We report here the demonstration of several new optical oscillator configurations including a unidirectional ring oscillator and a self‐pumped phase conjugate mirror. The gain medium is BaTiO_3, pumped by a 632.8‐nm He‐Ne laser at power levels down to 50 μW

Cronin-Golomb, Mark

Fischer, Baruch

White, Jeffrey O.

Yariv, Amnon

English

Caltech Authors - Main

Downloaded 24 May 2006 to 131.215.240.9. Redistribution subject to AIP license or copyright, see http://apl.aip.org/apl/copyright.jsp
Coherent oscillation by self-induced gratings in the photorefractive crystal 
BaTi03 
Jeffrey 0. White, Mark Cronin-Golomb, Baruch Fischer, and Am non Yariv 
Department of Applied Physics, California Institute of Technology, Pasadena, California 91125 
(Received 10 August 1981; accepted for publication 14 December 1981) 
We report here the demonstration of several new optical oscillator configurations including a 
unidirectional ring oscillator and a self-pumped phase conjugate mirror. The gain medium is 
BaTi03, pumped by a 632.8-nm He-Ne laser at power levels down to 50f-lW. 
PACS numbers: 42.65. - k, 42.55.Rz, 42.60.Da, 42.60.By 
The possibility of phase conjugate mirrors (PCM's) 
with reflectivities exceeding unity was predicted by Yariv 
and Pepper, 1 and subsequently observed by Bloom, Liao, 
and Economou,2 and by Pepper, Fekete, and Yariv. 3 The 
properties of resonators with PCM's have been studied ex-
perimentally and theoretically.4 Early work employed high 
intensity Q switched lasers as the pump beams.5 Because of 
formal similarities of four-wave mixing in third-order opti-
cally nonlinear media, and in real-time holographic media/' 
the considerations of these early works may be carried over 
to the analysis at PCM's based on real-time holography in 
photorefractive media. For example, in a recent experiment 
with BaTi03 as the holographic photorefractive medium, 
Feinberg and Hellwarth 7 demonstrated a PCM with reflec-
tivity exceeding unity using a cw argon laser at 514.5 nm for 
the pump beams. They used this mirror in conjunction with 
an ordinary mirror to demonstrate a cw oscillator of the type 
shown in Fig. l(a) and to examine its behavior with intraca-
vity spatial filters and phase distorters. 
In this letter, we report the demonstration of several 
new oscillator configurations involving BaTi03 : (i) an oscil-
lator using two conventional mirrors in which the gain is 
provided by two pumping beams [Fig. l(b)]; (ii) an oscillator 
pumped by a single beam using two conventional mirrors 
[Fig. l(c)]; (iii) a unidirectional ring oscillator pumped by a 
single beam [Fig. 1(d)]; (iv) a ring oscillator pumped with two 
counter-propagating beams [Fig. l(e)]. 
We pumped these oscillators, as well as the one shown 
in Fig. l(a) with a low-power ( -1 mW) He-Ne laser operat-
ing at 632.8 nm. This is the first demonstration of cw BaTi03 
oscillators with the He-Ne laser. This new oscillator technol-
ogy has thus been brought within the reach of even modestly 
equipped laboratories. 
The above oscillators do not operate by the simulated 
emission of radiation. Small signals experience gain when 
one or both pumps are diverted in the proper direction by 
Bragg reflection from self-induced index gratings as ex-
plained below. 
In the above oscillators, the amplified signal is fed back 
by one or more mirrors in the form of a resonator. Our reso-
nators were constructed with plane mirrors so that they were 
unstable to diffraction loss, except for any stabilization due 
to the phase-conjugating properties of the crystal. Experi-
ments are currently in progress to examine the effect of reso-
nator stabilization using spherical mirrors. We tried one res-
onator [Fig. 1(f)] in which the feedback was via a Bragg 
reflection inside the crystal, as in DFB lasers. No oscillation 
has been observed yet. 
