Abstract

For optimal χ (2) nonlinear interaction the phase matching condition must be satisfied. For type I and type II phase matched materials, this is generally achieved by controlling the temperature of the nonlinear media. We describe a technique to readout the phase-matching condition interferometrically, and experimentally demonstrate feedback control in a degenerate optical parametric amplifier (OPA) which is resonant at both the fundamental and harmonic frequencies. The interferometric readout technique is based on using the cavity resonances at the fundamental and harmonic frequencies to enable the readout of the phase mismatch. We achieve relatively fast temperature feedback using the photothermal effect, by modulating the amplitude of the OPA pump beam.

© 2006 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. J. A. Giordmaine and R. C. Miller, "Tunable coherent parametric oscillation in LiNbO3 at optical frequencies," Phys. Rev. Lett. 14, 973-976 (1965).
    [CrossRef]
  2. R. C. Eckardt, C. D. Nabors, W. J. Kozlovsky and R. L. Byer, "Optical parametric oscillator frequency tuning and control," J. Opt. Soc. Am. B 8, 646-667 (1991).
    [CrossRef]
  3. For Example: D. F. Walls and G. J. Milburn, Quantum Optics, (Springer-Verlag, Berlin, 1st ed., 1994).
  4. We define doubly resonant to be resonant at both the fundamental and harmonic frequencies.
  5. R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, H. Ward, "Laser phase and frequency stabilization using an optical resonator," Appl. Phys. B: Photophys. Laser Chem. 31, 97-105 (1983).
    [CrossRef]
  6. Assuming there is no differential phase shift on reflection between the harmonic and fundamental frequencies on the mirror coatings. In general, there will be a differential phase shift on reflection on the mirror coatings and on transmission through anti-reflective (AR) coatings. In our system we experimentally determine that the sum of the differential phase shifts per round trip of the cavity is close to an integer multiple times π. When the differential phase shift is significantly large we compensate for the dispersion by inserting a dichroic AR coated BK7 substrate placed in the cavity. This substrate is angled such that the dispersion on transmission through the substrate compensates for the differential phase shift on the mirror coatings and AR coatings, see Ref. [7].
  7. K. McKenzie, M. B. Gray, S. Goßler, P. K. Lam, and D. E. McClelland, "Squeezed State Generation for Interferometric Gravitational-Wave Detection," Class. Quantum Grav. 23, S245-S250 (2006).
  8. M. J. Collett and C.W. Gardiner, "Squeezing of intracavity and traveling-wave light fields produced in parametric amplification," Phys. Rev. A 30, 1386-1391 (1984).
    [CrossRef]
  9. A. E. Siegman, Lasers, (University Science Books, 1986).
  10. A. Yariv, Optical Electonics in Modern Communications, Fifth Edition, (Oxford University Press 1997).
  11. V. B. Braginsky, M. L. Gorodetsky, and S. P. Vyatchanin, "Thermodynamical fluctuations and photo-thermal shot noise in gravitational wave antennae," Phys. Lett. A 264, 1-10 (1999).
    [CrossRef]
  12. Y. T. Liu and K. S. Thorne, "Thermoelastic noise and homogeneous thermal noise in finite sized gravitationalwave test masses," Phys. Rev. D 62, 122002 (2000).
    [CrossRef]
  13. M. Cerdonio, L. Conti, A. Heidmann, and M. Pinard, "Thermoelastic effects at low temperatures and quantum limits in displacement measurements," Phys. Rev. D 63, 082003 (2001).
    [CrossRef]
  14. K. Goda, K. McKenzie, E. E. Mikhailov, P. K. Lam, D. E. McClelland, and N. Mavalvala, "Photothermal fluctuations as a fundamental limit to low-frequency squeezing in a degenerate optical parametric oscillator," Phys. Rev. A 72, 043819 (2005)
    [CrossRef]
  15. The SHG is a custom Diabolo model developed by Innolight GmbH.
  16. S. P. Tewari and G. S. Agarwal, "Control of phase matching and nonlinear generation in dense media by resonant fields," Phys. Rev. Lett. 171811-1814 (1986).

2006 (1)

K. McKenzie, M. B. Gray, S. Goßler, P. K. Lam, and D. E. McClelland, "Squeezed State Generation for Interferometric Gravitational-Wave Detection," Class. Quantum Grav. 23, S245-S250 (2006).

