Abstract

A phase-modulated RF current source is applied to an injection-locked diode laser operating at 780 nm. This produces tunable phase-modulated sidebands of the laser suitable for stabilizing the length of an optical transfer cavity using the Pound-Drever-Hall technique. The Pound-Drever-Hall signal is antisymmetric about the lock point, despite the presence of significant diode laser amplitude modulation. The stabilized optical transfer cavity is used to frequency stabilize a 776 nm external cavity diode laser. The stability and tunability of this transfer cavity locked laser is established by observation of the hyperfine components of the Rb87 5P3/25D5/2 transition in a vapor cell.

© 2012 Optical Society of America

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References

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  1. A. D. Ludlow, X. Huang, M. Notcutt, T. Zanon-Willette, S. M. Foreman, M. M. Boyd, S. Blatt, and J. Ye, “Compact, thermal-noise-limited optical cavity for diode laser stabilization at 1×10−15,” Opt. Lett. 32, 641–643 (2007).
    [CrossRef]
  2. B. Burghardt, W. Jitschin, and G. Meisel, “Precise rf tuning for cw dye lasers,” Appl. Phys. 20, 141–146 (1979).
    [CrossRef]
  3. T. C. Briles, D. C. Yost, A. Cingöz, J. Ye, and T. R. Schibli, “Simple piezoelectric-actuated mirror with 180 kHz servo bandwidth,” Opt. Express 18, 9739–9746 (2010).
    [CrossRef]
  4. D. F. Plusquellic, O. Votava, and D. J. Nesbitt, “Absolute frequency stabilization of an injection-seeded optical parametric oscillator,” Appl. Opt. 35, 1464–1472 (1996).
    [CrossRef]
  5. P. Bohlouli-Zanjani, K. Afrousheh, and J. D. D. Martin, “Optical transfer cavity stabilization using current-modulated injection-locked diode lasers,” Rev. Sci. Instrum. 77, 093105 (2006).
    [CrossRef]
  6. R. W. P. Drever, J. L. Hall, F. V. Kowalski, 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]
  7. R. W. Fox, C. W. Oates, and L. Hollberg, “Stabilizing diode lasers to high finesse cavities,” in Experimental Methods in the Physical Sciences; Cavity-Enhanced Spectroscopies, Vol. 40, R. D. V. Zee and J. Looney, eds. (Academic, 2002), Chap. 1, pp. 1–46.
  8. A. Siegman, Lasers (University Science Books, 1986).
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    [CrossRef]
  10. G. C. Bjorklund, “Frequency-modulation spectroscopy: a new method for measuring weak absorptions and dispersions,” Opt. Lett. 5, 15–17 (1980).
    [CrossRef]
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    [CrossRef]
  12. C. E. Liekhus-Schmaltz and J. D. D. Martin, “ Understanding Pound-Drever-Hall locking using voltage controlled radio-frequency oscillators: An undergraduate experiment,” Am. J. Phys. 80, 232–239 (2012).
    [CrossRef]
  13. S. H. Youn, M. Lu, U. Ray, and B. L. Lev, “Dysprosium magneto-optical traps,” Phys. Rev. A 82, 043425 (2010).
  14. S. Kobayashi, Y. Yamamoto, M. Ito, and T. Kimura, “Direct frequency modulation in AlGaAs semiconductor lasers,” IEEE J. Quantum Electron. 18, 582–595 (1982).
    [CrossRef]
  15. O. Lidoyne, P. Gallion, and D. Erasme, “Modulation properties of an injection-locked semiconductor laser,” IEEE J. Quantum Electron. 27, 344–351 (1991).
    [CrossRef]
  16. G. Watson, A Treatise on the Theory of Bessel Functions, Cambridge Mathematical Library (Cambridge University, 1995).
  17. T. T. Grove, V. Sanchez-Villicana, B. C. Duncan, S. Maleki, and P. L. Gould, “Two-photon two-color diode laser spectroscopy of the Rb 5D5/2 state,” Phys. Scr. 52, 271–276 (1995).
    [CrossRef]
  18. F. Nez, F. Biraben, R. Felder, and Y. Millerioux, “Optical frequency determination of the hyperfine components of the 5S1/2−5D3/2 two-photon transitions in rubidium,” Opt. Commun. 102, 432–438 (1993).
    [CrossRef]
  19. J. A. Petrus, P. Bohlouli-Zanjani, and J. D. D. Martin, “ac electric-field-induced resonant energy transfer between cold Rydberg atoms,” J. Phys. B 41, 245001 (2008).
  20. J. Helmcke, S. A. Lee, and J. L. Hall, “Dye laser spectrometer for ultrahigh spectral resolution: design and performance,” Appl. Opt. 21, 1686–1694 (1982).
    [CrossRef]

