We show theoretically that the incorporation of a frequency-dependent loss mechanism in a semiconductor laser can lead, in concert with the amplitude-to-phase coupling, to major reductions of the fundamental intensity and phase noise. A loss dispersion of the wrong sign, on the other hand, leads to an increase of the noise and, at a certain strength, to instability

Nabiev, Rashit

Vahala, Kerry

Yariv, Amnon

English

Caltech Authors - Main

December 1, 1990 / Vol. 15, No. 23 / OPTICS LETTERS 1359
Self-quenching of fundamental phase and amplitude noise in
semiconductor lasers with dispersive loss
Amnon Yariv, Rashit Nabiev,* and Kerry Vahala
California Institute of Technology, Pasadena, California 91125
Received June 18, 1990; accepted September 8, 1990; manuscript in hand October 12, 1990
We show theoretically that the incorporation of a frequency-dependent loss mechanism in a semiconductor laser can
lead, in concert with the amplitude-to-phase coupling, to major reductions of the fundamental intensity and phase
noise. A loss dispersion of the wrong sign, on the other hand, leads to an increase of the noise and, at a certain
strength, to instability.
The reduction of the linewidth of semiconductor (SC)
lasers is a topic of continuing interest. It was pro-
posedl 2 and demonstrated3' 4 that a narrowing of the
laser field spectrum can be accomplished by coupling
the SC laser to external resonators properly detuned.
In this Letter we show that the physical mechanism
responsible for spectral narrowing is fundamentally
the incorporation of a frequency-dependent loss (or
gain) mechanism.
To analyze the effect of the frequency-dependent
loss we use the Van der Pol oscillator laser model that
is driven by spontaneous emission. 5'6 The optical
field is taken as
E(t) = [AO + 6(t)]exp{i[kmt + ¢(t)]}, (1)
where 5(t) and OM(t) are the noise-driven amplitude and
phase excursions, respectively. The loss is represent-
ed by a frequency-dependent photon lifetime Tp,
1 1= - + 2Cg.
Tp rpo
(2)
with rnm the average frequency and xi(3 ) and Xr(3) the
imaginary and real part of the third-order amplifying
medium susceptibility so that a is the amplitude-to-
phase coupling factor. A formal solution of Eqs. (3)
with 6(0) = 0, q(0) = 0 leads to
6(t) = -I Ai'(X)exp[-,o 1 '(t - X)]dX
+ C J Ar(X)exp[-wl'(t - X)]d ,
10(t) = -2C0 - i: Ar(X)exp[-W1 '(t - X)]dX
+ t [aAi(X) -A,(X)]dX
+ 0
[aAi(X) - Ar(X)]exp[-,I'(t - M)]dX},
W' WI (1 + Ca),
This is the key ensatz. It stipulates that the loss rate
Tpr1 adjusts instantaneously to the (instantaneous)
frequency corm + W(t). The results of the analysis are
thus limited to frequencies below the inverse of the
(longest) propagation time involved in the system or
the response time of resonant elements. This limita-
tion still allows us to explore the spectral regions of
greatest interest, typically 0 to 100 MHz.
Solving, as in Ref. 6, for the steady-state amplitude
Ao and frequency corn leads one to the fluctuations'
equations,
4 + wlb + A0C,0 = -2Aw(t) Ao~ -w1ab = - dA(rt)A~k-cia6=-2wLm '
(3)
where Ai and Ar are, respectively, the real and imagi-
nary parts of the Langevin noise sources representing
the spontaneous emission, and
