Abstract

We experimentally and theoretically investigate the sources of phase noise for an injection-locked, diode-laser-pumped Nd:YAG laser. We use a fully quantum-mechanical model of the laser to describe the output phase noise of the laser explicitly in terms of the input noise sources. We compare the free-running and injection-locked output noise with the quantum-noise limit (QNL), and we find excellent quantitative agreement between the results of our experiments and theory. We show that the phase noise of the injection-locked laser can never be at the QNL for frequencies less than the injection-locking range. However, at frequencies well outside the linewidth of the slave laser, the phase noise can be at the QNL. We show that, although the technical noise of the laser system can be substantially reduced by injection locking, the influence of cavity-length fluctuations on the phase noise of an injection-locked laser is finite and much greater than the QNL. These fluctuations are the major impediment to achieving near-ideal performance for the injection-locked phase noise.

© 2000 Optical Society of America

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  1. C. D. Nabors, A. D. Farinas, T. Day, S. T. Yang, E. K. Gustafson, and R. L. Byer, “Injection locking of a 13-W cw Nd:YAG ring laser,” Opt. Lett. 14, 1189–1191 (1989).
    [CrossRef] [PubMed]
  2. D. Golla, I. Freitag, H. Zellmer, W. Schöne, I. Kröpke, and H. Welling, “15 W single-frequency operation of a cw, diode laser-pumped Nd:YAQ ring laser,” Opt. Commun. 98, 86–90 (1993).
    [CrossRef]
  3. O. Cregut, C. N. Man, D. Shoemaker, A. Brillet, A. Menhert, P. Peuser, N. P. Schmitt, P. Zeller, and K. Wallmeroth, “18 W single-frequency operation of an injection-locked, cw, Nd:YAG laser,” Phys. Lett. A 140, 294–298 (1989).
    [CrossRef]
  4. I. Freitag and H. Welling, “Investigation on amplitude and frequency noise of injection-locked diode-pumped Nd:YAG lasers,” Appl. Phys. B 58, 537–544 (1994).
    [CrossRef]
  5. R. J. Sandeman and D. E. McClelland, “Laser interferometers for gravitational wave detection—an overview,” J. Mod. Opt. 37, 1747–1760 (1990).
    [CrossRef]
  6. H.-A. Bachor, A Guide To Experiments in Quantum Optics (Wiley-VCH, Berlin, 1998).
  7. M. E. Hines, J. R. Collinet, and J. G. Ondria, “FM noise suppression of an injection phase-locked oscillator,” IEEE Trans. Microwave Theory Tech. MTT-16, 738–742 (1968).
    [CrossRef]
  8. A. D. Farinas, E. K. Gustafson, and R. L. Byer, “Frequency and intensity noise in an injection-locked, solid-state laser,” J. Opt. Soc. Am. B 12, 328–334 (1995).
    [CrossRef]
  9. R. Barillet, A. Brillet, R. Chiche, F. Cleva, L. Latrach, and C. N. Man, “Injection-locked Nd:YAG laser for the interferometric detection of gravitational waves,” Meas. Sci. Technol. 7, 162–169 (1996).
    [CrossRef]
  10. T. C. Ralph, C. C. Harb, and H.-A. Bachor, “Intensity noise of injection locked lasers: quantum theory using a linearized input-output method,” Phys. Rev. A 54, 4359–4369(1996); T. C. Ralph, “Low noise amplification and efficient squeezing from rate-matched lasers,” Phys. Rev. A 55, 2326–2333 (1997).
    [CrossRef] [PubMed]
  11. Dipole noise terms that could be important if higher order-terms in the fluctuations were considered have been neglected. See Ref. 10 for full expressions.
  12. B. C. Buchler, M. B. Gray, D. A. Shaddock, T. C. Ralph, and D. E. McClelland, “Suppression of classic and quantum ra-diation pressure noise by electro-optic feedback,” Opt. Lett. 24, 259–261 (1999).
    [CrossRef]
  13. D. F. Walls and G. J. Milburn, Quantum Optics, 2nd ed. (Springer-Verlag, Berlin, 1995).
  14. C. M. Caves, “Quantum limits on noise in linear amplifiers,” Phys. Rev. D 26, 1817–1839 (1982).
    [CrossRef]
  15. A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).
  16. A. L. Schawlow and C. H. Townes, “Infrared and optical masers,” Phys. Rev. 2, 1940–1949 (1958).
    [CrossRef]
  17. A. Yariv, Quantum Electronics (Wiley, New York, 1989).
  18. The more traditional way to express the frequency-noise spectrum that is due to the Schawlow–Townes linewidth is19 VSTL(ω)=(hν0Δνc2)/(2Pout), where h is Planck’s constant, ν0 is the average output frequency of the laser, Pout is the output power of the laser, and Δνc is the cold cavity linewidth of the laser. Translating these quantities into the notation used throughout this paper yields Δνc=2κ and Pout20. Substituting these quantities into the equation for VSTL gives VSTL=VΩ, a.
  19. Y.-J. Cheng, P. L. Mussche, and A. E. Siegman, “Measurement of laser quantum frequency fluctuations using a Pound–Drever stabilization system,” IEEE J. Quantum Electron. 30, 1498–1504 (1994).
    [CrossRef]
  20. C. C. Harb, T. C. Ralph, E. H. Huntington, I. Freitag, D. E. McClelland, and H.-A. Bachor, “Intensity-noise properties of injection-locked lasers,” Phys. Rev. A 54, 4370–4382 (1996).
    [CrossRef] [PubMed]
  21. M. S. Taubman, “The quantum mechanics of electro-optic feedback and second harmonic generation and their interaction,” Ph.D. dissertation (Australian National University, Canberra, 1995).
  22. R. W. P. Drever, J. L. Hall, F. V. Kawalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31, 97–105 (1983).
    [CrossRef]
  23. C. O’Brien, “Polarization effects and spatial mode mismatch in injection locked laser systems,” honours thesis (Australian National University, Canberra, 1997).
  24. W. Koechner, Solid-State Laser Engineering, 5th ed. (Springer-Verlag, Berlin, 1996).
  25. We expect the measured low-frequency part of the transfer function to be at 0 dB because, for VΦM≫H and V≫1, we find that V≅HVΦM, Vsub≅ηV≅ηHVΦM, and VΦM, det≅ηmVΦM. If we assume that we detect the same power, Pdet, in our measurements of VΦM and Vsub, we find that PdetmPm=ηP=ηHPm. This relationship between the two attenuations, when it is combined with our relationships above between the detected spectra, gives us Vsub≅VΦM in the low-frequency region. For more details, see Ref. 20.
  26. Measurements of the transfer functions of the PZT on both the master and the slave laser crystals show a frequency response that was similar to the spectra shown in Fig. 6.
  27. D. J. Ottaway, P. J. Veitch, M. W. Hamilton, C. Hollitt, D. Mudge, and J. Munch, “A compact injection-locked Nd:YAG laser for gravitational wave detection,” IEEE J. Quantum Electron. 34, 2006–2009 (1998).
    [CrossRef]

