Abstract

An important practical limitation to the sensitivity of frequency-modulation spectroscopy arises because of imperfect phase modulation of the laser beam. This imperfection manifests itself as residual amplitude modulation (RAM), and we present data from a careful series of experiments designed to elucidate its origin. Experimentally, we find two components to the RAM, one that depends for its intensity on the laser frequency and one that does not. We present a model that correctly accounts for the intensity and rf phase and frequency behavior of the laser-frequency-dependent component. We are unable to offer a plausible explanation for the laser-frequency-independent component.

© 1985 Optical Society of America

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References

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  1. G. C. Bjorklund, Opt. Lett. 5, 15 (1980); J. L. Hall, L. Hollberg, T. Baer, and H. G. Robinson, Appl. Phys. Lett. 39, 680 (1981).
    [CrossRef]
  2. R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, and A. J. Munley, Joint Institute for Laboratory Astrophysics, University of Colorado and National Bureau of Standards, Boulder, Colo. 80309 (personal communication, 1979); M. Prentiss, B. Peuse, G. Sanders, and S. Ezekiel, Research Laboratory of Electronics, Prog. Rep. No. 123 (Massachusetts Institute of Technology, Cambridge, Mass., 1981).
  3. E. A. Whittaker, P. Pokrowsky, W. Zapka, K. Roche, and G. C. Bjorklund, J. Quant. Spectrosc. Radiat. Transfer 30, 289 (1983).
    [CrossRef]
  4. M. D. Levenson, W. E. Moerner, and D. E. Horne, Opt. Lett. 8, 108 (1983).
    [CrossRef] [PubMed]
  5. E. A. Whittaker, B. J. Sullivan, G. C. Bjorklund, H. R. Wendt, and H. E. Hunziker, J. Chem. Phys. 80, 961 (1984).
    [CrossRef]
  6. E. A. Whittaker, H. R. Wendt, H. E. Hunziker, and G. C. Bjorklund, Appl. Phys. B 35, 105 (1984).
    [CrossRef]
  7. J. M. Jasinski, E. A. Whittaker, G. C. Bjorklund, R. W. Dreyfus, R. D. Estes, and R. E. Walkup, Appl. Phys. Lett. 44, 1155 (1984).
    [CrossRef]
  8. J. H. Bechtel and A. R. Chraplyvy, Proc. IEEE 70, 658 (1982).
    [CrossRef]
  9. A. Z. Genack, K. P. Leung, and A. Schenzle, Phys. Rev. 28, 308 (1983).
    [CrossRef]
  10. A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, New York, 1983), Chap. 8, pp. 288ff.
  11. A. Yariv, Quantum Electronics, 2nd ed. (Wiley, New York, 1975), Chap. 14, pp. 343ff.
  12. M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, 1975), Chap. 7, pp. 323ff.
  13. M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, 1975), Chap. 1, p. 40.
  14. W. L. Bond, J. Appl. Phys. 36, 1674 (1965).
    [CrossRef]
  15. D. J. Bernays, IBM Res. Rep. RJ4166, San Jose, Calif. (1984).
  16. G. C. Bjorklund, M. D. Levenson, W. Lenth, and C. Oritz, Appl. Phys. B32, 2565 (1983).
  17. This result is analytically obvious for Eq. (9) but not so for Eq. (15), where the phase shift (which is, of course, merely the arc-tangent of the ratio of the coefficients of sin ωmt and cos ωmt) is manifestly dependent on the laser frequency. Nonetheless, numerical calculations indicate that at the modulation frequencies used in this experiment, the phase shift is only weakly dependent on the laser frequency.
  18. M. Gehrtz, W. E. Moerner, and G. C. Bjorklund, Opt. Lett. (to be published).
  19. We have also considered the possibility that a dc-RAM effect could be induced by electro-optic modulation of the Fresnel coefficient. Our calculations indicate this effect to be rather small and of the order of 10−7equivalent differential absorption. There has also been the suggestion that scattering from defects in the electro-optic crystal may play a role in the dc-RAM (L. Hollberg and J. L. Hall, Joint Institute for Laboratory Astrophysics, University of Colorado and National Bureau of Standards, Boulder, Colo. 80309 (personal communication). Because of its somewhat unreproducible behavior we have concluded that several sources are probably involved. We thank the referees for their suggestions in this matter.

