CAVITY ENHANCEMENT OF RESONANT FREQUENCIES IN SEMICONDUCTOR-LASERS SUBJECT TO OPTICAL-INJECTION

PHYSICAL REVIEWA VOLUME 52, NUMBER 6 DECEMBER 1995

Cavity enhancement

of

resonant frequencies

in semiconductor

lasers

subject

to optical injection

T.

B.

Simpson

JAJ'COR,

P

0

Box.85154,San Diego, California 92186-5154

J.

M.

Liu

Department

of

Electrical Engineering, University

of

California, LosAngeles, LosAngeles, California 90095-159410

K. F.

Huang and

K.

Tai

Department

of

Electro Physic-s, National Chiao Tung University, Hsinchu, Taiwan

C. M.

Clayton, A. Gavrielides, and V.Kovanis

Nonlinear Optics Center, Phillips Laboratory, Kirtland AFB,New Mexico 87117-5776 (Received 21July 1995)

The injection of an optical signal into a semiconductor laser biased near or above the lasing threshold modifies the coupling between the free carriers and the intracavity field. The detuning between the frequency of the injected signal and the free-running oscillation frequency and the ratio of the photon lifetime to the carrier lifetime are key parameters in determining the enhancement of the carrier-field resonant coupling frequency and the stability ofthe output field. Experimental results using a vertical cavity surface emitting laser biased near threshold are in agreement with calculations using alumped-element oscillator model.

PACS number(s): 42.55.Px, 42.65.Hw, 42.65.Ky, 42.70.Nq

Semiconductor lasers subject to external optical injection are being studied asamodel nonlinear dynamical system and for their potential in optical communication and processing applications. External optical signals can induce stable and unstable injection locking

[1],

chaotic dynamics and multi-wave mixing

[2,

3],

and mode hopping

[4],

depending on the amplitudes

of

the injected and oscillating fields and the fre-quency offset between them. These various characteristics have all been recovered in lumped-circuit oscillator models

of

the semiconductor laser subject to external injection. Re-cently, an amplifier model was used to describe some novel features in the optical spectra

of

a vertical cavity surface emitting laser

(VCSEL)

under strong optical injection

[5].

The model predicted that stimulated emission and absorption due to the coherent transfer

of

energy significantly enhanced the semiconductor response and produced new resonances in the optical spectrum that are distinct from the relaxation resonances typically observed in semiconductor lasers. Here, we show that the laser cavity plays a major role in the gen-eration

of

the new resonances and in the enhancement

of

the modulation bandwidth, and that the new resonances can be related back to the relaxation resonances observed in free-running semiconductor lasers.

In our analysis we will consider a single-mode semicon-ductor laser subject to external optical injection

[3,

6,

7].

We will consider two external fields, a strong injection field that satisfies the conditions for injection locking, and aweak field that can be used as a linear probe. The interaction can be described by two coupled equations:

dA

_y,

dt

2 A

+

t(Co

—

to )A

dN

J

2epn

(2)

coL cop

—

—

b 6/2

—

b

U+

V.

Here, cop is the optical frequency

of

the free-running laser,

U=

r/IAt/Az~cos@z and

V=

r/IA, /Az~sin@z, where Az is the steady-state amplitude

of

the injection-locked circulating field. The gain defect

6=

_y,

—

I

_gp, where _gp is the steady-state gain

of

the free-running laser, can be important when the free-running laser is biased below threshold. The free-running coherent field amplitude, Ap, and AL are related

through

where A is the total complex intracavity field amplitude at the locked oscillating frequency

col,

y,

is the cavity decay rate, co, is the longitudinal mode frequency

of

the cold laser cavity, j. is the confinement factor, b is the linewidth en-hancement factor, _{g is the gain} coefficient, A& and A; are the

amplitudes

of

the strong injection signal at the locking fre-quency and the weak probe signal at cuL+A, respectively,

y

isthe coupling rate, N is the carrier density,

J

isthe injec-tion current density, eis the electronic charge, d is the active layer thickness,

y,

is the spontaneous carrier decay rate, and n is the refractive index

of

the semiconductor medium. The gain is assumed to obey alinear dependence ondeviations

of

both the carrier density and the circulating photon density about steady-state values

[7].

Steady-state values for the circulating field are obtained by setting the derivative terms and A; equal to zero. The locked phase

of

the oscillating field,

_@z,

relative to the phase

of

the injection field, is obtained through the relation

I

+

—

(1

—

ib)gA+

r/(At+A;e

'0'),

l

—

_Xy,

_/y

_I

iAO/Azi

=

(4)

'~~ ' ll _~IIII]~I 1 ~ If'llll ~

I IIII ii

52 CAVITY ENHANCEMENT OF RESONANT FREQUENCIES

IN.

. . R4349

where the factor,

X,

is given by

30

Here,

(2U

—

8)

y„L,

X=

(

y,

—

2 U) y 1.

+

y ylL.

20

lO C

10

2n2Ep

2'

2Ep

r.

