con-

vergence of the series (39) is very rapid near the axis because of the behaviour of

/k (v) for small values of v and most rapid near the focus (u=0, v=0) when

moreover the coefficients cok(u) tend to zero for k > 0, u--0 . This may be seen

from the relation (43) that involves the truncated Taylor expansion of the function

(u/4) cot (u/4) in the vicinity of the point u=0-cf . (48) .

Finally, let us mention two generalizations of the relation (43) . With respect

to (40), the imaginary part of wk(u) clearly represents the diffraction coefficients

due to the Fraunhofer pattern of the sine-shape filter g(t) = sin(ut/2) . Similarly,

we find the real part of wk(u) to be the diffraction coefficients yk(0) corresponding

to the filterg(t)= cos(ut/2), u now being the parameter of the filter transmissivities .

Another generalization can be obtained by differentiation of (43) in the variable

u/2 . From (40) it is not difficult to see that the mth derivative of (43) has the

meaning of the diffraction coefficients due to the Fresnel pattern u = const of the

filter (it)m . Hence, the filter

g(t) =tm

has the diffraction coefficients

yk(u) -

Z

-m

d(ul2)

These results enable us to express the diffraction coefficients associated with the

Fresnel diffraction field due to any polynomial filter transmissivity .

7 . The diffraction field in the focal planes

We shall discuss the focal diffraction pattern using the series representation

(18) . At first, let us examine the behaviour of the diffraction field for the increasing

number of periods M. As only the first term k = 0 representing the Airy pattern

contains M in the non-negative zero power, all other terms may be suppressed if

M is sufficiently large . Thus, we arrive at the results of paper [3] : the diffraction

field of any axially symmetrical filter approaches the Airy pattern in the focal

planes if the number of periods M increases .

Figures 3, 4 and 5 illustrate this for the examples of the Fresnel zone plates

with opaque and transparent central zone . The intensities in the Fraunhofer

plane L=0 are plotted in figure 3 . Figure 4 emphasizes the studied behaviour

illustrating the intensity ratios [CM/C.]2 . The graphs in figure 5 represent the

intensity distributions in the conjugate focal planes L = 1 of the Fresnel plates .

From the figures it may be seen how fast all curves tend to the limit case given by

the Airy pattern if M increasing .

Dow

nloa

ded

by [N

ation

al Ch

iao T

ung U

nivers

ity ]

at 07

:20 12

Nov

embe

r 201

5

2(C(0.V) 2

lCm(0 .v)

1

1

0

0

0 .20

0 .15

0 .10

0 .05

0

0

Axially symmetrical Fresnel diffraction

1

1

2

2

3 4 5 v --..

3 v ---- 4

563

Figure 3 . The intensities in the Franhofer plane u=0 calculated for the Fresnel plates

with transparent (the dashed curves) and obstacle central zone . They converge

with increasing number of the transparent zones M to the Airy pattern .

Figure 4. The ratios of the intensities from figure 3 to emphasize the convergence to the

Airy pattern C . for M increasing .

Now, let us examine the diffraction fields of the non-periodic pupil functions

M= 1 in the vicinity of the axis. From figure 2 it can be seen that the values of

the functions/ k v for a fixed value v are practically zero starting from certain k .

Thus, only a few first terms of the series representation 18 can bring a substantial

contribution to the total diffraction amplitude . Especially, in the very near

t

i

/

1

n

i

/

I

I

1

M=1 ,

,

2%

E = 1 /4

%

/

, 10:

/ /

20,%

50/

-

___=----

20

I

10

5

2

1

A _~l

---- E = 1/4

e = 3/4

0INII

~2

2 `50M=1

Dow

nloa

ded

by [N

ation

al Ch

iao T

ung U

nivers

ity ]

at 07

:20 12

Nov

embe

r 201

5

564

0 .12

ICM 4,rM,v I 2

0 .10

0.08

0 .06

0 .04

0 .02

0

0 1

M. Novotny

2

3

4 5 v --

Figure 5 . The intensity distributions in the principal conjugate focal planes u = 4rrM

of the Fresnel plates with opaque and transparent central zone, the full and dashed

lines respectively . For increasing number of transparent zones M they approach

the Airy distribution .

vicinity of the axis, the term k = 0 corresponding to the Airy pattern prevails .

The further terms k=1, 2, . . . begin to apply subsequently in wider and wider

environments . It is important that we are usually able to estimate the width of

the central intensity maximum from the first three diffraction coefficients . This

fact leads to the practical applications presented in [11] where the two-point

resolution is studied . In [11] is developed a simple graphical method for deter-

mining the two-point resolution achieved in the Fraunhofer focal plane L = 0 by

the filters with the real pupil functions .