1984 (3)

E. A. Whittaker, B. J. Sullivan, G. C. Bjorklund, H. R. Wendt, and H. E. Hunziker, J. Chem. Phys. 80, 961 (1984).
[CrossRef]

E. A. Whittaker, H. R. Wendt, H. E. Hunziker, and G. C. Bjorklund, Appl. Phys. B 35, 105 (1984).
[CrossRef]

J. M. Jasinski, E. A. Whittaker, G. C. Bjorklund, R. W. Dreyfus, R. D. Estes, and R. E. Walkup, Appl. Phys. Lett. 44, 1155 (1984).
[CrossRef]

1983 (4)

E. A. Whittaker, P. Pokrowsky, W. Zapka, K. Roche, and G. C. Bjorklund, J. Quant. Spectrosc. Radiat. Transfer 30, 289 (1983).
[CrossRef]

M. D. Levenson, W. E. Moerner, and D. E. Horne, Opt. Lett. 8, 108 (1983).
[CrossRef] [PubMed]

A. Z. Genack, K. P. Leung, and A. Schenzle, Phys. Rev. 28, 308 (1983).
[CrossRef]

G. C. Bjorklund, M. D. Levenson, W. Lenth, and C. Oritz, Appl. Phys. B32, 2565 (1983).

1982 (1)

J. H. Bechtel and A. R. Chraplyvy, Proc. IEEE 70, 658 (1982).
[CrossRef]

1980 (1)

1965 (1)

W. L. Bond, J. Appl. Phys. 36, 1674 (1965).
[CrossRef]

Bechtel, J. H.

J. H. Bechtel and A. R. Chraplyvy, Proc. IEEE 70, 658 (1982).
[CrossRef]

Bernays, D. J.

D. J. Bernays, IBM Res. Rep. RJ4166, San Jose, Calif. (1984).

Bjorklund, G. C.

E. A. Whittaker, H. R. Wendt, H. E. Hunziker, and G. C. Bjorklund, Appl. Phys. B 35, 105 (1984).
[CrossRef]

J. M. Jasinski, E. A. Whittaker, G. C. Bjorklund, R. W. Dreyfus, R. D. Estes, and R. E. Walkup, Appl. Phys. Lett. 44, 1155 (1984).
[CrossRef]

E. A. Whittaker, B. J. Sullivan, G. C. Bjorklund, H. R. Wendt, and H. E. Hunziker, J. Chem. Phys. 80, 961 (1984).
[CrossRef]

G. C. Bjorklund, M. D. Levenson, W. Lenth, and C. Oritz, Appl. Phys. B32, 2565 (1983).

E. A. Whittaker, P. Pokrowsky, W. Zapka, K. Roche, and G. C. Bjorklund, J. Quant. Spectrosc. Radiat. Transfer 30, 289 (1983).
[CrossRef]

G. C. Bjorklund, Opt. Lett. 5, 15 (1980); J. L. Hall, L. Hollberg, T. Baer, and H. G. Robinson, Appl. Phys. Lett. 39, 680 (1981).
[CrossRef]

M. Gehrtz, W. E. Moerner, and G. C. Bjorklund, Opt. Lett. (to be published).

Bond, W. L.

W. L. Bond, J. Appl. Phys. 36, 1674 (1965).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, 1975), Chap. 1, p. 40.

M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, 1975), Chap. 7, pp. 323ff.

Chraplyvy, A. R.

