Transcript Classical and Quantum behaviour of Optical Parametric

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
Introduction
Basic principles
Classical operation
Conclusion
Definition
An Optical Parametric Oscillator is a device that can
generate 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
Pump (0)
Signal (1)
Idler (2)
Introduction
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• Outdated between 1975-1990 due to the occurrence of dye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
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
Conclusion
Introduction
Basic principles
Basic
Principles
Conclusion
Classical operation
Optical nonlinearities
An electric field applied to an atomic medium displaces
the dipole :
+
+
As the electric field becomes large, one gets :
Introduction
Basic principles
Basic
Principles
Classical operation
Conclusion
Second order non linearity
In a non centrosymetric medium, one can get a non zero
O3
Nb
O3
Lithium Niobate
Li
Molecule
A
D
O3
Introduction
Basic principles
Basic
Principles
Classical operation
Conclusion
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 20 is generated
0
0
2
20
1
1+2
• Parametric down-conversion where two waves at
frequencies 1 and 2 are generated
0
1
2
Introduction
Basic principles
Basic
Principles
Classical operation
Conclusion
Energy 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
Introduction
Basic principles
Basic
Principles
Classical operation
Conclusion
Phase matching
Signal, idler  Pump
Output power
Pump  signal, idler
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.
k0
Crystal’s length
Introduction
Basic principles
Basic
Principles
Classical operation
Conclusion
Realisation of phase matching
The natural birefringence of the
crystal is generally used to ensure
phase matching
Index of
refraction
Extraordinary
axis
Input light
Ordinary
axis
Frequency
Introduction
Basic principles
Basic
Principles
Conclusion
Classical operation
Influence of temperature
The phase matching depends on the crystal temperature (and
angle)
Type I
Type II
Signal
Signal
Idler

Idler

Tmin
T
Tmin
T
Introduction
Basic principles
Basic
Principles
Classical operation
Conclusion
Quasi phase matching
The previous solution is not always chosen :
• the most efficient nonlinear coefficient is not always used
• some wavelength regions are not reachable
One can revert the sign of the non linearity after a length lc.
Single pass
output power
Crystal’s length
Introduction
Basic principles
Basic
Principles
Classical operation
Conclusion
Parametric 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.
Introduction
Basic Principles
principles
Basic
Classical operation
Conclusion
Notations
For a weak efficiency, we have a linear variation of the amplitudes
!
The variation depends on the relative phase !
Introduction
Basic principles
Basic
Principles
Classical operation
Conclusion
Laser vs OPO
Laser
• The pump creates a population inversion which generates gain through
stimulated emission
• The system depends on the pump intensity
Pump
OPO
• No population inversion, i.e. the medium is transparent
• The system depends on the pump amplitude
Signal (1)
Pump (0)
Idler (2)
Introduction
Basic principles
Classical
operation
Classical
operation
Conclusion
Different kind of cw OPOs
Singly resonant
Doubly resonant
Pump enhanced
singly resonant
Threshold
Frequency tuning
difficulty
Triply resonant
Introduction
Basic principles
Classical
operation
Classical
operation
Conclusion
Singly Resonant OPO
Only the signal (or idler) wave resonates inside the cavity.
Coupling mirror
is the free space round trip length
is the crystal length
is the amplitude reflection coefficient
Usual assumptions :
• Good cavity :
• close to resonance :
Finally, one gets :
with
with
Introduction
Basic principles
Classical
operation
Classical
operation
Conclusion
SROPO - Basic properties
• Pump threshold
4
which corresponds to optical powers on
the order of 1W
• Behaviour above threshold
Mean pump intensity constant
Signal field at resonance
Introduction
Basic principles
Classical
operation
Classical
operation
Conclusion
SROPO - Output Power
The output power is given by the implicit equation
100 % conversion efficiency at
times above threshold
E. Rosencher, C. Fabre JOSA B 19 1107 (2002)
Introduction
Basic principles
Classical
operation
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
Introduction
Basic principles
Classical
operation
Classical
operation
Conclusion
Doubly Resonant OPO
Signal and
pump Doubly
resonant :
Pump enhanced
singly resonant
Similar to a SROPO
Signal and idler
Doubly resonant
Specific behaviour
Introduction
Basic principles
Classical
operation
Classical
operation
Conclusion
PESROPO - Basic Properties
The pump threshold power is diminished with respect to the
SROPO case :
but the pump-cavity detuning, 0, must be taken into account.
The output power is also modified :
With
(normalised detuning)
Introduction
Basic principles
Classical
operation
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.
Introduction
Basic principles
Classical
operation
Classical
operation
Conclusion
DROPO - Basic Properties
The system forces the signal and idler detunings :1 = 2 = 
with
Output power :
Introduction
Basic principles
Classical
operation
Classical
operation
Conclusion
DROPO - Frequency tuning (1)
Since we have 1 = 2, the round trip phases are equal (modulo
2) :
which gives for the signal frequency
As opposed to the previous case, the variation depends on the
distance to frequency degeneracy
Introduction
Basic principles
Classical
operation
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
Introduction
Basic principles
Classical
operation
Classical
operation
Conclusion
Triply Resonant OPO
The threshold is again lower than for a DROPO :
It can be below 1 mW !
The output intensity now obeys a second degree equation :
the system can be monostable, bistable or even chaotic...
Introduction
Basic principles
TROPO - Stability
Classical
operation
Classical
operation
Conclusion
Introduction
Basic principles
Classical
operation
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 ...
Introduction
Basic principles
Classical operation
Conclusion
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.
Introduction
Basic principles
Classical operation
Conclusion
Conclusion
Summary
Singly resonant
Doubly resonant
Triply resonant
Threshold ~ 100s mW Threshold ~ 10s mW Threshold ~ 100s µW
Tuning like a laser
Tuning by mode hops Tuning by mode hops
Pump enhanced
singly resonant
Threshold ~ 100s mW
Tuning like a laser
Introduction
Basic principles
Classical operation
Conclusion
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