classical behaviour of cw optical parametric oscillators t. coudreau laboratoire kastler brossel,...
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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• Outdated between 1975-1990 due to the occurrence of dye lasers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
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