1999 (1)

1998 (1)

D. J. Ottaway, P. J. Veitch, M. W. Hamilton, C. Hollitt, D. Mudge, and J. Munch, “A compact injection-locked Nd:YAG laser for gravitational wave detection,” IEEE J. Quantum Electron. 34, 2006–2009 (1998).
[CrossRef]

1996 (2)

R. Barillet, A. Brillet, R. Chiche, F. Cleva, L. Latrach, and C. N. Man, “Injection-locked Nd:YAG laser for the interferometric detection of gravitational waves,” Meas. Sci. Technol. 7, 162–169 (1996).
[CrossRef]

C. C. Harb, T. C. Ralph, E. H. Huntington, I. Freitag, D. E. McClelland, and H.-A. Bachor, “Intensity-noise properties of injection-locked lasers,” Phys. Rev. A 54, 4370–4382 (1996).
[CrossRef] [PubMed]

1995 (1)

1994 (2)

I. Freitag and H. Welling, “Investigation on amplitude and frequency noise of injection-locked diode-pumped Nd:YAG lasers,” Appl. Phys. B 58, 537–544 (1994).
[CrossRef]

Y.-J. Cheng, P. L. Mussche, and A. E. Siegman, “Measurement of laser quantum frequency fluctuations using a Pound–Drever stabilization system,” IEEE J. Quantum Electron. 30, 1498–1504 (1994).
[CrossRef]

1993 (1)

D. Golla, I. Freitag, H. Zellmer, W. Schöne, I. Kröpke, and H. Welling, “15 W single-frequency operation of a cw, diode laser-pumped Nd:YAQ ring laser,” Opt. Commun. 98, 86–90 (1993).
[CrossRef]

1990 (1)

R. J. Sandeman and D. E. McClelland, “Laser interferometers for gravitational wave detection—an overview,” J. Mod. Opt. 37, 1747–1760 (1990).
[CrossRef]

1989 (2)

O. Cregut, C. N. Man, D. Shoemaker, A. Brillet, A. Menhert, P. Peuser, N. P. Schmitt, P. Zeller, and K. Wallmeroth, “18 W single-frequency operation of an injection-locked, cw, Nd:YAG laser,” Phys. Lett. A 140, 294–298 (1989).
[CrossRef]

C. D. Nabors, A. D. Farinas, T. Day, S. T. Yang, E. K. Gustafson, and R. L. Byer, “Injection locking of a 13-W cw Nd:YAG ring laser,” Opt. Lett. 14, 1189–1191 (1989).
[CrossRef] [PubMed]

1983 (1)

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

1982 (1)

C. M. Caves, “Quantum limits on noise in linear amplifiers,” Phys. Rev. D 26, 1817–1839 (1982).
[CrossRef]

1968 (1)

M. E. Hines, J. R. Collinet, and J. G. Ondria, “FM noise suppression of an injection phase-locked oscillator,” IEEE Trans. Microwave Theory Tech. MTT-16, 738–742 (1968).
[CrossRef]

1958 (1)

A. L. Schawlow and C. H. Townes, “Infrared and optical masers,” Phys. Rev. 2, 1940–1949 (1958).
[CrossRef]

Bachor, H.-A.

