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Spatial patterns in optical parametric oscillators with spherical mirrors: classical and quantum effects

M. Marte, H. Ritsch, K. I. Petsas, A. Gatti, L. A. Lugiato, C. Fabre, and D. Leduc

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Abstract

We investigate the formation of transverse patterns in a doubly resonant degenerate optical parametric oscillator. Extending previous work, we treat the more realistic case of a spherical mirror cavity with a finite–sized input pump field. Using numerical simulations in real space, we determine the conditions on the cavity geometry, pump size and detunings for which pattern formation occurs; we find multistability of different types of optical patterns. Below threshold, we analyze the dependence of the quantum image on the width of the input field, in the near and in the far field.

©1998 Optical Society of America

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Corrections

M. Marte, H. Ritsch, K. I. Petsas, A. Gatti, L. A. Lugiato, C. Fabre, and D. Leduc, "Spatial patterns in optical parametric oscillators with spherical mirrors: classical and quantum effects: errata," Opt. Express 3, 476-476 (1998)
https://opg.optica.org/oe/abstract.cfm?uri=oe-3-11-476
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Supplementary Material (1)

Media 1: MOV (2313 KB)     

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Figures (7)
Figure 1.
Figure 1. Modulus of steady-state field amplitude as a function of the transverse position: (a) input field, (b) intracavity pump field, (c) signal far field and (d) signal intracavity field for δp = δs = 0, we = 3wp , ξ = 0.05ks and Ep = 42ks . In the near field plots the length scale is wp ; in the far field diagram it is zλs /(2πwp ), where z is the distance from the cavity.

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Figure 2.
Figure 2. Same as in Fig. 1 except for we = wp , ξ = 0.2ks , δs = -2.5ks , Ep = 65ks : (a) input field, (b) intracavity pump field, (c) signal far field and (d) signal intracavity field.

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Figure 3.
Figure 3. Same as in Fig. 2 but for we = 4wp , ξ = 0.25ks , δs = -1.5ks : (a) input field, (b) intracavity pump field, (c) signal far field and (d) signal intracavity field.

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Figure 4.
Figure 4. The movie shows the formation of a spiral pattern for the same values of the parameters as in Fig. 3 but for E 0 = 70ks . [Media 1]

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Figure 5.
Figure 5. Various quasi–stationary patterns found for the signal field for we = 4wp , ξ = 0.25ks , δp = 0, δs = -1.5ks and E 0 = 85ks .

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Figure 6.
Figure 6. The correlation function F(r, Δϕ, ω) divided by F(r, Δϕ, ω = 0) is plotted as a function of Δϕ for a quadrature component with the phase ϕL specified in the text. We fix ω = 0 and the value of r as described in the text. We set Ēin = 0.9, ξ = 0.5ks , and δs = -1.5ks . (a) Near field, (b) Far field. Red curve: we → ∞, blue curve: we = 2.8ws , black curve: we = 14ws .

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Figure 7.
Figure 7. Same as in Fig. 6, but for Ēin = 0.6, ξ = 0.1ks , and δs = - 0.3ks .

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Equations (7)

Equations on this page are rendered with MathJax. Learn more.

∂ ∂ t A p r ϕ t = − ( k p + i δ p − iξ L p ) A p r ϕ t − χ 2 A s 2 r ϕ t + E p r ϕ + W p ( t ) ,
∂ ∂ t A s r ϕ t = − ( k s + i δ s − iξ L s ) A s r ϕ t + χ A p r ϕ t A s * r ϕ t + W s ( t ) ,
L k = w k 2 4 ∇ T 2 − r 2 w k 2 + 1 .
A p r ϕ = k s χ , A s 2 r ϕ = 2 χ [ E p r ϕ − k p k s χ ] .
A p r ϕ = E p r ϕ k p , A s r ϕ = 0 .
F ˜ ( r , Δ ϕ , ω ) = ∫ − ∞ + ∞ dt e − iωt F ( r , Δ ϕ , t ) .
φ L = r 2 w s 2 [ 1 + ( z z r ) 2 ] z z r − ( q c + 1 ) tan − 1 ( z z r ) + π 2 ,
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