Abstract

The beat frequencies between modes of an optical resonator with concave mirrors have been studied by the photomixing technique. Lasers operating on the 6328 Å He–Ne transition were used. The results agree well with theory. Removal of degeneracy by resonator irregularities and mode repulsion effects have been observed. The technique of partially blocking the external laser beam in order to observe all the beat frequencies is pointed out.

© 1964 Optical Society of America

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References

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  1. G. D. Boyd, J. P. Gordon, Bell System Tech. J. 40, 489 (1961).
  2. G. D. Boyd, H. Kogelnik, Bell System Tech. J. 41, 1347 (1962).
  3. A. G. Fox, T. Li, Bell System Tech. J. 40, 453 (1961).
  4. A. G. Fox, T. Li, Proc. IEEE 51, 80 (1963).
    [CrossRef]
  5. W. W. Rigrod, Appl. Phys. Letters 2, 51 (1963).
    [CrossRef]
  6. H. Kogelnik, W. W. Rigrod, Proc. Inst. Radio Engrs. 50, 220 (1962).
  7. A. Javan, W. R. Bennett, D. R. Herriott, Phys. Rev. Letters 6, 106 (1961).
    [CrossRef]
  8. G. Goubau, F. Schwering, Trans. IRE AP-9, 248 (1961).
  9. A. Yariv, J. P. Gordon, Proc. IEEE 51, 4 (1963).
    [CrossRef]
  10. A. D. White, J. D. Rigden, Proc. Inst. Radio Engrs. 50, 1697 (1962).
  11. W. W. Rigrod, H. Kogelnik, D. J. Brangoccio, D. R. Herriott, J. Appl. Phys. 33, 743 (1962).
    [CrossRef]
  12. G. H. Dieke, H. M. Crosswhite, J. Opt. Soc. Am. 42, 433 (1952).
    [CrossRef]
  13. W. R. Bennett, Appl. Opt. Suppl. on Opt. Masers 24 (1962).
  14. W. R. Bennett, Phys. Rev. 126, 580 (1962).
    [CrossRef]

1963

A. G. Fox, T. Li, Proc. IEEE 51, 80 (1963).
[CrossRef]

W. W. Rigrod, Appl. Phys. Letters 2, 51 (1963).
[CrossRef]

A. Yariv, J. P. Gordon, Proc. IEEE 51, 4 (1963).
[CrossRef]

1962

A. D. White, J. D. Rigden, Proc. Inst. Radio Engrs. 50, 1697 (1962).

W. W. Rigrod, H. Kogelnik, D. J. Brangoccio, D. R. Herriott, J. Appl. Phys. 33, 743 (1962).
[CrossRef]

W. R. Bennett, Appl. Opt. Suppl. on Opt. Masers 24 (1962).

W. R. Bennett, Phys. Rev. 126, 580 (1962).
[CrossRef]

H. Kogelnik, W. W. Rigrod, Proc. Inst. Radio Engrs. 50, 220 (1962).

G. D. Boyd, H. Kogelnik, Bell System Tech. J. 41, 1347 (1962).

1961

A. G. Fox, T. Li, Bell System Tech. J. 40, 453 (1961).

G. D. Boyd, J. P. Gordon, Bell System Tech. J. 40, 489 (1961).

A. Javan, W. R. Bennett, D. R. Herriott, Phys. Rev. Letters 6, 106 (1961).
[CrossRef]

G. Goubau, F. Schwering, Trans. IRE AP-9, 248 (1961).

1952

Bennett, W. R.

W. R. Bennett, Appl. Opt. Suppl. on Opt. Masers 24 (1962).

W. R. Bennett, Phys. Rev. 126, 580 (1962).
[CrossRef]

A. Javan, W. R. Bennett, D. R. Herriott, Phys. Rev. Letters 6, 106 (1961).
[CrossRef]

Boyd, G. D.

G. D. Boyd, H. Kogelnik, Bell System Tech. J. 41, 1347 (1962).

G. D. Boyd, J. P. Gordon, Bell System Tech. J. 40, 489 (1961).

Brangoccio, D. J.

W. W. Rigrod, H. Kogelnik, D. J. Brangoccio, D. R. Herriott, J. Appl. Phys. 33, 743 (1962).
[CrossRef]

Crosswhite, H. M.

Dieke, G. H.

Fox, A. G.