C. C. Harb, T. C. Ralph, E. H. Huntington, I. Freitag, D. E. McClelland, and H.-A. Bachor, “Intensity-noise properties of injection-locked lasers,” Phys. Rev. A 54, 4370–4382 (1996).
[CrossRef] [PubMed]

Barillet, R.

R. Barillet, A. Brillet, R. Chiche, F. Cleva, L. Latrach, and C. N. Man, “Injection-locked Nd:YAG laser for the interferometric detection of gravitational waves,” Meas. Sci. Technol. 7, 162–169 (1996).
[CrossRef]

Brillet, A.

R. Barillet, A. Brillet, R. Chiche, F. Cleva, L. Latrach, and C. N. Man, “Injection-locked Nd:YAG laser for the interferometric detection of gravitational waves,” Meas. Sci. Technol. 7, 162–169 (1996).
[CrossRef]

O. Cregut, C. N. Man, D. Shoemaker, A. Brillet, A. Menhert, P. Peuser, N. P. Schmitt, P. Zeller, and K. Wallmeroth, “18 W single-frequency operation of an injection-locked, cw, Nd:YAG laser,” Phys. Lett. A 140, 294–298 (1989).
[CrossRef]

Buchler, B. C.

Byer, R. L.

Caves, C. M.

C. M. Caves, “Quantum limits on noise in linear amplifiers,” Phys. Rev. D 26, 1817–1839 (1982).
[CrossRef]

Cheng, Y.-J.

Y.-J. Cheng, P. L. Mussche, and A. E. Siegman, “Measurement of laser quantum frequency fluctuations using a Pound–Drever stabilization system,” IEEE J. Quantum Electron. 30, 1498–1504 (1994).
[CrossRef]

Chiche, R.

R. Barillet, A. Brillet, R. Chiche, F. Cleva, L. Latrach, and C. N. Man, “Injection-locked Nd:YAG laser for the interferometric detection of gravitational waves,” Meas. Sci. Technol. 7, 162–169 (1996).
[CrossRef]

Cleva, F.

R. Barillet, A. Brillet, R. Chiche, F. Cleva, L. Latrach, and C. N. Man, “Injection-locked Nd:YAG laser for the interferometric detection of gravitational waves,” Meas. Sci. Technol. 7, 162–169 (1996).
[CrossRef]

Collinet, J. R.

M. E. Hines, J. R. Collinet, and J. G. Ondria, “FM noise suppression of an injection phase-locked oscillator,” IEEE Trans. Microwave Theory Tech. MTT-16, 738–742 (1968).
[CrossRef]

Cregut, O.

O. Cregut, C. N. Man, D. Shoemaker, A. Brillet, A. Menhert, P. Peuser, N. P. Schmitt, P. Zeller, and K. Wallmeroth, “18 W single-frequency operation of an injection-locked, cw, Nd:YAG laser,” Phys. Lett. A 140, 294–298 (1989).
[CrossRef]

Day, T.

Drever, R. W. P.

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

Farinas, A. D.

Ford, G. M.

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

Freitag, I.

C. C. Harb, T. C. Ralph, E. H. Huntington, I. Freitag, D. E. McClelland, and H.-A. Bachor, “Intensity-noise properties of injection-locked lasers,” Phys. Rev. A 54, 4370–4382 (1996).
[CrossRef] [PubMed]

I. Freitag and H. Welling, “Investigation on amplitude and frequency noise of injection-locked diode-pumped Nd:YAG lasers,” Appl. Phys. B 58, 537–544 (1994).
[CrossRef]

D. Golla, I. Freitag, H. Zellmer, W. Schöne, I. Kröpke, and H. Welling, “15 W single-frequency operation of a cw, diode laser-pumped Nd:YAQ ring laser,” Opt. Commun. 98, 86–90 (1993).
[CrossRef]

Golla, D.

D. Golla, I. Freitag, H. Zellmer, W. Schöne, I. Kröpke, and H. Welling, “15 W single-frequency operation of a cw, diode laser-pumped Nd:YAQ ring laser,” Opt. Commun. 98, 86–90 (1993).
[CrossRef]

Gray, M. B.

Gustafson, E. K.

Hall, J. L.

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

Hamilton, M. W.

D. J. Ottaway, P. J. Veitch, M. W. Hamilton, C. Hollitt, D. Mudge, and J. Munch, “A compact injection-locked Nd:YAG laser for gravitational wave detection,” IEEE J. Quantum Electron. 34, 2006–2009 (1998).
[CrossRef]

Harb, C. C.

C. C. Harb, T. C. Ralph, E. H. Huntington, I. Freitag, D. E. McClelland, and H.-A. Bachor, “Intensity-noise properties of injection-locked lasers,” Phys. Rev. A 54, 4370–4382 (1996).
[CrossRef] [PubMed]

Hines, M. E.