L=

&

IAil'g.

and

r,

L=

—

& IALI'I

g,

, (6)

AcoL flCOL

where

g„and

g~ are the differential gain and the nonlinear gain parameters defined as the derivatives

of

g with respect to the carrier density and the photon density, respectively, evaluated at the operating point

[7].

When the free-running laser is biased below threshold, lAo/A~l

=0.

With these definitions, we can find the perturbations in-duced by the weak injected field. The weak field induces a regeneratively amplified (RA) sideband,

_A„,

at the injection frequency,

cul+II,

and a four-wave-mixing (FWM) side-band,

_Af,

at ~L

—

A.

In the perturbation limit:

30

I

20

'U 05 C

10

-6 -2 Offset Frequency (GHz}

(a)

A„

—

=

—

_{((y, +}

_y„~

—

_if')[U

_{—i(II —v)]}

+

—,

'(1+ib)(y,

y„L+

y,

y _I,

—

2Uy„l

—

iAy„L)),

-6

-2 Offset Frequency (GHz)

(b}

Af

A;

(7)

D, {2(I

1

—

ib)(y,

y.

l.

+

y, y„l.

2Uy.

L+i&—y„L))/,

(8)

FIG.

l.

(a) Optical spectrum ofthe free-running VCSELbiased well above threshold and (b) the regeneratively amplified spectrum due tothe injection ofaweak optical probe. The experimental data are compared with the calculated spectra using the parameters listed

in the text.

D=(y,

+

_y„L

—

iA)[(U

—

iA)(U+

y„L

—

iA)

+

v(v+by„L)]+(U+bv

En)(r,

2U

— y„i)

r.

—

i.

—

(9)

Measured optical sidebands are proportional to the squared magnitude

_{of A„and Af}

. In addition, the spontaneous emis-sion noise spectrum is proportional to the sum

of

the squared magnitudes

of A„and

A&

[8].

This is because the

spontane-ous emission effectively acts as a weak external optical source. Therefore, this modeling also gives the sideband spectra due to amplified spontaneous emission and, for sim-plicity, we analyze the problem from this viewpoint. These equations simplify to the appropriate expressions for the free-running semiconductor laser when U

=

V=

0

and

AL

=Ap,

y„L

= y„,

and y~L=

y~,

the respective free-running values

[7].

Under optical injection,

D

can be

0

and unstable and chaotic dynamics can ensue

[2,3,

9].

Here, we are primarily interested in the offset frequencies

of

the reso-nances and defer consideration

of

their stability.

When the free-running laser isbiased below threshold and is subjected to a near-resonant locking field, the limit lAo/A~l(&1, and

_y,

&)2U and

_y,

, then

U=8/2+

y,

y„L,/

2

_y,

. Inthis limit,

Eqs. (7)

and

(8)

can be solved to show that the RA term dominates the FWM term.

If

the gain defect,

6', is large compared to

y,

and V, the central peak

of

the RA spectrum is shifted from the free-running spectrum. The dif-ference between the two in the weak-locking, large-6' limit has maxima and minima shifted from the free-running fre-quency by

—

y, [1/b

~

( I

+

I/b ₎'

_],

the same as predicted

by the amplifier analysis

[5].

However,

Eq. (7)

predicts that the amplifier analysis is inadequate when the free-running laser is near or above threshold and/or when V becomes comparable to

6.

As the injecting field isincreased sothat U and V dominate the other rates, except for

_y,

, the RA term

contains a strong resonance at

A

=

—

V=

~p

—

~L

—

_b

_y,

_{y„L/2y, . The corresponding resonance at positive}

fre-quencies ismuch weaker due to a canceling term,

A

—

V, in the numerator. The magnitude

of

the shift is proportional to the circulating locked power, due to the coherent field-induced _{carrier decay rate, y I .} Unlike the amplifier analy-sis, there is a multiplicative factor

of

approximately

by,

/2y, for the resonance shift. This factor is typically on the order

of

100—1000

in semiconductor lasers. When

y,

&&

_y,

_, a_{large shift in the resonance} can be observed _even

if y.

&y.

I.

To verify these predictions, we have investigated a

VCSEL.

Many

of

the characteristics

of

this

VCSEL

have

been described previously

[10,

11].

All measurements re-ported here were made at low output power, under operating conditions where the

VCSEL

displayed single-mode opera-tion and no significant transverse profile variations from ef-fects such as thermal lensing or spatial hole burning. Figure 1shows the output spectrum and the RA spectrum when the laser isbiased at

4.

8mA, well above the threshold value

of

4.

2 mA. Total output power at this injection current is

=0.

35

mW and the coherent output power is

=0.

3 mW. Here, we can determine the key dynamic parameters

of

the laser from its spectra. Figure 1shows the good agreement between data and model that isachieved using the determined parameters,

R4350 T,

B.

SIMPSON etal. 0.015 0.012 o 0.008 0$ C Ul 0.004

III«I rr v- w ««r«re .. «na «rrr«e-~a«raw'«r.en

-10 -5

Offset Frequency (GHz)

b=6,

y

=5.

5&&10"s

',

_y

=5&10

s

',

_y

=3.

5PX10

s

',

and _y

=

5.3P

X

10

s

'.