The crystal ofBaTi03 measured 7 X 4.5 X 4 mm. KIt was 
poled into a single domain so that the c axis was parallel to 
the 4 mm side. In order to make use of the large electro-optic 
coefficient r42 = 820X 10- 12 mv- \ 9 the crystal was orient-
ed so that the wave vectors of the holographic gratings were 
not parallel to any crystal axis. The polarization of all beams 
was extraordinary, and they were made to travel predomi-
nantly in the direction of the c axis. The polarity of the c axis 
determines the direction in which power is transferred in two 
beam coupling experiments. 10 The four beams shown in Fig. 
2 enter and leave the crystal through the two parallel crystal 
faces which measure 7X4.5 mm. Coupling is also possible 
when the beams 1 and 4 enter different faces, although the 
interaction region may be reduced. 
In each resonator pump A 1 is delivered directly from 
the laser and has a diameter of approximately 3 mm. Table I 
shows the steady-state intensities of the oscillation beams A.~ 
and A 4 as they left the cuvette, and the time taken to reach 
steady state. They were measured either directly or through 
1% transmitting mirrors. In each case, the angle a was 168° 
and,Bwas 160°. 
Ia! 
3 >..M2 
~ ,.:' 2 
M3'( 4 
(c) 
M4_.r ____ --A M2 
I ! ~ j"'y.if_3 
Ml~
I pr 1 
M3'*: 4 
lei 
FIG. 1. Arrangement of barium titanate crystal(rectangle) mirrors (labelled 
as referred to in text) and light beams of the oscillators (a)-(f) as described in 
the text. 
450 Appl. Phys. Lett. 40(6), 1 5 March 1982 0003-6951/82/060450-03$01 .00 © 1982 American Institute of Physics 450 
Downloaded 24 May 2006 to 131.215.240.9. Redistribution subject to AIP license or copyright, see http://apl.aip.org/apl/copyright.jsp
PUMP 
A2 I 
I 
/oSCILLATION 
I A4 
OSCILLATION 1 
A
3 
I 
I 
I 
I PUMP 
A, 
FIG. 2. Expanded view of the interaction region common to each of the 
oscillators, showing the pumping beams (solid), oscillation beams (dashed), 
and the relative orientation of the c axis of the crystal. 
In the resonator of F(g. l(a) pump A2 is provided by 
retroreflecting A 1 at the mirror M 1. It is known that in this 
configuration, beam A4 can experience amplified-phase con-
jugate reflection. 6 Oscillation can thus build up between M 2 
and the crystal, provided the reflection gain is sufficient to 
compensate for the losses in the oscillator. When the mirror 
M 2 is placed on the opposite side of the crystal 
(/J---+-/3 + 180"), the oscillation strength increases by a factor 
of 10. A preliminary check indicated that oscillation was 
possible in the angular ranges 155• <!3 < 160• and 
- 22• <!3 < - 12•. In terms of angles measured outside the 
crystal, the oscillation could be observed over a total range of 
so·. We note here that the threshold behavior of this device is 
quite different from that oflasers. It was observed that oscil-
lation builds up even with the very low pumping power of 50 
f.J-W in A,, although the time taken to reach steady state in 
that case was as long as eight minutes. 
Since one mirror on either side of the crystal is sufficient 
to obtain oscillation, it was not surprising to observe oscilla-
tion in the configuration of Fig. l(b). It was surprising, how-
ever, to see oscillation build up even when the counterpropa-
gating pumpA 2 is removed as in Fig. l(c). We believe that the 
mechanism of oscillation in this case is as follows: Beam A4 
interacts in the crystal with the pump A 1 to form a photore-
fractive grating. The direction of the c axis is such that, as 
explained by two-beam coupling theory, 10 beam A 4 is ampli-
fied by A 1• This amplified beam is fed back into the direction 
ofA4 by mirrorsM 3 andM2. If the gain exceeds the absorp-
tion and reflection losses, an oscillation will build up. 