M. E. Hines, J. R. Collinet, and J. G. Ondria, “FM noise suppression of an injection phase-locked oscillator,” IEEE Trans. Microwave Theory Tech. MTT-16, 738–742 (1968).
[CrossRef]

Hollitt, C.

D. J. Ottaway, P. J. Veitch, M. W. Hamilton, C. Hollitt, D. Mudge, and J. Munch, “A compact injection-locked Nd:YAG laser for gravitational wave detection,” IEEE J. Quantum Electron. 34, 2006–2009 (1998).
[CrossRef]

Hough, J.

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

Huntington, E. H.

C. C. Harb, T. C. Ralph, E. H. Huntington, I. Freitag, D. E. McClelland, and H.-A. Bachor, “Intensity-noise properties of injection-locked lasers,” Phys. Rev. A 54, 4370–4382 (1996).
[CrossRef] [PubMed]

Kawalski, F. V.

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

Kröpke, I.

D. Golla, I. Freitag, H. Zellmer, W. Schöne, I. Kröpke, and H. Welling, “15 W single-frequency operation of a cw, diode laser-pumped Nd:YAQ ring laser,” Opt. Commun. 98, 86–90 (1993).
[CrossRef]

Latrach, L.

R. Barillet, A. Brillet, R. Chiche, F. Cleva, L. Latrach, and C. N. Man, “Injection-locked Nd:YAG laser for the interferometric detection of gravitational waves,” Meas. Sci. Technol. 7, 162–169 (1996).
[CrossRef]

Man, C. N.

R. Barillet, A. Brillet, R. Chiche, F. Cleva, L. Latrach, and C. N. Man, “Injection-locked Nd:YAG laser for the interferometric detection of gravitational waves,” Meas. Sci. Technol. 7, 162–169 (1996).
[CrossRef]

O. Cregut, C. N. Man, D. Shoemaker, A. Brillet, A. Menhert, P. Peuser, N. P. Schmitt, P. Zeller, and K. Wallmeroth, “18 W single-frequency operation of an injection-locked, cw, Nd:YAG laser,” Phys. Lett. A 140, 294–298 (1989).
[CrossRef]

McClelland, D. E.

B. C. Buchler, M. B. Gray, D. A. Shaddock, T. C. Ralph, and D. E. McClelland, “Suppression of classic and quantum ra-diation pressure noise by electro-optic feedback,” Opt. Lett. 24, 259–261 (1999).
[CrossRef]

C. C. Harb, T. C. Ralph, E. H. Huntington, I. Freitag, D. E. McClelland, and H.-A. Bachor, “Intensity-noise properties of injection-locked lasers,” Phys. Rev. A 54, 4370–4382 (1996).
[CrossRef] [PubMed]

R. J. Sandeman and D. E. McClelland, “Laser interferometers for gravitational wave detection—an overview,” J. Mod. Opt. 37, 1747–1760 (1990).
[CrossRef]

Menhert, A.

O. Cregut, C. N. Man, D. Shoemaker, A. Brillet, A. Menhert, P. Peuser, N. P. Schmitt, P. Zeller, and K. Wallmeroth, “18 W single-frequency operation of an injection-locked, cw, Nd:YAG laser,” Phys. Lett. A 140, 294–298 (1989).
[CrossRef]

Mudge, D.

D. J. Ottaway, P. J. Veitch, M. W. Hamilton, C. Hollitt, D. Mudge, and J. Munch, “A compact injection-locked Nd:YAG laser for gravitational wave detection,” IEEE J. Quantum Electron. 34, 2006–2009 (1998).
[CrossRef]

Munch, J.

D. J. Ottaway, P. J. Veitch, M. W. Hamilton, C. Hollitt, D. Mudge, and J. Munch, “A compact injection-locked Nd:YAG laser for gravitational wave detection,” IEEE J. Quantum Electron. 34, 2006–2009 (1998).
[CrossRef]

Munley, A. J.

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

Mussche, P. L.

Y.-J. Cheng, P. L. Mussche, and A. E. Siegman, “Measurement of laser quantum frequency fluctuations using a Pound–Drever stabilization system,” IEEE J. Quantum Electron. 30, 1498–1504 (1994).
[CrossRef]

Nabors, C. D.

Ondria, J. G.

M. E. Hines, J. R. Collinet, and J. G. Ondria, “FM noise suppression of an injection phase-locked oscillator,” IEEE Trans. Microwave Theory Tech. MTT-16, 738–742 (1968).
[CrossRef]

Ottaway, D. J.

D. J. Ottaway, P. J. Veitch, M. W. Hamilton, C. Hollitt, D. Mudge, and J. Munch, “A compact injection-locked Nd:YAG laser for gravitational wave detection,” IEEE J. Quantum Electron. 34, 2006–2009 (1998).
[CrossRef]

Peuser, P.

O. Cregut, C. N. Man, D. Shoemaker, A. Brillet, A. Menhert, P. Peuser, N. P. Schmitt, P. Zeller, and K. Wallmeroth, “18 W single-frequency operation of an injection-locked, cw, Nd:YAG laser,” Phys. Lett. A 140, 294–298 (1989).
[CrossRef]

Ralph, T. C.