J. H. Bechtel and A. R. Chraplyvy, Proc. IEEE 70, 658 (1982).
[CrossRef]

Drever, R. W. P.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, and A. J. Munley, Joint Institute for Laboratory Astrophysics, University of Colorado and National Bureau of Standards, Boulder, Colo. 80309 (personal communication, 1979); M. Prentiss, B. Peuse, G. Sanders, and S. Ezekiel, Research Laboratory of Electronics, Prog. Rep. No. 123 (Massachusetts Institute of Technology, Cambridge, Mass., 1981).

Dreyfus, R. W.

J. M. Jasinski, E. A. Whittaker, G. C. Bjorklund, R. W. Dreyfus, R. D. Estes, and R. E. Walkup, Appl. Phys. Lett. 44, 1155 (1984).
[CrossRef]

Estes, R. D.

J. M. Jasinski, E. A. Whittaker, G. C. Bjorklund, R. W. Dreyfus, R. D. Estes, and R. E. Walkup, Appl. Phys. Lett. 44, 1155 (1984).
[CrossRef]

Ford, G. M.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, and A. J. Munley, Joint Institute for Laboratory Astrophysics, University of Colorado and National Bureau of Standards, Boulder, Colo. 80309 (personal communication, 1979); M. Prentiss, B. Peuse, G. Sanders, and S. Ezekiel, Research Laboratory of Electronics, Prog. Rep. No. 123 (Massachusetts Institute of Technology, Cambridge, Mass., 1981).

Gehrtz, M.

M. Gehrtz, W. E. Moerner, and G. C. Bjorklund, Opt. Lett. (to be published).

Genack, A. Z.

A. Z. Genack, K. P. Leung, and A. Schenzle, Phys. Rev. 28, 308 (1983).
[CrossRef]

Hall, J. L.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, and A. J. Munley, Joint Institute for Laboratory Astrophysics, University of Colorado and National Bureau of Standards, Boulder, Colo. 80309 (personal communication, 1979); M. Prentiss, B. Peuse, G. Sanders, and S. Ezekiel, Research Laboratory of Electronics, Prog. Rep. No. 123 (Massachusetts Institute of Technology, Cambridge, Mass., 1981).

We have also considered the possibility that a dc-RAM effect could be induced by electro-optic modulation of the Fresnel coefficient. Our calculations indicate this effect to be rather small and of the order of 10−7equivalent differential absorption. There has also been the suggestion that scattering from defects in the electro-optic crystal may play a role in the dc-RAM (L. Hollberg and J. L. Hall, Joint Institute for Laboratory Astrophysics, University of Colorado and National Bureau of Standards, Boulder, Colo. 80309 (personal communication). Because of its somewhat unreproducible behavior we have concluded that several sources are probably involved. We thank the referees for their suggestions in this matter.

Hollberg, L.

We have also considered the possibility that a dc-RAM effect could be induced by electro-optic modulation of the Fresnel coefficient. Our calculations indicate this effect to be rather small and of the order of 10−7equivalent differential absorption. There has also been the suggestion that scattering from defects in the electro-optic crystal may play a role in the dc-RAM (L. Hollberg and J. L. Hall, Joint Institute for Laboratory Astrophysics, University of Colorado and National Bureau of Standards, Boulder, Colo. 80309 (personal communication). Because of its somewhat unreproducible behavior we have concluded that several sources are probably involved. We thank the referees for their suggestions in this matter.

Horne, D. E.

Hough, J.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, and A. J. Munley, Joint Institute for Laboratory Astrophysics, University of Colorado and National Bureau of Standards, Boulder, Colo. 80309 (personal communication, 1979); M. Prentiss, B. Peuse, G. Sanders, and S. Ezekiel, Research Laboratory of Electronics, Prog. Rep. No. 123 (Massachusetts Institute of Technology, Cambridge, Mass., 1981).

Hunziker, H. E.