It was observed that a fourth beam, A 7, was generated, 
traveling backward relative to A 1, and having up to 10% of 
the intensity of A 1• This is presumably light scattered from 
A 3 by the grating formed between A, andA 4 • This device can 
be viewed as a signal-powered phase conjugate mirror, i.e., a 
"black box" which reflects a phase conjugate of an input 
TABLE I. Oscillation strengths and rise times for the resonators of Fig. I. 
(a) (b) (c) (d) (e) 
/ 3(mW) 0.1 0.8 0.3 0 0.2 
/ 4 (mW) 2 3 3 3 3 
'T(sec) 8 8 100 8 8 
451 Appl. Phys. Lett., Vol. 40, No. 6, 15 March 1982 
wave without a coherent pump source. It is important to 
note that the orientation of the parallel mirror/crystal com-
bination with respect to beam A 1 is not critical. The oscilla-
tor acts as a PCM with a wide spatial bandwidth; we have 
observed that two beams incident simultaneously on this 
PCM were reflected with 10% reflectivity, even though they 
were separated in angle by about ten degrees. 
In Fig. 1(d) we show a unidirectional ring oscillator. Its 
operation is due to the preferential amplification by A 1 of A 4 
compared to that ofA3• When two pumps are provided, Fig. 
1(e), oscillation is observed in both directions. 
The unidirectional ring oscillator can be described by 
two-beam coupling theory. 10 Consider coupling of A 1 and 
A4. Let e, be the angle between beam A I and the normal to 
the crystal surface. Let 04 be the corresponding angle for A4 • 
Let z be the distance measured along the normal to the crys-
tal surface, and Jet the length of the crystal along this direc-
tion be!. We write the total optical field as the sum of two 
beams 
E (r,t) =A 1(r)eilk,·r ~ '"' 1 +c. c. 
+ A4(r)eilk,·• ~ wtl + c.c. ( 1) 
The interference of these beams in a photorefractive 
crystal creates a phase grating which may be represented by 
a perturbation on the mean refractive index. 
(2) 
where k, the wave vector of the grating, is equal to k4 - k 1• 
The phase </J indicates the degree to which the grating is shift-
ed spatially with respect to the light interference pattern. In 
photorefractive media which can operate by diffusion only, 
e.g., BaTi03 , the magnitude of </J is 1r/2 with its sign depend-
ing on the direction of the c axis. / 0 is the intensity sum 
lA 11
2 + IA412• The parameter b depends on the magnitude of 
the grating vector and its direction and on the material prop-
erties of the crystal, e.g., the electro-optic coefficient. In Ba-
Ti03, holographic two-beam coupling was first observed in 
1980. 11 In that previous study the k vector of the holograph-
ic grating was arranged to be parallel to the c axis of the 
crystal, so that no advantage was taken at the large electro-
optic coefficient r42• 
Using the standard slowly varying field approximation 
and the scalar wave equation, we can obtain the following 
coupled wave equations for A 1 and A4 • 
6
•
10 
dA 1 ibwe,."' A,IA
4
I2, 
dz 2clo cos e 
(3) 
dA 4 ibwe ~ i<b A
4
jA ,j\ 
dz 2clo cos e 
where we have taken e, = 04 = Bforsimplicity. From (3) we 
can derive equations for the intensities / 1 = lA 1 1
2 and 
[4 = IA41 2• 
dl, 
= 
_ ri,I4 
dz Io 
d/4 
-= 
Fl.f4 
(4) 
dz Io 
White etal. 451 
Downloaded 24 May 2006 to 131.215.240.9. Redistribution subject to AIP license or copyright, see http://apl.aip.org/apl/copyright.jsp
where r = - bw sin¢> /c cos e, so that 14 exhibits gain for 
positive r. We note that the total energy is conserved, 
dl0 =O 
dz ' 
so that 10 is constant. The solution ofEqs. (3) and (4) is 
I.(z) = Io 
1 + [14(0)/11(0)]erz ' 
14(z) = Io 
1 + [11(0)/14(0)]e-rz 
(5) 
(6) 
When we apply boundary condition appropriate to a ring 
oscillator, 
(7) 
where L represents the cavity's fractional linear intensity 
loss per round trip we find that the ratio of oscillating energy 
to pumping energy is 
1- e-n 
-1. 