(5)
(6)
(7)
where co' is seen to play the role of the fundamental
decay rate of the fluctuations. This rate increases for
Ca >O. Taking first the case of col'> O we can neglect
the decaying terms, i.e., terms with cl'(t - X), in the
exponent of Eq. (6) when calculating the correlation
functions. The result is
(f't40t2)) = W (1 + a2) min(tl, t2),4 2rn 2 (1 + Ca)2
(02(r)) = W(1 ± a 2) 17r1I 2KIrI,
4AO2Wmr2(1 + Ca) 2
where we used 6
(Ar(ti)Ar(t 2)) = WD(t -t2),
(Ai(t1)Ai(t2)) = WD(t41 -
(Ar(ti)Ai(t 2 )) = 0,
(8)
AO2WrMXi(3 ) Xr(3)
°1 --- 2 a = (3)'
n X
(j3) <0,co>0 h( ~Xi(3  °1  °s 4 wm 3(N)
(4) = tAN
(4) EVTP,0
0146-9592/90/231359-03$2.00/0 © 1990 Optical Society of America
(9)
1360 OPTICS LETTERS / Vol. 15, No. 23 / December 1, 1990
and D(t) is the delta function of t. e is the dielectric
constant, V is the resonator volume, and (N2/AN) is
the inversion factor. Use of standard noise formal-
ism6 leads to
(E*(t)E(t + -r)) = AO2 exp[-112(k 2(-))]
X exp(iwzp-) + c.c., (10)
which, combined with Eq. (8) and after use of the
Wiener-Khintchine theorem, leads to the following
expression for the (optical) field spectral density:
WE(co) = f | (E*(t)E(t + T))exp(-iwr)d-r + c.c.
A2 
= °-~- I expf[-i(w -co) - K Irlldr
2 AO2K
(coa - co,,d' + K2- (11)
The laser field spectral width is thus (AP)Iaser = K/1r,
(AV)iaser NP.T 2(1 + Ca)l
l + a 2
= (AV)ST + a2
where Pe = (eAo2V)/Tpo is the total power emitted by
the electrons and (AVP)ST is the Schawlow-Townes
linewidth. The astounding result is that for Ca >> 1,
(A)laser- (Av)sT/C 2 so that the simultaneous pres-
ence of amplitude-to-phase coupling (a Ed 0) and dis-
persive loss (C $ 0) leads not only to relief from the
onerous (1 + a 2 ) factor but even to linewidth values
that are substantially smaller than the original Schaw-
low-Townes value with a = 0. This is illustrated in
Fig. 1, where, for example, the point a =-3, C = -30
shows a reduction from the (AV)S-T value by a factor of
900.
Turning our attention to the intensity fluctuations,
from Eqs. (5) and (9) we obtain
at °
40-18U~~~~~~~~~~~1-
-30
(M(t)6(t + T)) = 2 (I + Ca) exp(-wo1'TI). (13)
Use of AP = 2eVAo6(t)Irp and the Wiener-Khintchine
theorem yields
WAp() = 1 J (AP(t)AP(t + i))exp(-i9Zr)dr
7r _ c
hvp N2
m AN e
TpO2
(1 + C2)
[92 + cW12(1 + Ca) 2]
(14)
The low-frequency spectral density of the power fluc-
tuation AP is reduced relative to the case C = 0 by
WaP(O)(C $ 0)
W6(0)(C = 0)
(1 + C2 )
(1 + Ca)2
(15)
This reduction factor can approach a value of a- 2 in
the case C2 >> 1, Ca» >> 1. Since a is typically between
3 and 6, the reduction factor can become significant,
since C can be much larger than a-, though not so
dramatic as the reduction C-2 in linewidth. We note
that here, as well as in the case of the laser linewidth,
the noise reduction is brought about by the concerted
action of loss dispersion (C id 0) and amplitude-to-
phase coupling (a = 0).
In the case Ca -> -1 it follows directly from the
form of Eqs. (5) and (6) that the damping rate wl'- 0,
so that intensity and frequency fluctuations do not
decay. The laser field mode and its relation to insta-
bility and chaos are currently being investigated.
We have also considered the current modulation