E. A. Whittaker, B. J. Sullivan, G. C. Bjorklund, H. R. Wendt, and H. E. Hunziker, J. Chem. Phys. 80, 961 (1984).
[CrossRef]

E. A. Whittaker, H. R. Wendt, H. E. Hunziker, and G. C. Bjorklund, Appl. Phys. B 35, 105 (1984).
[CrossRef]

Jasinski, J. M.

J. M. Jasinski, E. A. Whittaker, G. C. Bjorklund, R. W. Dreyfus, R. D. Estes, and R. E. Walkup, Appl. Phys. Lett. 44, 1155 (1984).
[CrossRef]

Kowalski, F. V.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, and A. J. Munley, Joint Institute for Laboratory Astrophysics, University of Colorado and National Bureau of Standards, Boulder, Colo. 80309 (personal communication, 1979); M. Prentiss, B. Peuse, G. Sanders, and S. Ezekiel, Research Laboratory of Electronics, Prog. Rep. No. 123 (Massachusetts Institute of Technology, Cambridge, Mass., 1981).

Lenth, W.

G. C. Bjorklund, M. D. Levenson, W. Lenth, and C. Oritz, Appl. Phys. B32, 2565 (1983).

Leung, K. P.

A. Z. Genack, K. P. Leung, and A. Schenzle, Phys. Rev. 28, 308 (1983).
[CrossRef]

Levenson, M. D.

G. C. Bjorklund, M. D. Levenson, W. Lenth, and C. Oritz, Appl. Phys. B32, 2565 (1983).

M. D. Levenson, W. E. Moerner, and D. E. Horne, Opt. Lett. 8, 108 (1983).
[CrossRef] [PubMed]

Moerner, W. E.

M. D. Levenson, W. E. Moerner, and D. E. Horne, Opt. Lett. 8, 108 (1983).
[CrossRef] [PubMed]

M. Gehrtz, W. E. Moerner, and G. C. Bjorklund, Opt. Lett. (to be published).

Munley, A. J.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, and A. J. Munley, Joint Institute for Laboratory Astrophysics, University of Colorado and National Bureau of Standards, Boulder, Colo. 80309 (personal communication, 1979); M. Prentiss, B. Peuse, G. Sanders, and S. Ezekiel, Research Laboratory of Electronics, Prog. Rep. No. 123 (Massachusetts Institute of Technology, Cambridge, Mass., 1981).

Oritz, C.

G. C. Bjorklund, M. D. Levenson, W. Lenth, and C. Oritz, Appl. Phys. B32, 2565 (1983).

Pokrowsky, P.

E. A. Whittaker, P. Pokrowsky, W. Zapka, K. Roche, and G. C. Bjorklund, J. Quant. Spectrosc. Radiat. Transfer 30, 289 (1983).
[CrossRef]

Roche, K.

E. A. Whittaker, P. Pokrowsky, W. Zapka, K. Roche, and G. C. Bjorklund, J. Quant. Spectrosc. Radiat. Transfer 30, 289 (1983).
[CrossRef]

Schenzle, A.

A. Z. Genack, K. P. Leung, and A. Schenzle, Phys. Rev. 28, 308 (1983).
[CrossRef]

Sullivan, B. J.

E. A. Whittaker, B. J. Sullivan, G. C. Bjorklund, H. R. Wendt, and H. E. Hunziker, J. Chem. Phys. 80, 961 (1984).
[CrossRef]

Walkup, R. E.

J. M. Jasinski, E. A. Whittaker, G. C. Bjorklund, R. W. Dreyfus, R. D. Estes, and R. E. Walkup, Appl. Phys. Lett. 44, 1155 (1984).
[CrossRef]

Wendt, H. R.

E. A. Whittaker, B. J. Sullivan, G. C. Bjorklund, H. R. Wendt, and H. E. Hunziker, J. Chem. Phys. 80, 961 (1984).
[CrossRef]

E. A. Whittaker, H. R. Wendt, H. E. Hunziker, and G. C. Bjorklund, Appl. Phys. B 35, 105 (1984).
[CrossRef]

Whittaker, E. A.