L 
We note that 14(0)/11(0) is infinite for L = 0 and becomes 
zero, i.e., no oscillation at L = 1 - e-n. 
(8) 
In conclusion, we have demonstrated a number of new 
oscillators involving four-wave mixing in BaTi03 . Oscilla-
tion is achieved with a low-power He-Ne laser at 632.8 nm 
supplying the pump beam(s). 
This work was supported by the Air Force Office of 
Scientific Research and the Army Research Office. B. Fi-
scher would like to acknowledge the support of the Weiz-
mann post-doctoral fellowship and M. Cronin-Golomb the 
support of the University of Sydney. This paper is dedicated 
to Alice and Esther. 
1 A. Yariv and D. M. Pepper, Opt. Lett. 1, 16 ( 1977). 
'D. M. Bloom, P. F. Liao, and N. P. Economou, Opt. Lett. 2, 58(1978). 
'D. M. Pepper, D. Fekete, and A. Yariv, Appl. Phys. Lett. 33, 41 (1978). 
4 P. A. Belanger, A. Hardy, and A. E. Siegman, Appl. Opt. 19,602 (1980). 
'J. AuYeung, D. Fekete, D. M. Pepper, and A. Yariv, IEEE J. Quantum 
Electron. QE-15, 1180 ( 1979). 
"B. Fischer, M. Cronin-Golomb, J. 0. White, and A. Yariv, Opt. Lett. 6, 
519 (1981). 
71. Feinberg and R. W. Hellwarth, Opt. Lett. 5, 319 ( 1980). 
"Sanders Associates, 95 Canal Street, Nashua, NH 03060. 
9A. Yariv, Quantum Electronics, 2nd ed. (Wiley, New York, 1975). 
'"V. L. Vinetskii, N. V. Kukhtarev, S. G. Odu1ov, and M.S. Soskin, Usp. 
Fiz. Nauk 129, 113 (1979) [Sov. Phys. Usp. 22, 742 (1979)]. 
''J. Feinberg, D. Heiman, A. R. Tanguay, Jr., and R. W. Hellwarth, J. 
Appl. Phys. 51, 1297 (1980). 
Effect of superluminescence on the modulation response of semiconductor 
lasers 
KamY. Lau 
Ortel Corporation, Alhambra, California 91803 
Amnon Yariv 
California Institute of Technology, Pasadena, California 91125 
(Received 6 November 1981; accepted for publication 5 January 1982) 
The small-signal modulation response of semiconductor lasers with a very small mirror 
reflectivity is analyzed. Superluminescent effects inside the laser cavity provide yet another 
mechanism for damping relaxation oscillation resonance. These results can serve as useful 
guides in designing high frequency semiconductor lasers. 