2012 (1)

C. E. Liekhus-Schmaltz and J. D. D. Martin, “ Understanding Pound-Drever-Hall locking using voltage controlled radio-frequency oscillators: An undergraduate experiment,” Am. J. Phys. 80, 232–239 (2012).
[CrossRef]

2010 (2)

2008 (1)

J. A. Petrus, P. Bohlouli-Zanjani, and J. D. D. Martin, “ac electric-field-induced resonant energy transfer between cold Rydberg atoms,” J. Phys. B 41, 245001 (2008).

2007 (1)

2006 (1)

P. Bohlouli-Zanjani, K. Afrousheh, and J. D. D. Martin, “Optical transfer cavity stabilization using current-modulated injection-locked diode lasers,” Rev. Sci. Instrum. 77, 093105 (2006).
[CrossRef]

1996 (1)

1995 (1)

T. T. Grove, V. Sanchez-Villicana, B. C. Duncan, S. Maleki, and P. L. Gould, “Two-photon two-color diode laser spectroscopy of the Rb 5D5/2 state,” Phys. Scr. 52, 271–276 (1995).
[CrossRef]

1993 (1)

F. Nez, F. Biraben, R. Felder, and Y. Millerioux, “Optical frequency determination of the hyperfine components of the 5S1/2−5D3/2 two-photon transitions in rubidium,” Opt. Commun. 102, 432–438 (1993).
[CrossRef]

1991 (1)

O. Lidoyne, P. Gallion, and D. Erasme, “Modulation properties of an injection-locked semiconductor laser,” IEEE J. Quantum Electron. 27, 344–351 (1991).
[CrossRef]

1983 (1)

R. W. P. Drever, J. L. Hall, F. V. Kowalski, 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 (3)

S. Kobayashi and T. Kimura, “Optical phase modulation in an injection locked AlGaAs semiconductor laser,” IEEE J. Quantum Electron. 18, 1662–1669 (1982).
[CrossRef]

J. Helmcke, S. A. Lee, and J. L. Hall, “Dye laser spectrometer for ultrahigh spectral resolution: design and performance,” Appl. Opt. 21, 1686–1694 (1982).
[CrossRef]

S. Kobayashi, Y. Yamamoto, M. Ito, and T. Kimura, “Direct frequency modulation in AlGaAs semiconductor lasers,” IEEE J. Quantum Electron. 18, 582–595 (1982).
[CrossRef]

1980 (1)

1979 (1)

B. Burghardt, W. Jitschin, and G. Meisel, “Precise rf tuning for cw dye lasers,” Appl. Phys. 20, 141–146 (1979).
[CrossRef]

1946 (1)

R. V. Pound, “Electronic frequency stabilization of microwave oscillators,” Rev. Sci. Instrum. 17, 490–493 (1946).
[CrossRef]

Afrousheh, K.

P. Bohlouli-Zanjani, K. Afrousheh, and J. D. D. Martin, “Optical transfer cavity stabilization using current-modulated injection-locked diode lasers,” Rev. Sci. Instrum. 77, 093105 (2006).
[CrossRef]

Biraben, F.

F. Nez, F. Biraben, R. Felder, and Y. Millerioux, “Optical frequency determination of the hyperfine components of the 5S1/2−5D3/2 two-photon transitions in rubidium,” Opt. Commun. 102, 432–438 (1993).
[CrossRef]

Bjorklund, G. C.