2005 (1)

K. Goda, K. McKenzie, E. E. Mikhailov, P. K. Lam, D. E. McClelland, and N. Mavalvala, "Photothermal fluctuations as a fundamental limit to low-frequency squeezing in a degenerate optical parametric oscillator," Phys. Rev. A 72, 043819 (2005)
[CrossRef]

2001 (1)

M. Cerdonio, L. Conti, A. Heidmann, and M. Pinard, "Thermoelastic effects at low temperatures and quantum limits in displacement measurements," Phys. Rev. D 63, 082003 (2001).
[CrossRef]

2000 (1)

Y. T. Liu and K. S. Thorne, "Thermoelastic noise and homogeneous thermal noise in finite sized gravitationalwave test masses," Phys. Rev. D 62, 122002 (2000).
[CrossRef]

1999 (1)

V. B. Braginsky, M. L. Gorodetsky, and S. P. Vyatchanin, "Thermodynamical fluctuations and photo-thermal shot noise in gravitational wave antennae," Phys. Lett. A 264, 1-10 (1999).
[CrossRef]

1991 (1)

1986 (1)

S. P. Tewari and G. S. Agarwal, "Control of phase matching and nonlinear generation in dense media by resonant fields," Phys. Rev. Lett. 171811-1814 (1986).

1984 (1)

M. J. Collett and C.W. Gardiner, "Squeezing of intracavity and traveling-wave light fields produced in parametric amplification," Phys. Rev. A 30, 1386-1391 (1984).
[CrossRef]

1983 (1)

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, H. Ward, "Laser phase and frequency stabilization using an optical resonator," Appl. Phys. B: Photophys. Laser Chem. 31, 97-105 (1983).
[CrossRef]

1965 (1)

J. A. Giordmaine and R. C. Miller, "Tunable coherent parametric oscillation in LiNbO3 at optical frequencies," Phys. Rev. Lett. 14, 973-976 (1965).
[CrossRef]

Agarwal, G. S.

S. P. Tewari and G. S. Agarwal, "Control of phase matching and nonlinear generation in dense media by resonant fields," Phys. Rev. Lett. 171811-1814 (1986).

Braginsky, V. B.

V. B. Braginsky, M. L. Gorodetsky, and S. P. Vyatchanin, "Thermodynamical fluctuations and photo-thermal shot noise in gravitational wave antennae," Phys. Lett. A 264, 1-10 (1999).
[CrossRef]

Byer, R. L.

Cerdonio, M.

M. Cerdonio, L. Conti, A. Heidmann, and M. Pinard, "Thermoelastic effects at low temperatures and quantum limits in displacement measurements," Phys. Rev. D 63, 082003 (2001).
[CrossRef]

Collett, M. J.

M. J. Collett and C.W. Gardiner, "Squeezing of intracavity and traveling-wave light fields produced in parametric amplification," Phys. Rev. A 30, 1386-1391 (1984).
[CrossRef]

Conti, L.

M. Cerdonio, L. Conti, A. Heidmann, and M. Pinard, "Thermoelastic effects at low temperatures and quantum limits in displacement measurements," Phys. Rev. D 63, 082003 (2001).
[CrossRef]

Drever, R. W. P.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, H. Ward, "Laser phase and frequency stabilization using an optical resonator," Appl. Phys. B: Photophys. Laser Chem. 31, 97-105 (1983).
[CrossRef]

Eckardt, R. C.

Ford, G. M.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, H. Ward, "Laser phase and frequency stabilization using an optical resonator," Appl. Phys. B: Photophys. Laser Chem. 31, 97-105 (1983).
[CrossRef]

Gardiner, C.W.

M. J. Collett and C.W. Gardiner, "Squeezing of intracavity and traveling-wave light fields produced in parametric amplification," Phys. Rev. A 30, 1386-1391 (1984).
[CrossRef]

Giordmaine, J. A.

J. A. Giordmaine and R. C. Miller, "Tunable coherent parametric oscillation in LiNbO3 at optical frequencies," Phys. Rev. Lett. 14, 973-976 (1965).
[CrossRef]

Goda, K.

K. Goda, K. McKenzie, E. E. Mikhailov, P. K. Lam, D. E. McClelland, and N. Mavalvala, "Photothermal fluctuations as a fundamental limit to low-frequency squeezing in a degenerate optical parametric oscillator," Phys. Rev. A 72, 043819 (2005)
[CrossRef]

Gorodetsky, M. L.