A. G. Fox, T. Li, Proc. IEEE 51, 80 (1963).
[CrossRef]

A. G. Fox, T. Li, Bell System Tech. J. 40, 453 (1961).

Gordon, J. P.

A. Yariv, J. P. Gordon, Proc. IEEE 51, 4 (1963).
[CrossRef]

G. D. Boyd, J. P. Gordon, Bell System Tech. J. 40, 489 (1961).

Goubau, G.

G. Goubau, F. Schwering, Trans. IRE AP-9, 248 (1961).

Herriott, D. R.

W. W. Rigrod, H. Kogelnik, D. J. Brangoccio, D. R. Herriott, J. Appl. Phys. 33, 743 (1962).
[CrossRef]

A. Javan, W. R. Bennett, D. R. Herriott, Phys. Rev. Letters 6, 106 (1961).
[CrossRef]

Javan, A.

A. Javan, W. R. Bennett, D. R. Herriott, Phys. Rev. Letters 6, 106 (1961).
[CrossRef]

Kogelnik, H.

W. W. Rigrod, H. Kogelnik, D. J. Brangoccio, D. R. Herriott, J. Appl. Phys. 33, 743 (1962).
[CrossRef]

H. Kogelnik, W. W. Rigrod, Proc. Inst. Radio Engrs. 50, 220 (1962).

G. D. Boyd, H. Kogelnik, Bell System Tech. J. 41, 1347 (1962).

Li, T.

A. G. Fox, T. Li, Proc. IEEE 51, 80 (1963).
[CrossRef]

A. G. Fox, T. Li, Bell System Tech. J. 40, 453 (1961).

Rigden, J. D.

A. D. White, J. D. Rigden, Proc. Inst. Radio Engrs. 50, 1697 (1962).

Rigrod, W. W.

W. W. Rigrod, Appl. Phys. Letters 2, 51 (1963).
[CrossRef]

W. W. Rigrod, H. Kogelnik, D. J. Brangoccio, D. R. Herriott, J. Appl. Phys. 33, 743 (1962).
[CrossRef]

H. Kogelnik, W. W. Rigrod, Proc. Inst. Radio Engrs. 50, 220 (1962).

Schwering, F.

G. Goubau, F. Schwering, Trans. IRE AP-9, 248 (1961).

White, A. D.

A. D. White, J. D. Rigden, Proc. Inst. Radio Engrs. 50, 1697 (1962).

Yariv, A.

A. Yariv, J. P. Gordon, Proc. IEEE 51, 4 (1963).
[CrossRef]

Appl. Opt. Suppl. on Opt. Masers

W. R. Bennett, Appl. Opt. Suppl. on Opt. Masers 24 (1962).

Appl. Phys. Letters

W. W. Rigrod, Appl. Phys. Letters 2, 51 (1963).
[CrossRef]

Bell System Tech. J.

G. D. Boyd, J. P. Gordon, Bell System Tech. J. 40, 489 (1961).

G. D. Boyd, H. Kogelnik, Bell System Tech. J. 41, 1347 (1962).

A. G. Fox, T. Li, Bell System Tech. J. 40, 453 (1961).

J. Appl. Phys.

W. W. Rigrod, H. Kogelnik, D. J. Brangoccio, D. R. Herriott, J. Appl. Phys. 33, 743 (1962).
[CrossRef]

J. Opt. Soc. Am.

Phys. Rev.

W. R. Bennett, Phys. Rev. 126, 580 (1962).
[CrossRef]

Phys. Rev. Letters

A. Javan, W. R. Bennett, D. R. Herriott, Phys. Rev. Letters 6, 106 (1961).
[CrossRef]

Proc. IEEE

A. G. Fox, T. Li, Proc. IEEE 51, 80 (1963).
[CrossRef]

A. Yariv, J. P. Gordon, Proc. IEEE 51, 4 (1963).
[CrossRef]

Proc. Inst. Radio Engrs.

A. D. White, J. D. Rigden, Proc. Inst. Radio Engrs. 50, 1697 (1962).

H. Kogelnik, W. W. Rigrod, Proc. Inst. Radio Engrs. 50, 220 (1962).

Trans. IRE

G. Goubau, F. Schwering, Trans. IRE AP-9, 248 (1961).

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Figures (12)

Fig. 1
Fig. 1

Diagram indicating some of the possible beat frequencies between different transverse mode sets. The mode sets are labeled on the right by the sum of their transverse mode numbers. Note that when two transverse mode sets beat with each other that two frequencies whose sum is the fundamental frequency result. The numbers are the beat frequencies in Mc/sec for laser no. 2.

