Three methods for performing Hankel transforms with optical or digital processors are described. The first method is applicable when the input data is available in Cartesian (x-y) format and uses the close connection between generalized Hankel transform and the two dimensional Fourier transform in Cartesian coordinates. The second method is useful when the input data is in polar (r - theta) format and uses change of variables to perform the nth order Hankel transform as a correlation integral. The third method utilizes the von Neumann addition theorem for Bessel functions to extract the Hankel coefficients from a correlation between the radial part of the input and a Bessel function. Initial experimental results obtained for optical implementation of the first two methods are presented

Athale, R. A.

Lee, J. N.

Szu, H. H.

English

NASA Technical Reports Server

THREE METHODS FOR PERFORMING HANKEL TRANSFORMS 
R. A .  Athale, H. H. Szu and J. N. Lee 
Naval Research Laboratory, Washington, D. C. 20375 
ABSTRACT 
Generalized Hankel transforms are useful in analyzing the effect of circularly 
symmetric optical systems on arbitrary inputs. Some examples of such systems are 
complex laser resonators and space telescopes. Three methods for performing Hankel 
transforms with optical or digital processors are described. The first method is 
applicable when the input data is available in Cartesian (x-y) format and uses the 
close connection between generalized Hankel transform and the two-dimensional 
Fourier transform in Cartesian coordinates. The second method is useful when the 
input data is in polar (r - 8 )  format and uses change of variables to perform the 
nth order Hankel transform as a correlation integral. 
the von Neumann addition theorem for Bessel functions to extract the Hankel 
coefficients from a correlation between the radial part of the input and a Bessel 
function. Initial experimental results obtained for optical implementation of the 
first two methods are presented. 
The third method utilizes 
INTRODUCTION 
The analysis of complex optical systems is greatly facilitated by two- 
dimensional Fourier transform techniques. The effect of an optical system on 
arbitrary inputs is easily described by a transfer function in the Fourier domain. 
Generalized Hankel transforms are similarly useful when dealing with a circularly 
symmetric (or axisymmetric) system for arbitrary inputs. This situation is 
encountered in performing mode analysis on the output of a slightly misaligned 
laser resonator as well as in aligning space telescopes. 
performing mode analysis via generalized Hankel transform will have the unique 
advantage of preserving the phase of the wavefront to be analyzed. 
An optical method for 
It is well known that when a two-dimensional function has circular symmetry 
(i.e. it depends only on the radial variable, r), its Fourier transform is also 
circularly symmetric (i.e. it depends only on the radial variable, p ) .  It can be 
shown that in such a case the Fourier transform is equivalent to the 0th order 
Hankel transform. 
If 
Then 
where F2 { 1 indicates two-dimensional Fourier transformation. 
133 
https://ntrs.nasa.gov/search.jsp?R=19820008033 2020-03-21T09:51:36+00:00Z
For 
2 2  x+y , p =  
m 
- = Fo(p) = 2~ rdr f(r) Jo(p) 
0 
where Po(p)  is the Oth order Hankel transform and Jo(pr) is the Oth order Bessel 
function of the first kind. 
Oth order Hankel transform (which is a one-dimensional operation) can be used 
instead of the two-dimensional Fourier transform if the inputs also are circularly 
Thus in dealing with circularly symmetric systems, the 
symmetric. For arbitrary inputs, this technique can be extended by usingthe genera- 
lbed Hankel transform, which expands on the nth order transform (Fn(p) with kernel 
2n (PI in Eq. (1)). 
The generalized Hankel transform, Fnn(p), can be defined for an arbitrary 
function, f(r,e), as follows: 
W 
m 
0 
The generalized Hankel transform thus involves a Fourier series expansion in 0 ,  
followed by an nth order Hankel transform (with nth order Bessel function, Jn(pr), 
as the transformation kernel) on the nth coefficient of expansion, fn(r), This 
generalized Hankel transform is useful in analyzing systems with circular symmetry 
but when the input is not circularly symmetric. 
In the following sections we will describe three techniques for easy 
implementation of the generalized Hankel transform with optical or digital 
processors. These three techniques are each applicable in different circum- 
stances. Initial experimental results on the optical implementation of the 
first two methods will be presented. I n  conclusion, we will detail the course 
of future work in this area. 
