Abstract

This paper presents a theory of the transformation of the waist position and waist size of a Gaussian beam by a thick lens. We clarify the effects of lens thickness and shape on the transformed waist parameters. For equiconvex and equiconcavelenses, the effects of the thickness on the transformed parameters are significant when the Rayleigh length of the incident beam is small compared with the focal length of a thin positive lens. For plano-convex and plano-concave lenses, the waist parameters of the transmitted beam depend largely on whether the flat or curved side faces the incident beam. The conditions for locating the waist at a prescribed position in a spherical shell or a sphere are also given in terms of the waist parameters of the incident beam.

© 1990 Optical Society of America

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References

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  1. G. Goubau, “Optical Relations for Coherent Wave Beams,” in Electromagnetic Theory and Antennas, E. C. Jordan, Ed. (Pergamon, New York, 1963), p. 907.
  2. A. G. van Nie, “Rigorous Calculation of the Electromagnetic Field of Wave Beams,” Philips Res. Rep. 19, 378–394 (1964).
  3. H. Kogelnik, “Imaging of Optical Modes—Resonators with Internal Lenses,” Bell Syst. Tech. J. 44, 455–494 (1965).
  4. T. S. Chu, “Geometrical Representation of Gaussian Beam Propagation,” Bell Syst. Tech. J. 45, 287–299 (1966).
  5. L. D. Dickson, “Characteristics of a Propagating Gaussian Beam,” Appl. Opt. 9, 1854–1861 (1970).
    [CrossRef] [PubMed]
  6. S. R. Mallinson, G. Warnes, “Optimization of Thick Lenses for Single-Mode Optical-Fiber Microcomponents,” Opt. Lett. 10, 238–240 (1985).
    [CrossRef] [PubMed]
  7. K. S. Lee, F. S. Barnes, “Microlenses on the End of Single-Mode Optical Fibers for Laser Applications,” Appl. Opt. 24, 3134–3139 (1985).
    [CrossRef] [PubMed]
  8. K. S. Lee, “Focusing Characteristics of a Truncated and Aberrated Gaussian Beam through a Hemispherical Microlens,” Appl. Opt. 25, 3671–3676 (1986).
    [CrossRef] [PubMed]
  9. B. Hillerich, “Shape Analysis and Coupling Loss of Microlenses on Single-Mode Fiber Tips,” Appl. Opt. 27, 3102–3106 (1988).
    [CrossRef] [PubMed]
  10. H. Kogelnik, T. Li, “Laser Beams and Resonators,” Appl. Opt. 5, 1550–1567 (1966).
    [CrossRef] [PubMed]
  11. A. E. Siegman, An Introduction to Lasers and Masers (McGraw-Hill, New York, 1971), p. 295.
  12. A. Yariv, Quantum Electronics (Wiley, New York, 1975), p. 101.
  13. W. D. Gunter, A. De Young, “New Properties of Afocal Lens Pairs with Gaussian Beams,” Appl. Opt. 25, 1742–1744 (1986).
    [CrossRef] [PubMed]
  14. R. Ifflander, H. Weber, “Focusing of Multimode Laser Beams with Variable Beam Parameters,” Opt. Acta 33, 1083–1090 (1986).
    [CrossRef]
  15. S. Nemoto, “Waist Shift of a Gaussian Beam by Plane Dielectric Interfaces,” Appl. Opt. 27, 1833–1839 (1988).
    [CrossRef] [PubMed]
  16. S. Nemoto, “Waist Shift of a Gaussian Beam by a Dielectric Plate,” Appl. Opt. 28, 1643–1647 (1989).
    [CrossRef] [PubMed]

1989 (1)

1988 (2)

1986 (3)

1985 (2)

1970 (1)

1966 (2)

T. S. Chu, “Geometrical Representation of Gaussian Beam Propagation,” Bell Syst. Tech. J. 45, 287–299 (1966).

H. Kogelnik, T. Li, “Laser Beams and Resonators,” Appl. Opt. 5, 1550–1567 (1966).
[CrossRef] [PubMed]

1965 (1)

H. Kogelnik, “Imaging of Optical Modes—Resonators with Internal Lenses,” Bell Syst. Tech. J. 44, 455–494 (1965).

1964 (1)

A. G. van Nie, “Rigorous Calculation of the Electromagnetic Field of Wave Beams,” Philips Res. Rep. 19, 378–394 (1964).

Barnes, F. S.

Chu, T. S.

T. S. Chu, “Geometrical Representation of Gaussian Beam Propagation,” Bell Syst. Tech. J. 45, 287–299 (1966).

De Young, A.

