Abstract

We theoretically analyze the influence of the Gouy phase shift on the nonlinear interaction between waves of different frequencies. We focus on χ(2) interaction of optical fields, e.g. through birefringent crystals, and show that focussing, stronger than suggested by the Boyd-Kleinman factor, can further improve nonlinear processes. An increased value of 3.32 for the optimal focussing parameter for a single pass process is found. The new value builds on the compensation of the Gouy phase shift by a spatially varying, instead constant, wave vector phase mismatch. We analyze the single-ended, singly resonant standing wave nonlinear cavity and show that in this case the Gouy phase shift leads to an additional phase during backreflection. Our numerical simulations may explain ill-understood experimental observations in such devices.

© 2007 Optical Society of America

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  1. P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, "Generation of Optical Harmonics," Phys. Rev. Lett. 7, 118 (1961).
    [CrossRef]
  2. J. A. Giordmaine and R. C. Miller, "Tunable Coherent Parametric Oscillation in LiNbO3 at optical frequencies," Phys. Rev. Lett. 14, 973 (1965).
    [CrossRef]
  3. L. Wu, H. J. Kimble, J. L. Hall, and H. Wu, "Generation of squeezed states by parametric down conversion," Phys. Rev. Lett. 57, 2520 (1986).
    [CrossRef] [PubMed]
  4. R. Ghosh and L. Mandel, "Observation of nonclassical effects in the interference of two photons," Phys. Rev. Lett. 59, 1903 (1987).
    [CrossRef] [PubMed]
  5. C. M. Caves, "Quantum-mechanical noise in an interferometer," Phys. Rev. D 23, 1693 (1981).
    [CrossRef]
  6. H. Vahlbruch, S. Chelkowski, B. Hage, A. Franzen, K. Danzmann, and R. Schnabel, "Coherent control of vacuum squeezing in the gravitational-wave detection band," Phys. Rev. Lett. 97, 011101 (2006).
    [CrossRef] [PubMed]
  7. Y. Yamamoto and H. A. Haus, "Preparation, measurement and information capacity of optical quantum states," Rev. Mod. Phys. 58, 1001 (1986).
    [CrossRef]
  8. S. Feng and H. G. Winful, "Physical origin of the Gouy phase shift," Opt. Lett. 26, 485 (2001)
    [CrossRef]
  9. G. D. Boyd and D. A. Kleinman, "Parametric interaction of focused Gaussian light beams," J. Appl. Phys. 39, 3597 (1968).
    [CrossRef]
  10. R. Paschotta, K. Fiedler, P. Kurz, R. Henking, S. Schiller and J. Mlynek, "82% efficient continuous-wave frequency doubling of 1.06 μm with a monolithic MgO:LiNbO3 resonator," Opt. Lett. 19, 1325 (1994).
    [CrossRef] [PubMed]
  11. R. Paschotta, K. Fiedler, P. Kurz, and J. Mlynek, "Nonlinear mode coupling in doubly resonant frequency doublers," Appl. Phys. B 58, 117 (1994).
    [CrossRef]

2006 (1)

H. Vahlbruch, S. Chelkowski, B. Hage, A. Franzen, K. Danzmann, and R. Schnabel, "Coherent control of vacuum squeezing in the gravitational-wave detection band," Phys. Rev. Lett. 97, 011101 (2006).
[CrossRef] [PubMed]

2001 (1)

1994 (2)

1987 (1)

R. Ghosh and L. Mandel, "Observation of nonclassical effects in the interference of two photons," Phys. Rev. Lett. 59, 1903 (1987).
[CrossRef] [PubMed]

1986 (2)

Y. Yamamoto and H. A. Haus, "Preparation, measurement and information capacity of optical quantum states," Rev. Mod. Phys. 58, 1001 (1986).
[CrossRef]

L. Wu, H. J. Kimble, J. L. Hall, and H. Wu, "Generation of squeezed states by parametric down conversion," Phys. Rev. Lett. 57, 2520 (1986).
[CrossRef] [PubMed]

1981 (1)

