Classical behaviour of CW Optical Parametric Oscillators

T. Coudreau

Laboratoire Kastler Brossel, UMR CNRS 8552 et Université Pierre et Marie Curie, PARIS, France

also with Pôle Matériaux et Phénomènes Quantiques, Fédération de Recherche CNRS 2437 et Université Denis Diderot , PARIS, France

Introduction Basic principles Classical operation Conclusion

Definition

Pump (0)Signal (1)

Idler (2)

Introduction

An Optical Parametric Oscillator is a device that cangenerate two coherent waves (signal and idler) from a pump wave. It consists in :• an active medium• an optical cavity, Fabry Perot resonator, in which resonates one, two or three frequencies

Introduction Basic principles Classical operation Conclusion

History

• First realised in 1965 : Giordmaine & Miller, Phys. Rev. Lett 14, 973 (1965)

• Important development 1965 - 1975 as a tunable source of coherent radiation

• Outdated between 1975-1990 due to the occurrence of dye lasers

• Renewal since the 1990s due to • improvements in laser sources and crystals

• quantum properties

Introduction

Introduction Basic principles Classical operation Conclusion

Outline

•Introduction• Definition

• History

•Basic principles• Optical non linearities

• Second order non linearity

• Energy conservation and phase

matching

•Classical Operation• Singly resonant OPO

• Doubly resonant OPO

• Triply resonant OPO

•Conclusion

Introduction

Introduction Basic principles Classical operation Conclusion

Optical nonlinearities

An electric field applied to an atomic medium displaces the dipole :

+-

+

-

As the electric field becomes large, one gets :

Basic Principles

Introduction Basic principles Classical operation ConclusionBasic Principles

Second order non linearity

In a non centrosymetric medium, one can get a non zero

O3

O3

O3

Nb

Li

Lithium Niobate

Molecule A D

Introduction Basic principles Classical operation ConclusionBasic Principles

Second order non linearity

With a pump wave at frequency 0, on can get two kinds of behaviour :• Second Harmonic Generation (SHG) where a wave at frequency 20 is generated

• Parametric down-conversion where two waves at frequencies 1 and 2 are generated

0

0 20

0

1

2

1

2 1+2

Introduction Basic principles Classical operation ConclusionEnergy and momentum conservation

Two conditions must be fulfilled :

• Energy conservation

which must be always fulfilled exactly• Momentum conservation

which has to be fulfilled exactly only in the case of an infinite medium, the useful condition being

Basic Principles

Introduction Basic principles Classical operation Conclusion

Momentum conservation is often called phase matching : the generated signal and idler remain in phase with the waves generated before in the crystal.If , the phase shift is after a length called the coherence length.

Phase matching

k0

Crystal’s length

Output power

Basic Principles

Pum

p

sig

nal, idle

r

Sig

nal, idle

r

Pum

p

Introduction Basic principles Classical operation Conclusion

Realisation of phase matching

The natural birefringence of the crystal is generally used to ensure phase matching

Extraordinaryaxis

OrdinaryaxisInput light

Basic Principles

Frequency

Index of refractio

n

Introduction Basic principles Classical operation Conclusion

Influence of temperature

The phase matching depends on the crystal temperature (and angle)

TTmin

TTmin

Signal

Idler

Type II

Signal

Idler

Type I

Basic Principles

Introduction Basic principles Classical operation ConclusionBasic Principles

Quasi phase matching

The previous solution is not always chosen : • the most efficient nonlinear coefficient is not always used• some wavelength regions are not reachableOne can revert the sign of the non linearity after a length lc.

Crystal’s length

Single pass output power

Introduction Basic principles Classical operation ConclusionParametric down-conversion : basic eqns

where |i|2 is a number of photons and is a field envelope

These equations can be solved analytically in terms of elliptic functions.

Basic Principles

Introduction Basic principles Classical operation Conclusion

Notations

For a weak efficiency, we have a linear variation of the amplitudes

! The variation depends on the relative phase !