V. B. Braginsky, M. L. Gorodetsky, and S. P. Vyatchanin, "Thermodynamical fluctuations and photo-thermal shot noise in gravitational wave antennae," Phys. Lett. A 264, 1-10 (1999).
[CrossRef]

Goßler, S.

K. McKenzie, M. B. Gray, S. Goßler, P. K. Lam, and D. E. McClelland, "Squeezed State Generation for Interferometric Gravitational-Wave Detection," Class. Quantum Grav. 23, S245-S250 (2006).

Gray, M. B.

K. McKenzie, M. B. Gray, S. Goßler, P. K. Lam, and D. E. McClelland, "Squeezed State Generation for Interferometric Gravitational-Wave Detection," Class. Quantum Grav. 23, S245-S250 (2006).

Hall, J. L.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, H. Ward, "Laser phase and frequency stabilization using an optical resonator," Appl. Phys. B: Photophys. Laser Chem. 31, 97-105 (1983).
[CrossRef]

Heidmann, A.

M. Cerdonio, L. Conti, A. Heidmann, and M. Pinard, "Thermoelastic effects at low temperatures and quantum limits in displacement measurements," Phys. Rev. D 63, 082003 (2001).
[CrossRef]

Hough, J.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, H. Ward, "Laser phase and frequency stabilization using an optical resonator," Appl. Phys. B: Photophys. Laser Chem. 31, 97-105 (1983).
[CrossRef]

Kowalski, F. V.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, H. Ward, "Laser phase and frequency stabilization using an optical resonator," Appl. Phys. B: Photophys. Laser Chem. 31, 97-105 (1983).
[CrossRef]

Kozlovsky, W. J.

Lam, P. K.

K. McKenzie, M. B. Gray, S. Goßler, P. K. Lam, and D. E. McClelland, "Squeezed State Generation for Interferometric Gravitational-Wave Detection," Class. Quantum Grav. 23, S245-S250 (2006).

K. Goda, K. McKenzie, E. E. Mikhailov, P. K. Lam, D. E. McClelland, and N. Mavalvala, "Photothermal fluctuations as a fundamental limit to low-frequency squeezing in a degenerate optical parametric oscillator," Phys. Rev. A 72, 043819 (2005)
[CrossRef]

Liu, Y. T.

Y. T. Liu and K. S. Thorne, "Thermoelastic noise and homogeneous thermal noise in finite sized gravitationalwave test masses," Phys. Rev. D 62, 122002 (2000).
[CrossRef]

Mavalvala, N.

K. Goda, K. McKenzie, E. E. Mikhailov, P. K. Lam, D. E. McClelland, and N. Mavalvala, "Photothermal fluctuations as a fundamental limit to low-frequency squeezing in a degenerate optical parametric oscillator," Phys. Rev. A 72, 043819 (2005)
[CrossRef]

McClelland, D. E.

K. McKenzie, M. B. Gray, S. Goßler, P. K. Lam, and D. E. McClelland, "Squeezed State Generation for Interferometric Gravitational-Wave Detection," Class. Quantum Grav. 23, S245-S250 (2006).

K. Goda, K. McKenzie, E. E. Mikhailov, P. K. Lam, D. E. McClelland, and N. Mavalvala, "Photothermal fluctuations as a fundamental limit to low-frequency squeezing in a degenerate optical parametric oscillator," Phys. Rev. A 72, 043819 (2005)
[CrossRef]

McKenzie, K.

K. McKenzie, M. B. Gray, S. Goßler, P. K. Lam, and D. E. McClelland, "Squeezed State Generation for Interferometric Gravitational-Wave Detection," Class. Quantum Grav. 23, S245-S250 (2006).

K. Goda, K. McKenzie, E. E. Mikhailov, P. K. Lam, D. E. McClelland, and N. Mavalvala, "Photothermal fluctuations as a fundamental limit to low-frequency squeezing in a degenerate optical parametric oscillator," Phys. Rev. A 72, 043819 (2005)
[CrossRef]

Mikhailov, E. E.

K. Goda, K. McKenzie, E. E. Mikhailov, P. K. Lam, D. E. McClelland, and N. Mavalvala, "Photothermal fluctuations as a fundamental limit to low-frequency squeezing in a degenerate optical parametric oscillator," Phys. Rev. A 72, 043819 (2005)
[CrossRef]

Miller, R. C.

