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Photon correlations of a sub-threshold optical parametric oscillator

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Abstract

A microscopic multimode theory of collinear type-I spontaneous parametric downconversion in a cavity is presented. Single-mode and multimode correlation functions have been derived using fully quantized atom and electromagnetic field variables. From a first principles calculation the FWHM of the single-mode correlation function and the cavity enhancement factor have been obtained in terms of mirror reflectivities and the first-order crystal dispersion coefficient. The values obtained are in good agreement with recent experimental results [Phys. Rev. A 62 , 033804 (2000)].

©2002 Optical Society of America

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Figures (1)

Figure 1:
Figure 1: Schematic showing a cavity of length d bounded by an input mirror M1 and an exit mirror M2. The shaded area represents the nonlinear crystal which fills the cavity.

Equations (22)

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A ( 2 ) ( r 1 , r 2 , r 3 , t ) = { ( i h ) 5 t d t 1 t 1 d t 2 t 2 d t 3 t 3 d t 4 t 4 d t 5
× < a 1 , a 2 , s 3 μ a , ( 1 ) ( t 1 ) μ b , ( 2 ) ( t 2 ) μ c , ( 3 ) ( t 3 ) μ d , ( 3 ) ( t 4 ) μ e , ( 3 ) ( t 5 ) g 1 , g 2 , g 3 >
× < 0 , a k 0 λ 0 E a ( r 1 , t 1 ) E b ( r 2 , t 2 ) E c ( r 3 , t 3 ) E d ( r 3 , t 4 ) E e ( r 3 , t 5 ) 0 , α k 0 λ 0 > }
+ ( r 1 r 2 )
E ( r , t ) = i∫ d 3 k j = 1,2 ( ħ kc 16 π 3 ε 0 ) 1 2 ε j ( k ) U k j ( r ) a k j exp ( iωt )
U in , k j ( r ) = t 2 o exp ( i k ( ) · r ) D k t 2 o exp ( i k ( + ) · r + ikd ) D k
D k = 1 + r 2 o exp ( 2 ikd )
k ( ± ) = k ( sin θ cos ϕ , sin θ sin ϕ , ± cos θ )
d 3 k = 0 k 2 dk 0 π 2 sin θdθ 0 2 π d ϕ
U out , k j ( r ) = exp ( i k ( ) · r ) + R k j exp ( i k ( + ) · r )
R k j exp ( ikd ) + r 2 j exp ( ikd ) D k
G cav ( 2 ) ( z 1 , t 1 ; z 2 , t 2 ) = π ε 2 d ( t 2 o 2 t 1 p ) ω k 0 2 ω k 0 2 dx exp ( x ) sin ( dv x 2 2 ) ( dv x 2 2 ) 1 1 + r 2 o exp ( i 2 dvx ) 2
k i = k i * + k i ω k i ω k i = ω k i * ( ω k i ω k i * ) + 1 2 2 k i ω k i 2 ω k i = ω k i * ( ω k i ω k i * ) 2 +
ω k 1 * + ω k 2 * = ω k 0 * ; k 1 * + k 2 * = k 0 *
1 1 + r 2 o exp ( i 2 dvx ) 2 1 ( 1 + r 2 o ) 2 l = N l = N 1 [ n 1 2 ( vdx ) 2 + 1 ]
n 1 = 2 r 2 o 1 + r 2 o
G cav ( 2 ) = ( t 2 o 2 t 1 p ) n 1 ( 1 + r 2 o ) 2 sin [ ( 2 N + 1 2 ) π τ ˜ ] sin [ π τ ˜ 2 ] exp ( τ ˜ n 1 )
A SM = ( t 2 o 2 t 1 p ) n 1 ( 1 + r 2 o ) 2 exp ( τ ˜ n 1 )
[ sin [ ( 2 N + 1 2 ) π τ ˜ ] sin [ π τ ˜ 2 ] ] 2 = ( 2 N + 1 ) + 2 N ( 2 cos ( 2 π τ ˜ ) ) + ( 2 N 1 ) ( 2 cos ( 2 [ 2 π τ ˜ ] ) )
+ + 2 cos ( 2 N [ 2 π τ ˜ ] )
A MM ( 2 ) ~ 2 N + 1 ( t 2 o 2 t 1 p ) n 1 ( 1 + r 2 o ) 2 exp ( τ ˜ n 1 )
γ = ( count rate bandwidth ) cavity ( count rate bandwidth ) no cavity
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