Abstract

A model external-cavity system is investigated to describe the tuning characteristics appropriate to the various experimental procedures used in the operation of the spin-flip Raman laser. In particular a calculation is made of the amount by which internal subcavities pull the oscillation frequency away from the Raman frequency under the servocontrolled conditions used to achieve continuous tuning.

© 1978 Optical Society of America

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References

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  1. T. Scragg, S. D. Smith, Opt. Commun. 15, 166 (1975).
    [CrossRef]
  2. S. R. J. Brueck, A. Mooradian, IEEE J. Quantum Electron. QE-12, 201 (1976).
    [CrossRef]
  3. T. Scragg, C. Ironside, R. B. Dennis, S. D. Smith, Opt. Commun. 18, 456 (1976).
    [CrossRef]
  4. B. Walker, G. W. Chantry, D. G. Moss, C. C. Bradley, Opt. Commun. 17, 223 (1976).
    [CrossRef]
  5. W. J. Firth, B. S. Wherrett, D. Weaire, IEEE J. Quantum Electron. QE-12, 218 (1976);Opt. Commun. 15, 157 (1975).
    [CrossRef]
  6. D. A. Kleinman, P. P. Kisluik, Bell Syst. Tech. J. 41, 453 (1962).
  7. M. H. Dunn, Appl. Opt. 10, 1393 (1971).
    [CrossRef] [PubMed]
  8. See, for example S. R. Brueck, A. Mooradian, IEEE J. Quantum Electron. QE-10, 635 (1974);or Ref. 5.

1976 (4)

S. R. J. Brueck, A. Mooradian, IEEE J. Quantum Electron. QE-12, 201 (1976).
[CrossRef]

T. Scragg, C. Ironside, R. B. Dennis, S. D. Smith, Opt. Commun. 18, 456 (1976).
[CrossRef]

B. Walker, G. W. Chantry, D. G. Moss, C. C. Bradley, Opt. Commun. 17, 223 (1976).
[CrossRef]

W. J. Firth, B. S. Wherrett, D. Weaire, IEEE J. Quantum Electron. QE-12, 218 (1976);Opt. Commun. 15, 157 (1975).
[CrossRef]

1975 (1)

T. Scragg, S. D. Smith, Opt. Commun. 15, 166 (1975).
[CrossRef]

1974 (1)

See, for example S. R. Brueck, A. Mooradian, IEEE J. Quantum Electron. QE-10, 635 (1974);or Ref. 5.

1971 (1)

1962 (1)

D. A. Kleinman, P. P. Kisluik, Bell Syst. Tech. J. 41, 453 (1962).

Bradley, C. C.

B. Walker, G. W. Chantry, D. G. Moss, C. C. Bradley, Opt. Commun. 17, 223 (1976).
[CrossRef]

Brueck, S. R.

See, for example S. R. Brueck, A. Mooradian, IEEE J. Quantum Electron. QE-10, 635 (1974);or Ref. 5.

Brueck, S. R. J.

S. R. J. Brueck, A. Mooradian, IEEE J. Quantum Electron. QE-12, 201 (1976).
[CrossRef]

Chantry, G. W.

B. Walker, G. W. Chantry, D. G. Moss, C. C. Bradley, Opt. Commun. 17, 223 (1976).
[CrossRef]

Dennis, R. B.

T. Scragg, C. Ironside, R. B. Dennis, S. D. Smith, Opt. Commun. 18, 456 (1976).
[CrossRef]

Dunn, M. H.

Firth, W. J.

W. J. Firth, B. S. Wherrett, D. Weaire, IEEE J. Quantum Electron. QE-12, 218 (1976);Opt. Commun. 15, 157 (1975).
[CrossRef]

Ironside, C.

T. Scragg, C. Ironside, R. B. Dennis, S. D. Smith, Opt. Commun. 18, 456 (1976).
[CrossRef]

Kisluik, P. P.

D. A. Kleinman, P. P. Kisluik, Bell Syst. Tech. J. 41, 453 (1962).

Kleinman, D. A.

D. A. Kleinman, P. P. Kisluik, Bell Syst. Tech. J. 41, 453 (1962).

Mooradian, A.

S. R. J. Brueck, A. Mooradian, IEEE J. Quantum Electron. QE-12, 201 (1976).
[CrossRef]

See, for example S. R. Brueck, A. Mooradian, IEEE J. Quantum Electron. QE-10, 635 (1974);or Ref. 5.

Moss, D. G.

B. Walker, G. W. Chantry, D. G. Moss, C. C. Bradley, Opt. Commun. 17, 223 (1976).
[CrossRef]

Scragg, T.

T. Scragg, C. Ironside, R. B. Dennis, S. D. Smith, Opt. Commun. 18, 456 (1976).
[CrossRef]

T. Scragg, S. D. Smith, Opt. Commun. 15, 166 (1975).
[CrossRef]

Smith, S. D.

