Abstract

This paper presents a theory on the shift of the waist position of a Gaussian beam due to plane dielectric interfaces. The waist shift is defined as the distance between the waist positions of the incident and transmitted beams. We derive it for the beam passing through a single or two plane interfaces and examine how it depends on the waist position of the incident beam and on the refractive indices of dielectrics. The condition for locating the waist of the transmitted beam at a prescribed position is also clarified.

© 1988 Optical Society of America

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References

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  1. D. C. Sinclair, “Optical Loss and Thermal Distortion in Gas-Laser Brewster Windows,” Appl. Opt. 9, 797 (1970).
    [CrossRef] [PubMed]
  2. J. M. Harris, N. J. Dovichi, “Thermal Lens Calorimetry,” Anal. Chem. 52, 695A (1980).
    [CrossRef]
  3. G. A. Massey, A. E. Siegman, “Reflection and Refraction of Gaussian Light Beams at Tilted Ellipsoidal Surfaces,” Appl. Opt. 8, 975 (1969).
    [CrossRef] [PubMed]
  4. J. Turunen, “Astigmatism in Laser Beam Optical Systems,” Appl. Opt. 25, 2908 (1986).
    [CrossRef] [PubMed]
  5. J.-P. Taché, “Ray Matrices for Tilted Interfaces in Laser Resonators,” Appl. Opt. 26, 427 (1987).
    [CrossRef] [PubMed]
  6. W. H. Carter, “Focal Shift and Concept of Effective Fresnel Number for a Gaussian Laser Beam,” Appl. Opt. 21, 1989 (1982).
    [CrossRef] [PubMed]
  7. G. D. Sucha, W. H. Carter, “Focal Shift for a Gaussian Beam: an Experimental Study,” Appl. Opt. 23, 4345 (1984).
    [CrossRef] [PubMed]
  8. S. A. Self, “Focusing of Spherical Gaussian Beams,” Appl. Opt. 22, 658 (1983).
    [CrossRef] [PubMed]
  9. H. Kogelnik, T. Li, “Laser Beams and Resonators,” Appl. Opt. 5, 1550 (1966).
    [CrossRef] [PubMed]
  10. A. E. Siegman, An Introduction to Lasers and Masers (McGraw-Hill, New York, 1971), Chap. 8.
  11. A. Yariv, Quantum Electronics (Wiley, New York, 1975), Chap. 6.
  12. P. Laures, “Geometrical Approach to Gaussian Beam Propagation,” Appl. Opt. 6, 747 (1967).
    [CrossRef] [PubMed]
  13. D. C. Hanna, “Astigmatic Gaussian Beams Produced by Axially Asymmetric Laser Cavities,” IEEE J. Quantum Electron. QE-5, 483 (1969).
    [CrossRef]
  14. P. E. Dyer, “Influence of Brewster Angle Windows on Confocal Unstable Resonator Beam Properties,” Appl. Opt. 17, 687 (1978).
    [CrossRef] [PubMed]
  15. S. L. Chao, J. M. Forsyth, “Properties of High-Order Transverse Modes in Astigmatic Laser Cavities,” J. Opt. Soc. Am. 65, 867 (1975).
    [CrossRef]
  16. S. O. Park, S. S. Lee, “Forward Far-Field Pattern of a Laser Beam Scattered by a Water-Suspended Homogeneous Sphere Trapped by a Focused Laser Beam,” J. Opt. Soc. Am. A 4, 417 (1987).
    [CrossRef]
  17. S. Nemoto, “Determination of Waist Parameters of a Gaussian Beam,” Appl. Opt. 25, 3859 (1986). The term maximum below Eq. (B5) should be corrected to minimum.
    [CrossRef] [PubMed]
  18. J. E. Sollid, C. R. Phipps, S. J. Thomas, E. J. McLellan, “Lensless Method of Measuring Gaussian Laser Beam Divergence,” Appl. Opt. 17, 3527 (1978). This paper was brought to my attention after Ref. 17 was published.
    [CrossRef] [PubMed]
  19. H. W. Kogelnik, E. P. Ippen, A. Dienes, C. V. Shank, “Astigmatically Compensated Cavities for CW Dye Lasers,” IEEE J. Quantum Electron. QE-8, 373 (1972).
    [CrossRef]
  20. G. D. Boyd, J. P. Gordon, “Confocal Multimode Resonator for Millimeter Through Optical Wavelength Masers,” Bell Syst. Tech. J. 40, 489 (1961).
  21. T. R. Corle, J. T. Fanton, G. S. Kino, “Distance Measurements by Differential Confocal Optical Ranging,” Appl. Opt. 26, 2416 (1987).
    [CrossRef] [PubMed]

