A proposal and an analysis are presented for imaging through a distorting medium using field multiplication by four-wave mixing

Koch, Thomas L.

Yariv, Amnon

English

Caltech Authors - Main

March 1982 / Vol. 7, No. 3 / OPTICS LETTERS 113
One-way coherent imaging through a distorting medium
using four-wave mixing
Amnon Yariv and Thomas L. Koch
California Institute of Technology, Pasadena, California 91125
Received November 23,1981
A proposal and an analysis are presented for imaging through a distorting medium using field multiplication by
four-wave mixing.
The recent flourish of activity in phase-conjugation
optics has revived the interest in real-time imaging
through distorting media. Some demonstrations of
imaging using thin conventional holograms have been
reported earlierl' 3 as well as real-time imaging schemes
using the basic configuration of phase-conjugation op-
tics.4 -6 In this latter case, the image was obtained on
the same side of the distortion as the object, which is a
disadvantage, if not an impossibility, in many cases of
practical interest. An extension of these imaging ideas
to the case of real-time nonlinear mixing media has also
been considered. 7
The scheme proposed here is illustrated in Fig. 1. An
image f(x") is to be transmitted from plane (3) to plane
(1) through an intervening distorting medium that is
characterized by a transmittance exp[-i4'(x')]. The
proposed method of compensation consists of inter-
secting in a nonlinear medium (NLM) three waves:
(A) A wave Al produced by passing a plane wave
through the distortion and then imaging the latter on
the NLM.
(B) A picture-bearing wave A2 arriving at the NLM
from the direction opposite to that of Al.
(C) A spherical reference wave A3 counterpropa-
gating along the same axis as the picture-bearing wave
A2.
Picture-wave (B):
E2 (X, z) = A 2 eiz f d2xl )X(12 - Z)ei(x)
X expF- 2( 2 _) Ix - xI' 12j 
Reference-wave (C):
E3 (x, z) = A3(z) exp[-ikz 2(d ) 1x2 -
(1)
(2)
Wave (A) is given in plane s, just to the left of the lens,
by
El(s) = Al ' d 2 x' exp[-<P(x') -ik Ix' - s1 2
(3)
The effect of the lens is to multiply E1 (s) by
exp(ik I s I 2/2f). Next we propagate it to the point (x,
z) in the NLM, where it becomes
Ei(x, z) = A(l eikz d2 x' f d2s
X exp[-i4)(x') - ik IXI - s12
2exp12 f1 S 2(11 + z) (4)
While we assume that all three waves propagate along
the same axis, there will always be some tolerance for
wave A1 to be slightly off-axis. Our analysis of this
situation adopts the following plan: (a) We propagate
each of the three fields from their respective starting
planes to the NLM. (b) We obtain the nonlinear po-
larization PNL in the NLM that is proportional to the
product of the three fields. (c) We solve for the field at
plane (1) that is radiated by PNL. We will show that
this field is of the same form as that which would have
been produced at plane (1) by the propagating pic-
ture-bearing wave without any distortion.
Using the Fresnel-Kirchhoff diffraction integral and
dropping constant phase factors and the time factor
exp(iwt) throughout, we write the fields (B) and (C)
inside the NLM as:
U), 
A Ik-)
(A) :
RI plane
§ I V3 ' Plane
¾ plane
BS
XC(C)
( X "plane
I (r B)
z= C 1rr0sp.,en.y
1;X !
disto rting
-neaian-.
Fig. 1. The configuration for propagating an image from
plane (3) through a distorting medium to plane (1) with no
distortion.
0146-9592/82/030113-03$1.00/0 ©o 1982, Optical Society of America
_ dI __-4. 'eI  2 !
114 OPTICS LETTERS / Vol. 7, No. 3 / March 1982
If the NLM is short, i.e., z <« d2 , 11, 12, in the phase
factors, we can expand
1 1 z
d+z d2 d 2dZ + Z  
1 1 z
11 +Z 11 112'
12 Z 12 T22
and ignore the z dependence of the denominators pre-
ceding the integrals. These relations will be used in
what follows.
The intersection of the three waves in the NLM gives
rise to a polarization that is proportional to the product
of all three. The term of interest is that involving
Ei*E2E3 since it oscillates at the fundamental fre-
quency co and the term E1* signifies a reversal of the
sign of the distortion. Also this polarization radiates
in the desired -z direction toward plane (1). Thus the
nonlinear polarization induced in the NLM is given
by
PNL= X=3)El*E2E3, (6)
where X(3) is the third-order nonlinear susceptibility of
the medium. Using the expressions (1), (2), and (4) for
the fields as well as (6) and assuming that the distortion
is imaged by the lens at z = 0 so that f- 1 = di-' + 11-1,
we obtain
PNL(X, Z) = x(
3)Al*A2A3(z = 0)
X3 dl 1l12
x f fF dx d2x-d2xfd2Sf(x)ei'Wx)
X exp (- ik Ix 1 2 - ikz X 1 2)
