Abstract

When an input Gaussian beam is improperly aligned and mode-matched to a stable optical resonator, the electric field in the resonator couples to off-axis spatial eigenmodes. We show that a translation of the input axis or a mismatch of the beam waist to the resonator waist size causes a coupling of off-axis modes which is inphase with the input field. On the other hand, a tilt of the input beam or a mismatch of the beam waist position to cavity waist position couples to these modes in quadrature phase. We also propose a method to measure these coupling coefficients and thereby provide a means to align and mode-match a resonant optical cavity in real time.

© 1984 Optical Society of America

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References

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  1. A. Rudiger, R. Schilling, L. Schnupp, W. Winkler, H. Billing, K. Maischberger, “A Mode Selector to Suppress Fluctuations in Laser Beam Geometry,” Opt. Acta 28, 641 (1981).
    [CrossRef]
  2. H. Kogelnik, T. Li, “Laser Beams and Resonators,” Appl. Opt. 5, 1550 (1966).
    [CrossRef] [PubMed]
  3. K. E. Oughstun, “On the Completeness of the Stationary Transverse Modes in an Optical Cavity,” Opt. Commun. 42, 72 (1982).
    [CrossRef]
  4. Yu. V. Troitskii, “Optimization and Comparison of the Characteristics of Optical Interference Discriminators,” Sov. J. Quanum. Electron. 8, 628 (May1978).
    [CrossRef]
  5. See, for example, R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, H. Ward, “Laser Phase and Frequency Stabilization Using an Optical Resonator,” Appl. Phys. B31, 97 (1983).
  6. H. J. Baker, “Mode-Matching Techniques as an Aid to Laser Cavity Alignment,” Opt. Acta 27, 897 (1979).
    [CrossRef]
  7. S. A. Collins, “Analysis of Optical Resonators Involving Focusing Elements,” Appl. Opt. 3, 1263 (1964).
    [CrossRef]

1983 (1)

See, for example, R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, H. Ward, “Laser Phase and Frequency Stabilization Using an Optical Resonator,” Appl. Phys. B31, 97 (1983).

1982 (1)

K. E. Oughstun, “On the Completeness of the Stationary Transverse Modes in an Optical Cavity,” Opt. Commun. 42, 72 (1982).
[CrossRef]

1981 (1)

A. Rudiger, R. Schilling, L. Schnupp, W. Winkler, H. Billing, K. Maischberger, “A Mode Selector to Suppress Fluctuations in Laser Beam Geometry,” Opt. Acta 28, 641 (1981).
[CrossRef]

1979 (1)

H. J. Baker, “Mode-Matching Techniques as an Aid to Laser Cavity Alignment,” Opt. Acta 27, 897 (1979).
[CrossRef]

1978 (1)

Yu. V. Troitskii, “Optimization and Comparison of the Characteristics of Optical Interference Discriminators,” Sov. J. Quanum. Electron. 8, 628 (May1978).
[CrossRef]

1966 (1)

1964 (1)

Baker, H. J.

H. J. Baker, “Mode-Matching Techniques as an Aid to Laser Cavity Alignment,” Opt. Acta 27, 897 (1979).
[CrossRef]

Billing, H.

A. Rudiger, R. Schilling, L. Schnupp, W. Winkler, H. Billing, K. Maischberger, “A Mode Selector to Suppress Fluctuations in Laser Beam Geometry,” Opt. Acta 28, 641 (1981).
[CrossRef]

Collins, S. A.

Drever, R. W. P.

See, for example, R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, H. Ward, “Laser Phase and Frequency Stabilization Using an Optical Resonator,” Appl. Phys. B31, 97 (1983).

Ford, G. M.

See, for example, R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, H. Ward, “Laser Phase and Frequency Stabilization Using an Optical Resonator,” Appl. Phys. B31, 97 (1983).

Hall, J. L.

See, for example, R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, H. Ward, “Laser Phase and Frequency Stabilization Using an Optical Resonator,” Appl. Phys. B31, 97 (1983).

Hough, J.

See, for example, R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, H. Ward, “Laser Phase and Frequency Stabilization Using an Optical Resonator,” Appl. Phys. B31, 97 (1983).

Kogelnik, H.

Kowalski, F. V.

See, for example, R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, H. Ward, “Laser Phase and Frequency Stabilization Using an Optical Resonator,” Appl. Phys. B31, 97 (1983).

Li, T.

Maischberger, K.