E. A. Whittaker, H. R. Wendt, H. E. Hunziker, and G. C. Bjorklund, Appl. Phys. B 35, 105 (1984).
[CrossRef]

E. A. Whittaker, B. J. Sullivan, G. C. Bjorklund, H. R. Wendt, and H. E. Hunziker, J. Chem. Phys. 80, 961 (1984).
[CrossRef]

J. M. Jasinski, E. A. Whittaker, G. C. Bjorklund, R. W. Dreyfus, R. D. Estes, and R. E. Walkup, Appl. Phys. Lett. 44, 1155 (1984).
[CrossRef]

E. A. Whittaker, P. Pokrowsky, W. Zapka, K. Roche, and G. C. Bjorklund, J. Quant. Spectrosc. Radiat. Transfer 30, 289 (1983).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, 1975), Chap. 7, pp. 323ff.

M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, 1975), Chap. 1, p. 40.

Yariv, A.

A. Yariv, Quantum Electronics, 2nd ed. (Wiley, New York, 1975), Chap. 14, pp. 343ff.

A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, New York, 1983), Chap. 8, pp. 288ff.

Yeh, P.

A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, New York, 1983), Chap. 8, pp. 288ff.

Zapka, W.

E. A. Whittaker, P. Pokrowsky, W. Zapka, K. Roche, and G. C. Bjorklund, J. Quant. Spectrosc. Radiat. Transfer 30, 289 (1983).
[CrossRef]

Appl. Phys. (1)

G. C. Bjorklund, M. D. Levenson, W. Lenth, and C. Oritz, Appl. Phys. B32, 2565 (1983).

Appl. Phys. B (1)

E. A. Whittaker, H. R. Wendt, H. E. Hunziker, and G. C. Bjorklund, Appl. Phys. B 35, 105 (1984).
[CrossRef]

Appl. Phys. Lett. (1)

J. M. Jasinski, E. A. Whittaker, G. C. Bjorklund, R. W. Dreyfus, R. D. Estes, and R. E. Walkup, Appl. Phys. Lett. 44, 1155 (1984).
[CrossRef]

J. Appl. Phys. (1)

W. L. Bond, J. Appl. Phys. 36, 1674 (1965).
[CrossRef]

J. Chem. Phys. (1)

E. A. Whittaker, B. J. Sullivan, G. C. Bjorklund, H. R. Wendt, and H. E. Hunziker, J. Chem. Phys. 80, 961 (1984).
[CrossRef]

J. Quant. Spectrosc. Radiat. Transfer (1)

E. A. Whittaker, P. Pokrowsky, W. Zapka, K. Roche, and G. C. Bjorklund, J. Quant. Spectrosc. Radiat. Transfer 30, 289 (1983).
[CrossRef]

Opt. Lett. (2)

Phys. Rev. (1)

A. Z. Genack, K. P. Leung, and A. Schenzle, Phys. Rev. 28, 308 (1983).
[CrossRef]

Proc. IEEE (1)

J. H. Bechtel and A. R. Chraplyvy, Proc. IEEE 70, 658 (1982).
[CrossRef]

Other (9)

D. J. Bernays, IBM Res. Rep. RJ4166, San Jose, Calif. (1984).

This result is analytically obvious for Eq. (9) but not so for Eq. (15), where the phase shift (which is, of course, merely the arc-tangent of the ratio of the coefficients of sin ωmt and cos ωmt) is manifestly dependent on the laser frequency. Nonetheless, numerical calculations indicate that at the modulation frequencies used in this experiment, the phase shift is only weakly dependent on the laser frequency.