Fig. 2
Fig. 2

Schematic illustration showing necessity to block off part of beam to produce beats between TEMq10 and TEMq00. Since the field in one-half of TEMq10 is reversed the beat signals from the two sides are out of phase and cancel in the photomultiplier. If half of the beam is blocked off this symmetry is upset and a beat frequency can be observed.

Fig. 3
Fig. 3

The 59 and 94 Mc/sec beat frequencies from laser no. 2 appear when the mode pattern is blocked in a vertical plane. Only 152 and 304 Mc/sec were present when blocked in a horizontal plane. Frequencies are in Mc/sec.

Fig. 4
Fig. 4

(a) TEMq14 + TEMq02 modes from laser no. 2, ν0 = 152 Mc/sec. (b) Pure TEMq02 mode obtained by suppressing the TEMq14 with an aperture inside the cavity. Note the slight rotation of the x and the y axes between the two transverse modes.

Fig. 5
Fig. 5

A case similar to Fig. 4 except that angle between the mode axes is larger and more apparent.

Fig. 6
Fig. 6

Mode combinations of laser no. 2. In all cases a ν0 = 152 Mc/sec beat was present as it must be if two transverse frequencies whose sum is ν0 are present. In some cases, 304 Mc/sec was present also. The same applies to patterns shown in other figures also. The weaker mode assignment in cases (d) and (e) is not certain.

Fig. 7
Fig. 7

(a) TEMq01 + TEMq10 modes from laser no. 1. (b) Spectrum analyzer presentation of beats with beam partially blocked in vertical or horizontal plane. With unblocked beam side lobe frequencies were much weaker but still observable.

Fig. 8
Fig. 8

Beats between TEMq01, TEMq10, and TEMq00. (a) Vertically blocked beam. (b) Horizontally blocked beam. (c) Schematic diagram showing origin of different frequencies. Horizontal frequencies do not go to zero in (a) and (b) because of imperfect alignment of knife edge with modes. The apparent unequal spacing at 86 and 91 Mc/sec is due to nonlinearity in the spectrum analyzer.

Fig. 9
Fig. 9

(a) Longitudinal frequency signal showing audio beats present when longitudinal beats are present from two transverse mode sets. (b) “Clean” longitudinal beat from single transverse mode set.

Fig. 10
Fig. 10

Simultaneous oscillation on three transverse modes of laser no. 1. From the pattern, TEMq04 and TEMq51 are obviously present and the existence of TEMq14 is inferred from the presence of beat frequencies corresponding to Δ(m + n) = 1 and compatibility with the observed pattern.

Fig. 11
Fig. 11

Various superpositions of two modes from laser no. 1. Note rotation between the x and y axes of individual modes.

Fig. 12
Fig. 12

The existence of TEMq12 in (b) was confirmed by suspressing the higher order mode. (c) Output of laser no. 1 when adjusted for maximum output. Pattern is partially obscured by camera aperture.

Tables (1)

Tables Icon

Table I Beat Frequencies between Various Transverse Mode Setsa

Equations (8)

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ν = ( c / 2 d ) [ q + ( 1 + m + n ) f ] , f = π 1 cos 1 [ ( 1 d / b 1 ) ( 1 d / b 2 ) ] 1 / 2 .
δ ν Δ ( m + n ) = ( c / 2 d ) f Δ ( m + n ) .
0 ( d / b 1 1 ) ( d / b 2 1 ) 1.
E ( x , y ) = E 0 H m ( x 2 / w s ) H n ( y 2 / w s ) exp ( x 2 + y 2 ) / w s 2 ,
w s = ( λ b / π ) 1 / 2 ( 2 b / d 1 ) 1 / 4 .
E ( r , ϕ ) = E 0 ( r 2 / w s ) l L p l ( 2 r 2 / w s 2 ) exp ( r 2 / w s 2 ) cos l ϕ ,
ν = ( c / 2 d ) [ q + ( 1 + 2 p + l ) f ] .
[ TEM q 10 ( ν ) ] + [ a TEM q 01 ( ν + δ ) + [ b TEM q 10 ( ν + δ ) ] ,

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