Abstract

This paper consists of a theoretical discussion of maser action at optical frequencies with emphasis on the differences between stimulated emission and absorption in coherent and incoherent fields. Use is made of a geometrical representation of the equations of motion of the density matrix to illustrate the conditions for continuous-wave operation, amplification, production of large pulses, and for the appearance of relaxation oscillations.

© 1962 Optical Society of America

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References

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  1. W. E. Lamb, “Quantum Mechanical Amplifiers,” lecture presented at the Theoretical Physics Institute, University of Colorado, Summer, 1959 (with extensive bibliography).
  2. A. A. Vuylsteke, Elements of Maser Theory, (D. Van Nostrand Company, Inc., Princeton, 1961), references given in bibliography.
  3. Y. H. Pao and M. Resnikoff, “The Resonant Mode Structure of Solid-State Fabry-Perot Cavities” (to be published).
  4. T. H. Maiman, Phys. Rev. 123, 1145 (1961).
    [Crossref]
  5. R. W. Hellwarth, Phys. Rev. Letters 6, 9 (1961).
    [Crossref]
  6. J. R. Singer and S. Wang, Phys. Rev. Letters 6, 351 (1961).
    [Crossref]
  7. I. R. Senitsky, Phys. Rev. 111, 3 (1958).
    [Crossref]
  8. I. R. Senitsky, Phys. Rev. 115, 227 (1959).
    [Crossref]
  9. I. R. Senitsky, Phys. Rev. 119, 1807 (1960).
    [Crossref]
  10. I. R. Senitsky, Phys. Rev. 123, 1525 (1961).
    [Crossref]
  11. The relaxation times T1and T2 are inserted to represent the various effects of interaction with lattice, spontaneous emission, and exchange between maser molecules. These are discussed further in Secs. 3 and 5.
  12. R. P. Feynman, F. L. Vernon, and R. W. Hellwarth, J. Appl. Phys. 28, 49 (1957).
    [Crossref]
  13. R when not a vector and also when used without subscripts represents optical pumping.
  14. This point needs to be investigated further. This is not to say that R2is actually of a sinusoidal spatial nature but what is meant is that the effective R2is of this nature and departures of R2 from this dependence do not lead to emission which contributes to the excitation of this particular cavity mode.
  15. T1 represents the relaxation due to interaction with lattice as well as the action of spontaneous emission. In addition to these two relaxational mechanisms, T2 represents also the interaction between maser molecules, i.e., an exchange effect which is, however, negligible for low concentrations of these molecules.

1961 (4)

T. H. Maiman, Phys. Rev. 123, 1145 (1961).
[Crossref]

R. W. Hellwarth, Phys. Rev. Letters 6, 9 (1961).
[Crossref]

J. R. Singer and S. Wang, Phys. Rev. Letters 6, 351 (1961).
[Crossref]

I. R. Senitsky, Phys. Rev. 123, 1525 (1961).
[Crossref]

1960 (1)

I. R. Senitsky, Phys. Rev. 119, 1807 (1960).
[Crossref]

1959 (1)

I. R. Senitsky, Phys. Rev. 115, 227 (1959).
[Crossref]

1958 (1)

I. R. Senitsky, Phys. Rev. 111, 3 (1958).
[Crossref]

1957 (1)

R. P. Feynman, F. L. Vernon, and R. W. Hellwarth, J. Appl. Phys. 28, 49 (1957).
[Crossref]

Feynman, R. P.

R. P. Feynman, F. L. Vernon, and R. W. Hellwarth, J. Appl. Phys. 28, 49 (1957).
[Crossref]

Hellwarth, R. W.

R. W. Hellwarth, Phys. Rev. Letters 6, 9 (1961).
[Crossref]

R. P. Feynman, F. L. Vernon, and R. W. Hellwarth, J. Appl. Phys. 28, 49 (1957).
[Crossref]

Lamb, W. E.

W. E. Lamb, “Quantum Mechanical Amplifiers,” lecture presented at the Theoretical Physics Institute, University of Colorado, Summer, 1959 (with extensive bibliography).

Maiman, T. H.

T. H. Maiman, Phys. Rev. 123, 1145 (1961).
[Crossref]

Pao, Y. H.

Y. H. Pao and M. Resnikoff, “The Resonant Mode Structure of Solid-State Fabry-Perot Cavities” (to be published).

Resnikoff, M.

Y. H. Pao and M. Resnikoff, “The Resonant Mode Structure of Solid-State Fabry-Perot Cavities” (to be published).

Senitsky, I. R.

I. R. Senitsky, Phys. Rev. 123, 1525 (1961).
[Crossref]

I. R. Senitsky, Phys. Rev. 119, 1807 (1960).
[Crossref]

I. R. Senitsky, Phys. Rev. 115, 227 (1959).
[Crossref]

I. R. Senitsky, Phys. Rev. 111, 3 (1958).
[Crossref]

Singer, J. R.