Blatt, S.

Bohlouli-Zanjani, P.

J. A. Petrus, P. Bohlouli-Zanjani, and J. D. D. Martin, “ac electric-field-induced resonant energy transfer between cold Rydberg atoms,” J. Phys. B 41, 245001 (2008).

P. Bohlouli-Zanjani, K. Afrousheh, and J. D. D. Martin, “Optical transfer cavity stabilization using current-modulated injection-locked diode lasers,” Rev. Sci. Instrum. 77, 093105 (2006).
[CrossRef]

Boyd, M. M.

Briles, T. C.

Burghardt, B.

B. Burghardt, W. Jitschin, and G. Meisel, “Precise rf tuning for cw dye lasers,” Appl. Phys. 20, 141–146 (1979).
[CrossRef]

Cingöz, A.

Drever, R. W. P.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, 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]

Duncan, B. C.

T. T. Grove, V. Sanchez-Villicana, B. C. Duncan, S. Maleki, and P. L. Gould, “Two-photon two-color diode laser spectroscopy of the Rb 5D5/2 state,” Phys. Scr. 52, 271–276 (1995).
[CrossRef]

Erasme, D.

O. Lidoyne, P. Gallion, and D. Erasme, “Modulation properties of an injection-locked semiconductor laser,” IEEE J. Quantum Electron. 27, 344–351 (1991).
[CrossRef]

Felder, R.

F. Nez, F. Biraben, R. Felder, and Y. Millerioux, “Optical frequency determination of the hyperfine components of the 5S1/2−5D3/2 two-photon transitions in rubidium,” Opt. Commun. 102, 432–438 (1993).
[CrossRef]

Ford, G. M.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, 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]

Foreman, S. M.

Fox, R. W.

R. W. Fox, C. W. Oates, and L. Hollberg, “Stabilizing diode lasers to high finesse cavities,” in Experimental Methods in the Physical Sciences; Cavity-Enhanced Spectroscopies, Vol. 40, R. D. V. Zee and J. Looney, eds. (Academic, 2002), Chap. 1, pp. 1–46.

Gallion, P.

O. Lidoyne, P. Gallion, and D. Erasme, “Modulation properties of an injection-locked semiconductor laser,” IEEE J. Quantum Electron. 27, 344–351 (1991).
[CrossRef]

Gould, P. L.

T. T. Grove, V. Sanchez-Villicana, B. C. Duncan, S. Maleki, and P. L. Gould, “Two-photon two-color diode laser spectroscopy of the Rb 5D5/2 state,” Phys. Scr. 52, 271–276 (1995).
[CrossRef]

Grove, T. T.

T. T. Grove, V. Sanchez-Villicana, B. C. Duncan, S. Maleki, and P. L. Gould, “Two-photon two-color diode laser spectroscopy of the Rb 5D5/2 state,” Phys. Scr. 52, 271–276 (1995).
[CrossRef]

Hall, J. L.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, 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]

J. Helmcke, S. A. Lee, and J. L. Hall, “Dye laser spectrometer for ultrahigh spectral resolution: design and performance,” Appl. Opt. 21, 1686–1694 (1982).
[CrossRef]

Helmcke, J.

Hollberg, L.

R. W. Fox, C. W. Oates, and L. Hollberg, “Stabilizing diode lasers to high finesse cavities,” in Experimental Methods in the Physical Sciences; Cavity-Enhanced Spectroscopies, Vol. 40, R. D. V. Zee and J. Looney, eds. (Academic, 2002), Chap. 1, pp. 1–46.

Hough, J.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, 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]

Huang, X.

Ito, M.

S. Kobayashi, Y. Yamamoto, M. Ito, and T. Kimura, “Direct frequency modulation in AlGaAs semiconductor lasers,” IEEE J. Quantum Electron. 18, 582–595 (1982).
[CrossRef]

Jitschin, W.

