Alternative Solution of an Inverse Eigenvalues Problem in Optics
ANDREY OSTROVSKY, ALEXANDER ZEMLIAK, EDGAR HERNÁNDEZ-GARCÍA
Facultad de Ciencias Físico Matemáticas
Benemérita Universidad Autónoma de Puebla
A. P. 1152, 72000, Puebla, Puebla
MÉXICO
OLEG STAROSTENKO
Departamento de Ciencias Computacionales
Universidad de las Américas
72820, Cholula, Puebla
MÉXICO
Abstract: - The problem of eigenrepresentation of a planar optical source with unknown cross-spectral density
function is considered. It is proposed the alternative eigenrepresentation of a source which does not involve the
solution of the Fredholm integral equation but may be obtained as the solution of some optimization problem.
The basic concept is illustrated by the example of the alternative eigenrepresentation calculated for the 1-D
Lambertian source.
Key-Words: - Inverse eigenvalues problem; Enigenrepresentation; Radiance; Lambertian source
1
Introduction
As is well known, the cross-spectral density function
W x 1 , x 2  of a steady-state planar source of any state
of coherence localized in some finite domain D of
the plane z  0 and radiating into the half-space
z  0 may be represented by the Mercer expansion
[1]:
the problem, which does not involve the solution of
the Fredholm equation (2), is desired. In this paper,
we propose such an approach based on substituting
the original source with the unknown cross-spectral
density function for an alternative source with the
cross-spectral density function which may be
approximated according to the results of physical
measurements.

W x1 , x 2    n n* x1  n x 2  ,
(1)
n 0
where  n are the eigenvalues and  n x  are the
eigenfunctions of the Fredholm integral equation
 D W x1 , x 2  n x1 dx1  n n x 2  .
(2)
Besides, the eigenvalues  n are real and
nonnegative, and the eigenfunctions  n x  may be
chosen to form an orthonormal set.
The eigenrepresentation (1) is an essential tool in
describing the processes and system in optics [2-9].
However, the practical value of this representation is
essentially restricted, since the analytical expression
for cross-spectral density function W x1 , x 2  , as a
rule, is unknown. Clearly, an alternative approach to
2
Alternative eigenrepresentation of
a planar source
In radiometry the radiation generated by a planar
source is characterized completely by the radiance
function Bx, s  which is the surface-angular density
of the radiated energy. In accordance with Ref. [10] it
may be expressed in terms of the source coherent
modes as follows:
2

2
 k 
Bx, s   
 cos    n  n x   n ks  
 2 
n 0
where
 n ks    
 z 0 
 n x expiks   xdx ,
2
, (3)
(4)


k is the wave number, s  s x , s y , s z  0 is the unit
Substituting from Eq. (6) and making use of the
vector which points into the half-space z  0 ,
s   s x , s y ,0 is the projection of this vector onto the
relations d  ds  cos and cos  1  s 2 , one
may rewrite the optimization problem (8) in the
following explicit form:


plane z  0 , and  is the angle between s and the z
axis. It is important to stress here that the radiance
function Bx, s  represents the physically measurable
quantity.
Owing to the nonlinear dependence of Bx, s  on
eigenfunctions  n x  it may be founded some
alternative radiatively equivalent source which
occupies the same domain D in the plane z  0 , has
another cross-spectral density function, but radiates
into the half-space z  0 with the same radiance
function as does the original source. To describe such
a source, we will introduce the function
~ x , x  
W
1
2
M 1
  m x1  m x 2  ,
*
m
m 0
2
m ks    
2
, (6)
 m x expiks   xdx . (7)
 z 0 
Once the radiance function Bx, s  of the original
source has been measured in physical experiment, a
good approximation of the alternative radiatively
equivalent source may be obtained by solving the
following problem of the conditional optimization:
 2  z 0 Bx, s   B x, s 
~
2
dxd  min ,
m
 m  0,
(8)
where d is the element of solid angle around a
direction specified by s and the first integral extends
over 2 -solid angle subtended by a hemisphere in
the half-space into which the source radiates.
12
m  0 ,
where
(9)
Pml    m x   t x  dx
2


s 2 1
D
2 12
s
 1  
2
m ks  
 2 
Qm  

 k 
2
2
l ks   ds  , (10)
2
 s 1  D Bx, s 
2

  m x  m ks   dxds  .
2
M 1
2
 k 
Bx, s   
 cos    m  m x   m ks  
 2 
m 0

M 1
 1 M 1M 1

,
    m  t Pml    m Qm   min
m
m 0
 2 m 0 t 0

(5)
where  m are some arbitrary real positive values
bounded from above, and  m x  are some prescribed
orthonormal functions. This function may be
considered as the cross-spectral density function of
some alternative source in the domain D . Changing
formally  n and  n x  in Eqs. (4) and (5) for  m
and  m x  , respectively, one may obtain the
following expression for the radiance function of
such alternative source:
where

2
(11)
The problem (9) represents the classical quadratic
programming problem which may be solved by well
known methods.
As is well known, the only true physical quantity
that can be observed (or measured) in the output
plane of an optical system is the irradiance function
E x , s  defined as follows:
E x, s   
where
2 
Bx, s cos  d ,
B x , s   L Bx, s  ,
(12)
(13)
and L is an operator describing the propagation of
the primary source radiance function Bx, s  through
an optical system. It is obvious that two primary
sources with different cross-spectral density
functions, e.g., W x1 , x 2  and W x1 , x 2  , but with
the same radiance function, produce the same
irradiance in the output plane of an optical system.
Hence, when examining the processes going on in
some optical system, the original source with crossspectral density function W x1 , x 2  may be
substituted in a good approximation for the
alternative source with the appropriately chosen
cross-spectral density function W~ x1 , x 2  . In this
sense Eq. (5) with  m calculated as the solution of
problem (9) may be accepted as the alternative
eiegenrepresentation of the original source.
3
Example
To illustrate the proposed technique of constructing
the alternative eigenrepresentation of the original
source we realized a numerical experiment. In this
example we considered the homogeneous 1-D
Lambertian source with known radiance function