J. R. Singer and S. Wang, Phys. Rev. Letters 6, 351 (1961).
[Crossref]

Vernon, F. L.

R. P. Feynman, F. L. Vernon, and R. W. Hellwarth, J. Appl. Phys. 28, 49 (1957).
[Crossref]

Vuylsteke, A. A.

A. A. Vuylsteke, Elements of Maser Theory, (D. Van Nostrand Company, Inc., Princeton, 1961), references given in bibliography.

Wang, S.

J. R. Singer and S. Wang, Phys. Rev. Letters 6, 351 (1961).
[Crossref]

J. Appl. Phys. (1)

R. P. Feynman, F. L. Vernon, and R. W. Hellwarth, J. Appl. Phys. 28, 49 (1957).
[Crossref]

Phys. Rev. (5)

T. H. Maiman, Phys. Rev. 123, 1145 (1961).
[Crossref]

I. R. Senitsky, Phys. Rev. 111, 3 (1958).
[Crossref]

I. R. Senitsky, Phys. Rev. 115, 227 (1959).
[Crossref]

I. R. Senitsky, Phys. Rev. 119, 1807 (1960).
[Crossref]

I. R. Senitsky, Phys. Rev. 123, 1525 (1961).
[Crossref]

Phys. Rev. Letters (2)

R. W. Hellwarth, Phys. Rev. Letters 6, 9 (1961).
[Crossref]

J. R. Singer and S. Wang, Phys. Rev. Letters 6, 351 (1961).
[Crossref]

Other (7)

W. E. Lamb, “Quantum Mechanical Amplifiers,” lecture presented at the Theoretical Physics Institute, University of Colorado, Summer, 1959 (with extensive bibliography).

A. A. Vuylsteke, Elements of Maser Theory, (D. Van Nostrand Company, Inc., Princeton, 1961), references given in bibliography.

Y. H. Pao and M. Resnikoff, “The Resonant Mode Structure of Solid-State Fabry-Perot Cavities” (to be published).

The relaxation times T1and T2 are inserted to represent the various effects of interaction with lattice, spontaneous emission, and exchange between maser molecules. These are discussed further in Secs. 3 and 5.

R when not a vector and also when used without subscripts represents optical pumping.

This point needs to be investigated further. This is not to say that R2is actually of a sinusoidal spatial nature but what is meant is that the effective R2is of this nature and departures of R2 from this dependence do not lead to emission which contributes to the excitation of this particular cavity mode.

T1 represents the relaxation due to interaction with lattice as well as the action of spontaneous emission. In addition to these two relaxational mechanisms, T2 represents also the interaction between maser molecules, i.e., an exchange effect which is, however, negligible for low concentrations of these molecules.

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Figures (10)

Fig. 1
Fig. 1

Energy level scheme for two level optical maser.

Fig. 2
Fig. 2

Geometrical representation of the density matrix showing inverted population in the absence of cavity radiation.

Fig. 3
Fig. 3

Geometrical representation of stimulating radiation.

Fig. 4
Fig. 4

Motion of R for off-resonance conditions.

Fig. 5
Fig. 5

Motion of R at resonance.

Fig. 6
Fig. 6

Energy-level scheme for three-level optical maser.

Fig. 7
Fig. 7

Relaxation oscillations for two-level operation.

Fig. 8
Fig. 8

Relaxation oscillations or recurrent pulses in two-level operation.

Fig. 9
Fig. 9

Geometrical representation of conditions for the production of large-single pulses.

Fig. 10
Fig. 10

One set of conditions for continuous wave operation.

Equations (26)

