Laser diodes with phase conjugate feedback

Risø National Laboratory 

Laser diodes with phase conjugate feedback 

Paul M. Petersen 

Risø National Laboratory, Denmark 

Paul Michael Petersen, Laser Diodes with External Feedback (tutorial)


Generation of phase conjugate waves 

A 4 

A 1 

Nonlinear 

medium χ (3) 

A 2 

A 3 

The classic configuration for four-wave mixing 

The interaction between the three waves A 1 , A 2 and A 3 inside 

the nonlinear medium leads to the generation of a phase conjugate 

beam A 3 propagating exactly counteropposed to A 4 



Phase aberration correcting properties of 

phase conjugate waves 

original distorted corrected 

• Corrects for phase aberrations inside the laser cavity 

• Output beam of laser cavity close to the diffraction limit 



Phase conjugate mirrors 

normal mirror phase conjugate mirror 

The autoretracing property of phase conjugate mirrors 

is important for the basic understanding of how these 

mirrors work in laser cavities 



High-power laser diode arrays 

proton implantation 

metallization 

GaAs (p-type) 

GaAlAs (p-type) 

GaAlAs 

GaAlAs (n-type) 

GaAs (n-type) 

gain 

High output power (>50 W) 

Long lifetimes (>30.000 h) 

Very poor spatial and temporal coherence 



Optical phase conjugation using 

photorefractive materials 

BaTiO 3 

crystals 

Self-pumped phase conjugation 

using photorefractive materials 

•Self-organizing process leads to optical phase conjugation 

•No external pump beams are needed 

•High reflectivities (> 90 %) even at low optical power levels 



Improvement of the coherence properties 

of laser diode arrays 

Method: phase conjugate feedback 

-effective coupling of the laser elements 

-self starting feedback configuration 

-at high output power instabilities take place 



Phase conjugate feedback 

Løbel et al., Optics Lett. 23, 825 (1998) 

Frequency selective phase conjugate feedback 

eliminates instabilities 

Paul Michael Petersen, Laser Diodes with External Feedback (tutorial) 

.


Single-mode operation 

Intensity [arb.] 

FWHM=0.02 nm 

(10 GHz) 

PC-Feedback 

Freely run. 

0.63 

nm 

0.63 

nm 

810 811 812 813 814 815 816 817 

Wavelength [nm] 

100 µm 



Asymmetric phase conjugate feedback 

Single-lobe configuration 



Far-field 

output 

Diffraction 

limited output 

with singlelobe 

config. 

Intensity [arb. units] 

0 

0 

0 

1.4 times the 

diffraction limit 

-5 -4 -3 -2 -1 0 1 2 3 4 5 

Radiation angle [deg] 

Single-lobe 

Twin-lobe 

Runs freely 



Etalon setup 

Coherence degree γ 

1,0 

0,9 

0,8 

0,7 

0,6 

0,5 

0,4 

0,3 

0,2 

0,1 

Array runs freely (2I th ) 

Feedback applied (2I th ) 

Feedback applied (3I th ) 

Output sent 

to Michelson 

interferometer 

Coherence 

length is 

increased 

several orders 

of magnitude 

0,1 1 10 100 

Path difference [mm] 


.


Grating configuration 

Phase conjugate 

feedback only applied 

to one of two far-field 

lobes. 



Singlemode 

(a) 

Single-mode 

operation 

BW < 0.03 nm 

Wavelength 

is tunable 

over 20 nm ! 

Intensity [arbitrary units] 

807 808 809 810 811 812 813 814 

Wavelength [nm] 

(b) 

(c) 

(d) 

(e) 



Optical phase conjugation using 

semiconductor lasers 

P. Kürz, R. Nagar, and T. Mukai, Appl. 

Phys. Lett. 68, pp.1180-1182, 1996 

•Highly efficient phase conjugation is obtained (R>>100 %) 

•May be used for frequency stabilization of semiconductor 

lasers in external feedback configurations 



Four-wave mixing operation in high power diodes 

0 

L 

The nonlinear polarization is given by P= 

ε χ( N) 

E 

nc 

where χ( N) =− ( β+i)g(N) , g(N)=a(N-N 

0) 

ω 

Z 

R=R 1 Z=0 

R=0 

Gain and refractive 

index gratings 

A 1 

A 3 

Semiconductor 

amplifier 

A 2 

Electrode 

0 

Z=L 

dA1 

* * 

=−α0A1− γ( AA 

1 4 

+ A2A3) A4 

(1A) 

dz 

* 

dA2 

* * * * 

=+ α0A2 − γ( AA 

1 4 

+ A2A3) A3 

(1B) 

dz 

dA3 

* * 

=+ α0A3+ γ( AA 

1 4 

+ A2A3) A2 

(1C) 

dz 

* 

dA4 

* * * * 

=− α0A4 + γ( AA 

1 4 

+ A2A3) A1 

(1D) 

dz 

(1 − iβ 

) g0 

α0 

=− is the effective gain 

2(1 + P ) 