PACS numbers: 42.55.Px, 42.60.Fc, 42.80.Sa 
Semiconductor lasers are potentially important as light 
source for optical communication systems. Consequently, 
methods for improving the modulation response of these de-
vices are topics of great interest. The small-signal modula-
tion behavior of the laser is well established experimentally 
and theoretically. 1 The absence of a resonance peak in the 
modulation response of some lasers has been attributed to a 
variety of causes including the high spontaneous emission 
factor and/or lateral carrier diffusion. In this letter we point 
out yet another mechanism that can lead to a fiat frequency 
response. This mechanism involves the nonuniform distri-
bution of photons and carriers along the length of a laser 
cavity with a small mirror reflectivity. This effect is particu-
larly important when one attempts to increase the modula-
tion speed by reducing the end-facet reflectivities in order to 
reduce the lifetime of the cavity.2 
Previous considerations of nonuniform distribution of 
photons and carriers occurred in the context of super lu-
minescent light emitting diodes (LED's)-diodes with total-
ly suppressed mirror feedback to prevent lasing. 3- 7 Total 
suppression of mirror feedback is unattainable in practice. A 
very low reflectivity at one mirror can be obtained by leaving 
a long section of the laser unpumped3; nevertheless, scatter-
ing from material defects and waveguide irregularities make 
452 Appl. Phys. Lett. 40(6), 15 March 1982 0003-6951/82/060452-03$01.00 © 1982 American Institute of Physics 452 


Coherent oscillation by self‐induced gratings in the photorefractive crystal BaTiO_3

Downloaded 24 May 2006 to 131.215.240.9. Redistribution subject to AIP license or copyright, see http://apl.aip.org/apl/copyright.jsp
Coherent oscillation by self-induced gratings in the photorefractive crystal 
BaTi03 
Jeffrey 0. White, Mark Cronin-Golomb, Baruch Fischer, and Am non Yariv 
Department of Applied Physics, California Institute of Technology, Pasadena, California 91125 
(Received 10 August 1981; accepted for publication 14 December 1981) 
We report here the demonstration of several new optical oscillator configurations including a 
unidirectional ring oscillator and a self-pumped phase conjugate mirror. The gain medium is 
BaTi03, pumped by a 632.8-nm He-Ne laser at power levels down to 50f-lW. 
PACS numbers: 42.65. - k, 42.55.Rz, 42.60.Da, 42.60.By 
The possibility of phase conjugate mirrors (PCM's) 
with reflectivities exceeding unity was predicted by Yariv 
and Pepper, 1 and subsequently observed by Bloom, Liao, 
and Economou,2 and by Pepper, Fekete, and Yariv. 3 The 
properties of resonators with PCM's have been studied ex-
perimentally and theoretically.4 Early work employed high 
intensity Q switched lasers as the pump beams.5 Because of 
formal similarities of four-wave mixing in third-order opti-
cally nonlinear media, and in real-time holographic media/' 
the considerations of these early works may be carried over 
to the analysis at PCM's based on real-time holography in 
photorefractive media. For example, in a recent experiment 
with BaTi03 as the holographic photorefractive medium, 
Feinberg and Hellwarth 7 demonstrated a PCM with reflec-
tivity exceeding unity using a cw argon laser at 514.5 nm for 
the pump beams. They used this mirror in conjunction with 
an ordinary mirror to demonstrate a cw oscillator of the type 
shown in Fig. l(a) and to examine its behavior with intraca-
vity spatial filters and phase distorters. 
In this letter, we report the demonstration of several 
new oscillator configurations involving BaTi03 : (i) an oscil-
lator using two conventional mirrors in which the gain is 
provided by two pumping beams [Fig. l(b)]; (ii) an oscillator 
pumped by a single beam using two conventional mirrors 
[Fig. l(c)]; (iii) a unidirectional ring oscillator pumped by a 
single beam [Fig. 1(d)]; (iv) a ring oscillator pumped with two 
counter-propagating beams [Fig. l(e)]. 
We pumped these oscillators, as well as the one shown 
in Fig. l(a) with a low-power ( -1 mW) He-Ne laser operat-
ing at 632.8 nm. This is the first demonstration of cw BaTi03 
oscillators with the He-Ne laser. This new oscillator technol-
ogy has thus been brought within the reach of even modestly 
equipped laboratories. 
The above oscillators do not operate by the simulated 
emission of radiation. Small signals experience gain when 
one or both pumps are diverted in the proper direction by 
Bragg reflection from self-induced index gratings as ex-
plained below. 
In the above oscillators, the amplified signal is fed back 
by one or more mirrors in the form of a resonator. Our reso-
nators were constructed with plane mirrors so that they were 
unstable to diffraction loss, except for any stabilization due 
to the phase-conjugating properties of the crystal. Experi-
ments are currently in progress to examine the effect of reso-
nator stabilization using spherical mirrors. We tried one res-
onator [Fig. 1(f)] in which the feedback was via a Bragg 
reflection inside the crystal, as in DFB lasers. No oscillation 
has been observed yet. 