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Ψ j ( p ) = ν a ν j ( p ) ( t ) ϕ ν j ( p ) = ν c ν j ( p ) [ exp ( ћ / i E ν j t ) ] ϕ ν j ( p ) ,
ϕ 1 j = 1 ( n ! ) 1 2 = | ψ α j ( 1 ) ψ α j ( 2 ) ψ α j ( n ) ψ β j ( 1 ) ψ β j ( 2 ) ψ β j ( n ) ψ 1 j ( 1 ) ψ 1 j ( 2 ) ψ 1 j ( n ) | and ϕ 2 j = 1 ( n ! ) 1 2 = | ψ α j ( 1 ) ψ α j ( 2 ) ψ α j ( n ) ψ β j ( 1 ) ψ β j ( 2 ) ψ β j ( n ) ψ 2 j ( 1 ) ψ 2 j ( 2 ) ψ 2 j ( n ) |
ρ 11 ( r , ω , t ) = c 1 j * c 1 j δ ( r r j ) ρ 22 ( r , ω , t ) = c 2 j * c 2 j δ ( r r j ) ρ 12 ( r , ω , t ) = c 2 j * c 1 j δ ( r r j ) ρ 21 ( r , ω , t ) = c 1 j * c 2 j δ ( r r j ) ,
P ( r , ω , t ) = ( N / V ) g ( ω ) ( e r 12 ρ 21 + e r 21 ρ 12 ) ,
g ( ω ) d ω = 1 , r 12 = ψ 1 * | r | ψ 2 ,
P ( r , t ) = N V g ( ω ) ( e r 12 ρ 21 + e r 21 ρ 12 ) d ω .
i ћ ( ϱ / t ) = [ ϱ , H ] ,
H = ( i e ћ / m ) A · grad ,
H 12 = i ω A · ( e r 12 ) = E 0 e z · ( e r 12 ) ( e i ω 0 t + e i ω 0 t ) sin ( n π y / L ) ,
( / t ) ( ρ 12 + ρ 21 ) = ( 1 / T 2 ) ( ρ 12 + ρ 21 ) ( i / ћ ) [ H 12 ( exp i ω t ) H 21 ( exp i ω t ) ] ( ρ 22 ρ 11 ) , ( / t ) i ( ρ 12 ρ 21 ) = ( i / T 2 ) ( ρ 12 ρ 21 ) + ( 1 / ћ ) [ H 12 ( exp i ω t ) + H 21 ( exp i ω t ) ] ( ρ 22 ρ 11 ) , ( / t ) ( ρ 22 ρ 11 ) = R 1 / T 1 R + 1 / T 1 ) ( ρ 22 ρ 11 ) ( 1 / ћ ) [ H 12 ( exp i ω t ) + H 21 ( exp i ω t ) ] i ( ρ 12 ρ 21 ) + ( i / h ) [ H 12 ( exp i ω t ) H 21 ( exp i ω t ) ] ( ρ 12 + ρ 21 ) ,
( / t ) R 1 = ( 1 / T 2 ) R 1 + ω 2 R 3 ω 3 R 2 , ( / t ) R 2 = ( 1 / T 2 ) R 2 + ω 3 R 1 ω 1 R 3 , ( / t ) R 3 = [ R ( 1 / T 1 ) ] [ R + ( 1 / T 1 ) ] R 3 + ω 1 R 2 ω 2 R 1 ,
R 1 = ρ 12 + ρ 21 , R 2 = i ( ρ 12 ρ 21 ) , R 3 = ρ 22 ρ 11 ,
ω 1 = ( 1 / h ) [ H 12 ( exp i ω t ) + H 21 ( exp i ω t ) ] = ( i / ћ ) E 0 e z · ( e r ) [ sin ( n π y / L ) ] cos [ ( ω 0 ω ) t + ϕ ] , ω 2 = ( i / ћ ) [ H 12 exp ( i ω t ) H 21 exp ( i ω t ) ] = ( 1 / ћ ) E 0 e z · ( e r ) [ sin ( n π y / L ) ] sin [ ( ω 0 ω ) t + ϕ ] , ω 3 = 0.
d R / d t = A B · R + ω × R
A = ( R 1 / T 1 ) e 3 , B = ( 1 / T 2 0 0 0 1 / T 2 0 0 0 R + 1 / T 1 ) , R = R 1 e 1 + R 2 e 2 + R 3 e 3 , ω = ω 1 e 1 + ω 2 e 2 + ω 3 e 3 .
g ( ω ) Δ ω N V A 0 L ( ω 1 R 2 ω 2 R 1 ) d y = m { t V 4 E 0 2 + q V 4 E 0 2 } ,
t R 2 = 1 T 2 R 2 + ( e r ћ ) E 0 sin n π y L R 3 t R 3 = ( R 1 T 1 ) ( R + 1 T 1 ) R 3 ( e r ћ ) E 0 sin n π y L R 2 ,
D L 0 L ( e r ћ ) E 0 R 2 sin n π y L d y = 4 V ( t E 0 2 + q E 0 2 ) ,
D ( e r / ћ ) E 0 R 2 = 1 2 V [ ( / t ) E 0 2 + q E 0 2 ] ,
E 0 / t = ( D / V ) ( e r / ћ ) R 2 1 2 q E 0 .
ω 1 = ( e r / ћ ) E 0 cos ( δ t + ϕ ) sin ( n π y / L )
ω 2 = ( e r / ћ ) E 0 sin ( δ t + ϕ ) sin ( n π y / L ) ,
R 1 / t = ( 1 / T 2 ) R 1 + ω 2 R 3 ω 3 R 2 , R 2 / t = ( 1 / T 2 ) R 2 + ω 3 R 1 ω 1 R 3 , R 3 / t = R ( 1 / T 1 ) ( 1 + R 3 ) + ω 1 R 2 ω 2 R 1 ,
E / t = ( D / V ) ( e r / ћ ) R 2 q E 0 .
E 0 ( t = t 2 ) = D V q ( e r / ћ ) R 2 [ 1 e q ( t 2 t 1 ) ] + E 0 ( t = t 2 ) e q ( t 2 t 1 ) .
E 0 ( D / V q ) ( e r / ћ ) R 2 ,