Dickson, L. D.

Goubau, G.

G. Goubau, “Optical Relations for Coherent Wave Beams,” in Electromagnetic Theory and Antennas, E. C. Jordan, Ed. (Pergamon, New York, 1963), p. 907.

Gunter, W. D.

Hillerich, B.

Ifflander, R.

R. Ifflander, H. Weber, “Focusing of Multimode Laser Beams with Variable Beam Parameters,” Opt. Acta 33, 1083–1090 (1986).
[CrossRef]

Kogelnik, H.

H. Kogelnik, T. Li, “Laser Beams and Resonators,” Appl. Opt. 5, 1550–1567 (1966).
[CrossRef] [PubMed]

H. Kogelnik, “Imaging of Optical Modes—Resonators with Internal Lenses,” Bell Syst. Tech. J. 44, 455–494 (1965).

Lee, K. S.

Li, T.

Mallinson, S. R.

Nemoto, S.

Siegman, A. E.

A. E. Siegman, An Introduction to Lasers and Masers (McGraw-Hill, New York, 1971), p. 295.

van Nie, A. G.

A. G. van Nie, “Rigorous Calculation of the Electromagnetic Field of Wave Beams,” Philips Res. Rep. 19, 378–394 (1964).

Warnes, G.

Weber, H.

R. Ifflander, H. Weber, “Focusing of Multimode Laser Beams with Variable Beam Parameters,” Opt. Acta 33, 1083–1090 (1986).
[CrossRef]

Yariv, A.

A. Yariv, Quantum Electronics (Wiley, New York, 1975), p. 101.

Appl. Opt. (8)

Bell Syst. Tech. J. (2)

H. Kogelnik, “Imaging of Optical Modes—Resonators with Internal Lenses,” Bell Syst. Tech. J. 44, 455–494 (1965).

T. S. Chu, “Geometrical Representation of Gaussian Beam Propagation,” Bell Syst. Tech. J. 45, 287–299 (1966).

Opt. Acta (1)

R. Ifflander, H. Weber, “Focusing of Multimode Laser Beams with Variable Beam Parameters,” Opt. Acta 33, 1083–1090 (1986).
[CrossRef]

Opt. Lett. (1)

Philips Res. Rep. (1)

A. G. van Nie, “Rigorous Calculation of the Electromagnetic Field of Wave Beams,” Philips Res. Rep. 19, 378–394 (1964).

Other (3)

G. Goubau, “Optical Relations for Coherent Wave Beams,” in Electromagnetic Theory and Antennas, E. C. Jordan, Ed. (Pergamon, New York, 1963), p. 907.

A. E. Siegman, An Introduction to Lasers and Masers (McGraw-Hill, New York, 1971), p. 295.

A. Yariv, Quantum Electronics (Wiley, New York, 1975), p. 101.

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Figures (22)

Fig. 1
Fig. 1

Top: three dielectrics with spherical interfaces. Middle: lenses to be considered [(i) equiconvex, (ii) equiconcave, (iii), (v) plano-convex, (iv), (vi) plano-concave]. Bottom: waist parameters of the incident and transmitted beams for lenses (i) and (ii).

Fig. 2
Fig. 2

Parameters for estimating the focal length f0 of a thin positive lens from the shape of a thick lens.

Fig. 3
Fig. 3

Equiconvex lenses for several values of d/ρ along with the values of d/nf0 for n = 1.5.

Fig. 4
Fig. 4

Transformation of the waist position by an equiconvex lens when z1 = f0.

Fig. 5
Fig. 5

Transformation of the waist size by an equiconvex lens when z = f0.

Fig. 6
Fig. 6

Transformation of the waist parameters by an equiconvex lens when z1 = 0.

Fig. 7
Fig. 7

Drum lens with the property that the waist size of the transmitted beam is identical with that of the incident beam irrespective of the lens position.

Fig. 8
Fig. 8

Drum lens with d = 4nf0 and the equivalent afocal lens system.

Fig. 9
Fig. 9

Plano-convex lenses for several values of d/ρ along with the values of d/2nf0 for n = 1.5.

Fig. 10
Fig. 10

Transformation of the waist parameters by a plano-convex lens when the curved side faces the incident beam and z1 = 2f0.

Fig. 11
Fig. 11

Transformation of the waist parameters by a plano-convex lens when the flat side faces the incident beam and z1 = 2f0.

Fig. 12
Fig. 12

Transformation of the waist parameters by a plano-convex lens when the curved side faces the incident beam and z1 = 0.