1987

1986

1984

1983

1982

1980

J. M. Harris, N. J. Dovichi, “Thermal Lens Calorimetry,” Anal. Chem. 52, 695A (1980).
[CrossRef]

1978

1975

1972

H. W. Kogelnik, E. P. Ippen, A. Dienes, C. V. Shank, “Astigmatically Compensated Cavities for CW Dye Lasers,” IEEE J. Quantum Electron. QE-8, 373 (1972).
[CrossRef]

1970

1969

G. A. Massey, A. E. Siegman, “Reflection and Refraction of Gaussian Light Beams at Tilted Ellipsoidal Surfaces,” Appl. Opt. 8, 975 (1969).
[CrossRef] [PubMed]

D. C. Hanna, “Astigmatic Gaussian Beams Produced by Axially Asymmetric Laser Cavities,” IEEE J. Quantum Electron. QE-5, 483 (1969).
[CrossRef]

1967

1966

1961

G. D. Boyd, J. P. Gordon, “Confocal Multimode Resonator for Millimeter Through Optical Wavelength Masers,” Bell Syst. Tech. J. 40, 489 (1961).

Boyd, G. D.

G. D. Boyd, J. P. Gordon, “Confocal Multimode Resonator for Millimeter Through Optical Wavelength Masers,” Bell Syst. Tech. J. 40, 489 (1961).

Carter, W. H.

Chao, S. L.

Corle, T. R.

Dienes, A.

H. W. Kogelnik, E. P. Ippen, A. Dienes, C. V. Shank, “Astigmatically Compensated Cavities for CW Dye Lasers,” IEEE J. Quantum Electron. QE-8, 373 (1972).
[CrossRef]

Dovichi, N. J.

J. M. Harris, N. J. Dovichi, “Thermal Lens Calorimetry,” Anal. Chem. 52, 695A (1980).
[CrossRef]

Dyer, P. E.

Fanton, J. T.

Forsyth, J. M.

Gordon, J. P.

G. D. Boyd, J. P. Gordon, “Confocal Multimode Resonator for Millimeter Through Optical Wavelength Masers,” Bell Syst. Tech. J. 40, 489 (1961).

Hanna, D. C.

D. C. Hanna, “Astigmatic Gaussian Beams Produced by Axially Asymmetric Laser Cavities,” IEEE J. Quantum Electron. QE-5, 483 (1969).
[CrossRef]

Harris, J. M.

J. M. Harris, N. J. Dovichi, “Thermal Lens Calorimetry,” Anal. Chem. 52, 695A (1980).
[CrossRef]

Ippen, E. P.

H. W. Kogelnik, E. P. Ippen, A. Dienes, C. V. Shank, “Astigmatically Compensated Cavities for CW Dye Lasers,” IEEE J. Quantum Electron. QE-8, 373 (1972).
[CrossRef]

Kino, G. S.

Kogelnik, H.

Kogelnik, H. W.

H. W. Kogelnik, E. P. Ippen, A. Dienes, C. V. Shank, “Astigmatically Compensated Cavities for CW Dye Lasers,” IEEE J. Quantum Electron. QE-8, 373 (1972).
[CrossRef]

Laures, P.

Lee, S. S.

Li, T.

Massey, G. A.

McLellan, E. J.

Nemoto, S.

Park, S. O.

Phipps, C. R.

Self, S. A.

Shank, C. V.