B. C. Buchler, M. B. Gray, D. A. Shaddock, T. C. Ralph, and D. E. McClelland, “Suppression of classic and quantum ra-diation pressure noise by electro-optic feedback,” Opt. Lett. 24, 259–261 (1999).
[CrossRef]

C. C. Harb, T. C. Ralph, E. H. Huntington, I. Freitag, D. E. McClelland, and H.-A. Bachor, “Intensity-noise properties of injection-locked lasers,” Phys. Rev. A 54, 4370–4382 (1996).
[CrossRef] [PubMed]

Sandeman, R. J.

R. J. Sandeman and D. E. McClelland, “Laser interferometers for gravitational wave detection—an overview,” J. Mod. Opt. 37, 1747–1760 (1990).
[CrossRef]

Schawlow, A. L.

A. L. Schawlow and C. H. Townes, “Infrared and optical masers,” Phys. Rev. 2, 1940–1949 (1958).
[CrossRef]

Schmitt, N. P.

O. Cregut, C. N. Man, D. Shoemaker, A. Brillet, A. Menhert, P. Peuser, N. P. Schmitt, P. Zeller, and K. Wallmeroth, “18 W single-frequency operation of an injection-locked, cw, Nd:YAG laser,” Phys. Lett. A 140, 294–298 (1989).
[CrossRef]

Schöne, W.

D. Golla, I. Freitag, H. Zellmer, W. Schöne, I. Kröpke, and H. Welling, “15 W single-frequency operation of a cw, diode laser-pumped Nd:YAQ ring laser,” Opt. Commun. 98, 86–90 (1993).
[CrossRef]

Shaddock, D. A.

Shoemaker, D.

O. Cregut, C. N. Man, D. Shoemaker, A. Brillet, A. Menhert, P. Peuser, N. P. Schmitt, P. Zeller, and K. Wallmeroth, “18 W single-frequency operation of an injection-locked, cw, Nd:YAG laser,” Phys. Lett. A 140, 294–298 (1989).
[CrossRef]

Siegman, A. E.

Y.-J. Cheng, P. L. Mussche, and A. E. Siegman, “Measurement of laser quantum frequency fluctuations using a Pound–Drever stabilization system,” IEEE J. Quantum Electron. 30, 1498–1504 (1994).
[CrossRef]

Townes, C. H.

A. L. Schawlow and C. H. Townes, “Infrared and optical masers,” Phys. Rev. 2, 1940–1949 (1958).
[CrossRef]

Veitch, P. J.

D. J. Ottaway, P. J. Veitch, M. W. Hamilton, C. Hollitt, D. Mudge, and J. Munch, “A compact injection-locked Nd:YAG laser for gravitational wave detection,” IEEE J. Quantum Electron. 34, 2006–2009 (1998).
[CrossRef]

Wallmeroth, K.

O. Cregut, C. N. Man, D. Shoemaker, A. Brillet, A. Menhert, P. Peuser, N. P. Schmitt, P. Zeller, and K. Wallmeroth, “18 W single-frequency operation of an injection-locked, cw, Nd:YAG laser,” Phys. Lett. A 140, 294–298 (1989).
[CrossRef]

Ward, H.

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

Welling, H.

I. Freitag and H. Welling, “Investigation on amplitude and frequency noise of injection-locked diode-pumped Nd:YAG lasers,” Appl. Phys. B 58, 537–544 (1994).
[CrossRef]

D. Golla, I. Freitag, H. Zellmer, W. Schöne, I. Kröpke, and H. Welling, “15 W single-frequency operation of a cw, diode laser-pumped Nd:YAQ ring laser,” Opt. Commun. 98, 86–90 (1993).
[CrossRef]

Yang, S. T.

Zeller, P.

O. Cregut, C. N. Man, D. Shoemaker, A. Brillet, A. Menhert, P. Peuser, N. P. Schmitt, P. Zeller, and K. Wallmeroth, “18 W single-frequency operation of an injection-locked, cw, Nd:YAG laser,” Phys. Lett. A 140, 294–298 (1989).
[CrossRef]

Zellmer, H.

D. Golla, I. Freitag, H. Zellmer, W. Schöne, I. Kröpke, and H. Welling, “15 W single-frequency operation of a cw, diode laser-pumped Nd:YAQ ring laser,” Opt. Commun. 98, 86–90 (1993).
[CrossRef]

Appl. Phys. B (2)

I. Freitag and H. Welling, “Investigation on amplitude and frequency noise of injection-locked diode-pumped Nd:YAG lasers,” Appl. Phys. B 58, 537–544 (1994).
[CrossRef]

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

IEEE J. Quantum Electron. (2)

D. J. Ottaway, P. J. Veitch, M. W. Hamilton, C. Hollitt, D. Mudge, and J. Munch, “A compact injection-locked Nd:YAG laser for gravitational wave detection,” IEEE J. Quantum Electron. 34, 2006–2009 (1998).
[CrossRef]