Fig. 13
Fig. 13

Transformation of the waist parameters by a plano-convex lens when the flat side faces the incident beam and z1 = 0.

Fig. 14
Fig. 14

Equiconcave and plano-concave lenses for several values of d/ρ along with the values of d/nf0 and d/2nf0 for n = 1.5.

Fig. 15
Fig. 15

Transformation of the waist parameters by an equiconcave lens when z1 = f0.

Fig. 16
Fig. 16

Transformation of the waist parameters by a plano-concave lens when the curved side faces the incident beam and z1 = 2f0.

Fig. 17
Fig. 17

Transformation of the waist parameters by a plano-concave lens when the flat side faces the incident beam and z1 = 2f0.

Fig. 18
Fig. 18

Transformation of the waist parameters by a plano-concave lens when the curved side faces the incident beam and z1 = 0.

Fig. 19
Fig. 19

Transformation of the waist parameters by a plano-concave lens when the flat side faces the incident beam and z1 = 0.

Fig. 20
Fig. 20

Top: spherical shell with the beam waist located at its center; bottom: two configurations for locating the waist at the center of a sphere.

Fig. 21
Fig. 21

Waist parameters of the incident beam for locating the waist at the center of the shell shown in Fig. 20 for n1 = 1, n2 = 1.5, a = 5 cm, and b = 5.2 cm.

Fig. 22
Fig. 22

Waist parameters of the incident beam for locating the waist at the center of the sphere shown in Fig. 20.

Equations (61)

