A method and apparatus are disclosed for generation of second-rank tensors using a photorefractive crystal to perform the outer-product between two vectors via four-wave mixing, thereby taking 2n input data to a control n squared output data points. Two orthogonal amplitude modulated coherent vector beams x and y are expanded and then parallel sides of the photorefractive crystal in exact opposition. A beamsplitter is used to direct a coherent pumping beam onto the crystal at an appropriate angle so as to produce a conjugate beam that is the matrix product of the vector beam that propagates in the exact opposite direction from the pumping beam. The conjugate beam thus separated is the tensor output xy (sup T)

Liu, Hua-Kuang

English

NASA Technical Reports Server

United States Patent [191 pi] Patent Number: 5,005,954 
Liu [45] Date of Patent: Apr. 9, 1991 
[54] METHOD AND APPARATUS FOR 
SECONDRANK TENSOR GENERATION 
[75] Inventor: 
[73] Assignee: 
[21] Appl. No.: 
[22] Filed: 
Hun-Ku~ng Liu, South Pasadena, 
Calif. 
The United States of America JIS 
represented by the Administrator of 
the National Aeronautics and Space 
Administration, Washington, D.C. 
310,992 
Feb. 16,1989 
[51] 
[52] 
Int. C l . 5  ......................... GO2B 5/23; GO6G 7/16; 
G06G 7/162; G06G 9/00 
U.S. Cl. .................................... 350/354; 350/353; 
350/358; 364/713; 364/822; 364/837; 364/841; 
364/845; 364/846 
[58] Field of Search ................... 350/354, 162.12, 353, 
350/358; 364/845, 841, 822, 837, 846, 713 
(561 References Cited 
U.S. PATENT DOCUMENTS 
4,009,380 2/1977 Bocker et al. ....................... 235/181 
4,508,431 4/1985 Henshaw ............................. 350/354 
4,569,033 2/1986 Collins et al. ....................... 364/845 
4,592,004 5/1986 Bocker et al. ....................... 364/713 
4,620,293 10/1986 Schlunt et al. ...................... 364/845 
4,686,647 8/1987 Soffer et al. ........................ 364/850 
4,767,197 8/1988 Yeh ..................................... 350/354 
OTHER PUBLICATIONS 
Pepper, D. M. et al.; “Spatial Convolution and Correla- 
tion of Optical Fields via Degenerate Four-Wave Mix- 
ing”; Jul. 1978; p. 79; Optic Letters. 
Shkunov et al.; “Optical Phase Conjugation”; Dec. 
1985; pp. 54-59; Scientific American. 
Primary Examiner-Eugene R. LaRoche 
Assisrant Examiner-Evelyn Lester 
Attorney, Agent, or Firm-Thomas H. Jones; Harold W. 
Adams; John R. Manning 
P71 ABSTRACT 
A method and apparatus for generation of second-rank 
tensors using a photorefractive crystal to perform the 
outer-product between two vectors via four-wave mix- 
ing, thereby taking 2n input data to a control n2 output 
data points. Two orthogonal amplitude modulated co- 
herent vector beams x and y are expanded and then 
collimated before directing them onto two opposing 
parallel sides of the photorefractive crystal in exact 
opposition. A beamsplitter is used to direct a coherent 
pumping beam onto the crystal at an appropriate angle 
so as to produce a conjugate beam that is the matrix 
product of the vector beams x and y, and to separate the 
resulting conjugate beam that propagates in the exact 
opposite direction from the pumping beam. The conju- 
gate beam thus separated is the tensor output xyT 
6 Claims, 1 Drawing Sheet 
n 
22 
TENSOR 
OUTPUT 
XYT 
https://ntrs.nasa.gov/search.jsp?R=19910017604 2020-03-24T07:00:43+00:00Z
U.S. Patent Apr. 9, 1991 Sheet 1 of 1 5,005,954 
1 
5,005.954 
METHOD AND APPARATUS FOR SECOND-RANK 
TENSOR GENERATION 
ORIGIN OF THE INVENTION 5 
The invention described herein was made in the per- 
formance of work under a NASA contract, and is sub- 
ject to the provisions of Public Law 96-517 (35 USC 
202) in which the Contractor has elected not to retain 1o 
title. 
TECHNICAL FIELD 
This invention relates to a method and apparatus for 