Y.-J. Cheng, P. L. Mussche, and A. E. Siegman, “Measurement of laser quantum frequency fluctuations using a Pound–Drever stabilization system,” IEEE J. Quantum Electron. 30, 1498–1504 (1994).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

M. E. Hines, J. R. Collinet, and J. G. Ondria, “FM noise suppression of an injection phase-locked oscillator,” IEEE Trans. Microwave Theory Tech. MTT-16, 738–742 (1968).
[CrossRef]

J. Mod. Opt. (1)

R. J. Sandeman and D. E. McClelland, “Laser interferometers for gravitational wave detection—an overview,” J. Mod. Opt. 37, 1747–1760 (1990).
[CrossRef]

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

Meas. Sci. Technol. (1)

R. Barillet, A. Brillet, R. Chiche, F. Cleva, L. Latrach, and C. N. Man, “Injection-locked Nd:YAG laser for the interferometric detection of gravitational waves,” Meas. Sci. Technol. 7, 162–169 (1996).
[CrossRef]

Opt. Commun. (1)

D. Golla, I. Freitag, H. Zellmer, W. Schöne, I. Kröpke, and H. Welling, “15 W single-frequency operation of a cw, diode laser-pumped Nd:YAQ ring laser,” Opt. Commun. 98, 86–90 (1993).
[CrossRef]

Opt. Lett. (2)

Phys. Lett. A (1)

O. Cregut, C. N. Man, D. Shoemaker, A. Brillet, A. Menhert, P. Peuser, N. P. Schmitt, P. Zeller, and K. Wallmeroth, “18 W single-frequency operation of an injection-locked, cw, Nd:YAG laser,” Phys. Lett. A 140, 294–298 (1989).
[CrossRef]

Phys. Rev. (1)

A. L. Schawlow and C. H. Townes, “Infrared and optical masers,” Phys. Rev. 2, 1940–1949 (1958).
[CrossRef]

Phys. Rev. A (1)

C. C. Harb, T. C. Ralph, E. H. Huntington, I. Freitag, D. E. McClelland, and H.-A. Bachor, “Intensity-noise properties of injection-locked lasers,” Phys. Rev. A 54, 4370–4382 (1996).
[CrossRef] [PubMed]

Phys. Rev. D (1)

C. M. Caves, “Quantum limits on noise in linear amplifiers,” Phys. Rev. D 26, 1817–1839 (1982).
[CrossRef]

Other (12)

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).

D. F. Walls and G. J. Milburn, Quantum Optics, 2nd ed. (Springer-Verlag, Berlin, 1995).

M. S. Taubman, “The quantum mechanics of electro-optic feedback and second harmonic generation and their interaction,” Ph.D. dissertation (Australian National University, Canberra, 1995).

A. Yariv, Quantum Electronics (Wiley, New York, 1989).

The more traditional way to express the frequency-noise spectrum that is due to the Schawlow–Townes linewidth is19 VSTL(ω)=(hν0Δνc2)/(2Pout), where h is Planck’s constant, ν0 is the average output frequency of the laser, Pout is the output power of the laser, and Δνc is the cold cavity linewidth of the laser. Translating these quantities into the notation used throughout this paper yields Δνc=2κ and Pout20. Substituting these quantities into the equation for VSTL gives VSTL=VΩ, a.

H.-A. Bachor, A Guide To Experiments in Quantum Optics (Wiley-VCH, Berlin, 1998).

T. C. Ralph, C. C. Harb, and H.-A. Bachor, “Intensity noise of injection locked lasers: quantum theory using a linearized input-output method,” Phys. Rev. A 54, 4359–4369(1996); T. C. Ralph, “Low noise amplification and efficient squeezing from rate-matched lasers,” Phys. Rev. A 55, 2326–2333 (1997).
[CrossRef] [PubMed]

Dipole noise terms that could be important if higher order-terms in the fluctuations were considered have been neglected. See Ref. 10 for full expressions.

C. O’Brien, “Polarization effects and spatial mode mismatch in injection locked laser systems,” honours thesis (Australian National University, Canberra, 1997).

W. Koechner, Solid-State Laser Engineering, 5th ed. (Springer-Verlag, Berlin, 1996).

We expect the measured low-frequency part of the transfer function to be at 0 dB because, for VΦM≫H and V≫1, we find that V≅HVΦM, Vsub≅ηV≅ηHVΦM, and VΦM, det≅ηmVΦM. If we assume that we detect the same power, Pdet, in our measurements of VΦM and Vsub, we find that PdetmPm=ηP=ηHPm. This relationship between the two attenuations, when it is combined with our relationships above between the detected spectra, gives us Vsub≅VΦM in the low-frequency region. For more details, see Ref. 20.

Measurements of the transfer functions of the PZT on both the master and the slave laser crystals show a frequency response that was similar to the spectra shown in Fig. 6.

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Figures (8)

Fig. 1
Fig. 1

Schematic of the laser model, showing the atomic energy-level scheme and the various noise sources. Noise inputs to the quadrature phase-noise spectrum are the noise incident upon the output coupler of the laser cavity, Vm-; the noise from losses within the cavity, Vl-; the noise from the atomic dipole fluctuations, Vdip-; and the noise from fluctuations of the slave laser cavity length, Vcav.