C. M. Caves, "Quantum-mechanical noise in an interferometer," Phys. Rev. D 23, 1693 (1981).
[CrossRef]

1968 (1)

G. D. Boyd and D. A. Kleinman, "Parametric interaction of focused Gaussian light beams," J. Appl. Phys. 39, 3597 (1968).
[CrossRef]

1965 (1)

J. A. Giordmaine and R. C. Miller, "Tunable Coherent Parametric Oscillation in LiNbO3 at optical frequencies," Phys. Rev. Lett. 14, 973 (1965).
[CrossRef]

1961 (1)

P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, "Generation of Optical Harmonics," Phys. Rev. Lett. 7, 118 (1961).
[CrossRef]

Boyd, G. D.

G. D. Boyd and D. A. Kleinman, "Parametric interaction of focused Gaussian light beams," J. Appl. Phys. 39, 3597 (1968).
[CrossRef]

Caves, C. M.

C. M. Caves, "Quantum-mechanical noise in an interferometer," Phys. Rev. D 23, 1693 (1981).
[CrossRef]

Chelkowski, S.

H. Vahlbruch, S. Chelkowski, B. Hage, A. Franzen, K. Danzmann, and R. Schnabel, "Coherent control of vacuum squeezing in the gravitational-wave detection band," Phys. Rev. Lett. 97, 011101 (2006).
[CrossRef] [PubMed]

Danzmann, K.

H. Vahlbruch, S. Chelkowski, B. Hage, A. Franzen, K. Danzmann, and R. Schnabel, "Coherent control of vacuum squeezing in the gravitational-wave detection band," Phys. Rev. Lett. 97, 011101 (2006).
[CrossRef] [PubMed]

Feng, S.

Fiedler, K.

Franken, P. A.

P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, "Generation of Optical Harmonics," Phys. Rev. Lett. 7, 118 (1961).
[CrossRef]

Franzen, A.

H. Vahlbruch, S. Chelkowski, B. Hage, A. Franzen, K. Danzmann, and R. Schnabel, "Coherent control of vacuum squeezing in the gravitational-wave detection band," Phys. Rev. Lett. 97, 011101 (2006).
[CrossRef] [PubMed]

Ghosh, R.

R. Ghosh and L. Mandel, "Observation of nonclassical effects in the interference of two photons," Phys. Rev. Lett. 59, 1903 (1987).
[CrossRef] [PubMed]

Giordmaine, J. A.

J. A. Giordmaine and R. C. Miller, "Tunable Coherent Parametric Oscillation in LiNbO3 at optical frequencies," Phys. Rev. Lett. 14, 973 (1965).
[CrossRef]

Hage, B.

H. Vahlbruch, S. Chelkowski, B. Hage, A. Franzen, K. Danzmann, and R. Schnabel, "Coherent control of vacuum squeezing in the gravitational-wave detection band," Phys. Rev. Lett. 97, 011101 (2006).
[CrossRef] [PubMed]

Hall, J. L.

L. Wu, H. J. Kimble, J. L. Hall, and H. Wu, "Generation of squeezed states by parametric down conversion," Phys. Rev. Lett. 57, 2520 (1986).
[CrossRef] [PubMed]

Haus, H. A.

Y. Yamamoto and H. A. Haus, "Preparation, measurement and information capacity of optical quantum states," Rev. Mod. Phys. 58, 1001 (1986).
[CrossRef]

Hill, A. E.

P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, "Generation of Optical Harmonics," Phys. Rev. Lett. 7, 118 (1961).
[CrossRef]

Kimble, H. J.

L. Wu, H. J. Kimble, J. L. Hall, and H. Wu, "Generation of squeezed states by parametric down conversion," Phys. Rev. Lett. 57, 2520 (1986).
[CrossRef] [PubMed]

Kleinman, D. A.

G. D. Boyd and D. A. Kleinman, "Parametric interaction of focused Gaussian light beams," J. Appl. Phys. 39, 3597 (1968).
[CrossRef]

Mandel, L.