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q = z + j n k s 0 2 ,
q 2 = ( A q 1 + B ) / ( C q 1 + D ) .
[ 1 + ( C / D ) q 1 ] [ 1 - ( C / A ) q 2 ] = 1 - B C / A D ,
q i = z i + j u i ,             u i = n i k s i 2 ,             i = 1 - 3.
[ A B C D ] = [ 1 0 1 / f 3 n 2 / n 3 ] [ 1 d 0 1 ] [ 1 0 1 / f 2 n 1 / n 2 ] ,
f 2 = n 2 ρ 1 / ( n 2 - n 1 ) ,             f 3 = n 3 ρ 2 / ( n 3 - n 2 ) ,
( 1 + q 1 / f 1 ) ( 1 - q 2 / f 2 ) = 1 ,
( 1 + q 2 / f 2 ) ( 1 - q 3 / f 3 ) = 1 ,
q 2 = q 2 + d ,
f 1 = n 1 ρ 1 / ( n 2 - n 1 ) , f 2 = n 2 ρ 2 / ( n 3 - n 2 ) ,
( α + γ q 1 / f 1 ) ( β - γ q 3 / f 3 ) = δ ,
α = 1 + δ ( d / f 2 ) ,             β = δ ( 1 + d / f 2 ) ,
γ = 1 + δ ( 1 + d / f 2 ) ,             δ = f 2 / f 2 .
( i ) ρ 1 = - ρ 2 = - ρ ,             ( ii ) ρ 1 = - ρ 2 = ρ ,
( iii ) ρ 1 = , ρ 2 = ρ ,             ( iv ) ρ 1 = , ρ 2 = - ρ ,
( v ) ρ 1 = - ρ , ρ 2 = ,             ( vi ) ρ 1 = ρ , ρ 2 = ,
( i ) ( σ p - q 1 / f p ) ( σ p + q 3 / f p ) = 1 ,
( ii ) ( σ n - q 1 / f n ) ( σ n + q 3 / f n ) = 1 ,
( iii ) ( σ p - q 1 / 2 f 0 ) ( 1 + q 3 / 2 f 0 ) = 1 ,
( iv ) ( σ n + q 1 / 2 f 0 ) ( 1 - q 3 / 2 f 0 ) = 1 ,
( v ) ( 1 - q 1 / 2 f 0 ) ( σ p + q 3 / 2 f 0 ) = 1 ,
( vi ) ( 1 + q 1 / 2 f 0 ) ( σ n - q 3 / 2 f 0 ) = 1 ,
1 / f p = ( 1 - d / 4 n f 0 ) / f 0 ,             1 / f n = - ( 1 + d / 4 n f 0 ) / f 0 ,
1 / f 0 = 2 ( n - 1 ) / ρ ,
σ p = 1 - d / 2 n f 0 ,             σ n = 1 + d / 2 n f 0 ,
( i ) τ p + Z 3 = v p ( τ p - Z 1 ) / [ ( τ p - Z 1 ) 2 + U 1 2 ] ,
( ii ) τ n - Z 3 = v n ( τ n + Z 1 ) / [ ( τ n + Z 1 ) 2 + U 1 2 ] ,
( iii ) 1 + Z 3 = ( σ p - Z 1 ) / [ ( σ p - Z 1 ) 2 + U 1 2 ] ,
( iv ) 1 - Z 3 = ( σ n + Z 1 ) / [ ( σ n + Z 1 ) 2 + U 1 2 ] ,
( v ) σ p + Z 3 = ( 1 - Z 1 ) / [ ( 1 - Z 1 ) 2 + U 1 2 ] ,
( vi ) σ n - Z 3 = ( 1 + Z 1 ) / [ ( 1 + Z 1 ) 2 + U 1 2 ] ,
( s 3 / s 1 ) 2 = ( τ p + Z 3 ) / ( τ p - Z 1 ) ( i ) ,
= ( τ n - Z 3 ) / ( τ n + Z 1 ) ( ii ) ,
= ( 1 + Z 3 ) / ( σ p - Z 1 ) ( iii ) ,
= ( 1 - Z 3 ) / ( σ n + Z 1 ) ( iv ) ,
= ( σ p + Z 3 ) / ( 1 - Z 1 ) ( v ) ,
= ( σ n - Z 3 ) / ( 1 + Z 1 ) ( vi ) ,
Z 1 = z 1 / f 0 ,             Z 3 = z 3 / f 0 ,             U 1 = u 1 / f 0 ,             for ( i ) , ( ii ) ,
Z 1 = z 1 / 2 f 0 ,             Z 3 = z 3 / 2 f 0 ,             U 1 = u 1 / 2 f 0 ,             for ( iii ) - ( vi ) ,
τ p = 2 σ p / ( σ p + 1 ) ,             v p = [ 2 / ( σ p + 1 ) ] 2 ,
τ n = 2 σ n / ( σ n + 1 ) ,             v n = [ 2 / ( σ n + 1 ) ] 2 ,
ρ = [ ( d - d 1 ) 2 + d 0 2 ] / [ 4 ( d - d 1 ) ] ( i ) ,
ρ = [ ( d 2 - d ) 2 + d 0 2 ] / [ 4 ( d 2 - d ) ] ( ii ) ,
ρ = [ 4 ( d - d 3 ) 2 + d 0 2 ] / [ 8 ( d - d 3 ) ] ( iii ) ,
ρ = [ 4 ( d 4 - d ) 2 + d 0 2 ] / [ 8 ( d 4 - d ) ] ( iv ) ,
z 3 = z 1 - 4 f 0 ,             s 3 = s 1 .
- z 3 - f 2 = - ( f 2 / f 1 ) 2 ( z 1 - f 1 ) ,             s 3 = ( f 2 / f 1 ) s 1 ,
α + γ z 1 / f 1 = δ ( β - γ z 3 / f 3 ) × [ ( β - γ z 3 / f 3 ) 2 + ( γ u 3 / f 3 ) 2 ] - 1 ,
( s 1 / s 3 ) 2 = δ ( α + γ z 1 / f 1 ) / ( β - γ z 3 / f 3 ) ,
α = 1 + ( n 2 - n 3 ) ( b - a ) / n 2 a ,
β = δ [ 1 + ( n 1 - n 2 ) ( b - a ) / n 2 b ] ,
γ = 1 + β , δ = ( n 2 - n 3 ) b / ( n 1 - n 2 ) a ,
f 1 = n 1 b / ( n 1 - n 2 ) , f 3 = n 3 a / ( n 2 - n 3 ) .
1 - η 1 z 1 / a = ( 1 + η 3 z 3 / a ) × [ ( 1 + η 3 z 3 / a ) 2 + ( η 3 u 3 / a ) 2 ] - 1 ,
( s 1 / s 3 ) 2 = ( 1 - η 1 z 1 / a ) / ( 1 + η 3 z 3 / a ) ,
η 1 = n 3 / n 1 - 1 ,             η 3 = 1 - n 1 / n 3 .
z 1 / a = - ( 1 - η 1 U 2 ) / ( 1 + η 1 2 U 2 ) ,
( s 1 / s 3 ) 2 = ( 1 + η 1 ) 2 / ( 1 + η 1 2 U 2 ) ,
U = u 3 / a = n 3 k s 3 2 / a .
τ p + Z 3 = v p / ( τ p - Z 1 ) .
Z 3 = 2 [ 2 ( 1 - σ p ) + σ p Z 1 ] / [ 2 σ p - ( σ p + 1 ) Z 1 ] .

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