Fig. 2
Fig. 2

Relation of the unnormalized frequency noise VΩ(ω) to the normalized frequency noise VΩ(ω) and the normalized quadrature phase noise V-(ω). Here we show the specific example of a free-running laser. At low frequencies the frequency noise is given by the STL, and at high frequencies it is given by the QNL. Here we have assumed an output power of 1 mW at 1 µm and a cavity decay rate of κ=12.2×107 s-1.

Fig. 3
Fig. 3

Transfer functions for the sources of quadrature phase or frequency noise for the injection-locked laser. The transfer functions for the noise sources that are fundamentally optical in origin, Tm(ω), Tdip(ω), and Tl(ω), are independent of which units one measures the noise of the laser in. We simply redefine the input noise spectra into those units. For the optical noise induced by the length fluctuations of the cavity we keep the input noise spectrum Vcav(ω) fixed and alter the transfer function, depending on the measurement units. (a) Transfer functions Tm(ω), Tcav(ω), Tdip(ω), and Tl(ω) when we measure the normalized quadrature phase noise or the normalized frequency noise of the laser. (b) Transfer functions Tm,Ω(ω)=Tm(ω), Tcav,Ω(ω), Tdip,Ω(ω)=Tdip(ω), and Tl,Ω(ω)=Tl(ω) for the unnormalized frequency noise. FSR, free spectral range.

Fig. 4
Fig. 4

Schematic diagram of the experiment: RM’s, removable mirrors; BS, 50:50 nonpolarizing beam splitter; PBS’s, polarizing beam splitters; PZT, piezoelectric transducer; EOM, resonant electro-optic modulator; EOM BB, broadband EOM; SIG, sinusoidal signal generator at 12 MHz; MML, mode-matching lens; λ/2’s, half-wave plates at 1.064 µm; λ/4’s, quarter-wave plates at 1.064 µm; VA, variable attenuator; PT, partially transmitting mirror with 17% tap-off for Pound–Drever measurement; LPZT, PZT input to laser; NA, Anritsu MS4630A network analyzer; SA, Tektronix 2753P spectrum analyzer; PDL, resonant photodetector used for Pound–Drever frequency-locking servo; PD’s, broadband photodetectors used for homodyne measurements.

Fig. 5
Fig. 5

Measured and predicted transfer function for master phase modulation to injection-locked slave output for the high-power, high-gain case. The measurements were taken with a power ratio of HHP=21 and a measured locking range of ΔLhp=4.5 MHz. The theory is based on the measured parameters for the experiment given in Table 2. The large signal at 12 MHz is due to the phase modulation sidebands imposed on the master laser for the Pound–Drever locking loop. These sidebands were larger than Vm (12 MHz) imposed by the EOM BB and so appear in the measured transfer function. The other resonant structures in the figure were due to resonances in the EOM BB that were sufficiently sharp that they could not be completely normalized out of the measurement of Tm.

Fig. 6
Fig. 6

Measurement of phase noise in the injection-locked laser for a master field with fluctuations at the QNL. (i) The low-power, low-gain case with no attenuation and HLP=7. (ii) The high-power, high-gain case with attenuation of η2=0.005 and HHP=21. We have allowed for the attenuation in the high-power case, using the equation VHP=[(Vdet-1)/η2HP]+1. In both cases the dominant source of noise is from the length fluctuations of the slave laser cavity. The roll-off of these noise sources at high frequencies is observable. In both cases the total detected power was due primarily to the local-oscillator power of 18.2 mW. The very large signal at 12 MHz is again a measurement of the phase modulation signal imposed on the master laser for the Pound–Drever frequency-locking circuit.

Fig. 7
Fig. 7

Measurements of the phase modulation of the injection-locked laser relative to the QNL as a function of the modulation of the injection-locking field. (a) Modulations imposed at a frequency less than the locking range. HHP=21 for the high-power, high-gain case and HLP=7 for the low-power, low-gain case. These results show that the slave laser amplifies the modulations by the power amplification factor and that excess noise is added to the output. (b) Modulations imposed at a frequency greater than the locking range. Shown are the results for the high- and low-power cases as well as the theoretical prediction of Vout=Vm. This shows that Vout=Vm independently of the power amplification factor and the master variance.

Fig. 8
Fig. 8

Transfer function for quadrature phase noise introduced to laser output as a result of intracavity losses. We show the transfer function for the free-running laser Tl,free(ω) and the injection-locked laser Tl,lock(ω) along with the change between the free-running and injection-locked lasers. We quantify this change by plotting Tl,free(ω)/Tl,lock(ω).