OPTICAL IMPLEMENTATION OF GENERALIZED HANKEL TRANSFORM 
In  optical processors a two-dimensional Fourier transform with respect to the 
Cartesian coordinates (x,y) is performed very easily with the help of a simple 
spherical lens. The equivalence of two-dimensional Fourier transform and 0th 
order Hankel transform for circularly symmetric functions was established in the 
Introduction. A similar connection exists between two-dimensional Fourier trans- 
form with respect to Cartesian coordinates (x,y) and nth order Hankel transform 
of the radial part if the function is of a’ form f(r) ejne, 
134 
j n0 = 2aFn(p) e - j  n$ 
where 
1, 
2 2 '2 -1 
P = (u +v 1 , $ = t a n  (v /u ) .  
and by d e f i n i t i o n  
r d r  
This  connect ion arises ou t  of t h e  i n t e g r a l  r e p r e s e n t a t i o n  of J n ( p r ) .  
r e s u l t  i n  t h e  d e f i n i t i o n  of gene ra l i zed  Hankel t ransform given  i n  E q .  (2) t h e  
fo l lowing  r e l a t i o n  i s  ob ta ined  between t h e  two-dimensional Four i e r  t ransform and 
t h e  gene ra l i zed  Hankel t ransform:  
Using t h i s  
W 
t h  Thus t h e  gene ra l i zed  Hankel t ransform Fnn(p) i s  e q u i v a l e n t  t o  t h e  n 
c o e f f i c i e n t  of Four i e r  series expansion of F ( p , $ )  i n  v a r i a b l e  $.  
t h i s  r e s u l t ,  an o p t i c a l  system shown i n  Fig.  1 i s  designed t o  perform 
gene ra l i zed  Hankel t ransform on an a r b i t r a r y  i n p u t  t h a t  is  a v a i l a b l e  i n  
Car t e s i an  format  ( f ( x , y ) ) .  The s p h e r i c a l  l e n s  performs a two-dimensional 
Four i e r  t ransform on f (x ,y)  gene ra t ing  F ( u , v ) .  
genera ted  hologram then  performs t h e  Car tes ian- to-polar  (u,v+p,$) coord ina te  
t r ans fo rma t ion  on F ( u , v )  gene ra t ing  , F ( p  ,+) . 
c a l  l e n s  which t a k e s  a one-dimensional Four i e r  t ransform wi th  r e s p e c t  t o  v a r i a b l e  
$ gene ra t ing  t h e  d e s i r e d  ou tpu t ,  Fnn(p). 
dimensional  Four i e r  t ransform and nth o r d e r  Hankel t ransform f o r  a f u n c t i o n  
of t h e  form f ( r ) e j n O .  The schematic  diagram of t h e  o p t i c a l  system i s  shown 
i n  F ig .  2.  
of t h e  inpu t .  The ou tpu t  w a s  de t ec t ed  by a TV camera, which measures t h e  l i g h t  
i n t e n s i t y  i n  t h e  F o u r i e r  p l ane  of t h e  inpu t .  The i n p u t  used i n  t h e s e  exper i -  
ments had an r dependence g iven  by 6 ( r - a ) ,  t hus  corresponding t o  a t h i n  r i n g  
of r a d i u s  I r a r c  i n  t h e  C a r t e s i a n  (x,y)  plane.  
6(r-a)  i s  aJn (ap ) ,  making t h e  ou tpu t  e a s i l y  unders tandable .  
r e s u l t s  corresponding t o  Oth o r d e r  Hankel t ransform (i .e.  ejne = 1 ) .  