B. Burghardt, W. Jitschin, and G. Meisel, “Precise rf tuning for cw dye lasers,” Appl. Phys. 20, 141–146 (1979).
[CrossRef]

Kimura, T.

S. Kobayashi, Y. Yamamoto, M. Ito, and T. Kimura, “Direct frequency modulation in AlGaAs semiconductor lasers,” IEEE J. Quantum Electron. 18, 582–595 (1982).
[CrossRef]

S. Kobayashi and T. Kimura, “Optical phase modulation in an injection locked AlGaAs semiconductor laser,” IEEE J. Quantum Electron. 18, 1662–1669 (1982).
[CrossRef]

Kobayashi, S.

S. Kobayashi, Y. Yamamoto, M. Ito, and T. Kimura, “Direct frequency modulation in AlGaAs semiconductor lasers,” IEEE J. Quantum Electron. 18, 582–595 (1982).
[CrossRef]

S. Kobayashi and T. Kimura, “Optical phase modulation in an injection locked AlGaAs semiconductor laser,” IEEE J. Quantum Electron. 18, 1662–1669 (1982).
[CrossRef]

Kowalski, F. V.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, 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]

Lee, S. A.

Lev, B. L.

S. H. Youn, M. Lu, U. Ray, and B. L. Lev, “Dysprosium magneto-optical traps,” Phys. Rev. A 82, 043425 (2010).

Lidoyne, O.

O. Lidoyne, P. Gallion, and D. Erasme, “Modulation properties of an injection-locked semiconductor laser,” IEEE J. Quantum Electron. 27, 344–351 (1991).
[CrossRef]

Liekhus-Schmaltz, C. E.

C. E. Liekhus-Schmaltz and J. D. D. Martin, “ Understanding Pound-Drever-Hall locking using voltage controlled radio-frequency oscillators: An undergraduate experiment,” Am. J. Phys. 80, 232–239 (2012).
[CrossRef]

Lu, M.

S. H. Youn, M. Lu, U. Ray, and B. L. Lev, “Dysprosium magneto-optical traps,” Phys. Rev. A 82, 043425 (2010).

Ludlow, A. D.

Maleki, S.

T. T. Grove, V. Sanchez-Villicana, B. C. Duncan, S. Maleki, and P. L. Gould, “Two-photon two-color diode laser spectroscopy of the Rb 5D5/2 state,” Phys. Scr. 52, 271–276 (1995).
[CrossRef]

Martin, J. D. D.

C. E. Liekhus-Schmaltz and J. D. D. Martin, “ Understanding Pound-Drever-Hall locking using voltage controlled radio-frequency oscillators: An undergraduate experiment,” Am. J. Phys. 80, 232–239 (2012).
[CrossRef]

J. A. Petrus, P. Bohlouli-Zanjani, and J. D. D. Martin, “ac electric-field-induced resonant energy transfer between cold Rydberg atoms,” J. Phys. B 41, 245001 (2008).

P. Bohlouli-Zanjani, K. Afrousheh, and J. D. D. Martin, “Optical transfer cavity stabilization using current-modulated injection-locked diode lasers,” Rev. Sci. Instrum. 77, 093105 (2006).
[CrossRef]

Meisel, G.

B. Burghardt, W. Jitschin, and G. Meisel, “Precise rf tuning for cw dye lasers,” Appl. Phys. 20, 141–146 (1979).
[CrossRef]

Millerioux, Y.

F. Nez, F. Biraben, R. Felder, and Y. Millerioux, “Optical frequency determination of the hyperfine components of the 5S1/2−5D3/2 two-photon transitions in rubidium,” Opt. Commun. 102, 432–438 (1993).
[CrossRef]

Munley, A. J.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, 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]

Nesbitt, D. J.

Nez, F.

F. Nez, F. Biraben, R. Felder, and Y. Millerioux, “Optical frequency determination of the hyperfine components of the 5S1/2−5D3/2 two-photon transitions in rubidium,” Opt. Commun. 102, 432–438 (1993).
[CrossRef]

Notcutt, M.

Oates, C. W.