Tables (2)

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Table 1 Parameters of the Laser Modela

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Table 2 Values of the Parameters of the Nd:YAG Laser

Equations (53)

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aˆ˙=G2Jˆ3-κ+iδXˆcavaˆ+2κmAˆm+2κlδAˆl-G4Jˆ3δXˆdip,
Jˆ˙3=-GJˆ3aˆaˆ-γtJˆ3+Γ+γtJ3δXˆt-ΓδXˆpump+GJˆ3δXˆdip.
a˙=G2J3α-κα+2κmA¯m,
J˙3=-GJ3|α|2-γtJ3+Γ.
δXˆ˙a-=2κmδXˆAm-+2κlδXˆAl--GJ3δXˆdip-+2αδXˆcav.
δXA-(ω)=2κmδXa--δXAm-.
δXA-(ω)=Fm(ω)δXAm-(ω)+Fcav(ω)δXcav(ω)+Fdip(ω)δXdip-(ω)+Fl(ω)δXAl-(ω),
Fm(ω)=[κm-κl+(GJ3/2)]-ιωκ-(GJ3/2)+ιω,
Fcav(ω)=2κmακ-(GJ3/2)+ιω
Fdip(ω)=(2κmGJ3)1/2κ-(GJ3/2)+ιω,
Fl(ω)=2κmκlκ-(GJ3/2)+ιω, Δ=2κmA¯m2α21/2.
V-(ω)=|δXA-(ω)|2=i|Fi(ω)δXi-(ω)|2,
V-(ω)=Tm(ω)Vm-(ω)+Tcav(ω)Vcav(ω)+Tdip(ω)Vdip-(ω)+Tl(ω)Vl-(ω),
Tm=|Fm|2HΔ2+ω2Δ2+ω2,
Tcav=|Fcav|22κmα2Δ2+ω2,
Tdip=|Fdip|24κm(κ-Δ)Δ2+ω2,
Tl=|Fl|24κmκlΔ2+ω2,
H=Pm+PsPmPsPm1,PmPs.
V-(ω)HVm-(ω)+H-1+A¯2Δ2Vcav(ω),ωΔ.
V-(ω)=Vm-(ω),ωΔ.
δXˆa-(t)=2αδϕa(t).
δΩa(t)=δϕ˙a(t),
δΩa(ω)=ωϕa(ω)=ω δXa-(ω)2α[Hz/Hz].
VΩ,a(ω)=|δΩa(ω)|2=ω2 |δXa-|24α2=ω2 Va-4α2[Hz2/Hz].
VΩ(ω)=δΩ2(ω)=ω2 V-4A¯2.
VΩQNL(ω)=ω2 VQNL-4A¯2=ω24A¯2.
VΩ,a(ω)=δΩa2(ω)=2κ[δXm-(ω)-δXdip-(ω)]2α2=2×2κ4α2=12n¯tc=STL,
VΩ(ω)=8κmκ+ω24A¯2[Hz2/Hz]
2κm2κm4κ4α2=12n¯tc=STL,ω28κmκ,
ω24A¯2=QNL,ω28κmκ,
VΩ(ω)=VΩ(ω)VΩ,QNL(ω)=V-(ω).
VΩ(ω)=Tm(ω)Vm,Ω(ω)+Tcav,Ω(ω)Vcav(ω)+Tdip(ω)Vdip,Ω(ω)+Tl(ω)Vl,Ω(ω),
Vm,Ω(ω)=Vm-(ω)×ω2/4A¯2,
Tcav,Ω(ω)=Tcav(ω)×ω2/4A¯2,
Vdip,Ω(ω)=Vdip-(ω)×ω2/4A¯2,
Vl,Ω(ω)=Vl-(ω)×ω2/4A¯2.
V-(ω)=Tm(ω)Vm-(ω)+VN-(ω),
VN-(ω)=Tcav(ω)Vcav(ω)+Tl(ω)Vl-(ω)+Tdip(ω)Vdip-(ω).
nsub(ω, θ)=n¯sub(θ)+δnsub(ω, θ),
δnsub(ω, θ)=A¯LOδXθ(ω)+A¯δXLO-θ(ω),
n¯sub(θ)=A¯LOA ¯cos(θ),
Vsub(ω, θ)Vθ(ω),
VsubTm[(1-η)ηdetVin-θ+ηηdetVv-θ+ηdetVΦM-θ]+ηdetVN-θ+(1-η)ηdetHVinθ+(1-η)ηdetηHVvθ+(1-ηdet)Vv2-θ,
Vsub(ω)Tm(ω)[Vv-(ω)+VΦM-(ω)]+VN-(ω).
Vsub(ω)=VN-(ω)+TmVv-.
Vsub(ω)=Tm(ω)VΦM-(ω).
Tm(ω)=Vsub(ω)VΦM-(ω).
Vf-=A¯2ω2Vcav(ω)+1ω2VQuant-(ω)+1ω2VEXT(ω),
VQuant-(ω)=4κm(κ+κl),
VEXT(ω)=2κm+ω2.
VΩ,f(ω)=14Vcav(ω)+14A¯2VQuant+14A¯2VEXT.
V-(ω)=A¯2ω2+Δ2Vcav+1ω2+Δ2VQuant(ω)+Tm(ω)Vm-(ω).
VΩ=14ω2Δ2+ω2Vcav(ω)+Tm(ω)×ω24A¯2Vm-(ω)+14ω2ω2+Δ2VQuant(ω).

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