J. A. Giordmaine and R. C. Miller, "Tunable coherent parametric oscillation in LiNbO3 at optical frequencies," Phys. Rev. Lett. 14, 973-976 (1965).
[CrossRef]

Munley, A. J.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, H. Ward, "Laser phase and frequency stabilization using an optical resonator," Appl. Phys. B: Photophys. Laser Chem. 31, 97-105 (1983).
[CrossRef]

Nabors, C. D.

Pinard, M.

M. Cerdonio, L. Conti, A. Heidmann, and M. Pinard, "Thermoelastic effects at low temperatures and quantum limits in displacement measurements," Phys. Rev. D 63, 082003 (2001).
[CrossRef]

Tewari, S. P.

S. P. Tewari and G. S. Agarwal, "Control of phase matching and nonlinear generation in dense media by resonant fields," Phys. Rev. Lett. 171811-1814 (1986).

Thorne, K. S.

Y. T. Liu and K. S. Thorne, "Thermoelastic noise and homogeneous thermal noise in finite sized gravitationalwave test masses," Phys. Rev. D 62, 122002 (2000).
[CrossRef]

Vyatchanin, S. P.

V. B. Braginsky, M. L. Gorodetsky, and S. P. Vyatchanin, "Thermodynamical fluctuations and photo-thermal shot noise in gravitational wave antennae," Phys. Lett. A 264, 1-10 (1999).
[CrossRef]

Ward, H.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, H. Ward, "Laser phase and frequency stabilization using an optical resonator," Appl. Phys. B: Photophys. Laser Chem. 31, 97-105 (1983).
[CrossRef]

Appl. Phys. B: Photophys. Laser Chem. (1)

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, H. Ward, "Laser phase and frequency stabilization using an optical resonator," Appl. Phys. B: Photophys. Laser Chem. 31, 97-105 (1983).
[CrossRef]

Class. Quantum Grav. (1)

K. McKenzie, M. B. Gray, S. Goßler, P. K. Lam, and D. E. McClelland, "Squeezed State Generation for Interferometric Gravitational-Wave Detection," Class. Quantum Grav. 23, S245-S250 (2006).

J. Opt. Soc. Am. B (1)

Phys. Lett. A (1)

V. B. Braginsky, M. L. Gorodetsky, and S. P. Vyatchanin, "Thermodynamical fluctuations and photo-thermal shot noise in gravitational wave antennae," Phys. Lett. A 264, 1-10 (1999).
[CrossRef]

Phys. Rev. A (2)

K. Goda, K. McKenzie, E. E. Mikhailov, P. K. Lam, D. E. McClelland, and N. Mavalvala, "Photothermal fluctuations as a fundamental limit to low-frequency squeezing in a degenerate optical parametric oscillator," Phys. Rev. A 72, 043819 (2005)
[CrossRef]

M. J. Collett and C.W. Gardiner, "Squeezing of intracavity and traveling-wave light fields produced in parametric amplification," Phys. Rev. A 30, 1386-1391 (1984).
[CrossRef]

Phys. Rev. D (2)

Y. T. Liu and K. S. Thorne, "Thermoelastic noise and homogeneous thermal noise in finite sized gravitationalwave test masses," Phys. Rev. D 62, 122002 (2000).
[CrossRef]

M. Cerdonio, L. Conti, A. Heidmann, and M. Pinard, "Thermoelastic effects at low temperatures and quantum limits in displacement measurements," Phys. Rev. D 63, 082003 (2001).
[CrossRef]

Phys. Rev. Lett. (2)

S. P. Tewari and G. S. Agarwal, "Control of phase matching and nonlinear generation in dense media by resonant fields," Phys. Rev. Lett. 171811-1814 (1986).

J. A. Giordmaine and R. C. Miller, "Tunable coherent parametric oscillation in LiNbO3 at optical frequencies," Phys. Rev. Lett. 14, 973-976 (1965).
[CrossRef]

Other (6)

The SHG is a custom Diabolo model developed by Innolight GmbH.

