Abstract

A packet of light rays is defined that is equivalent to a Gaussian beam of light. The transformation of the Gaussian beam as it passes through any combination of perfect lenses and flat dielectric interfaces can be found by applying geometric optics to the equivalent ray packet. The envelope of the ray packet gives the Gaussian beam spot size and the curves perpendicular to the average ray slope give the Gaussian beam phase fronts.

© 1966 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. H. Kogelnik, Bell System Tech. J. 44, 455 (1965).
  4. G. Goubau, F. Schwering, Trans. Inst. Radio Engrs. AP-9, 248 (1961).
  5. P. K. Tien, J. P. Gordon, J. R. Whinnery, Proc. IEEE 53, 129 (1965).
    [CrossRef]
  6. H. E. Rowe (private communication), Nov.1964.
  7. W. K. Kahn, Appl. Opt. 4, 758 (1965).
    [CrossRef]
  8. V. P. Bykov, L. A. Vainshtein, Soviet Phys.—JETPx, 20, 338 (1965).
  9. J. Hirano, Y. Fukatsu, Proc. IEEE 52, 1284 (1964).
    [CrossRef]
  10. M. Born, E. Wolf, Principles of Optics (Pergamon Press, Inc., New York, 1964).
  11. W. H. Steier, Bell System Tech. J. 45, 451 (1966).

1966

W. H. Steier, Bell System Tech. J. 45, 451 (1966).

1965

W. K. Kahn, Appl. Opt. 4, 758 (1965).
[CrossRef]

P. K. Tien, J. P. Gordon, J. R. Whinnery, Proc. IEEE 53, 129 (1965).
[CrossRef]

V. P. Bykov, L. A. Vainshtein, Soviet Phys.—JETPx, 20, 338 (1965).

H. Kogelnik, Bell System Tech. J. 44, 455 (1965).

1964

J. Hirano, Y. Fukatsu, Proc. IEEE 52, 1284 (1964).
[CrossRef]

1962

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

1961

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

G. Goubau, F. Schwering, Trans. Inst. Radio Engrs. AP-9, 248 (1961).

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon Press, Inc., New York, 1964).

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).

Bykov, V. P.

V. P. Bykov, L. A. Vainshtein, Soviet Phys.—JETPx, 20, 338 (1965).

Fukatsu, Y.

J. Hirano, Y. Fukatsu, Proc. IEEE 52, 1284 (1964).
[CrossRef]

Gordon, J. P.

P. K. Tien, J. P. Gordon, J. R. Whinnery, Proc. IEEE 53, 129 (1965).
[CrossRef]

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

Goubau, G.

G. Goubau, F. Schwering, Trans. Inst. Radio Engrs. AP-9, 248 (1961).

Hirano, J.

J. Hirano, Y. Fukatsu, Proc. IEEE 52, 1284 (1964).
[CrossRef]

Kahn, W. K.

Kogelnik, H.

H. Kogelnik, Bell System Tech. J. 44, 455 (1965).

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

Rowe, H. E.

H. E. Rowe (private communication), Nov.1964.

Schwering, F.

G. Goubau, F. Schwering, Trans. Inst. Radio Engrs. AP-9, 248 (1961).

Steier, W. H.

W. H. Steier, Bell System Tech. J. 45, 451 (1966).

Tien, P. K.

P. K. Tien, J. P. Gordon, J. R. Whinnery, Proc. IEEE 53, 129 (1965).
[CrossRef]

Vainshtein, L. A.

V. P. Bykov, L. A. Vainshtein, Soviet Phys.—JETPx, 20, 338 (1965).

Whinnery, J. R.

P. K. Tien, J. P. Gordon, J. R. Whinnery, Proc. IEEE 53, 129 (1965).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon Press, Inc., New York, 1964).

Appl. Opt.

Bell System Tech. J.

W. H. Steier, Bell System Tech. J. 45, 451 (1966).

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).

H. Kogelnik, Bell System Tech. J. 44, 455 (1965).

Proc. IEEE

P. K. Tien, J. P. Gordon, J. R. Whinnery, Proc. IEEE 53, 129 (1965).
[CrossRef]

J. Hirano, Y. Fukatsu, Proc. IEEE 52, 1284 (1964).
[CrossRef]

Soviet Phys.—JETPx

V. P. Bykov, L. A. Vainshtein, Soviet Phys.—JETPx, 20, 338 (1965).

Trans. Inst. Radio Engrs.

G. Goubau, F. Schwering, Trans. Inst. Radio Engrs. AP-9, 248 (1961).

Other

M. Born, E. Wolf, Principles of Optics (Pergamon Press, Inc., New York, 1964).

H. E. Rowe (private communication), Nov.1964.

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Figures (4)

Fig. 1
Fig. 1

Development of a Gaussian mode inside a cavity.

Fig. 2
Fig. 2

Development of the equivalent ray packet on a beam waveguide.

Fig. 3
Fig. 3

Gaussian beam transformation through a general optical medium.

Fig. 4
Fig. 4

Gaussian beam injected into a beam waveguide.