T. Scragg, C. Ironside, R. B. Dennis, S. D. Smith, Opt. Commun. 18, 456 (1976).
[CrossRef]

T. Scragg, S. D. Smith, Opt. Commun. 15, 166 (1975).
[CrossRef]

Walker, B.

B. Walker, G. W. Chantry, D. G. Moss, C. C. Bradley, Opt. Commun. 17, 223 (1976).
[CrossRef]

Weaire, D.

W. J. Firth, B. S. Wherrett, D. Weaire, IEEE J. Quantum Electron. QE-12, 218 (1976);Opt. Commun. 15, 157 (1975).
[CrossRef]

Wherrett, B. S.

W. J. Firth, B. S. Wherrett, D. Weaire, IEEE J. Quantum Electron. QE-12, 218 (1976);Opt. Commun. 15, 157 (1975).
[CrossRef]

Appl. Opt. (1)

Bell Syst. Tech. J. (1)

D. A. Kleinman, P. P. Kisluik, Bell Syst. Tech. J. 41, 453 (1962).

IEEE J. Quantum Electron. (3)

See, for example S. R. Brueck, A. Mooradian, IEEE J. Quantum Electron. QE-10, 635 (1974);or Ref. 5.

S. R. J. Brueck, A. Mooradian, IEEE J. Quantum Electron. QE-12, 201 (1976).
[CrossRef]

W. J. Firth, B. S. Wherrett, D. Weaire, IEEE J. Quantum Electron. QE-12, 218 (1976);Opt. Commun. 15, 157 (1975).
[CrossRef]

Opt. Commun. (3)

T. Scragg, S. D. Smith, Opt. Commun. 15, 166 (1975).
[CrossRef]

T. Scragg, C. Ironside, R. B. Dennis, S. D. Smith, Opt. Commun. 18, 456 (1976).
[CrossRef]

B. Walker, G. W. Chantry, D. G. Moss, C. C. Bradley, Opt. Commun. 17, 223 (1976).
[CrossRef]

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Figures (7)

Fig. 1
Fig. 1

The model cavity system under investigation. Region 1: air; no absorption, refractive index = 1. Region 2: InSb sample; α2,n2 = 4.0, Lorentzian Raman gain. Region 3: antireflection coating; α3 = 0, n3 ≏ 2, quarterwave thickness. Region 4: air-gap; α4 = 0, n4 = 1. Region 5: reflective mirror, approximated by a single surface of purely real reflection coefficient r45.

Fig. 2
Fig. 2

Mode tuning as a function of magnetic field, showing regions of tuning over modes of a treble cavity and hops for fixed cavity length (see table I). Point A: chosen such that ωc,Q = ωc2,Q2 = ωc4,Q4, there being exactly (Q2) 16,000 ½-wavelengths in the sample cavity, etc. ωs is shown to equal ωcQ at this treble-resonance point. Point B: as for A but Q2 = 16,000—1. With the above data this requires Q4 = 160,000—10, Q = 176,000—11. There are eleven modes of the full cavity between A and B. Other than at A and B, Q2,Q4 will not be integers when ωQ = ωs. Point C: as ωQ = πcQ4/n4L4 and this figure is for fixed L4, on modes of difference Q value equal frequencies correspond to equal Q4 values. Q4 changes by unity in 1/10th of the range AB. If A is labeled (0,0) indicating (Q2—16,000, Q4—160,000), C is (+1,−1). Point D: the point (0,1) is displaced from D slightly, AD itself being 1/11th of the range AB. Note that Q = Q2 + Q4 is a constant integer along any single mode. Dots on the figure are positions of various double-resonances for which Q2 and Q4 pass through integer values together. Point E: double-resonance points also occur when Q2 and Q4 are both half-integers. For the mode E,Q = 176,000—5, and the double resonance (−½, −9/2) lies close to, but not exactly at, ωQ = ωs.

Fig. 3
Fig. 3

Fine detail of mode tuning as a function of ωs, using the data of Table I. Points A, E, and B are as per Fig. 2. A and B are adjacent treble resonances. E lies approximately at a treble resonance for which Q2 and Q4 are half integers. ν ( A ) = 59958.500 GHz ν ( E ) = 59956.795 GHz ν ( B ) = 59954.753 GHzThe dashed lines indicate the linear tuning applicable for a perfect coating. These have the same gradient for each mode, Γc/(Γs + Γc). The full lines are nonlinear and vary mode to mode. That is, between A and E there are four modes; the gradients of their tuning curves move from that of A across the dashed line to that of E. The converse happens on the way to B. This pattern is repeated over further modes. The modes hop at the extremes of the tuning curves shown. In this and the following figures based on numerical data we employ the units GHz, etc. rather than the slightly less conventional rad per sec; ν = ωQ/2π, etc.

Fig. 4
Fig. 4

Variation of threshold gain across a mode, for modes centered at A and E. Intermediate modes have intermediate gains. See Fig. 2 and Table I.

Fig. 5
Fig. 5

Modulation of the oscillation frequency under synchronous tuning conditions.