A. Rudiger, R. Schilling, L. Schnupp, W. Winkler, H. Billing, K. Maischberger, “A Mode Selector to Suppress Fluctuations in Laser Beam Geometry,” Opt. Acta 28, 641 (1981).
[CrossRef]

Munley, A. J.

See, for example, R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, H. Ward, “Laser Phase and Frequency Stabilization Using an Optical Resonator,” Appl. Phys. B31, 97 (1983).

Oughstun, K. E.

K. E. Oughstun, “On the Completeness of the Stationary Transverse Modes in an Optical Cavity,” Opt. Commun. 42, 72 (1982).
[CrossRef]

Rudiger, A.

A. Rudiger, R. Schilling, L. Schnupp, W. Winkler, H. Billing, K. Maischberger, “A Mode Selector to Suppress Fluctuations in Laser Beam Geometry,” Opt. Acta 28, 641 (1981).
[CrossRef]

Schilling, R.

A. Rudiger, R. Schilling, L. Schnupp, W. Winkler, H. Billing, K. Maischberger, “A Mode Selector to Suppress Fluctuations in Laser Beam Geometry,” Opt. Acta 28, 641 (1981).
[CrossRef]

Schnupp, L.

A. Rudiger, R. Schilling, L. Schnupp, W. Winkler, H. Billing, K. Maischberger, “A Mode Selector to Suppress Fluctuations in Laser Beam Geometry,” Opt. Acta 28, 641 (1981).
[CrossRef]

Troitskii, Yu. V.

Yu. V. Troitskii, “Optimization and Comparison of the Characteristics of Optical Interference Discriminators,” Sov. J. Quanum. Electron. 8, 628 (May1978).
[CrossRef]

Ward, H.

See, for example, R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, H. Ward, “Laser Phase and Frequency Stabilization Using an Optical Resonator,” Appl. Phys. B31, 97 (1983).

Winkler, W.

A. Rudiger, R. Schilling, L. Schnupp, W. Winkler, H. Billing, K. Maischberger, “A Mode Selector to Suppress Fluctuations in Laser Beam Geometry,” Opt. Acta 28, 641 (1981).
[CrossRef]

Appl. Opt. (2)

Appl. Phys. (1)

See, for example, R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, H. Ward, “Laser Phase and Frequency Stabilization Using an Optical Resonator,” Appl. Phys. B31, 97 (1983).

Opt. Acta (2)

H. J. Baker, “Mode-Matching Techniques as an Aid to Laser Cavity Alignment,” Opt. Acta 27, 897 (1979).
[CrossRef]

A. Rudiger, R. Schilling, L. Schnupp, W. Winkler, H. Billing, K. Maischberger, “A Mode Selector to Suppress Fluctuations in Laser Beam Geometry,” Opt. Acta 28, 641 (1981).
[CrossRef]

Opt. Commun. (1)

K. E. Oughstun, “On the Completeness of the Stationary Transverse Modes in an Optical Cavity,” Opt. Commun. 42, 72 (1982).
[CrossRef]

Sov. J. Quanum. Electron. (1)

Yu. V. Troitskii, “Optimization and Comparison of the Characteristics of Optical Interference Discriminators,” Sov. J. Quanum. Electron. 8, 628 (May1978).
[CrossRef]

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Figures (4)

Fig. 1
Fig. 1

Optical axis of a two-mirrored cavity is the line which intersects the centers of curvature of the two mirrors. This figure defines the sense of positive curvature for each mirror (after Kogelnik and Li2).

Fig. 2
Fig. 2

Possible misalignments of the input axis with respect to the cavity axis: (a) transverse displacement in the x direction; (b) tilt through an angle ax; (c) waist size mismatch; and (d) axial waist displacement. Equivalent to (a) and (b) are transverse displacements and tilts in the y dimension.

Fig. 3
Fig. 3

Definition of waist distances t1 and t2; here both are shown positive (after Kogelnik and Li2).

Fig. 4
Fig. 4

Apparatus for controlling alignment. Instrumentation for controlling alignment in one dimension only is shown. Quadrant detector is fixed either physically or electronically to the left cavity mirror. The alignment signals are derived from the rf components of the detector outputs.