Basic Principles

Introduction Basic principles Classical operation Conclusion

Pump

Pump (0)Signal (1)

Idler (2)

Laser• The pump creates a population inversion which generates gain through stimulated emission• The system depends on the pump intensity

OPO• No population inversion, i.e. the medium is transparent• The system depends on the pump amplitude

Laser vs OPOBasic Principles

Introduction Basic principles Classical operation Conclusion

Singly resonant Doubly resonant

Pump enhancedsingly resonant

Triply resonant

Classical operation

Different kind of cw OPOs

ThresholdFrequency tuning

difficulty

Introduction Basic principles Classical operation Conclusion

Singly Resonant OPO

Only the signal (or idler) wave resonates inside the cavity. Coupling mirror

Usual assumptions :• Good cavity : with

• close to resonance : with

Finally, one gets :

Classical operation

is the free space round trip lengthis the crystal lengthis the amplitude reflection coefficient

Introduction Basic principles Classical operation Conclusion

SROPO - Basic properties

Signal field at resonance

Mean pump intensity constant

which corresponds to optical powers on the order of 1W

• Pump threshold

• Behaviour above threshold

Classical operation

4

Introduction Basic principles Classical operation Conclusion

SROPO - Output Power

100 % conversion efficiency at times above threshold

The output power is given by the implicit equation

E. Rosencher, C. Fabre JOSA B 19 1107 (2002)

Classical operation

Introduction Basic principles Classical operation Conclusion

SROPO - Frequency tuning

There is a linear variation of the frequency (for small variations of ).The SROPO is• tunable like a standard laser• has a bandwidth limited by phase-matching, and/or mirror bandwidth

Classical operation

Introduction Basic principles Classical operation Conclusion

Doubly Resonant OPO

Signal and idler Doubly resonant

Signal and pump Doubly

resonant :Pump

enhancedsingly

resonant

Similar to a SROPO Specific behaviour

Classical operation

Introduction Basic principles Classical operation Conclusion

PESROPO - Basic Properties

but the pump-cavity detuning, 0, must be taken into account.The output power is also modified :

With (normalised detuning)

The pump threshold power is diminished with respect to the SROPO case :

Classical operation

Introduction Basic principles Classical operation Conclusion

PESROPO - Frequency tuning

As in a SROPO, the frequency depends linearly on the cavity length. However, the cavity length region is limited by the pump resonance width.

Classical operation

Introduction Basic principles Classical operation Conclusion

DROPO - Basic Properties

The system forces the signal and idler detunings :1 = 2 =

with

Output power :

Classical operation

Introduction Basic principles Classical operation Conclusion

Since we have 1 = 2, the round trip phases are equal (modulo 2) :

which gives for the signal frequency

DROPO - Frequency tuning (1)

As opposed to the previous case, the variation depends on the distance to frequency degeneracy

Classical operation

Introduction Basic principles Classical operation Conclusion

DROPO - Frequency tuning (2)

m m+1

The resonance width is the signal resonance width which is very narrow : it is almost impossible to tune by length without mode hops

Classical operation

Introduction Basic principles Classical operation Conclusion

Triply Resonant OPO

The output intensity now obeys a second degree equation :

the system can be monostable, bistable or even chaotic...

The threshold is again lower than for a DROPO :

It can be below 1 mW !

Classical operation

Introduction Basic principles Classical operation Conclusion

TROPO - StabilityClassical operation

Introduction Basic principles Classical operation Conclusion

TROPO - Frequency tuning

The behaviour is similar to a DROPO with a limitation due to the pump resonance width.

m m+1 m+2 ...

Classical operation

Introduction Basic principles Classical operation Conclusion

Frequency of emission

OPOs draw their advantage from their very broad tunability since it is not limited by the proximity of a resonance in the active medium. What then limits this tunability ?• The nonlinear coefficient and the reflection coefficients of the mirrors• Phase matching which can be varied using temperature (or orientation)• Recycling of one or more waves inside the cavity

The system oscillates on frequency corresponding to the lowest threshold and only on this frequency (in a cw laser) as an homogeneously broadened laser.

Conclusion

Introduction Basic principles Classical operation Conclusion

Summary

Singly resonant Doubly resonant

Pump enhancedsingly resonant

Triply resonant

Threshold ~ 10s mWTuning by mode hops

Threshold ~ 100s mWTuning like a laser

Threshold ~ 100s mWTuning like a laser

Threshold ~ 100s µWTuning by mode hops

Conclusion

Introduction Basic principles Classical operation Conclusion

Conclusion

The OPO• is a coherent source of radiation• can be tuned over large domains of wavelength• can have a very low threshold• can have a very small linewidth

Conclusion