Ck 2 , when x  X 2, s  1,
B  x, s   
when x  X 2, s  1,

0,
(14)
problem (9) using the modified gradient method. To
simplify calculations we chose kX  10 (typically
this value may be of the order of 10 3 and larger). The
obtained solutions for different values of M and the
corresponding values of the relative error of
approximation  are given to four decimal places in
the Table I. As can be seen from this table, the error
of approximation decreases rapidly with the increase
of M and, when M  30 , it becomes less than 10%
that is quite admissible in the majority of practical
applications.
where C is a constant (further, for the sake of
simplicity, we equate it to unity). As the basis
functions  m  x  we chose the Hermitian functions
 m x  
  
 x
H m 
x  exp 
2
 2X
 X 
2
1
X 2 m!
12
m
M
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29


 , (15)


where H n u  is the Hermitian polynomial of the
integral order n and the scale factor attached to the
argument is chosen to provide the approximate
orthonormality of functions  m  x  within the source.
It may be readily shown that
12
 kX 
 2X 
m ks  i m  m  H m 
s
  
 2 m! 


2
2 2
 k X s 
.
 exp 

2



(16)
For this example the coefficients of the problem (9)
given by Eqs. (10) and (11) take the following
values:
16
Pml 
2
2
m
k 2 m! 2 l l!


 
 2
H m u 2 H l u 2 exp 2u du

kX
0
0


v 2 
1 
2 
 k X  

12
H m v 2

 H l v  exp  2v 2 dv ,
2
Qm  
(17)
 2
H m u 2 exp u 2 du

0
m
k 2 m!
kX 
H m v 2 exp v 2 dv.

(18)
0
8
2
We realized the numerical calculation of
coefficients (17) and (18) and solved the optimization
4
Table I. Results of Calculations.
5
10
20
0.2723
0.1335
0.0000
0.5899
0.2938
0.0000
1.0718
0.4544
0.0000
2.1906
0.6467
0.0000
3.0898
0.9426
0.0106
1.1818
0.0008
1.8589
0.0117
2.0979
0.0318
4.4068
0.0374
5.4077
0.1575
0.2035
0.5844
1.0153
2.1614
3.8866
6.2033
9.4392
10.5465
11.0982
6.9331
0.6595
0.4586
0.1087
30
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.2039
1.2738
2.7781
5.0672
7.6670
8.9493
8.8142
5.1981
1.1903
0.0008
1.6739
5.2415
5.2495
1.5986
0.0000
1.9317
4.8002
3.1132
0.0993
Conclusions
We have defined an alternative eigenrepresentation of
an original planar source with unknown crossspectral density function in terms of the
eigenfunctions of some alternative source which
generates radiation with nearly the same radiance
function as does the original source. Such a
representation may be obtained by approximating the
measured radiance function of the original source by
the radiance function of an alternative source
calculated in the prescribed orthonormal basis. We
wish
to
stress
that
an
alternative
eigenrepresentatation does not reproduce the true
eigenrepresentation of the original source but may
substitute it in practice when examining the processes
going on in some optical system.
This work was supported by the National Council for
Science and Technology (CONACyT) from Mexico
under the research project No. 36875-E.
References:
[1] L. Mandel, E. Wolf, Optical Coherence and
Quantum Optics, Cambridge University Press,
Cambridge, 1995.
[2] A. S. Ostrovsky, Paraxial approximation of
modern radiometry for beam-like wave
propagation, Opt. Rev., Vol. 3, No. 2, 1996, pp.
83-88.
[3] A. S. Ostrovsky, M. V. Rodriguez-Solis, Freespace propagation of the generalized radiance
defined in terms of the coherent-mode
representation of cross-spectral density function,
Opt. Rev., Vol. 7, 2000, pp. 112-114.
[4] A. S. Ostrovsky, G. Martínez-Niconoff, J. C.
Ramírez-San-Juan,
Coherent
mode
representation of propagation-invariant fields,
Opt. Commun., Vol. 195, 2001, pp. 27-34
[5] A. S. Ostrovsky, G. Martínez-Niconoff, J. C.
Ramírez-San-Juan, Generation of light string and
light capillary beams, Opt. Commun., Vol. 207,
2002, pp. 131-138.
[6] A. S. Ostrovsky, G. Martínez-Niconoff, J. C.
Ramírez-San-Juan, Propagation-invariant fields
of the third kind and its optical generarion, J.
Opt. A.: Pure and Appl. Opt., Vol. 5, 2003, pp.
276-279.
[7] A. S. Ostrovsky, O. Ramos-Romero, M. V.
Rodríguez-Solis, Coherent-mode representation
of partially coherent imagery, Opt. Rev., Vol. 3,
1996, pp. 111-115.
[8] A. S. Ostrovsky, O. Ramos-Romero, G. MartínezNiconoff, J. C. Ramírez-San-Juan, On the modal
representation of partially coherent imagery,
Revista Mexicana de Física, Vol. 47, 2001, pp.
404-407.
[9] A. S. Ostrovsky, O. Ramos-Romero, G. MartínezNiconoff, Fast algorithm for bilinear transforms
in optics, Revista Mexicana de Física, Vol. 48,
2001, pp. 186-191.
[10] R. Martínez-Herrero, P. M. Mejias, Radiometric
definitions from second order coherence
characteristics of planar sources, J. Opt. Soc. Am.
A., Vol. 3, 1986, pp. 1055-1058.