The crystal ofBaTi03 measured 7 X 4.5 X 4 mm. KIt was 
poled into a single domain so that the c axis was parallel to 
the 4 mm side. In order to make use of the large electro-optic 
coefficient r42 = 820X 10- 12 mv- \ 9 the crystal was orient-
ed so that the wave vectors of the holographic gratings were 
not parallel to any crystal axis. The polarization of all beams 
was extraordinary, and they were made to travel predomi-
nantly in the direction of the c axis. The polarity of the c axis 
determines the direction in which power is transferred in two 
beam coupling experiments. 10 The four beams shown in Fig. 
2 enter and leave the crystal through the two parallel crystal 
faces which measure 7X4.5 mm. Coupling is also possible 
when the beams 1 and 4 enter different faces, although the 
interaction region may be reduced. 
In each resonator pump A 1 is delivered directly from 
the laser and has a diameter of approximately 3 mm. Table I 
shows the steady-state intensities of the oscillation beams A.~ 
and A 4 as they left the cuvette, and the time taken to reach 
steady state. They were measured either directly or through 
1% transmitting mirrors. In each case, the angle a was 168° 
and,Bwas 160°. 
Ia! 
3 >..M2 
~ 
,.:' 2 
M3'( 4 
(c) 
M4_.r ____ --A M2 
I ! ~ j"'y.if_3 Ml~
I pr 1 
M3'*: 4 
lei 
FIG. 1. Arrangement of barium titanate crystal(rectangle) mirrors (labelled 
as referred to in text) and light beams of the oscillators (a)-(f) as described in 
the text. 
450 Appl. Phys. Lett. 40(6), 1 5 March 1982 0003-6951/82/060450-03$01 .00 © 1982 American Institute of Physics 450 
Downloaded 24 May 2006 to 131.215.240.9. Redistribution subject to AIP license or copyright, see http://apl.aip.org/apl/copyright.jsp
PUMP 
A2 I 
I 
/oSCILLATION 
I A4 
OSCILLATION 1 
A 3 I I 
I 
I PUMP 
A, 
FIG. 2. Expanded view of the interaction region common to each of the 
oscillators, showing the pumping beams (solid), oscillation beams (dashed), 
and the relative orientation of the c axis of the crystal. 
In the resonator of F(g. l(a) pump A2 is provided by 
retroreflecting A 1 at the mirror M 1. It is known that in this 
configuration, beam A4 can experience amplified-phase con-
jugate reflection. 6 Oscillation can thus build up between M 2 
and the crystal, provided the reflection gain is sufficient to 
compensate for the losses in the oscillator. When the mirror 
M 2 is placed on the opposite side of the crystal 
(/J---+-/3 + 180"), the oscillation strength increases by a factor 
of 10. A preliminary check indicated that oscillation was 
possible in the angular ranges 155• <!3 < 160• and 
- 22• <!3 < - 12•. In terms of angles measured outside the 
crystal, the oscillation could be observed over a total range of 
so·. We note here that the threshold behavior of this device is 
quite different from that oflasers. It was observed that oscil-
lation builds up even with the very low pumping power of 50 
f.J-W in A,, although the time taken to reach steady state in 
that case was as long as eight minutes. 
Since one mirror on either side of the crystal is sufficient 
to obtain oscillation, it was not surprising to observe oscilla-
tion in the configuration of Fig. l(b). It was surprising, how-
ever, to see oscillation build up even when the counterpropa-
gating pumpA 2 is removed as in Fig. l(c). We believe that the 
mechanism of oscillation in this case is as follows: Beam A4 
interacts in the crystal with the pump A 1 to form a photore-
fractive grating. The direction of the c axis is such that, as 
explained by two-beam coupling theory, 10 beam A 4 is ampli-
fied by A 1• This amplified beam is fed back into the direction 
ofA4 by mirrorsM 3 andM2. If the gain exceeds the absorp-
tion and reflection losses, an oscillation will build up. 
