Abstract

We present a time-dependent theory that describes the continuous-wave (cw) Raman laser in H2. The time-dependent theory is compared with existing theories, and threshold measurements are taken. The relative intensity noise of the pump and the Stokes beams is measured and found to be in agreement with the predictions of the presented theory. The Raman laser decreases the relative intensity noise of the pump beam by 34 dB/Hz at a frequency of 30 kHz. In addition, the spectral heterodyne beat-note linewidth of the continuous-wave Raman laser is measured to be 8 kHz.

© 1999 Optical Society of America

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References

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  1. J. K. Brasseur, K. S. Repasky, and J. L. Carlsten, Opt. Lett. 23, 367 (1998).
    [CrossRef]
  2. K. S. Repasky, J. K. Brasseur, L. Meng, and J. L. Carlsten, J. Opt. Soc. Am. B 15, 1667 (1998).
    [CrossRef]
  3. K. S. Repasky, L. Meng, J. K. Brasseur, J. L. Carlsten, and R. C. Swanson, J. Opt. Soc. Am. B 16, 717 (1999).
    [CrossRef]
  4. K. S. Repasky, L. E. Watson, and J. L. Carlsten, Appl. Opt. 34, 2615 (1995).
    [CrossRef] [PubMed]
  5. K. S. Repasky, J. G. Wessel, and J. L. Carlsten, Appl. Opt. 35, 609 (1996).
    [CrossRef] [PubMed]
  6. S. Rebic, A. S. Parkins, and D. F. Walls, Opt. Commun. 156, 426 (1998).
    [CrossRef]
  7. P. Peterson, A. Gavrielides, and M. P. Sharma, Opt. Commun. 160, 80 (1999).
    [CrossRef]
  8. R. G. Harrison, Weiping Lu, and P. K. Gupta, Phys. Rev. Lett. 63, 1372 (1989).
    [CrossRef] [PubMed]
  9. 13 and ρ˙31 from Ref. 8 are set to zero. The population difference of the ground state and the vibration state is assumed to be 1, since in 10 atm (7.6×103 Torr) of H2, only ~1 molecule in 104 of the molecules available participate in the Raman process. In addition, the Raman linewidth is of the order of 510 MHz for 10 atm of H2, which gives a coherence decay of ~2 ns, whereas the Raman cavity has a build-up time of ~1–10 μs; thus coherence effects can be ignored.
  10. D. C. MacPherson, R. C. Swanson, and J. L. Carlsten, IEEE J. Quantum Electron. 25, 1741 (1989).
    [CrossRef]
  11. G. D. Boyd, W. D. Johnston, and I. P. Kaminow, IEEE J. Quantum Electron. QE-4, 203 (1969).
    [CrossRef]
  12. Since for our system the gain per pass of the Stokes beam is ~1×10−4, the gain of the system can be treated as if it were linear [exp(G)→1+G]; thus a spatial average of the pump’s field inside the Raman laser cavity is sufficient to calculate the gain. The spatial average reduces the gain by a factor of 2. This factor of one-half is absent in Refs. 1 and 2 and should be included.
  13. We have an inhomogeneous differential equation, which is solved by a linear combination of the homogeneous solution and the particular solution such that the following boundary conditions are met. At time equal to zero the pump inside the cavity is zero, and at time equal to infinity our solution limits to T/(1−R), the result of a discrete sum of the fields inside an interferometer.
  14. All the fits used the following parameters: λp(S)=532 nm (683 nm), α=3.45×10−9 cm/W, Rp(f )=Rp(b)= 0.99980, Rs(f )=Rs(b)=0.99977, Tp(f )=156 ppm, l= 7.68 cm, β=0.001 [for Eq. (13)], b=18 cm. The equations were integrated numerically by use of the Bulstoer algorithm.
  15. To conserve energy the areas for the pump and the Stokes beams, used to calculate power, need to be identical and are normalized to the pump beam. The wavelength dependence of the area for the Stokes beam is included in the mode-filling parameter of Ref. 11.
  16. J. L. Hall and T. W. Hänsch, Opt. Lett. 9, 502 (1984).
    [CrossRef] [PubMed]
  17. This is the optical power that is coupled into the TEM00 mode of the HFC; the actual power at the entrance of the cavity was 1.1 mW.
  18. The measured photon conversion efficiency is smaller than the efficiency reported in Ref. 1 owing to additional exposures to atmospheric conditions; this increases the absorption of the mirrors of the HFC.
  19. The values for the pump and the Stokes mirror reflectivies were measured by a cavity ring-down. The values are Rp(S)=0.99979±0.00001 (0.99977±0.00001). The transmissions were Tp=(153±8) ppm, and TS=(150± 20) ppm.
  20. W. K. Bischel and M. J. Dyer, Phys. Rev. A 33, 3113 (1986).
    [CrossRef] [PubMed]
  21. The calculated value for the plane-wave gain coefficient is 2.94×10−9 cm/W, and the experimental values are (2.5± 0.4)×10−9 cm/W. [See W. K. Bischel and M. J. Dyer, J. Opt. Soc. Am. B 3, 677 (1986).]
    [CrossRef]
  22. The absence of the EOM does not affect the cw Raman laser linewidth, since the narrowed pump linewidth is of the order of ~1 kHz, owing to instabilities in the HFC, and the Raman linewidth for the vibrational transition is 510 MHz. However, the HFC transforms frequency noise into amplitude noise, so that an EOM was added to increase the stability at frequencies near or above the cavity half-width for the RIN measurements.
  23. Vibrations on the optical table are of the order of 30 μgrms, which occur at a frequency of 90 Hz.