t h e  f u n c t i o n  I Jo(ap)  l 2  is  p l o t t e d ,  which i s  then  compared t o  a l i n e  scan  through 
Using 
A s u i t a b l y  designed computer 
This  i s  followed by a c y l i n d r i -  
I n i t i a l  o p t i c a l  experiments  e s t a b l i s h e d  t h e  connec t ion  between two- 
A computer genera ted  hologram w a s  used t o  encode eJne dependence 
The nth o r d e r  Hankel t ransform of  
F igu re  3 shows t h e  
I n  F ig .  3(b)  
135 
t h e  o r i g i n  (Fig.  3 ( c ) )  of t h e  output  shown i n  F ig .  3 ( a ) .  Very good q u a l i t a t i v e  
agreement between t h e  t h e o r e t i c a l  and exper imenta l  r e s u l t s  i s  obta ined .  F igures  
4(a-c) p r e s e n t  t h e  r e s u l t s  f o r  lSt o rde r  Hankel t ransform f o r  t h e  same f ( r ) ,  
b u t  h e r e  the  8 dependence i s  e''. Again good q u a l i t a t i v e  agreement is 
seen  between theory  and experiment.  A l i n e  scan  through t h e  o r i g i n  of t h e - o u t p u t  
corresponding t o  2nd o rde r  Hankel t ransform of 6(r-a)  (with 8 dependence eJ2e)  i s  
shown i n  F ig .  5. The zero a t  t h e  o r i g i n  w a s  broader  and t h e  s i d e  lobes  w e r e  seen  
t o  f a l l  s lower i n d i c a t i n g  t h a t  w e  indeed have I J (ap) l 2  as expected.  2 
OPTICAL IMPLEMENTATION OF NTH ORDER HANKEL TRANSFORM 
The previous  method is  a p p l i c a b l e  when t h e  i n p u t  i s  a v a i l a b l e  i n  Car t e s i an  
(x,y)  format ,  s i n c e  i t  invo lves  performing two-dimensional Four i e r  t r ans fo rms  
wi th  r e s p e c t  t o  t h e  Car t e s i an  v a r i a b l e s .  I f  t h e  i n p u t  i s  p o l a r  ( r y e )  fo rma t t ed ,  
a more d i r e c t  approach o u t l i n e d  i n  Eq. ( 3 )  has  t o  be followed i n  o b t a i n i n g  
g e n e r a l i z e d  Hankel t ransform.  The f i r s t  p a r t  of t he  ope ra t ion ,  which invo lves  a 
Four i e r  series expansion i n  v a r i a b l e  8 ,  i s  e a s i l y  performed o p t i c a l l y  u s i n  a 
c y l i n d r i c a l  l e n s .  
c o e f f i c i e n t  of expansion is  less s t r a i g h t f o r w a r d  
space-var ian t  ope ra t ion .  So t h e  main a i m  of t h e  next  two methods i s  t o  perform 
t h e  space-var ian t  o p e r a t i o n  of  nth o r d e r  Hankel t ransform o p t i c a l l y .  
The c a l c u l a t i o n  of nth o r d e r  Hankel t ransform of t h e  n tf? 
s i n c e  i t  corresponds t o  a 
a.  Method Using Change of Var i ab le s  
One s t anda rd  procedure used i n  conver t ing  a space-var ian t  o p e r a t i o n  i n t o  a 
s h i f t - i n v a r i a n t  o p e r a t i o n  i s  t o  employ a p p r o p r i a t e  change of v a r i a b l e s .  I n  t h e  
case  of n t h  o r d e r  Hankel t ransforms t h e  fo l lowing  procedure w a s  desc r ibed  by 
Siegman f o r  implementing t h e  space-var ian t  o p e r a t i o n  as a c o r r e l a t i o n  i n t e g r a l  
on a d i g i t a l  p rocesso r .4  From t h e  d e f i n i t i o n  
F (PI = n 
0 
us ing  
ax w 
r = r  0 e > P = p o e  
A A A 
where Fn(y) = pFn(p),  f (x) = r f ( r )  , and Jn(x+y) = arp J n ( r p > .  The a lgo r i thm,  
t h e r e f o r e ,  c o n s i s t s  of f i r s t  l i n e a r l y  weight ing  t h e  i n p u t  f ( r )  and perform- 
i n g  r-tx ( l o g a r i t h m i c  r e l a t e d )  coord ina te  t ransform.  This  d i s t o r t e d  i n p u t  
i s  then  c o r r e l a t e d  wi th  a s i m i l a r l y  weighted and coordinate- t ransformed n t h  
o r d e r  Bessel f u n c t i o n  t o  g ive  t h e  d e s i r e d  Hankel t ransform as w e l l  as l i n e a r l y  
weighted and coord ina te  d i s t o r t e d  forms. I n  any p h y s i c a l  system t h e  c o r r e l a -  
t i o n  i n t e g r a l  w i l l  be performed over  a f i n i t e  interval ,  g iv ing  r i se  t o  
t r u n c a t i o n  e r r o r s .  Also i f  t h e  i n p u t  is  sampled i n  t h e  x-domain, t h e  sampling 
rate should be  adequate  t o  r e p r e s e n t  t h e  f u n c t i o n  a c c u r a t e l y  i n  x-domain. 
136 
These f a c t o r s  and o t h e r s  are d i scussed  a t  l e n g t h ,  e s p e c i a l l y  f o r  d i g i t a l  
implementation, i n  Refs.  4 and 5.  
S ince  t h e  ope ra t ions  of coord ina te  t r ans fo rma t ion  and c o r r e l a t i o n  can be  
performed by an  o p t i c a l  p rocesso r ,  t h e  o p t i c a l  system o u t l i n e d  i n  F ig .  6 can  
c a l c u l a t e  nth o rde r  Hankel t ransforms.  Computer genera ted  holograms are used 
t o  perform r-tx coord ina te  t r ans fo rma t ion  as w e l l  as t o  encode t h e  Four i e r  p l ane  
f i l t e r  w i t h  impulse response  i n ( x ) .  The second dimension of t h e  o p t i c a l  p rocesso r  
can be  used t o  perform d i f f e r e n t  o rde r  Hankel t ransforms on d i f f e r e n t  i n p u t s ,  t hus  
ach iev ing  mul t ichannel  ope ra t ion .  