H. W. Kogelnik, E. P. Ippen, A. Dienes, C. V. Shank, “Astigmatically Compensated Cavities for CW Dye Lasers,” IEEE J. Quantum Electron. QE-8, 373 (1972).
[CrossRef]

Siegman, A. E.

Sinclair, D. C.

Sollid, J. E.

Sucha, G. D.

Taché, J.-P.

Thomas, S. J.

Turunen, J.

Yariv, A.

A. Yariv, Quantum Electronics (Wiley, New York, 1975), Chap. 6.

Anal. Chem.

J. M. Harris, N. J. Dovichi, “Thermal Lens Calorimetry,” Anal. Chem. 52, 695A (1980).
[CrossRef]

Appl. Opt.

G. A. Massey, A. E. Siegman, “Reflection and Refraction of Gaussian Light Beams at Tilted Ellipsoidal Surfaces,” Appl. Opt. 8, 975 (1969).
[CrossRef] [PubMed]

J. Turunen, “Astigmatism in Laser Beam Optical Systems,” Appl. Opt. 25, 2908 (1986).
[CrossRef] [PubMed]

J.-P. Taché, “Ray Matrices for Tilted Interfaces in Laser Resonators,” Appl. Opt. 26, 427 (1987).
[CrossRef] [PubMed]

W. H. Carter, “Focal Shift and Concept of Effective Fresnel Number for a Gaussian Laser Beam,” Appl. Opt. 21, 1989 (1982).
[CrossRef] [PubMed]

G. D. Sucha, W. H. Carter, “Focal Shift for a Gaussian Beam: an Experimental Study,” Appl. Opt. 23, 4345 (1984).
[CrossRef] [PubMed]

S. A. Self, “Focusing of Spherical Gaussian Beams,” Appl. Opt. 22, 658 (1983).
[CrossRef] [PubMed]

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

P. Laures, “Geometrical Approach to Gaussian Beam Propagation,” Appl. Opt. 6, 747 (1967).
[CrossRef] [PubMed]

P. E. Dyer, “Influence of Brewster Angle Windows on Confocal Unstable Resonator Beam Properties,” Appl. Opt. 17, 687 (1978).
[CrossRef] [PubMed]

S. Nemoto, “Determination of Waist Parameters of a Gaussian Beam,” Appl. Opt. 25, 3859 (1986). The term maximum below Eq. (B5) should be corrected to minimum.
[CrossRef] [PubMed]

J. E. Sollid, C. R. Phipps, S. J. Thomas, E. J. McLellan, “Lensless Method of Measuring Gaussian Laser Beam Divergence,” Appl. Opt. 17, 3527 (1978). This paper was brought to my attention after Ref. 17 was published.
[CrossRef] [PubMed]

D. C. Sinclair, “Optical Loss and Thermal Distortion in Gas-Laser Brewster Windows,” Appl. Opt. 9, 797 (1970).
[CrossRef] [PubMed]

T. R. Corle, J. T. Fanton, G. S. Kino, “Distance Measurements by Differential Confocal Optical Ranging,” Appl. Opt. 26, 2416 (1987).
[CrossRef] [PubMed]

Bell Syst. Tech. J.

G. D. Boyd, J. P. Gordon, “Confocal Multimode Resonator for Millimeter Through Optical Wavelength Masers,” Bell Syst. Tech. J. 40, 489 (1961).

IEEE J. Quantum Electron.

H. W. Kogelnik, E. P. Ippen, A. Dienes, C. V. Shank, “Astigmatically Compensated Cavities for CW Dye Lasers,” IEEE J. Quantum Electron. QE-8, 373 (1972).
[CrossRef]

D. C. Hanna, “Astigmatic Gaussian Beams Produced by Axially Asymmetric Laser Cavities,” IEEE J. Quantum Electron. QE-5, 483 (1969).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Other

A. E. Siegman, An Introduction to Lasers and Masers (McGraw-Hill, New York, 1971), Chap. 8.

A. Yariv, Quantum Electronics (Wiley, New York, 1975), Chap. 6.

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Figures (12)

Fig. 1
Fig. 1

Spot-size variation and complex beam parameters at three positions.