ACKNOWLEDGMENTS

The author wishes to thank Mrs. Z. Kucerova for her help with numerical

calculations and with preparation of the manuscript . Thanks are due to to Dr. J .

Komrska for his encouragement during the preparation of the paper .

On etablit une nouvelle representation par une serie du champ diffracts G u, v du

aux filtres a symetrie axiale . Les coefficients de la serie sont formes par certains produits

scalaires de la fonction pupillaire et des polynomes de Bernoulli . Contrairement aux

precedentes representations du champ diffracts de Fresnel, qui contiennent des fonctions

de Lommel de deux variables, la presente representation opere uniquement avec des fonctions

spsciales d une seule variable . L application de la theorie aux figures de diffraction au foyer

G 4nrML, v L entier, dues a des filtres a transmissivites periodiques avec le carre de la

distance a l axe, on retrouve le resultat precedent [3] : les figures de diffraction au foyer

tendent vers la figure d Airy lorsque le nombre de periodes M augmente. On obtient

les coefficients de la serie pour la diffraction de Fresnel due a une lentille ideale, ainsi

que les transmissivites du filtre polynomial . Les resultats sont utiles pour les champs

diffractes par les lames zones de Fresnel et de Gabor .

Fur achsensymmetrische Filter wird eine neue Reihendarstellung fur das Beugungsfeld

G u, v abgeleitet . Die Reihenkoeffizienten werden durch bestimmte skalare Produkte

der Pupillenfunktion and der Bernoulli-Polynome gebildet . Im Gegensatz zu fruheren

Darstellungen des Fresnelschen Beugungsfeldes, die Lommel-Funktionen von zwei

E = 1 /4

e = 3/4

1-

hL

alk

,\` `

\5`.

1 2 500.

Dow

nloa

ded

by [N

ation

al Ch

iao T

ung U

nivers

ity ]

at 07

:20 12

Nov

embe

r 201

5

Axially symmetrical Fresnel diffraction

565

Variablen enthielten, arbeitet these Darstellung nur mit speziellen Funktionen von einer

Variablen . Wendet man die Theorie an auf das fokale Beugungsbild G 47TML, v , L

ganze Zahl, von Filtern deren Transmission periodisch mit dem Quadrat des Abstandes von

der Achse ist, so folgen direkt die frtiheren Ergebnisse [3] : Fur jedes Filter n ahern sich mit

wachsender Periodenzahl M die fokalen Beugungsbilder dem Airyschen Beugungsbild .

Die Reihenkoeffizienten fur die Fresnelschen Beugungsbilder der idealen Linse sowie

fur die polynomische Pupillenfunktionen werden abgeleitet . Die Ergebnisse fur die Beu-

gungsfelder von Fresnel- and Gabor-Zonenplatten werden dargestellt .

REFERENCES

[1] BolviN, A ., 1964, Theorie et Calcul des Figures de Diffraction de Revolution Paris :

Gauthier-Villars , Chap . VII .

[2] ARSENAULT, H ., 1968, .7. opt. Soc. Am., 58, 871 .

[3] NovoTNf, M ., 1973, Optica Acta, 20, 217 .

[4] ARSENAULT, H., and BoIVIN, A., 1967, Y. appl. Phys ., 38, 3988 .

[5] LUIS, Y. L., 1962, Integrals of Bessel Functions New York : McGraw-Hill Book

Company, Inc. .

[6] WATSON, G. N., 1958, A Treatise on the Theory of Bessel Functions Cambridge

University Press .

[7] COURANT, R., and HILBERT, D ., 1953, Methods of Mathematical Physics, Vol . 1 New

York: Interscience Publishers .

[8] NovoTNY, M., 1977, Optica Acta, 24 577 .

[9] LINFOOT, E. H ., 1958, Recent Advances in Optics Oxford: Clarendon Press , p . 39 .

[10] BORN, M., and WOLF, E ., 1965, Principles of Optics Oxford : Pergamon Press , p . 437 .

[11] NovoTNf, M ., 1977, Optica Acta, 24, 567 .

Dow

nloa

ded

by [N

ation

al Ch

iao T

ung U

nivers

ity ]

at 07

:20 12

Nov

embe

r 201

5

page 1page 2page 3page 4page 5page 6page 7page 8page 9page 10page 11page 12page 13page 14page 15