response of the laser by complementing Eqs. (3) (with
Ar = 0, Ai = 0) with a third equation for the carrier
density and a driving sinusoidal current.6 The results
show that the relaxation resonance frequency CwR,
which is often taken as the maximum practical modu-
lation frequency, increases for Ca > 0 by (1 + Ca)1/2,
while the low-frequency portion of the response is
unchanged. We also find that the attendant chirp
broadening of the spectrum is reduced by a factor (1 +
Ca).
To estimate the all-important C parameter, we con-
sidered two cases. The first involves placing an atom-
ic vapor absorption cell inside the resonator. Sub-
stantial values of C - 10 are indicated.
The second example involves the weak coupling of
the semiconductor laser to an external high-finesse
Fabry-Perot 6talon. This configuration was demon-
strated by Dahmani et al. 4 to lead to great (100OX)
reduction in the laser linewidth. If we-consider the
external resonator, properly detuned, as a frequency-
dependent loss element, it is an easy and short task to
show that the C parameter [see Eq. (2)] representing
the extra loss coupled by the resonator into the SC
laser is accounted for by
(16)
C= (l - Rd)
2 7rnLdAVex
= (1 + a2)/Fig. 1. Reduction factor (AV)laser/(AP)S-T
(1 + Cax)2 as a function of C and ae. where f is the intensity spatial feedback factor (not
December 1, 1990 / Vol. 15, No. 23 / OPTICS LETTERS 1361
including the SC laser facet reflectivity), Rd is the SC
laser facet reflectivity, Ld and Lex are the SC laser
length and Fabry-Perot cavity length, respectively, n
is the SC laser index of refraction, and Av5x is the
frequency width of the reflection peak of the external
resonator. Using the reported values of / - 10-5, APex
5 X 106 Hz, Ld - 3 X 10-2 cm, and Rd - 0.3, we
calculate C - 42. The expected linewidth narrowing
according to Eq. (12) is _C2 _ 1750. The observed
narrowing of 1000 is consistent with this estimate.
A recent detailed analysis by Laurent et al.7 of the
Dahmani et al. configuration using a coupled-cavity
formalism as in Ref. 2 leads to a similar prediction.
This shows that the operative physical principle in the
configuration of Ref. 4 is the frequency-dependent
loss.
A major challenge would be to incorporate a narrow-
linewidth frequency-dependent loss in a monolithic
fashion with the SC laser.
This research was supported by the Defense Ad-
vanced Research Projects Agency, the U.S. Air Force
Office of Scientific Research, and the Office of Naval
Research.
* Permanent address, Lebedev Physics Institute,
Moscow, USSR.
References
1. E. M. Belenov, V. L. Velichanskii, A. S. Zibrov, V. V.
Nikitin, V. A. Sautenkov, and A. V. Uskov, Sov. J. Quan-
tum Electron. 13, 792 (1983).
2. K. Vahala and A. Yariv, Appl. Phys. Lett. 45, 501 (1984).
3. K. Vahala, J. Paslaski, and A. Yariv, Appl. Phys. Lett. 46,
1025 (1985).
4. B. Dahmani, L. Hollberg, and R. Drullinger, Opt. Lett. 12,
876 (1987).
5. A. Yariv and W. Caton, IEEE J. Quantum Electron. QE-
10, 509 (1974).
6. K. Vahala and A. Yariv, IEEE J. Quantum Electron. QE-
19, 1096 (1983).
7. P. H. Laurent, A. Clairon, and C. H. Breaut, IEEE J.
Quantum Electron. 25, 1131 (1989).