M. Gehrtz, W. E. Moerner, and G. C. Bjorklund, Opt. Lett. (to be published).

We have also considered the possibility that a dc-RAM effect could be induced by electro-optic modulation of the Fresnel coefficient. Our calculations indicate this effect to be rather small and of the order of 10−7equivalent differential absorption. There has also been the suggestion that scattering from defects in the electro-optic crystal may play a role in the dc-RAM (L. Hollberg and J. L. Hall, Joint Institute for Laboratory Astrophysics, University of Colorado and National Bureau of Standards, Boulder, Colo. 80309 (personal communication). Because of its somewhat unreproducible behavior we have concluded that several sources are probably involved. We thank the referees for their suggestions in this matter.

A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, New York, 1983), Chap. 8, pp. 288ff.

A. Yariv, Quantum Electronics, 2nd ed. (Wiley, New York, 1975), Chap. 14, pp. 343ff.

M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, 1975), Chap. 7, pp. 323ff.

M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, 1975), Chap. 1, p. 40.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, and A. J. Munley, Joint Institute for Laboratory Astrophysics, University of Colorado and National Bureau of Standards, Boulder, Colo. 80309 (personal communication, 1979); M. Prentiss, B. Peuse, G. Sanders, and S. Ezekiel, Research Laboratory of Electronics, Prog. Rep. No. 123 (Massachusetts Institute of Technology, Cambridge, Mass., 1981).

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Figures (11)

Fig. 1
Fig. 1

Diagram illustrating the principle of FM spectroscopy and the basic apparatus needed to carry out a typical measurement. A single-frequency (ωc) laser beam is phase modulated at radio frequency (ωm) by an electro-optic modulator (EOM) and is then sent through an absorbing sample and detected by a high-speed photodiode. Differential absorption of the high-frequency sidebands at ωc ± ωm converts the laser phase modulation to amplitude modulation, which is detected by mixing the amplified (rf Amp) detector photosignal with an appropriately phase-shifted (ϕ Shifter) portion of the rf signal used to drive the EOM.

Fig. 2
Fig. 2

Sketch illustrating the residual amplitude-modulation background problem of FM spectroscopy. The figure shows the idealized frequency spectrum of the photodiode signal in FM spectroscopy in the absence of any sample absorption. The peak at the modulation frequency, ωm, comes from amplitude modulation imparted by the imperfect phase modulator.

Fig. 3
Fig. 3

Residual amplitude-modulation fringes obtained by scanning the laser frequency 25 GHz with no sample present. (a) Laser orthogonal to crystal endface and ωm = 333 MHz; (b) laser orthogonal to crystal endface, ωm = 309 MHz (equivalent to π/2 phase shift on our system); (c) laser angled 3° from normal to crystal endface, ωm = 309 MHz.

Fig. 4
Fig. 4

Physical situation assumed in calculation of properties of a multipass electro-optic modulator. The laser and modulation waves are assumed to have the same phase velocity in the forward direction and equal and opposite phase velocities on the backward direction. The lossless crystal has length l, index of refraction n, and reflectivity R.

Fig. 5
Fig. 5

Physical situation assumed for calculation of the effect of a Fabry–Perot modulator on a pure FM laser spectrum. The Fabry–Perot is assumed to have identical characteristics to the modulator crystal in Fig. 4.

Fig. 6
Fig. 6

Peak-to-peak fringe amplitude of RAM arising from the two situations depicted in Figs. 4 and 5 and calculated in the text, as a function of the modulation frequency. The solid curves are in phase with the modulation, the broken curves are in quadrature with the modulation, and both are normalized with respect to laser intensity and modulation index. (a) RAM predicted by Eq. (9) and Fig. 4; (b) RAM predicted by Eq. (15) and Fig. 5.

Fig. 7
Fig. 7

Calculated laser-frequency-dependent RAM, assuming ωm/νFSR = 0.616, (a) Eq. (9) and in phase with modulation; (b) Eq. (9) and in quadrature with modulation; (c) Eq. (15) and in phase with modulation; (d) Eq. (15) and in quadrature with modulation.