R. Ghosh and L. Mandel, "Observation of nonclassical effects in the interference of two photons," Phys. Rev. Lett. 59, 1903 (1987).
[CrossRef] [PubMed]

Miller, R. C.

J. A. Giordmaine and R. C. Miller, "Tunable Coherent Parametric Oscillation in LiNbO3 at optical frequencies," Phys. Rev. Lett. 14, 973 (1965).
[CrossRef]

Paschotta, R.

Peters, C. W.

P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, "Generation of Optical Harmonics," Phys. Rev. Lett. 7, 118 (1961).
[CrossRef]

Schnabel, R.

H. Vahlbruch, S. Chelkowski, B. Hage, A. Franzen, K. Danzmann, and R. Schnabel, "Coherent control of vacuum squeezing in the gravitational-wave detection band," Phys. Rev. Lett. 97, 011101 (2006).
[CrossRef] [PubMed]

Vahlbruch, H.

H. Vahlbruch, S. Chelkowski, B. Hage, A. Franzen, K. Danzmann, and R. Schnabel, "Coherent control of vacuum squeezing in the gravitational-wave detection band," Phys. Rev. Lett. 97, 011101 (2006).
[CrossRef] [PubMed]

Weinreich, G.

P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, "Generation of Optical Harmonics," Phys. Rev. Lett. 7, 118 (1961).
[CrossRef]

Winful, H. G.

Wu, H.

L. Wu, H. J. Kimble, J. L. Hall, and H. Wu, "Generation of squeezed states by parametric down conversion," Phys. Rev. Lett. 57, 2520 (1986).
[CrossRef] [PubMed]

Wu, L.

L. Wu, H. J. Kimble, J. L. Hall, and H. Wu, "Generation of squeezed states by parametric down conversion," Phys. Rev. Lett. 57, 2520 (1986).
[CrossRef] [PubMed]

Yamamoto, Y.

Y. Yamamoto and H. A. Haus, "Preparation, measurement and information capacity of optical quantum states," Rev. Mod. Phys. 58, 1001 (1986).
[CrossRef]

Appl. Phys. B (1)

R. Paschotta, K. Fiedler, P. Kurz, and J. Mlynek, "Nonlinear mode coupling in doubly resonant frequency doublers," Appl. Phys. B 58, 117 (1994).
[CrossRef]

J. Appl. Phys. (1)

G. D. Boyd and D. A. Kleinman, "Parametric interaction of focused Gaussian light beams," J. Appl. Phys. 39, 3597 (1968).
[CrossRef]

Opt. Lett. (2)

Phys. Rev. D (1)

C. M. Caves, "Quantum-mechanical noise in an interferometer," Phys. Rev. D 23, 1693 (1981).
[CrossRef]

Phys. Rev. Lett. (5)

H. Vahlbruch, S. Chelkowski, B. Hage, A. Franzen, K. Danzmann, and R. Schnabel, "Coherent control of vacuum squeezing in the gravitational-wave detection band," Phys. Rev. Lett. 97, 011101 (2006).
[CrossRef] [PubMed]

P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, "Generation of Optical Harmonics," Phys. Rev. Lett. 7, 118 (1961).
[CrossRef]

J. A. Giordmaine and R. C. Miller, "Tunable Coherent Parametric Oscillation in LiNbO3 at optical frequencies," Phys. Rev. Lett. 14, 973 (1965).
[CrossRef]

L. Wu, H. J. Kimble, J. L. Hall, and H. Wu, "Generation of squeezed states by parametric down conversion," Phys. Rev. Lett. 57, 2520 (1986).
[CrossRef] [PubMed]

R. Ghosh and L. Mandel, "Observation of nonclassical effects in the interference of two photons," Phys. Rev. Lett. 59, 1903 (1987).
[CrossRef] [PubMed]

Rev. Mod. Phys. (1)

Y. Yamamoto and H. A. Haus, "Preparation, measurement and information capacity of optical quantum states," Rev. Mod. Phys. 58, 1001 (1986).
[CrossRef]

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Figures (3)

Fig. 1.
Fig. 1.