It was observed that a fourth beam, A 7, was generated, 
traveling backward relative to A 1, and having up to 10% of 
the intensity of A 1• This is presumably light scattered from 
A 3 by the grating formed between A, andA 4 • This device can 
be viewed as a signal-powered phase conjugate mirror, i.e., a 
"black box" which reflects a phase conjugate of an input 
TABLE I. Oscillation strengths and rise times for the resonators of Fig. I. 
(a) (b) (c) (d) (e) 
/ 3(mW) 0.1 0.8 0.3 0 0.2 
/ 4 (mW) 2 3 3 3 3 
'T(sec) 8 8 100 8 8 
451 Appl. Phys. Lett., Vol. 40, No. 6, 15 March 1982 
wave without a coherent pump source. It is important to 
note that the orientation of the parallel mirror/crystal com-
bination with respect to beam A 1 is not critical. The oscilla-
tor acts as a PCM with a wide spatial bandwidth; we have 
observed that two beams incident simultaneously on this 
PCM were reflected with 10% reflectivity, even though they 
were separated in angle by about ten degrees. 
In Fig. 1(d) we show a unidirectional ring oscillator. Its 
operation is due to the preferential amplification by A 1 of A 4 
compared to that ofA3• When two pumps are provided, Fig. 
1(e), oscillation is observed in both directions. 
The unidirectional ring oscillator can be described by 
two-beam coupling theory. 10 Consider coupling of A 1 and 
A4. Let e, be the angle between beam A I and the normal to 
the crystal surface. Let 04 be the corresponding angle for A4 • 
Let z be the distance measured along the normal to the crys-
tal surface, and Jet the length of the crystal along this direc-
tion be!. We write the total optical field as the sum of two 
beams 
E (r,t) =A 1(r)eilk,·r ~ '"' 1 +c. c. 
+ A4(r)eilk,·• ~ wtl + c.c. ( 1) 
The interference of these beams in a photorefractive 
crystal creates a phase grating which may be represented by 
a perturbation on the mean refractive index. 
(2) 
where k, the wave vector of the grating, is equal to k4 - k 1• 
The phase </J indicates the degree to which the grating is shift-
ed spatially with respect to the light interference pattern. In 
photorefractive media which can operate by diffusion only, 
e.g., BaTi03 , the magnitude of </J is 1r/2 with its sign depend-
ing on the direction of the c axis. / 0 is the intensity sum 
lA 112 + IA412• The parameter b depends on the magnitude of 
the grating vector and its direction and on the material prop-
erties of the crystal, e.g., the electro-optic coefficient. In Ba-
Ti03, holographic two-beam coupling was first observed in 
1980. 11 In that previous study the k vector of the holograph-
ic grating was arranged to be parallel to the c axis of the 
crystal, so that no advantage was taken at the large electro-
optic coefficient r42• 
Using the standard slowly varying field approximation 
and the scalar wave equation, we can obtain the following 
coupled wave equations for A 1 and A4 • 6• 10 
dA 1 ibwe,."' A,IA4I2, dz 2clo cos e 
(3) 
dA 4 ibwe ~ i<b A4jA ,j\ dz 2clo cos e 
where we have taken e, = 04 = Bforsimplicity. From (3) we 
can derive equations for the intensities / 1 = lA 1 12 and 
[4 = IA41 2• 
dl, 
= 
_ ri,I4 
dz Io 
d/4 
-= 
Fl.f4 (4) 
dz Io 
White etal. 451 
Downloaded 24 May 2006 to 131.215.240.9. Redistribution subject to AIP license or copyright, see http://apl.aip.org/apl/copyright.jsp
where r = - bw sin¢> /c cos e, so that 14 exhibits gain for 
positive r. We note that the total energy is conserved, 
dl0 =O 
dz ' 
so that 10 is constant. The solution ofEqs. (3) and (4) is 
I.(z) = Io 
1 + [14(0)/11(0)]erz ' 
14(z) = Io 
1 + [11(0)/14(0)]e-rz 
(5) 
(6) 
When we apply boundary condition appropriate to a ring 
oscillator, 
(7) 
where L represents the cavity's fractional linear intensity 
loss per round trip we find that the ratio of oscillating energy 
to pumping energy is 
1- e-n 
-1. 