Self-quenching of fundamental phase and amplitude noise in semiconductor lasers with dispersive loss

December 1, 1990 / Vol. 15, No. 23 / OPTICS LETTERS 1359
Self-quenching of fundamental phase and amplitude noise in
semiconductor lasers with dispersive loss
Amnon Yariv, Rashit Nabiev,* and Kerry Vahala
California Institute of Technology, Pasadena, California 91125
Received June 18, 1990; accepted September 8, 1990; manuscript in hand October 12, 1990
We show theoretically that the incorporation of a frequency-dependent loss mechanism in a semiconductor laser can
lead, in concert with the amplitude-to-phase coupling, to major reductions of the fundamental intensity and phase
noise. A loss dispersion of the wrong sign, on the other hand, leads to an increase of the noise and, at a certain
strength, to instability.
The reduction of the linewidth of semiconductor (SC)
lasers is a topic of continuing interest. It was pro-
posedl 2 and demonstrated3' 4 that a narrowing of the
laser field spectrum can be accomplished by coupling
the SC laser to external resonators properly detuned.
In this Letter we show that the physical mechanism
responsible for spectral narrowing is fundamentally
the incorporation of a frequency-dependent loss (or
gain) mechanism.
To analyze the effect of the frequency-dependent
loss we use the Van der Pol oscillator laser model that
is driven by spontaneous emission. 5'6 The optical
field is taken as
E(t) = [AO + 6(t)]exp{i[kmt + ¢(t)]}, (1)
where 5(t) and OM(t) are the noise-driven amplitude and
phase excursions, respectively. The loss is represent-
ed by a frequency-dependent photon lifetime Tp,
1 1
= - + 2Cg.
Tp rpo
(2)
with rnm the average frequency and xi(3 ) and Xr(3) the
imaginary and real part of the third-order amplifying
medium susceptibility so that a is the amplitude-to-
phase coupling factor. A formal solution of Eqs. (3)
with 6(0) = 0, q(0) = 0 leads to
6(t) = -I Ai'(X)exp[-,o 1 '(t - X)]dX
+ C J Ar(X)exp[-wl'(t - X)]d ,
10(t) = -2C0 - i: Ar(X)exp[-W1 '(t - X)]dX
+ t [aAi(X) -A,(X)]dX
+ 0
[aAi(X) - Ar(X)]exp[-,I'(t - M)]dX},
W' WI (1 + Ca),
This is the key ensatz. It stipulates that the loss rate
Tpr1 adjusts instantaneously to the (instantaneous)
frequency corm + W(t). The results of the analysis are
thus limited to frequencies below the inverse of the
(longest) propagation time involved in the system or
the response time of resonant elements. This limita-
tion still allows us to explore the spectral regions of
greatest interest, typically 0 to 100 MHz.
Solving, as in Ref. 6, for the steady-state amplitude
Ao and frequency corn leads one to the fluctuations'
equations,
4 + wlb + A0C,0 = -2Aw(t) Ao~ -w1ab = - dA(rt)A~k-cia6=-2wLm '
(3)
where Ai and Ar are, respectively, the real and imagi-
nary parts of the Langevin noise sources representing
the spontaneous emission, and
(5)
(6)
(7)
where co' is seen to play the role of the fundamental
decay rate of the fluctuations. This rate increases for
Ca >O. Taking first the case of col'> O we can neglect
the decaying terms, i.e., terms with cl'(t - X), in the
exponent of Eq. (6) when calculating the correlation
functions. The result is
(f't40t2)) = W (1 + a2) min(tl, t2),4 2rn 2 (1 + Ca)2
(02(r)) = W(1 ± a 2) 17r1I 2KIrI,
4AO2Wmr2(1 + Ca) 2
where we used 6
(Ar(ti)Ar(t 2)) = WD(t -t2),
(Ai(t1)Ai(t2)) = WD(t41 -
(Ar(ti)Ai(t 2 )) = 0,
(8)
AO2WrMXi(3 ) Xr(3)
°1 --- 2 a = (3)'
n X
(j3) <0,co>0 h( ~Xi(3  °1  °s 4 wm 3(N)
(4) = tAN(4) EVTP,0
0146-9592/90/231359-03$2.00/0 © 1990 Optical Society of America
(9)
1360 OPTICS LETTERS / Vol. 15, No. 23 / December 1, 1990
and D(t) is the delta function of t. e is the dielectric
constant, V is the resonator volume, and (N2/AN) is
the inversion factor. Use of standard noise formal-
ism6 leads to
(E*(t)E(t + -r)) = AO2 exp[-112(k 2(-))]
X exp(iwzp-) + c.c., (10)
which, combined with Eq. (8) and after use of the
Wiener-Khintchine theorem, leads to the following
expression for the (optical) field spectral density:
WE(co) = f | (E*(t)E(t + T))exp(-iwr)d-r + c.c.
A2 
= °-~- I expf[-i(w -co) - K Irlldr
2 AO2K
(coa - co,,d' + K2- (11)
The laser field spectral width is thus (AP)Iaser = K/1r,
(AV)iaser NP.T 2(1 + Ca)l l + a 2= (AV)ST + a2
where Pe = (eAo2V)/Tpo is the total power emitted by
the electrons and (AVP)ST is the Schawlow-Townes
linewidth. The astounding result is that for Ca >> 1,
(A)laser- (Av)sT/C 2 so that the simultaneous pres-
ence of amplitude-to-phase coupling (a Ed 0) and dis-
persive loss (C $ 0) leads not only to relief from the
onerous (1 + a 2 ) factor but even to linewidth values
that are substantially smaller than the original Schaw-
low-Townes value with a = 0. This is illustrated in