R. W. Fox, C. W. Oates, and L. Hollberg, “Stabilizing diode lasers to high finesse cavities,” in Experimental Methods in the Physical Sciences; Cavity-Enhanced Spectroscopies, Vol. 40, R. D. V. Zee and J. Looney, eds. (Academic, 2002), Chap. 1, pp. 1–46.

Petrus, J. A.

J. A. Petrus, P. Bohlouli-Zanjani, and J. D. D. Martin, “ac electric-field-induced resonant energy transfer between cold Rydberg atoms,” J. Phys. B 41, 245001 (2008).

Plusquellic, D. F.

Pound, R. V.

R. V. Pound, “Electronic frequency stabilization of microwave oscillators,” Rev. Sci. Instrum. 17, 490–493 (1946).
[CrossRef]

Ray, U.

S. H. Youn, M. Lu, U. Ray, and B. L. Lev, “Dysprosium magneto-optical traps,” Phys. Rev. A 82, 043425 (2010).

Sanchez-Villicana, V.

T. T. Grove, V. Sanchez-Villicana, B. C. Duncan, S. Maleki, and P. L. Gould, “Two-photon two-color diode laser spectroscopy of the Rb 5D5/2 state,” Phys. Scr. 52, 271–276 (1995).
[CrossRef]

Schibli, T. R.

Siegman, A.

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

Votava, O.

Ward, H.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, 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]

Watson, G.

G. Watson, A Treatise on the Theory of Bessel Functions, Cambridge Mathematical Library (Cambridge University, 1995).

Yamamoto, Y.

S. Kobayashi, Y. Yamamoto, M. Ito, and T. Kimura, “Direct frequency modulation in AlGaAs semiconductor lasers,” IEEE J. Quantum Electron. 18, 582–595 (1982).
[CrossRef]

Ye, J.

Yost, D. C.

Youn, S. H.

S. H. Youn, M. Lu, U. Ray, and B. L. Lev, “Dysprosium magneto-optical traps,” Phys. Rev. A 82, 043425 (2010).

Zanon-Willette, T.

Am. J. Phys. (1)

C. E. Liekhus-Schmaltz and J. D. D. Martin, “ Understanding Pound-Drever-Hall locking using voltage controlled radio-frequency oscillators: An undergraduate experiment,” Am. J. Phys. 80, 232–239 (2012).
[CrossRef]

Appl. Opt. (2)

Appl. Phys. (1)

B. Burghardt, W. Jitschin, and G. Meisel, “Precise rf tuning for cw dye lasers,” Appl. Phys. 20, 141–146 (1979).
[CrossRef]

Appl. Phys. B (1)

R. W. P. Drever, J. L. Hall, F. V. Kowalski, 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. (3)

S. Kobayashi and T. Kimura, “Optical phase modulation in an injection locked AlGaAs semiconductor laser,” IEEE J. Quantum Electron. 18, 1662–1669 (1982).
[CrossRef]

S. Kobayashi, Y. Yamamoto, M. Ito, and T. Kimura, “Direct frequency modulation in AlGaAs semiconductor lasers,” IEEE J. Quantum Electron. 18, 582–595 (1982).
[CrossRef]

O. Lidoyne, P. Gallion, and D. Erasme, “Modulation properties of an injection-locked semiconductor laser,” IEEE J. Quantum Electron. 27, 344–351 (1991).
[CrossRef]

J. Phys. B (1)

J. A. Petrus, P. Bohlouli-Zanjani, and J. D. D. Martin, “ac electric-field-induced resonant energy transfer between cold Rydberg atoms,” J. Phys. B 41, 245001 (2008).