1999 (2)

1998 (3)

1996 (1)

1995 (1)

1989 (2)

R. G. Harrison, Weiping Lu, and P. K. Gupta, Phys. Rev. Lett. 63, 1372 (1989).
[CrossRef] [PubMed]

D. C. MacPherson, R. C. Swanson, and J. L. Carlsten, IEEE J. Quantum Electron. 25, 1741 (1989).
[CrossRef]

1986 (2)

1984 (1)

1969 (1)

G. D. Boyd, W. D. Johnston, and I. P. Kaminow, IEEE J. Quantum Electron. QE-4, 203 (1969).
[CrossRef]

Bischel, W. K.

Boyd, G. D.

G. D. Boyd, W. D. Johnston, and I. P. Kaminow, IEEE J. Quantum Electron. QE-4, 203 (1969).
[CrossRef]

Brasseur, J. K.

Carlsten, J. L.

Dyer, M. J.

Gavrielides, A.

P. Peterson, A. Gavrielides, and M. P. Sharma, Opt. Commun. 160, 80 (1999).
[CrossRef]

Gupta, P. K.

R. G. Harrison, Weiping Lu, and P. K. Gupta, Phys. Rev. Lett. 63, 1372 (1989).
[CrossRef] [PubMed]

Hall, J. L.

Hänsch, T. W.

Harrison, R. G.

R. G. Harrison, Weiping Lu, and P. K. Gupta, Phys. Rev. Lett. 63, 1372 (1989).
[CrossRef] [PubMed]

Johnston, W. D.

G. D. Boyd, W. D. Johnston, and I. P. Kaminow, IEEE J. Quantum Electron. QE-4, 203 (1969).
[CrossRef]

Kaminow, I. P.

G. D. Boyd, W. D. Johnston, and I. P. Kaminow, IEEE J. Quantum Electron. QE-4, 203 (1969).
[CrossRef]

Lu, Weiping

R. G. Harrison, Weiping Lu, and P. K. Gupta, Phys. Rev. Lett. 63, 1372 (1989).
[CrossRef] [PubMed]

MacPherson, D. C.

D. C. MacPherson, R. C. Swanson, and J. L. Carlsten, IEEE J. Quantum Electron. 25, 1741 (1989).
[CrossRef]

Meng, L.

Parkins, A. S.

S. Rebic, A. S. Parkins, and D. F. Walls, Opt. Commun. 156, 426 (1998).
[CrossRef]

Peterson, P.

P. Peterson, A. Gavrielides, and M. P. Sharma, Opt. Commun. 160, 80 (1999).
[CrossRef]

Rebic, S.

S. Rebic, A. S. Parkins, and D. F. Walls, Opt. Commun. 156, 426 (1998).
[CrossRef]

Repasky, K. S.