I n  t h e  i n i t i a l  experiment ,  bo th  t h e  inpu t  and t h e  Four i e r  p lane  f i l t e r  w e r e  
encoded by computer genera ted  holograms. 
t r ans fo rma t ion  of  t h e  i n p u t  w a s  performed by t h e  d i g i t a l  computer b e f o r e  doing t h e  
holographic  encoding. The computer generated holograms used t h e  Lee-Burckhardt 
technique6 and conta ined  128 p i x e l s .  The o p t i c a l  system is  depic ted  i n  F ig .  7 .  
This system w a s  then  used t o  perform Oth  o r d e r  Hankel t ransform on two d i f f e r e n t  
i n p u t s ,  f l ( r )  = p l J o ( p l r )  and f 2 ( r )  = p2Jo(p2r) .  The r e s u l t  of Oth o rde r  Hankel 
t ransform on f l ( r )  and f 2 ( r )  should be  6(p-p1) and 6(p-p2) r e s p e c t i v e l y .  
r e s u l t s  of t h e  computer s imula t ion  of t h i s  a lgo r i thm are dep ic t ed  i n  Fig.  8. The 
f i n i t e  width of t h e  peak and t h e  s i d e l o b e s  are due t o  t h e  f i n i t e  l i m i t s  of i n t e r g r a -  
t i o n .  To f a c i l i t a t e  easy  comparison wi th  t h e  exper imenta l  r e s u l t s ,  t h e  Oth o r d e r  
Hankel t ransform wi th  l i n e a r  weight ing and coord ina te  d i s t o r t i o n  ( i . e .  I g o ( Y >  1 2, 
w a s  p l o t t e d  v e r s u s  y i n s t e a d  of F (p) ver sus  p .  Figure  9 shows t h e  exper imenta l  
r e s u l t s  ob ta ined .  
l i n e a r  photodiode a r r a y .  
peak h e i g h t s  (due t o  t h e  l i n e a r  weight)  are e v i d e n t ,  i n d i c a t i n g  good q u a l i t a t i v e  
agreement w i t h  t h e  computer s imula t ion  r e s u l t s .  
The l i n e a r  weight ing  and t h e  coord ina te  
The 
The o p t i c a l  ou?put w a s  de t ec t ed  by a 1024-element Ret icon  
The s h i f t  i n  t h e  peak p o s i t i o n  and t h e  d i f f e r e n c e  i n  t h e  
b .  Method Using Neumann Addi t ion  Theorem 
This  method i n v e s t i g a t e s  an approach based on t h e  s p e c i a l  p r o p e r t i e s  o f  Bessel 
 function^.^ 
c o e f f i c i e n t  F (p) i s  ob ta ined  a t  t h e  o r i g i n  of t h e  c o r r e l a t i o n  p l ane  f o r  t h e  
p a r t i c u l a r  v a l u e  of p encoded i n  t h e  Bessel f u n c t i o n  k e r n e l .  
I f  t h e  i n p u t  f ( r )  is c o r r e l a t e d  wi th  Jm(pr )  then  the  mth o r d e r  Hankel 
m 
Fm(P) = r d r  f ( r )  J m ( p ( r + r ' ) )  
0 
r ' = O  
The Neumann a d d i t i o n  theorem f o r  Bessel f u n c t i o n s  s t a t e s  t h a t  
co 
J (p( r4- r ' ) )  = J ( p r ' )  J n ( d  m m-n (7) 
S u b s t i t u t i n g  f o r  J m ( p ( r + r ' ) )  from E q .  ( 7 )  i n t o  Eq .  (6)  we  g e t  
137 
03 m m 
rdr f(r) Jm(p(r+r')) = rdr f(r) J (m') Jn(pr> m-n 
0 0 n=- 03 
m 
Thus it is seen that the correlation plane contains an infinite sum of Bessel 
functions of different order weighted by Hankel transform coefficients of 
different order. So in principle it is possible to extract Hankel transform 
coefficients of different order out of a single one-dimensional correlation 
operation. 
The other dimension of optical system can be used to perform correlations 
with Jm(pr) with different values of p to obtain complete Hankel transform. 
approach is currently being evaluated further to determine the situations in which 
this method will be suitable. 