Fig. 1, where, for example, the point a =-3, C = -30
shows a reduction from the (AV)S-T value by a factor of
900.
Turning our attention to the intensity fluctuations,
from Eqs. (5) and (9) we obtain
at °
40-18U~~~~~~~~~~~1-
-30
(M(t)6(t + T)) = 2 (I + Ca) exp(-wo1'TI). (13)
Use of AP = 2eVAo6(t)Irp and the Wiener-Khintchine
theorem yields
WAp() = 1 J (AP(t)AP(t + i))exp(-i9Zr)dr
7r _ c
hvp N2
m AN eTpO2
(1 + C2)
[92 + cW12(1 + Ca) 2]
(14)
The low-frequency spectral density of the power fluc-
tuation AP is reduced relative to the case C = 0 by
WaP(O)(C $ 0)
W6(0)(C = 0)
(1 + C2 )
(1 + Ca)2 (15)
This reduction factor can approach a value of a- 2 in
the case C2 >> 1, Ca» >> 1. Since a is typically between
3 and 6, the reduction factor can become significant,
since C can be much larger than a-, though not so
dramatic as the reduction C-2 in linewidth. We note
that here, as well as in the case of the laser linewidth,
the noise reduction is brought about by the concerted
action of loss dispersion (C id 0) and amplitude-to-
phase coupling (a = 0).
In the case Ca -> -1 it follows directly from the
form of Eqs. (5) and (6) that the damping rate wl'- 0,
so that intensity and frequency fluctuations do not
decay. The laser field mode and its relation to insta-
bility and chaos are currently being investigated.
We have also considered the current modulation
response of the laser by complementing Eqs. (3) (with
Ar = 0, Ai = 0) with a third equation for the carrier
density and a driving sinusoidal current.6 The results
show that the relaxation resonance frequency CwR,
which is often taken as the maximum practical modu-
lation frequency, increases for Ca > 0 by (1 + Ca)1/2,
while the low-frequency portion of the response is
unchanged. We also find that the attendant chirp
broadening of the spectrum is reduced by a factor (1 +
Ca).
To estimate the all-important C parameter, we con-
sidered two cases. The first involves placing an atom-
ic vapor absorption cell inside the resonator. Sub-
stantial values of C - 10 are indicated.
The second example involves the weak coupling of
the semiconductor laser to an external high-finesse
Fabry-Perot 6talon. This configuration was demon-
strated by Dahmani et al. 4 to lead to great (100OX)
reduction in the laser linewidth. If we-consider the
external resonator, properly detuned, as a frequency-
dependent loss element, it is an easy and short task to
show that the C parameter [see Eq. (2)] representing
the extra loss coupled by the resonator into the SC
laser is accounted for by
(16)C= (l - Rd)
2 7rnLdAVex
= (1 + a2)/Fig. 1. Reduction factor (AV)laser/(AP)S-T
(1 + Cax)2 as a function of C and ae. where f is the intensity spatial feedback factor (not
December 1, 1990 / Vol. 15, No. 23 / OPTICS LETTERS 1361
including the SC laser facet reflectivity), Rd is the SC
laser facet reflectivity, Ld and Lex are the SC laser
length and Fabry-Perot cavity length, respectively, n
is the SC laser index of refraction, and Av5x is the
frequency width of the reflection peak of the external
resonator. Using the reported values of / - 10-5, APex
5 X 106 Hz, Ld - 3 X 10-2 cm, and Rd - 0.3, we
calculate C - 42. The expected linewidth narrowing
according to Eq. (12) is _C2 _ 1750. The observed
narrowing of 1000 is consistent with this estimate.
A recent detailed analysis by Laurent et al.7 of the
Dahmani et al. configuration using a coupled-cavity
formalism as in Ref. 2 leads to a similar prediction.
This shows that the operative physical principle in the
configuration of Ref. 4 is the frequency-dependent
loss.
A major challenge would be to incorporate a narrow-
linewidth frequency-dependent loss in a monolithic
fashion with the SC laser.
This research was supported by the Defense Ad-
vanced Research Projects Agency, the U.S. Air Force
Office of Scientific Research, and the Office of Naval
Research.
* Permanent address, Lebedev Physics Institute,
Moscow, USSR.
References
1. E. M. Belenov, V. L. Velichanskii, A. S. Zibrov, V. V.
Nikitin, V. A. Sautenkov, and A. V. Uskov, Sov. J. Quan-
tum Electron. 13, 792 (1983).
2. K. Vahala and A. Yariv, Appl. Phys. Lett. 45, 501 (1984).
3. K. Vahala, J. Paslaski, and A. Yariv, Appl. Phys. Lett. 46,
1025 (1985).
4. B. Dahmani, L. Hollberg, and R. Drullinger, Opt. Lett. 12,
876 (1987).
5. A. Yariv and W. Caton, IEEE J. Quantum Electron. QE-
10, 509 (1974).
6. K. Vahala and A. Yariv, IEEE J. Quantum Electron. QE-
19, 1096 (1983).
7. P. H. Laurent, A. Clairon, and C. H. Breaut, IEEE J.
Quantum Electron. 25, 1131 (1989).


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Self-quenching of fundamental phase and amplitude noise in semiconductor lasers with dispersive loss

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