P

in the formulas refers to the coherent output power in milliwatts, and the uncertainty for the parameters is

~20%.

The enhancement factor for the frequency shifts, relative to the amplifier case, is approxi-mately

330.

For

the injection-locking measurements, the bias current was set to

3.9

mA and the output from a tunable, narrow linewidth, low noise, external cavity laser (New Focus Model

6126)

was injected into the

VCSEL.

Output power from the free-running

VCSEL

was

=0.

02

mW. Below threshold operation insured that lAo/AL &&1.Here, we con-centrate on optical injection at the free-running frequency. Consistent results were obtained for detuned injection. Fig-ure 2shows free-running and injection-locked optical spectra taken with ahigh finesse optical spectrum analyzer (Newport Model

SR-240C)

with

=60

MHz resolution. The resonance feature shifts to lower frequencies as the injection power increases.

Features from these spectra can be compared with the predictions

of

the model. The injection-locked output power

0.08 o 005 CL. CL 0.04

0

0.02 o 12 N K CQ 8 O 4 0 0.1 1 10 100

Injected Power (arb.units)

0 1000

FIG.3.The dependence ofthe injection-locked power, ( )data

and (

—

)model, and the shift ofthe resonance from the free-running frequency,

(6)

data and ₍

——

) model, for the VCSELbiased just below threshold when subject to an injected signal at the

free-running oscillation frequency.

FIG. 2. Representative optical spectra of the free-running VCSEL biased just below threshold (a), and under increasing in-jected power from a narrow-band laser tuned to the free-running oscillation frequency (b)—(e). The injected power ratio of (b):(c):(d):(e)is

1:10:100:1000.

L

r,

r.

l.

+

r,

r„l.

+

(

r,

—

r.

L)U,

(10)

YrL

Ys+

YnL+ YpL,

+

U.

Both A„Land _Y,

_I

show the direct modification due to the enhancement

of

the circulating coherent power under injection-locked operation. In addition, there are the terms proportional to U. These terms dominate the changes over a wide range

of

injection levels. More generally, when

_$1

40,

there is a complicated dependence on both U and V which can lead to the destabilization

of

the laser as well as the enhancement

of

the modulation bandwidth. It is this complex coupling, involving both phase and amplitude modifications due to the strong injection-locking field, that dominates the shift in the resonance frequencies and their damping characteristics, not a simple enhancement

of

stimu-lated emission and absorption rates.

The lumped-element analysis

of

laser oscillation assumes that spatial effects can be averaged over the mode profiles. It has given excellent quantitative agreement with awide vari-ety

of

single-mode semiconductor lasers, including Fabry-Perot edge emitting lasers with large output coupling

[3,

7].

The linearized treatment, like that given here, fails to accu-rately reproduce the central linewidth

of

the free-running la-ser and, more generally, the dynamics whenever aresonance can be determined from the spectra and the measured output power. Figure 3 shows the dependence

of

the injection-locked coherent power and the frequency shift

of

the reso-nance on the power injected into the

VCSEL.

Because we are unable to independently determine the coupling param-eter, rg, we can only make a relative measurement

of

the

injection power, and the relative uncertainty is

50%.

Using the experimentally determined parameters, model calcula-tions for both the injection-locked coherent power and the frequency shift are in good agreement with the data. At high injection power levels, the offset frequency scales linearly with the injection-locked coherent power, and both scale with the cube root

of

the injected power. Even at the highest injected power measured, where the injection-locked coher-ent power is

=

0.

07

mW, the field-induced enhancements

of

the decay rate are

Y„L=0.

05Y,

and

_Y„I=0.

_{075y, .}

These values coincide with the expected values at the lower oper-ating power.

The modulation and noise spectra

of

the injection-locked laser aremore complicated than the free-running spectra due, in part, to the dependence

of

the field amplitude on the locked phase.

If

the locking frequency is detuned from the free-running frequency so that

@L=0,

the amplitude be-comes decoupled from the phase

of

the locking field, as is the case in the free-running laser. The correspondence be-tween the new resonance peaks and the resonance peaks

of

an above threshold, free-running laser is more direct. The resonance peaks

of

the free-running laser are determined by the relaxation resonance frequency,

A„=(y,

_y„+

y,

y„)",

and the damping rate,

y„=

y,

+ y„+

y„[7].

For the

_$1

=

0

locked laser,

52 CAVITY ENHANCEMENT OF RESONANT FREQUENCIES

IN.

. . R4351

becomes unstable, but it shows good accuracy in the predic-tion

of

the positions and shifts

of

the resonance frequencies. The laser cavity strongly enhances the frequency shifts in-duced by the injection field beyond what is expected from the increased stimulated emission due to the stronger oscil-lating field. Phase and amplitude characteristics must be

ana-lyzed for adetailed quantitative understanding

of

the spectral features.

The authors would like to thank Dr. Tim Day forthe loan

of

the New Focus tunable laser. The work

of

T.

B.

S.

and

J.

M.

L.

was supported by the

U.

S.

Air

Force's

Phillips Labo-ratory under Contract No.

F29601-94-C-0166.

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