X exp(- 2Ž Ix - x"12 - ikz IX - X"Ip 2 12 2122
X exp hI12+i 
2
ik +dlx ikZ I_ S2 7
-dix'±%x)*s 2 1 1 2 sI2j. (7)
The field at the s' plane that is due to PNL is obtained
using a Green function solution
E(S') = 4d fPNL(xz)e d2xdz, (8)
A D ~~r
where r = [lx - s1I2 + (1I + Z)2] 112. In the Fresnel ap-
proximation limit this becomes
E (s') = 2 4' d 2xdzPNL(x, z)
X exp -ikz---Ix - s' 1 2 + ihZIX - S' 2). (9)21, 2112 I '
The effect of the lens again is to multiply E(s') by
exp(ik s' 1 2/2f). The resulting field propagates to the
X plane (2) and is there multiplied by the distortion
function exp[-Ui(77)]. If we recall the imaging condi-
tion f1 I = dv'l + 1I-I we obtain after some algebra the
desired output field
E(Q) = .4"2x1(3)A*A 2A 3 -1'(q)
X ST Si 55 dzd2 xd2x'd 2x"d 2 sd2s'
X e""(>)f(x") exp (- -i jq,2 - d lxI 2X e ~ ~ \2d1 2d2
2d 2 1 2 x 2)
X exp -- (X' + di X .s
Xexp Wt & + qix) . sj
X 1exp 2id2 2 1x12 - i2z2 IX -xI' l 2
+ I2 } x-s'I 2 -i x -I s2)} (10)
For sufficiently thin NLM such that
kz Ix or x or s or 2 << 
(l11Or1 2 ord2)2
the last factor in the curly brackets is very nearly unity.
Ignoring finite aperature effects, the integrations over
s and s'yield
(d1X)2 6 (X'+ x) and (J1X)26 x+ 77)
respectively. These collapse the x' and x integrations
to get x' - , x - -(lI/dl), and then Eq. (10) be-
comes
E(n) = [41r2X(3)Al*A3t/XI exp ( 2d2d 2 In 12)
X .42 d2x'f(Xm ) exp (- 4 1/ + x"| 2)
(11)
where t is the thickness (along z) of the NLM.
If d2 - A, i.e., the reference wave (C) is a plane wave,
then we have according to Eq. (11) in the v plane an
enlarged (by d1/ll) and inverted version of the original
object field having propagated from an effective plane
a distance 12 to the right without distortion. For finite
d2 the resulting field is the same as in Eq. (11) with the
additional effect of a lens with focal length d2d1 2/112 in
the , plane. This completes the proof.
The factor in the square brackets preceding the in-
tegral in Eq. (11) represents the change in the amplitude
of the image field between planes (3) and (1) that is due
to nonlinear mixing. This factor is therefore the ef-
fective amplitude gain
G - 47r2X(3)Al*A3t/X (12)
and can exceed unity for sufficiently strong fields. The
proposed scheme can thus be used for image amplifi-
cation as well as correction.
In conclusion, a method for a one-way imaging
through a turbulent medium using four-wave mixing in
a nonlinear medium has been proposed and analyzed.
March 1982 / Vol. 7, No. 3 / OPTICS LETTERS 115
The authors would like to acknowledge especially
insightful discussions with and valuable suggestions of
B. Fischer, J. White, and M. Cronin-Golomb.
This work was supported by the U.S. Army Research
Office, Durham, North Carolina.
References
1. E. N. Leith and J. Upatnieks, "Holographic imagery
through diffusing media," J. Opt. Soc. Am. 56, 523
(1966).
2. H. Kogelnik, "Holographic image projection through in-
homogenious media," Bell Syst. Tech. J. 44, 2451 (1965).
3. J. W. Goodman et al., "Wavefront reconstruction imaging
through random media," Appl. Phys. Lett. 8, 311 (1966).
4. B. I. Stepanov, E. V. Ivakin, and A. S. Rubanov, "On reg-
istration of plane wave and volume dynamic holograms in
bleachable substances," Dokl. Akad. Nauk SSSR 196, 567
(1971).
5. J. P. Woerdman, "Some optical and electrical properties
of a laser generated free carrier plasma in Si," Ph.D. Thesis,
U. of Amsterdam (Dec. 1971). Also republished in Philips
Res. Dept. Suppl. 1971, No. 7.
6. A. Yariv, "On transmission and recovery of three dimen-
sional image information in optical waveguides," J. Opt.
Soc. Am. 66, 301 (1976).
7. T. R. O'Meara, "Applications of nonlinear phase conju-
gation in compensated active media," Proc. Soc. Photo-
Opt. Instrum. Eng. 168, 542 (1979).


One-way coherent imaging through a distorting medium using four-wave mixing

Applications of nonlinear phase conjugation in compensated active media,&quot;

Holographic image projection through inhomogenious media,&quot;

Holographic imagery through diffusing media,&quot;

On registration of plane wave and volume dynamic holograms in bleachable substances,&quot;

On transmission and recovery of three dimensional image information in optical waveguides,&quot;

Some optical and electrical properties of a laser generated free carrier plasma in Si,&quot;

Wavefront reconstruction imaging through random media,&quot;

http://authors.library.caltech.edu/9803/1/YARol82.pdf

One-way coherent imaging through a distorting medium using four-wave mixing

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