real-time generation of second-rank tensors using non- 15 
linear photorefractive crystals. 
BACKGROUND ART 
A tensor is an element of an abstract system used to 
denote position determined within the context of more 2o 
than one coordinate system, a special case of which is a 
vector that is determined in a single coordinate system. 
Before presenting the optical apparatus of the present 
invention for generating second-rank tensors, the defini- 
tion of a second-rank tensor will be reviewed, and then 25 
properties of nonlinear photorefractive materials used 
in the apparatus will be reviewed. 
Assume a given group G of linear transformations in 
the n-dimensional space R,. A vector x in the space has 
components X I ,  . . . Xn. The transformation A of the 
group G transforms x into x': 
x'=&x[=a- fzxi (1) 
where i=l ,  2, . . . , n. 
Taking the product of x and y (XCRn, yrRn), and 
applying the transformation Equation (l), the set of 
tensor quantities is 
35 
The n2 quantities of XNj transform according to A x A. 
A set of nz quantities T'V whose law of transformation 
is 
fl/=alw,lvrkl (3) 45 
form a tensor T of rank two. 
Recently, nonlinear photorefractive materials such as 
GaAs, BaTiO3, LiNbO3, Bi12 Si2003 (BSO), and Srl--x 
Ba, Nbz Ob (SBN) have been used in two-wave, three- 50 
wave and four-wave mixing schemes. The present in- 
vention uses a four-wave mixing scheme for the archi- 
tecture of an optical tensor generator. 
The fundamental principle of four-wave mixing illus- 
trated in FIG. 1 is to apply three waves El, E2 and Ep 55 
as inputs to the nonlinear photorefractive crystal 10. An 
output conjugate wave Ec proportional to the multipli- 
cation of the two input waves can be obtained through 
the third-order nonlinear interaction of the three input 
waves and the photorefractive crystal 10. The resultant 
polarization can be written as 
pour= f~3)E~(r )E2(r )Ep*(~)  
e[@J IOJ2UP)t - t I + k2- kpkl +cc. (4) 
where 01, 0 2  and up are the frequencies of the three 
input waves, and 
65 
E l ( r , r ) = E l ( r ) ~ ~ l r - k l ~ + c . c .  
E2(c t)= E2(r)dw2r-k23+ C.C. ( 5 )  
Ep(r,t)= Ep(r)dofl-kp3+c.c. 
are the electric fields of the three input waves, and X(3) 
(originally a tensor quantity) is taken as a scalar quantity 
based on the assumption that the waves are copolarized. 
The third-order nonlinear polarization in Equation 
(4) radiates the conjugate wave E, of frequency 
0,=01+02--0p 16) 
where if ol=oz=op=o,  then oc=o. 
When a plane wave is selected for Ep the conjugate 
wave E, will be propagating in the opposite direction of 
the pumping plane wave. The amplitude of the conju- 
gate wave E, will be proportional to the multiplied 
value of E1 and Ez. This is the basic principle used in the 
second-rank tensor generator of the present invention. 
In summary of the basic principle utilized in this 
invention, the nonlinear refractive crystal 10 provides 
four-wave mixing of a coherent incident beam E,, with 
coherent input beams E1 and Ez. The beams E1 and E2 
are arranged to pass through the crystal 10 in exact 
opposition, and the beam Ep is so oriented at an appro- 
priate angle as to pass through the crystal 10 and pro- 
duce self-induced diffraction gratings in the crystal. The 
interaction of beams E1 and Ez with this diffraction 
grating produces the conjugate beam E, that is propor- 
tional to the product of beams E1 and E2. 
STATEMENT OF THE INVENTION 
In accordance with the present invention, a real-time 
tensor generator utilizes means for generating first and 
second amplitude modulated coherent vector beams 
orthogonally disposed in space, and incident in exact 
opposition on parallel sides of a nonlinear refractive 
crystal. The first vector beam is expanded using a first 
cylindrical lens, and then collimated using a second 