This 
FUTURE WORK 
In this paper we have presented brief outlines of three different approaches 
for performing generalized Hankel transforms by optical or digital processors. 
We also presented initial experimental results on two of the three approaches. 
The future work will concentrate on carrying out the coordinate transformation 
required by the first two methods optically via computer generated holograms. A 
more quantitative analysis o f  the performance of the optical processor will be 
carried out. These three methods will be compared with each other with respect 
to space-bandwidth requirements as well as pre- and post-processing requirements. 
Finally, generalized Hankel transforms will be applied to specific problems, such 
as analysis of complex laser resonators, and optical systems will be developed for 
those problems. 
ACKNOWLEDGEMENT 
We wish to thank Dr. J. R. Leger for producing the computer generated 
holograms using the facilities at University of California, San Diego. 
138 
n 
Figure 1. - Schematic diagram of o p t i c a l  p rocessor  f o r  performing gene ra l i zed  
Hankel t r ans fo rm v ia  a two-dimensional Four i e r  t ransform w i t h  r e s p e c t  t o  
C a r t e s i a n  coord ina te s  (x,y) . 
U 
Figure  2. - Schematic diagram of exper imenta l  s e t u p  f o r  e s t a b l i s h i n g  t h e  
r e l a t i o n  between nth o r d e r  Hankel t ransform of f ( r )  and two-dimensional 
Four i e r  t r ans fo rm of f ( r )  ejne w i t h  r e s p e c t  t o  Car t e s i an  coord ina te s  (X,Y)  * 
139 
(a)  Photograph of t h e  ou tpu t .  
(c)  Line scan  through t h e  o r i g i n  of t h e  
p a t t e r n  i n  f i g u r e  3 ( a ) ,  g iv ing  
2 I F ~ ( P )  I ver sus  P. 
f o r  Oth o r d e r  Hankel t r ans fo rm 
2 
(b) P l o t  of I Jo(ap)  I v e r s u s  p ,  which i s  
t h e  t h e o r e t i c a l l y  expec ted  r e s u l t .  
F igu re  3. - Experimental  r e s u l t s  
Fo(p) of 8 ( r - a ) .  
140 
(a) Photograph of t h e  output .  
1 512 
(e)  Line scan  through o r i g i n  of t h e  2 
t h e  t h e o r e t i c a l l y  expec ted  r e s u l t .  
(b) P l o t  of I Jo(ap)  I v e r s u s  p, which is  bottom i n  f i g u r e  4 (a),  giving 
/ F ~ ( ~ ) I *  versus  p. 
s t  Figure  4 .  - Experimental  r e s u l t s  f o r  1 o r d e r  Hankel t ransform 
F1(p) of d ( r - a ) .  
141 
1 512 
Figure 5. - Line s c a n  through t h e  o r i g i n  of t h e  two-dimensional 
2 
F o u r i e r  t ransform of 6 ( r  - a )  g i v i n g  IF2(P)/  f o r  
6 ( r  - a ) .  
t h  F igure  6.  - Schematic diagram of  a n  o p t i c a l  p r o c e s s o r  f o r  performing n 
o r d e r  Hankel t ransform on f ( r )  employing change of v a r i a b l e s .  
142 
Figure  7 .  - Schematic diagram of t h e  experimental s e t u p  f o r  performing t h e  
one-dimensional c o r z e l a t i o n  between l i n e a r l y  weighted and coord ina te  
transform2d i n p u t ,  f ( X ) ,  and s i m i l a r l y  weighted and transformed Bessel 
f u n c t i o n  J (X) .  
0 
Y 
t h  F igure  8. - R e s u l t s  of computer s i m u l a t i o n  of 0 o r d e r  Hankel t r ans fo rm 
of  p lJo(p , r )  and p2  J o ( p 2 r ) .  
t ransformed Hankel c o e f f i c i e n t s ,  I Fo (y) I 2, are p l o t t e d  ve r sus  y . 
The l i n e a r l y  weighted and c o o r d i n a t e  
I\ 
P2 = 2P1. 
143 
( a )  Input  f l ( r )  = p J ( p  r ) .  l o 1  
(b) Inpu t  f ( r )  = p J (p r). 2 2 0 2  
Figure  9. - Osci l loscope  traces of ou tpu t  of t h e  o p t i c a l  p rocesso r  
performing the Oth o rde r  Hankel t ransform.  
t o   IF^(^) 1 
Traces correspond 
A 
ver sus  y f o r  the two i n p u t s .  
144 


Three methods for performing Hankel transforms

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