Sharma, M. P.

P. Peterson, A. Gavrielides, and M. P. Sharma, Opt. Commun. 160, 80 (1999).
[CrossRef]

Swanson, R. C.

K. S. Repasky, L. Meng, J. K. Brasseur, J. L. Carlsten, and R. C. Swanson, J. Opt. Soc. Am. B 16, 717 (1999).
[CrossRef]

D. C. MacPherson, R. C. Swanson, and J. L. Carlsten, IEEE J. Quantum Electron. 25, 1741 (1989).
[CrossRef]

Walls, D. F.

S. Rebic, A. S. Parkins, and D. F. Walls, Opt. Commun. 156, 426 (1998).
[CrossRef]

Watson, L. E.

Wessel, J. G.

Appl. Opt. (2)

IEEE J. Quantum Electron. (2)

D. C. MacPherson, R. C. Swanson, and J. L. Carlsten, IEEE J. Quantum Electron. 25, 1741 (1989).
[CrossRef]

G. D. Boyd, W. D. Johnston, and I. P. Kaminow, IEEE J. Quantum Electron. QE-4, 203 (1969).
[CrossRef]

J. Opt. Soc. Am. B (3)

Opt. Commun. (2)

S. Rebic, A. S. Parkins, and D. F. Walls, Opt. Commun. 156, 426 (1998).
[CrossRef]

P. Peterson, A. Gavrielides, and M. P. Sharma, Opt. Commun. 160, 80 (1999).
[CrossRef]

Opt. Lett. (2)

Phys. Rev. A (1)

W. K. Bischel and M. J. Dyer, Phys. Rev. A 33, 3113 (1986).
[CrossRef] [PubMed]

Phys. Rev. Lett. (1)

R. G. Harrison, Weiping Lu, and P. K. Gupta, Phys. Rev. Lett. 63, 1372 (1989).
[CrossRef] [PubMed]

Other (10)

13 and ρ˙31 from Ref. 8 are set to zero. The population difference of the ground state and the vibration state is assumed to be 1, since in 10 atm (7.6×103 Torr) of H2, only ~1 molecule in 104 of the molecules available participate in the Raman process. In addition, the Raman linewidth is of the order of 510 MHz for 10 atm of H2, which gives a coherence decay of ~2 ns, whereas the Raman cavity has a build-up time of ~1–10 μs; thus coherence effects can be ignored.

Since for our system the gain per pass of the Stokes beam is ~1×10−4, the gain of the system can be treated as if it were linear [exp(G)→1+G]; thus a spatial average of the pump’s field inside the Raman laser cavity is sufficient to calculate the gain. The spatial average reduces the gain by a factor of 2. This factor of one-half is absent in Refs. 1 and 2 and should be included.

We have an inhomogeneous differential equation, which is solved by a linear combination of the homogeneous solution and the particular solution such that the following boundary conditions are met. At time equal to zero the pump inside the cavity is zero, and at time equal to infinity our solution limits to T/(1−R), the result of a discrete sum of the fields inside an interferometer.

All the fits used the following parameters: λp(S)=532 nm (683 nm), α=3.45×10−9 cm/W, Rp(f )=Rp(b)= 0.99980, Rs(f )=Rs(b)=0.99977, Tp(f )=156 ppm, l= 7.68 cm, β=0.001 [for Eq. (13)], b=18 cm. The equations were integrated numerically by use of the Bulstoer algorithm.

To conserve energy the areas for the pump and the Stokes beams, used to calculate power, need to be identical and are normalized to the pump beam. The wavelength dependence of the area for the Stokes beam is included in the mode-filling parameter of Ref. 11.

This is the optical power that is coupled into the TEM00 mode of the HFC; the actual power at the entrance of the cavity was 1.1 mW.

The measured photon conversion efficiency is smaller than the efficiency reported in Ref. 1 owing to additional exposures to atmospheric conditions; this increases the absorption of the mirrors of the HFC.

The values for the pump and the Stokes mirror reflectivies were measured by a cavity ring-down. The values are Rp(S)=0.99979±0.00001 (0.99977±0.00001). The transmissions were Tp=(153±8) ppm, and TS=(150± 20) ppm.