P 

0 

= 

E 

0 

P 

s 

2 

0 

0 

is the average intercavity pump intensity 

2C 

γ=-i α0 

is the coupling factor 

1+ 

P 



DFWM in diodes using total internal reflection 

0 

Gain and refractive 


A 1 

High reflective 

coating 

Z=0 

Semiconductor 

amplifier 

Electrode 

L 

Z 

A 3 

A 2 

Z=L 

Antireflection 

coating 

Spatial- and temporal 

filtering 

Output 

beam 

M 



Generation of carriers is governed by the rate Equation: 

dN I N 2 

= − + D ∇ N − g ( N ) E 

0 

dt qV τ 

s 

(1) 

where N(x,y,z) is the excited carrier population, 

s 

is the carrier recombination time,I is the injected current 

D the ambipolar diffusivity, E 0 is the total optical field 

g(N)=a(N-N 0 ) is the gain. 

Assuming that: 

N = N +∆nexp( i∆ky) 

(2) 

We find that the third order susceptibility is given by: 

χ 

FWM 

⎛ 

⎞ 

⎜ 

⎟ 

χ 

FWM ,max 

1 

= 

⎜ 

⎟ 

2 2 

E ⎜ IkD 

0 

s 

τ ⎟ 

s 2 

1+ 1+ 

2 θ 

I ⎜ I 

s 

s 

E ⎟ 

⎝ + 

0 ⎠ 

2 

(3) 



The nonlinear susceptibility χ (3) 

χ (3) 

The third order nonlinear susceptibility versus the angle [in units of 1/( k(D 

a 

 

R 

)-½)] 

 

The nonlinear susceptibility is reduced to half the maximum at an angle 

corresponding to: 

2 

E0 

1+ 

1 Is 

θ½ 

= 

k Dτ 


s


The mode suppression factor 

0 

Z=0 

Semiconductor 

amplifier 

Electrode 

L 

Z 

Z=L 

The output beam is a result of diffraction in the induced gratings inside 

the semiconductor amplifier. The diffraction angle ’ inside the 

semiconductor is given by: sin θ’ = λ/(Λn). For small angles the phase 

difference between a wave diffracted at the front facet and the back facet 

of the amplifier is given by: 

δ = 2πnL/λ -2πnLcos θ’/λ = 2πn(1-cosθ’)/λ 

Since 1-cosθ’= 2sin 2 (θ’/2) and n sin’ = sin we obtain: 

4π 

L 

δ = Sin 

λn 

2 

θ 

2 



The mode suppression factor 

8 

The mode suppression factor δ 

6 

4 

2 

1 2 3 4 5 

 

The mode suppression factor δ of the induce 

grating versus the angle between the interfering 

laser beams in the four-wave interaction (L=1 mm, 

n=3.4, •=810 nm) 

In practice the factor must be somewhat larger 2 π to have effectively 

suppression of different axial modes in the broad area amplifier. 

The critical angle for which mode suppression occurs depends on the length of 

the amplifier and is determined by : 

θ = 

crit 

nλ 

2Arcsin( ) 

2L 



Fundamental condition for laser action 

In general, the laser system needs mode suppression from the 

gratings and a large nonlinear susceptibility. This condition leads to 

the condition that the angle must be chosen according to: 

E0 

1+ 

⎛ nλ 

⎞ 1 Is 

2arcsin 

⎜ 

θ 

2L ⎟ < < 

⎝ ⎠ k Dτ 

s 

2 



Nonlinear gratings and mode suppression 

(Numerical example) 

Mode suppression 

Inserting L=1 mm, n=3.4, •=810 nm for GaAlAs we 

obtain crit 

= 4.2 

Nonlinear gratings 

For GaAlAs the parameters are D a 

=13cm 2 /s and R 

=1 

ns and consequently we obtain at 810 nm ½ 

= 6.5 

Mode suppression and strong gratings are obtained 

when 4.2 < < 6.5° 


Dynfrac amic tivein dexg gain andre rating -s 

A1 

Sem icond uctor 

E lectro 

amp lifier 

A3 A2 

O utpu t 

b eam 


Monolithic dynamic grating laser 

Permanent grating 

A 1 

Dynamic gain and refractive 


Semiconductor 

amplifier 

A 3 

A 2 

Electrode 

Output 

beam

Laser diodes with phase conjugate feedback

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