Fig. 2
Fig. 2

Two dielectrics with a plane interface. Spot-size variations are drawn for n1 < n2.

Fig. 3
Fig. 3

Three dielectrics with plane-parallel interfaces.

Fig. 4
Fig. 4

Relation between z1 and z3. Three regions I, II, and III are defined for z1.

Fig. 5
Fig. 5

Spot-size variations and waist shifts for case (i). Upper, middle, and lower traces are drawn for z1 in regions I, II, and III, respectively (this applies also to Figs. 6 and 7).

Fig. 6
Fig. 6

Spot-size variations and waist shifts for case (ii).

Fig. 7
Fig. 7

Spot-size variations and waist shifts for case (iii).

Fig. 8
Fig. 8

Dependence of (Δz)c, and (Δz)f on |z1|. Broken lines in (a) and (b) represent (Δz)b and (Δz)e, respectively.

Fig. 9
Fig. 9

Beam focused at a prescribed position in the material contained in a cell. Spot-size variation is drawn for the case when the beam is focused at the midplane of the cell.

Fig. 10
Fig. 10

Spot-size variations when (Δz)a = 0 in (a) and (Δz)e = 0 in (b).

Fig. 11
Fig. 11

Beam volume V formed by the two interfaces and a hyperboloid r = s(z).

Fig. 12
Fig. 12

Refraction of a ray due to a single or two plane interfaces of dielectrics.

Tables (3)

Tables Icon

Table I Signs of Waist Shifts for Three Distinct Media

Tables Icon

Table II Examples of |z1| for Locating the Waist at a Prescribed Position, and Values of Waist Shifts (Δz)c and (Δz)f (n1 = 1, n2 = 1.5, d = 1 mm, |z3| = 10 mm)

Tables Icon

Table III Comparison of dα and dβ with d

Equations (51)