Equations (28)

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k w o 2 = d ( 4 - d / f d / f ) ½ , k = 2 π / λ , λ = wavelength .
r n - m = w o cos ( n - m ) θ + ( d / k w o ) sin ( n - m ) θ r n - m = - ( 2 / k w o ) sin ( n - m ) θ ,
r n - m ( a ) = w o cos ( n - m ) θ - ( 2 z o / k w o ) sin ( n - m ) θ r n - m ( a ) = - ( 2 / k w o ) sin ( n - m ) θ .
r ( a ) / w o = cos φ - ( 2 z o / k w o 2 ) sin φ r ( a ) / w o = - ( 2 / k w o 2 ) sin φ ,
r a / w o = [ 1 + ( 2 z o / k w o 2 ) 2 ] ½ .
r = 4 z o r / ( k w o 2 ) 2 ( 2 z o / k w o 2 ) 2 + 1 .
R = r / r .
R = z o + [ ( k w o 2 ) 2 / 4 z o ] .
r / w = [ ( 2 R cos φ - k w 2 sin φ ) / ( 4 R 2 + k 2 w 4 ) ½ ] ; r / w = - [ ( 4 R 2 + k 2 w 4 ) ½ / R k w 2 ] sin φ
[ r 2 r 2 ] = [ A B C D ] [ r 1 r 1 ] + [ E F ] .
r 1 / w 1 = [ ( 2 R 1 cos φ - k w 1 2 sin φ ) / ( 4 R 1 2 + k 2 w 1 4 ) ½ ] + s 1 / w 1 ; r 1 / w 1 = - [ ( 4 R 1 2 + k 2 w 1 4 ) ½ / ( R 1 k w 1 2 ) ] sin φ + γ 1 / w 1 ;
r 2 w 1 = 2 R 1 A cos φ ( 4 R 1 2 + k 2 w 1 4 ) ½ - [ A k w 1 2 ( 4 R 1 2 + k 2 w 1 4 ) ½ + B ( 4 R 1 2 + k 2 w 1 4 ) ½ R 1 k w 1 2 ] sin φ + A s 1 w 1 + B γ 1 w 1 + E w 1 ,
r 2 w 1 = 2 R 1 C cos φ ( 4 R 1 2 + k 2 w 1 4 ) ½ - [ C k w 1 2 ( 4 R 1 2 + k 2 w 1 4 ) ½ + D ( 4 R 1 2 + k 2 w 1 4 ) ½ R 1 k w 1 2 ] sin φ + C s 1 w 1 + D γ 1 w 1 + F w 1 .
w 2 2 w 1 2 = 4 R 1 2 A 2 4 R 1 2 + k 2 w 1 4 + [ A k w 1 2 ( 4 R 1 2 + k 2 w 1 4 ) ½ + B ( 4 R 1 2 + k 2 w 1 4 ) ½ R 1 k w 1 2 ] 2 .
w 2 2 / w 1 2 = ( A + B / R 1 ) 2 + ( 2 B / k w 1 2 ) 2 .
r 2 av = r 2 ( C + D / R 1 ) ( A + B / R 1 ) + D B ( 2 / k w 1 2 ) 2 ( A + B / R 1 ) 2 + ( 2 B / k w 1 2 ) .
1 / R 2 = ( C + D / R 1 ) ( A + B / R 1 ) + D B ( 2 / k w 1 2 ) 2 ( A + B / R 1 ) 2 + ( 2 B / k w 1 2 ) 2 .
q 2 = ( A q 1 + B ) / ( C q 1 + D )
r o w 1 = 2 R 1 cos φ - k w 1 2 sin φ ( 4 R 1 2 + k 2 w 1 4 ) ½ r o w 1 = - ( 4 R 1 2 + k 2 w 1 4 ) ½ R 1 k w 1 2 sin φ ,
r n = [ ( L / 2 f sin θ ) sin n θ + cos n θ ] r o + ( L sin n θ / sin θ ) r o
r n = [ k w m 2 / 4 f sin n θ + cos n θ ] r o + ( k w m 2 / 2 ) r o sin n θ
r n / w 1 = [ ( k w m 2 / 4 f ) sin n θ + cos n θ ] 2 R 1 cos φ ( 4 R 1 2 + k 2 w 1 4 ) ½ - { k w m 2 / 2 [ 1 / 2 f + ( 4 R 1 / k 2 w 1 4 ) + 1 / R 1 ] sin n θ + cos n θ } k w 1 2 sin φ ( 4 R 1 2 + k 2 w 1 4 ) ½ ,
w n 2 / w 1 2 = 1 2 [ 1 + w m 4 / w 1 4 + ( k w m 2 / 2 ) 2 ( 1 / R 1 + 1 / 2 f ) 2 ] + 1 2 [ 1 - w m 4 / w 1 4 - ( k w m 2 / 2 ) 2 ( 1 / R 1 + 1 / 2 f ) 2 ] cos 2 n θ + k w m 2 / 2 ( 1 / R 1 + 1 / 2 f ) sin 2 n θ .
n = n 0 - n 2 r 2 / 2 ,
( d 2 r / d z 2 ) = - ( n 2 / n 0 ) r ,
r ( z ) = r o cos ( n 2 / n o ) ½ z + ( n o / n 2 ) ½ r o sin ( n 2 / n o ) ½ z ,
r ( z ) / w 1 = { 2 R 1 ( 4 R 1 2 + k 2 w 1 4 ) ½ cos ( n 2 / n o ) ½ z } cos φ - { cos ( n 2 / n o ) ½ z + ( n o / n 2 ) ½ [ 1 / R 1 + ( 4 R 1 / k 2 w 1 4 ) ] sin ( n 2 / n o ) ½ z } × k w 1 2 sin φ ( 4 R 1 2 + k 2 w 1 4 ) ½ ,
w 2 ( z ) / w 1 2 = 1 2 [ 1 + ( w M 4 / w 4 ) + ( 1 / R 1 2 ) ( n o / n 2 ) ] + 1 2 [ 1 - ( w M 4 / w 4 ) - ( 1 / R 1 2 ) ( n o / n 2 ) ] cos 2 ( n 2 / n o ) ½ z + ( n o / n 2 ) ½ / R 1 sin 2 ( n 2 / n o ) ½ z ,

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