L 
We note that 14(0)/11(0) is infinite for L = 0 and becomes 
zero, i.e., no oscillation at L = 1 - e-n. 
(8) 
In conclusion, we have demonstrated a number of new 
oscillators involving four-wave mixing in BaTi03 . Oscilla-
tion is achieved with a low-power He-Ne laser at 632.8 nm 
supplying the pump beam(s). 
This work was supported by the Air Force Office of 
Scientific Research and the Army Research Office. B. Fi-
scher would like to acknowledge the support of the Weiz-
mann post-doctoral fellowship and M. Cronin-Golomb the 
support of the University of Sydney. This paper is dedicated 
to Alice and Esther. 
1 A. Yariv and D. M. Pepper, Opt. Lett. 1, 16 ( 1977). 
'D. M. Bloom, P. F. Liao, and N. P. Economou, Opt. Lett. 2, 58(1978). 
'D. M. Pepper, D. Fekete, and A. Yariv, Appl. Phys. Lett. 33, 41 (1978). 
4 P. A. Belanger, A. Hardy, and A. E. Siegman, Appl. Opt. 19,602 (1980). 
'J. AuYeung, D. Fekete, D. M. Pepper, and A. Yariv, IEEE J. Quantum 
Electron. QE-15, 1180 ( 1979). 
"B. Fischer, M. Cronin-Golomb, J. 0. White, and A. Yariv, Opt. Lett. 6, 
519 (1981). 
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Effect of superluminescence on the modulation response of semiconductor 
lasers 
KamY. Lau 
Ortel Corporation, Alhambra, California 91803 
Amnon Yariv 
California Institute of Technology, Pasadena, California 91125 
(Received 6 November 1981; accepted for publication 5 January 1982) 
The small-signal modulation response of semiconductor lasers with a very small mirror 
reflectivity is analyzed. Superluminescent effects inside the laser cavity provide yet another 
mechanism for damping relaxation oscillation resonance. These results can serve as useful 
guides in designing high frequency semiconductor lasers. 
PACS numbers: 42.55.Px, 42.60.Fc, 42.80.Sa 
Semiconductor lasers are potentially important as light 
source for optical communication systems. Consequently, 
methods for improving the modulation response of these de-
vices are topics of great interest. The small-signal modula-
tion behavior of the laser is well established experimentally 
and theoretically. 1 The absence of a resonance peak in the 
modulation response of some lasers has been attributed to a 
variety of causes including the high spontaneous emission 
factor and/or lateral carrier diffusion. In this letter we point 
out yet another mechanism that can lead to a fiat frequency 
response. This mechanism involves the nonuniform distri-
bution of photons and carriers along the length of a laser 
cavity with a small mirror reflectivity. This effect is particu-
larly important when one attempts to increase the modula-
tion speed by reducing the end-facet reflectivities in order to 
reduce the lifetime of the cavity.2 
Previous considerations of nonuniform distribution of 
photons and carriers occurred in the context of super lu-
minescent light emitting diodes (LED's)-diodes with total-
ly suppressed mirror feedback to prevent lasing. 3- 7 Total 
suppression of mirror feedback is unattainable in practice. A 
very low reflectivity at one mirror can be obtained by leaving 
a long section of the laser unpumped3; nevertheless, scatter-
ing from material defects and waveguide irregularities make 
452 Appl. Phys. Lett. 40(6), 15 March 1982 0003-6951/82/060452-03$01.00 © 1982 American Institute of Physics 452 


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Coherent oscillation by self‐induced gratings in the photorefractive crystal BaTiO_3

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