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ψ = ( s 0 / s ) exp ( j n k r 2 / 2 q ) exp ( j ϕ ) ,
1 / q = 1 / R j / n k s 2 ,
s = s 0 [ 1 + ( z / n k s 0 2 ) 2 ] 1 / 2 ,
R = z [ 1 + ( n k s 0 2 / z ) 2 ] ,
ϕ = n k z tan 1 ( z / n k s 0 2 ) ,
q = z + j n k s 0 2 ,
q 2 = ( A q 1 + B ) / ( C q 1 + D ) ,
q 1 = z 1 + j n 1 k s 1 2 , q 2 = z 2 + j n 2 k s 2 2 ,
q 2 = ( n 2 / n 1 ) q 1 .
z 2 = ( n 2 / n 1 ) z 1 , s 2 = s 1 .
( Δ z ) a z 2 z 1 = ( n 2 / n 1 1 ) z 1 , z 1 > 0 ,
( Δ z ) b | z 2 | | z 1 | = ( n 2 / n 1 1 ) | z 1 | , z 1 < 0 .
q i = z i + j n i k s i 2 , i = 1 3 ,
q 2 = ( n 2 / n 1 ) q 1 , q 2 = q 2 + d , q 3 = ( n 3 / n 2 ) q 2 ,
q 3 = ( n 3 / n 1 ) q 1 + ( n 3 / n 2 ) d .
q 3 = ( A q 1 + B ) / ( C q 1 + D ) ,
[ A B C D ] = [ 1 0 0 n 2 / n 3 ] [ 1 d 0 1 ] [ 1 0 0 n 1 / n 2 ] .
z 3 = ( n 3 / n 1 ) z 1 + ( n 3 / n 2 ) d , s 3 = s 1 .
I : z 1 > 0 , II : ( n 1 / n 2 ) d < z 1 < 0 , III : z 1 < ( n 1 / n 2 ) d .
( i ) n 1 < n 3 < n 2 or n 3 < n 1 < n 2 , ( ii ) n 1 < n 2 < n 3 or n 3 < n 2 < n 1 , ( iii ) n 2 < n 1 < n 3 or n 2 < n 3 < n 1 ,
( Δ z ) a d + z 1 z 3 , = ( 1 n 3 / n 2 ) d + ( 1 n 3 / n 1 ) z 1 ,
( Δ z ) b d | z 1 | z 3 , = ( 1 n 3 / n 2 ) d ( 1 n 3 / n 1 ) | z 1 | ,
( Δ z ) c d | z 1 | + | z 3 | , = ( 1 n 3 / n 2 ) d ( 1 n 3 / n 1 ) | z 1 | .
( Δ z ) d z 3 z 1 d , = ( n 3 / n 2 1 ) d + ( n 3 / n 1 1 ) z 1 ,
( Δ z ) e d | z 1 | z 3 , = ( n 3 / n 1 1 ) | z 1 | ( n 3 / n 2 1 ) d ,
( Δ z ) f d | z 1 | + | z 3 | , = ( n 3 / n 1 1 ) | z 1 | ( n 3 / n 2 1 ) d ,
( Δ z ) g z 3 z 1 d , = ( n 3 / n 2 1 ) d + ( n 3 / n 1 1 ) z 1 ,
( Δ z ) h | z 1 | + z 3 d , = ( n 3 / n 2 1 ) d ( n 3 / n 1 1 ) | z 1 | ,
( Δ z ) i | z 1 | | z 3 | d , = ( n 3 / n 2 1 ) d ( n 3 / n 1 1 ) | z 1 | .
( Δ z ) a = ( Δ z ) b = ( Δ z ) c = ( 1 n 1 / n 2 ) d > 0 ,
( Δ z ) g = ( Δ z ) h = ( Δ z ) i = ( n 1 / n 2 1 ) d > 0 ,
| z 1 | = ( n 1 / n 3 ) | z 3 | + ( n 1 / n 2 ) d ,
| z 2 | = ( n 2 / n 1 ) d β = [ ( n 2 n 3 ) / ( n 1 n 3 ) ] d .
s ( z ) = s 2 [ 1 + ( z / n 2 k s 2 2 ) 2 ] 1 / 2 ,
V = π s 2 2 { d + [ | z 2 | 3 + ( d | z 2 | ) 3 ] / 3 ( n 2 k s 2 2 ) 2 } .
V = 2 π s 2 2 d e when | z 2 | = d / 2 , 2 π s e 2 d when s 2 = s e , 2 π s m 2 d when | z 2 | = d / 2 , s 2 = s m ,
d e = ( d / 2 ) [ 1 + ( d / 2 3 n 2 k s 2 2 ) 2 ] ,
s e = { [ | z 2 | 3 + ( d | z 2 | ) 3 ] / 3 ( n 2 k ) 2 d } 1 / 4 ,
s m = ( d / 2 3 n 2 k ) 1 / 2 = 0 . 21435 ( λ d / n 2 ) 1 / 2 .
s ( d / 2 ) = s 2 [ 1 + ( d / 2 n 2 k s 2 2 ) 2 ] 1 / 2 ,
θ lim z s ( z ) / z = 1 / n k s 0 ,
θ 1 = 1 / n 1 k s 1 , θ 3 = 1 / n 3 k s 3 .
r 1 = r 1 d 1 , r 2 = r 2 d 2 .
[ r 2 r 2 ] = [ 1 0 0 n 1 / n 2 ] [ r 1 r 1 ] .
d 2 = ( n 2 / n 1 ) d 1 ,
Δ d d 2 d 1 = ( n 2 / n 1 1 ) d 1 .
r 1 = r 1 d 1 , r 1 r 2 = r 2 d , r 3 = r 3 d 3 ,
[ r 2 r 2 ] = [ 1 d 0 1 ] [ 1 0 0 n 1 / n 2 ] [ r 1 r 1 ] ,
[ r 3 r 3 ] = [ 1 0 0 n 2 / n 3 ] [ r 2 r 2 ] .
d 3 = ( n 3 / n 1 ) d 1 ( n 3 / n 2 ) d ,
Δ d d + d 3 d 1 , = ( 1 n 3 / n 2 ) d ( 1 n 3 / n 1 ) d 1 .

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