Fig. 8
Fig. 8

Detail of the optical path through the electro-optic modulator crystal. E0, collimated input laser beam, 0.25-μm pinhole spatial filter; L (lenses), f1 = 152 mm (5-mm-diameter collimation), f2 = 254 mm (focusing), f3 = 105 mm (collimation), f4 = 63 mm; EOM, LiTaO3 electro-optic modulator; BS, microscope slide beam splitter; SF, 50-μm pinhole spatial filter; PD, FND-100 photodiode. The dashed items were usually removed for the RAM measurements.

Fig. 9
Fig. 9

(a) Iodine absorption line in the vicinity of 570 nm measured by direct transmission of laser through 11-cm-length cell at room temperature; (b) output from 2-GHz confocal Fabry–Perot interferometer for laser light sampled after the electro-optic modulator. The scan provides an accurate calibration of the approximately 8-GHz laser scan and also shows the FM sidebands of the 250-MHz modulated laser.

Fig. 10
Fig. 10

FM signal for the same laser scan and modulation conditions of Fig. 8: (a) in phase with modulation and with iodine cell in beam; (b) in quadrature with modulation and with iodine cell in beam; (c) same as (a) without iodine cell; (d) same as (b) without iodine cell. For (a)–(d), the vertical axis is the FM signal in arbitrary units and the horizontal axis is an 8-GHz laser frequency scan.

Fig. 11
Fig. 11

Summary of data taken at 250 MHz for the electro-optic modulator fringes and the passive Fabry–Perot fringes. The figure is composed to facilitate comparison with Fig. 7. (a) and (b) in-phase and in-quadrature outputs, respectively, of the EOM aligned perpendicular to the laser beam; (c) and (d) in-phase and in-quadrature FM signal from the passive Fabry–Perot crystal with the EOM tilted about 3° from the beam axis. The two traces in (c) were obtained at two different phase settings and are the extrema obtained by varying the phase 180°. The vertical scale for (a) and (b) is not commensurate with that for (c) and (d).

Equations (22)

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

ϕ RT = ϕ 0 + ϕ F + ϕ B ,
ϕ 0 = 2 n l ω c = ω ν F S R
ϕ F = M sin ω m t
ϕ B = M sin ϕ m ϕ m sin ( ω m t + ϕ m )
E M P = E 0 e i ϕ 1 ( 1 - R ) 1 - R e i ϕ RT ,
ϕ 1 = ϕ 0 / 2 + ϕ F
I M P = I 0 | 1 - R 1 - R e i ϕ RT | 2
= I 0 ( 1 - R ) 2 1 + R 2 - 2 R cos ϕ RT ,
I M P = I 0 ( 1 - R ) 2 1 + R 2 - 2 R cos ϕ 0 [ 1 - 2 R M sin ϕ 0 1 + R 2 - 2 R cos ϕ 0 f m ( t ) ] + O ( M 2 ) .
f m ( t ) = sin ω m t ( 1 + cos ϕ m sin ϕ m ϕ m ) + cos ω m t ( sin 2 ϕ m ϕ m ) .
E FP = E 0 ( 1 - R ) 1 - R e i ϕ 0
= E 0 exp [ - δ ( ω ) + i ϕ ( ω ) ]
δ ( ω ) = - 2 ln ( 1 - R ) + ln ( 1 + R 2 - 2 R cos ϕ 0 ) ,
ϕ ( ω ) = arctan ( R sin ϕ 0 1 - R cos ϕ 0 ) .
I FP = I 0 e - 2 δ 0 [ 1 - M ( - Δ δ FP cos ω m t + Δ ϕ FP sin ω m t ) ] ,
Δ δ FP = δ + - δ -
Δ ϕ FP = 2 ϕ 0 - ϕ + - ϕ - .
ω m 2 π ν FSR = 250 MHz 2.55 GHz = 0.0980
V FM = I 0 G ( ω m ) M Δ δ ω m .
V MP = G ( ω m ) I MP ,
V FP = G ( ω m ) I FP ,
T X ( ω m ) = V X V FM ω m .

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