Opt. Commun. (1)

F. Nez, F. Biraben, R. Felder, and Y. Millerioux, “Optical frequency determination of the hyperfine components of the 5S1/2−5D3/2 two-photon transitions in rubidium,” Opt. Commun. 102, 432–438 (1993).
[CrossRef]

Opt. Express (1)

Opt. Lett. (2)

Phys. Rev. A (1)

S. H. Youn, M. Lu, U. Ray, and B. L. Lev, “Dysprosium magneto-optical traps,” Phys. Rev. A 82, 043425 (2010).

Phys. Scr. (1)

T. T. Grove, V. Sanchez-Villicana, B. C. Duncan, S. Maleki, and P. L. Gould, “Two-photon two-color diode laser spectroscopy of the Rb 5D5/2 state,” Phys. Scr. 52, 271–276 (1995).
[CrossRef]

Rev. Sci. Instrum. (2)

R. V. Pound, “Electronic frequency stabilization of microwave oscillators,” Rev. Sci. Instrum. 17, 490–493 (1946).
[CrossRef]

P. Bohlouli-Zanjani, K. Afrousheh, and J. D. D. Martin, “Optical transfer cavity stabilization using current-modulated injection-locked diode lasers,” Rev. Sci. Instrum. 77, 093105 (2006).
[CrossRef]

Other (3)

R. W. Fox, C. W. Oates, and L. Hollberg, “Stabilizing diode lasers to high finesse cavities,” in Experimental Methods in the Physical Sciences; Cavity-Enhanced Spectroscopies, Vol. 40, R. D. V. Zee and J. Looney, eds. (Academic, 2002), Chap. 1, pp. 1–46.

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

G. Watson, A Treatise on the Theory of Bessel Functions, Cambridge Mathematical Library (Cambridge University, 1995).

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Figures (4)

Fig. 1.
Fig. 1.

PDH transfer cavity locking of a 776 nm laser using a 780 nm modulated slave laser. The 776 nm and 780 nm beams have orthogonal polarizations and are combined and separated using polarizing beam splitter cubes. The fiber is polarization preserving. AMP: amplifier, BS: nonpolarizing beam splitter, ECDL: external cavity diode laser, FM-SAS: frequency-modulated saturated absorption spectroscopy, LPF: low-pass filter, MX: mixer, PBS: polarizing beam splitter, PD: photodiode, PM: phase modulation, PZT: piezo-electric transducer.

Fig. 2.
Fig. 2.

(a) Fabry–Perot transmission spectra, and (b) PDH error signal for δ / ( 2 π ) = 100 MHz observed using a scanning Fabry-Perot cavity with a free spectral range (FSR) of 1.6 GHz and finesse ( ) of 2500. No offset has been added to this signal. (c) Calculated PDH error signal shown for comparison (see, for example [12]).

Fig. 3.
Fig. 3.

Scheme to monitor 776 nm ECDL tunability and frequency stability using the Rb 87 , 5 P 3 / 2 5 D 5 / 2 transitions. LIA: lock-in amplifier MX: mixer, PD: photodiode.

Fig. 4.
Fig. 4.

(a) Absorption spectrum of the 5 P 3 / 2 5 D 5 / 2 transition of Rb 87 . (b) FM absorption spectrum [see Fig. 3]. (c) Frequency stability of the locked 776 nm laser as a function of time, monitored using the FM absorption signal (0.3 s time constant). The RF frequency is set to place the 776 nm laser at the 5 P 3 / 2 , F = 3 5 D 5 / 2 , F = 4 transition and the FM signal is used as a frequency discriminator. Periodically stepping δ / ( 2 π ) up and down by 1 MHz allows conversion of the FM signal to frequency deviation.

Equations (4)

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

E ~ = E ~ 0 exp { j ( ω 0 t + k Δ I ( t ) ) } ,
E ~ = E ~ 0 exp { j ( ω 0 t + α sin [ δ t + β sin ( Ω t ) ] ) } .
E ~ E 0 ~ = ( 1 + A sin [ ϕ ] + B cos [ ϕ ] ) exp { j ( ω 0 t + α sin [ ϕ ] ) } ,
E ~ ω 0 δ + Ω E 0 ~ = J 1 ( α ) J 1 ( β ) ( A j + B ) 2 J 1 ( β ) J 0 ( α ) + ( A j + B ) 2 J 2 ( α ) = J ( β ) J 1 ( 2 β ) ,

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