cylindrical lens. The second vector beam is expanded 
using a third cylindrical lens, and then collimated using 
a fourth cylindrical lens. A coherent pumping beam is 
so directed onto one of the parallel sides of the nonlin- 
ear refractive crystal at an appropriate angle to the 
common axis of the first and second vector beams so as 
to perform matrix multiplication of the first and second 
vector beams using the nonlinear photorefractive crys- 
tal as a four-wave mixer to produce a conjugate beam as 
the matrix multiplication product of the frrst and second 
vector beams. A beam-splitter separates the conjugate 
beam from the pumping beam while reflecting the 
pumping beam onto the nonlinear photorefractive crys- 
tal, thereby to provide an output tensor beam. 
The novel features that are considered characteristic 
of this invention are set forth with particularity in the I 
appended claims. The invention will best be understood 
from the following description when read in connection 
with the accompanying drawings. 
BRIEF DESCRIPTION OF THE DRAWINGS 
FIG. 1 illustrates basic four-wave mixing of three 
input waves in a crystal of nonlinear photorefractive 
material. 
FIG. 2 illustrates the architecture of a tensor genera- 
tor using a crystal of nonlinear photorefractive material 
in accordance with the present invention. 
t h m v , c =  
0 0 0 0 0 0 0 0  pumping beam onto said one side of said crystal, 
0 0 0 0 0 0 0 0  thereby to produce four-wave mixing in order to 
l l l o l l o l  generate a conjugated beam from said nonlinear 
0 0 0 0 0 0 0 0  photorefractive crystal that represents said second- 
11101101 rank tensor. 
11101101 65 2. A real-time optical second-rank tensor generator as 
60 
1 1  1 0 1  101 
5,005,954 
5 6 
3. A real-time optical second-rank generator as de- 
fined in claim 2 wherein said first and second vector 
beams are spatially modulated in amplitude to set values 
of vector components. 
second-rank tensor by multiplication of two vectors 
using a nonlinear refractive crystal having two parallel 
sides, comprising the steps of 
generating separately a first and a second orthogo- 
nally disposed spatially modulated vector beam of 10 
coherent light representing linear arrays of compo- 
nents of respective ones of said two vectors, 
expanding one of said two vectors in a direction per- 
pendicular to its linear array of vector components, 
and expanding the other of said two vectors in a I5 
direction perpendicular to its linear array of vector 
components, said pumping beam. 
collimating said two vector beams after expansion 
onto said parallel sides of said nonlinear refractive 
crystal with one vector beam on one side and the 20 values of vector components. 
other vector beam on the other side of said two 
parallel sides, and with the two vector beams in 
exact opposition, 
generating a coherent plane wave pumping beam for 
all of said two vector beams expanded and reflect- 
ing said pumping beam onto one of said two paral- 
lel sides of said nonlinear refractive crystal using a 
beamsplitter, said pumping beam being reflected 
onto said one of said two parallel sides of said crys- 
tal at an angle with said two vector beams, thereby 
to provide four-wave mixing of said pumping beam 
with said two vector beams to produce a conjugate 
beam that represents said second-rank tensor prop- 
agating in exact opposition with said pumping 
beam. 
5. A method as defined in claim 4 wherein said two 
vector beams are generated at the same frequency as 
6. A method as defined in claim 5 wherein said two 
vector beams are spatially modulated in amplitude to set 
4. A method for real-time optical generation of a 5 
* * * * *  
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Method and apparatus for second-rank tensor generation

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