Fig. 6
Fig. 6

(a) The modal plane for a perfect cavity. The plane between the parallel lines a and b contains those frequencies for which cavity oscillation is possible on a single mode. Line c is an example of mode tuning for fixed cavity length, and line d describes length tuning for a fixed magnetic-field strength. The center of the mode (minimum gain) follows the almost straight line e for which ωs = ωcQ = πc[(Q2 +Q4)/(n2L4 + n4L4)]. The mode hops occur along a and b, for which ωs = ωcQ ± (Δωc)/2. (Q2 + Q4) is a constant over the entire plane. The adjacent mode to higher ωs is described by a similar, angled plane alongside the one shown and displaced upward by a frequency Δωc. Thus, as ωs is increased one moves along a line such as c, increasing ωQ, then jumps onto the adjacent plane, increases ωQ again, etc. As L4 is increased one moves along d, decreasing ωQ, jumps to the adjacent higher plane, decreasing ωQ again, etc. The adjacent mode to lower ωs is displaced downward. (b) Gain profile on adjacent modes. Lines e, a, and b are at the same ωs,L4 values as in (a). The threshold gain gQ is constant along these lines. Mode tuning at constant ωs one cuts across acutely from a to b (along the projection of line d). The parabolic edge shown would correspond to tuning at a fixed L4 value. For an imperfect cavity the lines a and b would be warped in the horizontal and vertical directions. The mode center e is also warped and no longer follows ωs = ωcQ.

Fig. 7
Fig. 7

The mode-hop syndrome. Referring to Fig. 6(b), if the nonlinearities involved are sufficient that for an adjacent mode and gain (off-center) is equal to the mode-center gain of the initial mode, point X, a mode hop will occur, and tuning will continue along the adjacent mode, from the point at the same magnetic field as X but at which the gain is minimized.

Tables (1)

Tables Icon

Table I Data Used in the Numerical Calculations (Figs. 35)

Equations (20)

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1 + r 12 r 23 exp ( i 2 K 2 L 2 ) + r 12 r 34 exp [ i 2 ( K 2 L 2 + K 3 L 3 ) ] + r 12 r 45 exp [ i 2 ( K 2 L 2 + K 3 L 3 + K 4 L 4 ) ] + r 23 r 34 exp ( i 2 K 3 L 3 ) + r 23 r 45 exp [ i 2 ( K 3 L 3 + K 4 L 4 ) ] + r 34 r 45 exp ( i 2 K 4 L 4 ) + r 12 r 23 r 34 r 45 exp [ i 2 ( K 2 L 2 + K 4 L 4 ) ] = 0 .
2 Im ( K 2 ) = g [ ( Γ s / 2 ) 2 ( ω ω s ) 2 + ( Γ s / 2 ) 2 ] α 2 .
2 R e ( K 2 ) = 2 ω n 2 c + g [ ( ω ω s ) Γ s / 2 ( ω ω s ) 2 + ( Γ s / 2 ) 2 ] .
ω Q = ω c Q Γ s + ω s Γ c Γ s + Γ c .
Γ c = [ α 2 L 2 ln ( | r 12 r 45 | ) ] c / L .
( g Q L 4 ) ω s = 0 .
g Q = Γ c c L L 2 { 1 + ( 2 ( ω c Q ω s ) Γ c + Γ s ) 2 } ,
Δ ω = Γ s Γ s + Γ c Δ L 4 L 4 ω Q .
ω Q = Q 4 π c / n 4 L 4 ,
exp [ g ( ω ) α 2 ] L 2 = ( 1 ρ c r 45 ) / r 12 ( r 45 ρ c ) ,
ρ c = r 23 r 34 1 r 23 r 34 .
f 1 ( g , ω ) + ρ c f 2 ( g , ω ) = 0 ,
f 3 ( g , ω ) + ρ c f 4 ( g , ω ) = 0 ;
g Q ( ω s , L 4 ) ( g Q ) 0 1 L 2 Γ s Γ s + Γ c ρ c ( r 45 1 r 45 ) cos { 2 n 4 L 4 c ( ω Q ) 0 } ,
ω Q ( ω s , L 4 ) ( ω Q ) 0 c L Γ s Γ s + Γ c ρ c ( r 45 1 r 45 2 ) sin { 2 n 4 L 4 c ( ω Q ) 0 } .
ω Q ( sync ) = ω s + ρ c ( r 1 r 45 ) ( n 2 L 2 + n 4 L 4 n 4 L 4 Γ s + Γ c Γ s 1 ) × sin { 2 n 4 L 4 c ( ω Q ) 0 } Γ s 2 Γ c .
Δ g Q | 2 ρ c ( r 45 1 r 45 ) | ;
| ρ c ( r 45 1 r 45 ) Γ s 2 Γ c | 4 π Δ ω c .
Γ s Γ s + Γ c n 4 L 4 n 2 L 2 + n 4 L 4 ( r 1 r 45 ) ρ c .
ν ( A ) = 59958.500 GHz ν ( E ) = 59956.795 GHz ν ( B ) = 59954.753 GHz

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