The absence of the EOM does not affect the cw Raman laser linewidth, since the narrowed pump linewidth is of the order of ~1 kHz, owing to instabilities in the HFC, and the Raman linewidth for the vibrational transition is 510 MHz. However, the HFC transforms frequency noise into amplitude noise, so that an EOM was added to increase the stability at frequencies near or above the cavity half-width for the RIN measurements.

Vibrations on the optical table are of the order of 30 μgrms, which occur at a frequency of 90 Hz.

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Figures (13)

Fig. 1
Fig. 1

Growth of the pump (solid curve) and the Stokes (dotted curve) fields shown as a function of time. A relaxation oscillation occurs at ∼12 µs but is damped out in one oscillation.

Fig. 2
Fig. 2

Comparison of the predictions of Ref. 2 and 3 (dashed curve) with the steady-state limit of the presented time-dependent theory (solid curve). The percent difference of the theories is plotted (dashed–dotted curve). On average the theories differ by less than 1 part per 1000.

Fig. 3
Fig. 3

Normalized RIN’s of the Stokes (dashed curve) and pump (solid curve) beams are shown as a function of noise frequency at four times the threshold of the Raman laser.

Fig. 4
Fig. 4

Relaxation-oscillation frequency (solid curve) is shown as a function of pump power. The Stokes cavity linewidth (dashed line) is overlaid for reference. The threshold of the Raman laser is 640 µW.

Fig. 5
Fig. 5

Experimental apparatus used to measure the threshold and the RIN of the cw Raman laser is shown.

Fig. 6
Fig. 6

Stokes output versus pump power for the cw Raman laser is shown with the theory overlaid. The threshold was measured to be 640±30 µW.

Fig. 7
Fig. 7

Photon-number conversion efficiency versus pump power for the cw Raman laser is shown with the theory overlaid. The maximum photon-number conversion efficiency was measured to be 27±3% at 2.6 mW of pump power.

Fig. 8
Fig. 8

RIN for the pump beam is shown before (dashed curve) and after (solid curve) the HFC as a function of frequency. The calculated shot-noise level is added for reference.

Fig. 9
Fig. 9

(a) Normalized Stokes RIN and (b) the normalized pump RIN are shown as a function of noise frequency. Theoretical predictions of Eqs. (13)–(16) are overlaid (dashed curves).

Fig. 10
Fig. 10

Measured normalized RIN’s of the pump (circles) and the Stokes (squares) beams at 30 kHz as a function of pump power is shown. The data are fitted by the time-dependent theory. The pump RIN is reduced by as much as 35 dB/Hz for higher pump powers. The threshold of the Raman laser is 640 µW.

Fig. 11
Fig. 11

Experimental apparatus used to measure the free-running linewidth and stability of the cw Raman laser is shown. NB denotes narrow band.

Fig. 12
Fig. 12

Heterodyne beat-note linewidth measured by a rf spectrum analyzer. The linewidth of the cw Raman laser was measured to be 4 kHz.

Fig. 13
Fig. 13

Allan variance versus integration time is shown. A peak that is due to table vibrations is seen to occur at ∼100 Hz.

Equations (18)

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E˙p=-LpEp-ωpωsG|Es|2Ep+K(Epin, t),
E˙s=-LsEs+G|Ep|2Es.
G=1218αc0μ01/22λpλp+λs,
Lp(s)=-c2lln Rp(s)f Rp(s)b,
K(Epin, t)=clTpf Epin,
Epss=(LS/G)1/2,
ESss=ωS(K-LpEp)ωpGEp1/2.
Epin,thres=lLpcLSTpfG1/2.
Epssf=-Rpf Epin+½TpfEpss,
Epssb=½TpbEpss,
ESssf=½TSf ESss,
ESssb=½TSbESss,
=120μ01/2|E|2A,
A=lλp4 tan-1(l/b),
E˙p=-LpEp-ωpωSG|ES|2Ep+K[1+β sin(ωt)],
E˙S=-LSES+G|Ep|2ES,
RINoutput=40 log10δErmsEss,
ΔP(S)=RINoutputP(S)-40 log10(βrms)-20 log10[Cp(ω)],

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