(Color online) Left: For weak focussing into the nonlinear medium (ξ =0.18) the Gouy phase shift can be compensated by choosing Δk = 1/zR , and perfect phase matching can be realized over the full crystal length. Right: For stronger focussing (ξ = 2.03) a constant Δk can not provide perfect phase matching. (a) Gouy phase shift Δϕ, (b) compensating phase Δkz, (c) overall phase ϕ 0, where a constant value describes perfect phase matching.

Fig. 2.
Fig. 2.

(Color online) Change of refractive index along the crystal to compensate for the Gouy phase shift, for three different strengths of focussing. The temperature scale on the right corresponds to a MgO(7%) :LiNbO3 crystal that has been used in [6]. (a) ξ = 0.55, (b) ξ = 1.14, (c) ξ = 3.32.

Fig. 3.
Fig. 3.

(Color online) Effective nonlinearity κdp (normalized) versus the differential phase introduced by the back reflecting surface Δφ. For Gaussian beams with waist position on the reflecting surface, Δφ = 0 generally provides the highest κdp , (a) (here with strong focussing ξ = 2.84, z 0 = L); the same result is found for plane waves. Contrary, for ξ = 2.84 and z 0 = L/2 we find Δφπ (c). The result for weaker focussing is shown in (b) (ξ = 0.775, z 0 = L/2).

Equations (24)

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ξ : = L 2 z R = 2.84 ,
z E 0,1 ( z ) E 0,1 * ( z ) E 0,2 ( z ) g * ( z ) ,
z E 0,2 ( z ) E 0,1 2 ( z ) g ( z ) ,
g ( z ) : = e i Δ kz 1 + i z z 0 z R = w 0 w ( z ) e i ( Δ kz + Δϕ ( z ) ) ,
Δ ϕ ( z ) = arctan ( z z 0 z R ) ,
w ( z ) = w 0 1 + ( z z 0 z R ) 2 ,
ϕ ˜ G ( ω i ) := ϕ G ω i = ( m + n + 1 ) ω i arctan ( z z 0 z R ) ,
Δϕ ω 2 = ϕ ˜ G ( ω 1 ) ϕ ˜ G ( ω 2 ) .
Δ ϕ = ϕ G .
κ := dz g ( z ) w 0 2 ,
κ sp = 0 L dz e i ( Δ kz + ϕ G ( z ) ) w ( z ) 2 .
Δ kz + Δ ϕ ( z ) = ϕ 0 = const .
Δ n sp ( z ) = λ 4 π Δ k ( z ) = λ 4 π ϕ 0 + arctan ( z z 0 z R ) z ,
ϕ 0 = arctan ( z 0 z R ) = ϕ G ( 0 ) .
κ sp = 1 ξ ln 2 ( 1 + ξ 2 + ξ 1 + ξ 2 + ξ ) ,
ξ opt = 3.32 .
κ dp , pw = sin 2 ( Δ kL 2 ) ( Δ kL 2 ) 2 cos 2 ( Δ kL 2 + Δφ 2 ) .
k dp = 1 w 0 2 0 L dzg ( z , z 0 ) + L 2 L dzg ( z , z 0 ) e i Δφ 2
= 1 w 0 2 0 L dz [ g ( z , z 0 ) + g ( z + L , z 0 ) e i Δφ ] 2
= 0 L dz e i ( Δ kz + ϕ G ( z ) ) w ( z ) × ( 1 + w ( z ) w ( z ) e i ( ϕ G ( z ) ϕ G ( z ) + Δ kL + Δφ ) ) 2 ,
κ dp = sin 2 ( Δ k L 2 ) ( Δ k L 2 ) 2 cos 2 ( Δ k L 2 + Δφ 2 ) ,
Δφ := Δφ + 2 ( L z 0 ) z R .
κ dp = 4 cos 2 ( Δ kL + Δφ 2 ) 0 L dz e i ( Δ kz + ϕ G ( z ) ) w ( z ) 2 .
Δφ = 3 ξ 2 [ ( 1 + ξ 2 ) arctan ( ξ ) ξ ] .

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