A. E. Siegman, Lasers, (University Science Books, 1986).

A. Yariv, Optical Electonics in Modern Communications, Fifth Edition, (Oxford University Press 1997).

Assuming there is no differential phase shift on reflection between the harmonic and fundamental frequencies on the mirror coatings. In general, there will be a differential phase shift on reflection on the mirror coatings and on transmission through anti-reflective (AR) coatings. In our system we experimentally determine that the sum of the differential phase shifts per round trip of the cavity is close to an integer multiple times π. When the differential phase shift is significantly large we compensate for the dispersion by inserting a dichroic AR coated BK7 substrate placed in the cavity. This substrate is angled such that the dispersion on transmission through the substrate compensates for the differential phase shift on the mirror coatings and AR coatings, see Ref. [7].

For Example: D. F. Walls and G. J. Milburn, Quantum Optics, (Springer-Verlag, Berlin, 1st ed., 1994).

We define doubly resonant to be resonant at both the fundamental and harmonic frequencies.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (5)

Fig. 1.
Fig. 1.

Examples of nonlinear gain (Eq. 13) vs temperature offset from phase matching. Trace (a) is a DROPA with harmonic cavity held on resonance (Δ b =0) and the fundamental detuning (Δla) given by Eq. 10. Trace (b) is a SROPA with the fundamental cavity held on resonance (Δla=0). The dashed curve gives the nonlinear gain envelope of the DROPA found with both fundamental and harmonic cavities are held on resonance (Δ b =Δla=0). Parameters: Pinb =0.35 W, λ a =2λ b =1064 nm, n=2.23, L=.65 m, Lc =.0065 m, d n a d T = 3.3 × 10 6 1 K , d n b d T = 37.0 × 10 6 1 K , αa =αb =5×10-6 1/K, κina =1×104 1/s, κa =2.26×107 1/s, κinb =6.77×106 1/s, κb =6.82×106 1/s, ξ=749 1/m/K, ε 0=30 1/s. Parameter mostly taken from [14]

Fig. 2.
Fig. 2.

The experimental layout. Most of the laser power is used to produce the harmonic beam for the OPA. The harmonic beam is passed though a broad band amplitude modulator (AM) and a resonant (70MHz) phase modulator (PM) before being incident on the OPA. The cavity length error signal is derived from the reflected harmonic field and fed to the PZT1. The phase matching error signal is derived from the transmitted fundamental field and sent to the AM on the harmonic field. The temperature of the crystal oven is actively controlled to 63°C.

Fig. 3.
Fig. 3.

(a) The transmitted fundamental power from the OPA as the temperature of the crystal is varied by sweeping the oven temperature. This is normalized to the resonant transmitted power through the OPA without parametric gain. The nonlinear gain is being dithered at approx. 1kHz, which shows the nonlinear gain envelope (amplification and deamplification). (b) the corresponding error signal of the phase matching condition. The dither signal of the nonlinear gain is filtered out of the error signal.

Fig. 4.
Fig. 4.

(a) The transmitted fundamental power of the OPA as a function of time. This is normalized to the resonant transmitted power through the OPA without parametric gain. (b) the phase matching error signal. At 12.5 seconds the control loop is closed and the nonlinear gain is optimized.

Fig. 5.
Fig. 5.

Measured (solid line) and theoretical (dashed line) photothermal response of the nonlinear crystal. (Top) the amplitude response and (bottom) the phase response.

Equations (13)

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

a ˙ = ( κ a + i Δ a ) a + ε * a * b + 2 κ in a A in ,
b ˙ = ( κ b + i Δ b ) b ε a 2 2 + 2 κ in b B in ,
a = 2 κ in a A in ( ( κ a + i Δ a ) ε * b * ) ( κ a ) 2 + ( Δ a ) 2 ε b 2 ,
b 2 κ in b B in κ b i Δ b .
Δ j = δ p j λ j ω f sr
δ p j = δ L + n L c ( 1 n d n j d T + α j ) δ T
Δ j = Δ fs + Δ cr j ( δ T )
Δ a = Δ fs + Δ cr a ( δ T )
= Δ cr a ( δ T ) Δ cr b ( δ T )
Δ l a = Δ cr a ( δ T ) Δ cr b ( δ T ) Δ θ l ω fsr ,
a l = 2 κ in a A in [ ( κ a + i Δ l a ) + ε * b * e i ϕ l ] ( κ a ) 2 + ( Δ l a ) 2 ε b 2
ε = ε 0 z e i Δ k L c 2 sinc Δ k L c 2 ,
P out P out | b = 0 = a l 2 a l b = 0 2

Metrics