Tables (1)

Tables Icon

Table I Couplings due to Possible Misalignments of an Input Beam with Respect to the Cavity Fundamental Mode for a Two-Mirrored Cavity

Equations (40)

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U 0 ( x ) = ( 2 π x 0 2 ) 1 / 4 exp [ - ( x x 0 ) 2 ] ,
U 1 ( x ) = ( 2 π x 0 2 ) 1 / 4 2 x x 0 exp [ - ( x x 0 ) 2 ] ,
x 0 4 = ( λ π ) 2 d ( R 1 - d ) ( R 2 - d ) ( R 1 + R 2 - d ) ( R 1 + R 2 - 2 d ) 2 .
Ψ ( x ) = A U 0 ( x - a x ) = A ( 2 π x 0 2 ) 1 / 4 exp [ - ( x - a x ) 2 x 0 2 ] .
Ψ ( x ) A ( 2 π x 0 2 ) 1 / 4 ( 1 + 2 a x x x 0 2 ) exp [ - ( x x 0 ) 2 ]
Ψ ( x ) A [ U 0 ( x ) + a x x 0 U 1 ( x ) ] .
Ψ ( x ) = Ψ ( x ) ( cos α x ) - 1 = Ψ ( x ) ( 1 + α x 2 + higher order terms ) .
φ ( x ) = ( 2 π λ ) x sin α x ( 2 π / λ ) α x x .
Ψ ( x ) A U 0 ( x ) exp ( i 2 π α x x λ ) .
Ψ ( x ) A U 0 ( x ) ( 1 + 2 π i α x x 0 λ + higher order terms )
Ψ ( x ) A [ U 0 ( x ) + π i α x x 0 λ U 1 ( x ) ] .
V 0 ( r ) = 2 π 1 w 0 exp ( - r 2 w 0 2 ) ,
V 1 ( r ) = 2 π 1 w 0 ( 1 - 2 r 2 w 0 2 ) exp ( - r 2 w 0 2 ) ,
Ψ ( r ) = A 2 π 1 w 0 ( 1 + ɛ ) exp [ - r 2 w 0 2 ( 1 + ɛ ) 2 ] .
Ψ ( r ) A [ V 0 ( r ) + ɛ V 1 ( r ) ] .
V 0 ( r , z ) = 2 π 1 w exp [ - r 2 ( 1 w 2 + i π λ R ) ] ,
V 1 ( r , z ) = 2 π 1 w ( 1 - 2 r 2 w 2 ) exp [ - r 2 ( 1 w 2 - i π λ R ) ] .
R ( z ) = z [ 1 + ( π w 0 2 λ z ) 2 ] .
w 2 ( z ) = w 0 2 [ 1 + ( λ z π w 0 2 ) 2 ] .
Ψ ( r , b ) = A 2 π 1 w exp [ - r 2 w 0 2 ( 1 - i κ ) ] .
Ψ ( r , z ) = A ( V 0 + i λ b 2 π w 0 2 V 1 ) .
δ ν = k ν 0 ,
ν 0 = c 2 d 1 π cos - 1 [ ( 1 - d R 1 ) ( 1 - d R 2 ) ] 1 / 2 .
t 1 = d ( R 2 - d ) R 1 + R 2 - 2 d ,
t 2 = d ( R 1 - d ) R 1 + R 2 - 2 d .
Θ 2 = R 1 R 2 Θ 1
a = R 1 sin Θ 1
sin Θ 2 = ( 1 - t 2 R 2 ) ( 1 - t 1 R 1 ) sin Θ 1
sin α = sin Θ 1 - t 1 R 1 .
E 1 t = A 1 E 1 U 1 ( x ) exp { - i [ ( ν + ν 0 ) t + φ 1 + φ 0 ] } ,
A 1 = [ ( a x / x 0 ) 2 + ( π α x x 0 λ ) 2 ] 1 / 2 ,
φ = tan - 1 π α x 0 2 λ a .
I ( x ) = E 0 U 0 ( x ) 2 + A 1 E 1 U 1 ( x ) 2 + 2 A 1 E 0 E 1 U 0 ( x ) U 1 ( x ) cos ( ν 0 t + φ + φ 0 ) .
V I A 1 E 0 E 1 cos φ = a x x 0 E 0 E 1 ,
V Q A 1 E 0 E 1 sin φ = π α x x 0 λ E 0 E 1 ,
I ( r ) = E 0 2 = A 2 E 2 2 + 2 A 2 E 0 E 2 V 0 ( r ) V 1 ( r ) cos ( 2 ν 0 t + φ + φ 0 ) ,
A 2 = [ ɛ 2 + ( λ b 2 π w 0 2 ) 2 ] 1 / 2 ,
φ = λ b 2 π w 0 ɛ 2 .
V I A 2 E 0 E 2 cos φ = ɛ E 0 E 2 ,
V Q A 2 E 0 E 2 sin φ = λ b 2 π w 0 2 E 0 E 2 .

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