Abstract

We demonstrate a method for real time alignment of a Gaussian beam to an optical resonator. While the frequency of a source laser is stabilized to a fundamental cavity mode resonance, phase modulation sidebands are applied at the off-axis mode frequencies. Asymmetrical transmission of the sideband at the frequency of each off-axis mode produces amplitude modulated optical signals and indicates the extent of the misalignments. Phase sensitive detection of these optical signals provides the error signals which are minimized by a control system that steers the input beam. In this way, optimum coupling of an injected source beam can be maintained to the fundamental mode of the resonator. This active alignment technique has demonstrated a sensitivity to tilts of 0.01nrad/Hz and to lateral beam displacements of 0.08nm/Hz in the ~1-Hz–1-kHz frequency range. These values correspond to 2 parts in 107/Hz for both the far-field divergence angle and the beam waist size. Such performance is within a factor of 2 of the shot noise limitation of the error signal measurement for a detected power of 160 μW.

© 1990 Optical Society of America

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References

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  1. G. A. Sanders, M. G. Prentiss, S. Ezekiel, “Passive Ring Resonator Method for Sensitive Inertial Rotation Measurements in Geophysics and Relativity,” Opt. Lett. 6, 569–571 (1981).
    [CrossRef] [PubMed]
  2. J. Hough, D. Hils, M. D. Rayman, L.-S. Ma, L. Hollberg, J. L. Hall, “Dye Laser Stabilization Using Optical Resonators,” Appl. Phys. B 33, 179 (1984).
    [CrossRef]
  3. A. Rudiger, R. Schulling, L. Schnupp, W. Winkler, H. Billing, K. Maischberger, “A Mode Selector to Suppress Laser Beam Fluctuations in Laser Beam Geometry,” Opt. Acta 28, 641 (1981).
    [CrossRef]
  4. S. Grafstrom, U. Harbarth, J. Kowalski, R. Neumann, S. Noehte, “Fast Laser Beam Position Control with Sub-Microradian Precision,” Opt. Commun. 65, 121 (1988).
    [CrossRef]
  5. D. Z. Anderson, “Alignment of Resonant Optical Cavities,” Appl. Opt. 23, 2944–2949 (1984).
    [CrossRef] [PubMed]
  6. W. R. P. Drever, R. Spero, California Institute of Technology; private communication (1984).
  7. M. R. Sayeh, H. R. Bilger, T. Habib, “Optical Resonator with an External Source: Excitation of the Hermite-Gaussian Modes,” Appl. Opt. 24, 3756–3761 (1985).
    [CrossRef] [PubMed]
  8. H. Kogelnik, T. Li, “Laser Beams and Resonators,” Appl. Opt. 5, 1550–1567 (1960).
    [CrossRef]
  9. K. E. Oughstun, “On the Completeness of Stationary Transverse Modes on an Optical Cavity,” Opt. Commun. 42, 72 (1982).
    [CrossRef]
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    [CrossRef]
  11. A. E. Siegman, Lasers (University Science Books, Mill Valley, Ca, 1986), pp. 642–652.
  12. S. A. Collins “Analysis of Optical Resonators Involving Focusing Elements,” Appl. Opt. 3, 1263–1275 (1964).
    [CrossRef]
  13. 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. B 31, 97 (1983).
    [CrossRef]
  14. J. L. Hall, L-S. Ma, G. Kramer, “Principles of Optical Phase Locking: With Application to Internal Mirror He–Ne Lasers Phase Locked via Fast Control of the Discharge Current,” IEEE J. Quantum Electron. (Special issue on Applications of Q.E. to Frequency Standards, Clocks, and Inertial Rotation Sensors) QE-23, 427 (1987).
    [CrossRef]
  15. R. E. Meyer, G. A. Sanders, S. Ezekiel, “Observation of Spatial Variations in the Resonance Frequency of an Optical Resonator,” J. Opt. Soc. Am. 73, 939–942 (1983).
    [CrossRef]

1988 (1)

S. Grafstrom, U. Harbarth, J. Kowalski, R. Neumann, S. Noehte, “Fast Laser Beam Position Control with Sub-Microradian Precision,” Opt. Commun. 65, 121 (1988).
[CrossRef]

1987 (1)

J. L. Hall, L-S. Ma, G. Kramer, “Principles of Optical Phase Locking: With Application to Internal Mirror He–Ne Lasers Phase Locked via Fast Control of the Discharge Current,” IEEE J. Quantum Electron. (Special issue on Applications of Q.E. to Frequency Standards, Clocks, and Inertial Rotation Sensors) QE-23, 427 (1987).
[CrossRef]

1985 (1)

1984 (2)

D. Z. Anderson, “Alignment of Resonant Optical Cavities,” Appl. Opt. 23, 2944–2949 (1984).
[CrossRef] [PubMed]

J. Hough, D. Hils, M. D. Rayman, L.-S. Ma, L. Hollberg, J. L. Hall, “Dye Laser Stabilization Using Optical Resonators,” Appl. Phys. B 33, 179 (1984).
[CrossRef]

1983 (2)

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. B 31, 97 (1983).
[CrossRef]

R. E. Meyer, G. A. Sanders, S. Ezekiel, “Observation of Spatial Variations in the Resonance Frequency of an Optical Resonator,” J. Opt. Soc. Am. 73, 939–942 (1983).
[CrossRef]

1982 (1)

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

1981 (2)

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

G. A. Sanders, M. G. Prentiss, S. Ezekiel, “Passive Ring Resonator Method for Sensitive Inertial Rotation Measurements in Geophysics and Relativity,” Opt. Lett. 6, 569–571 (1981).
[CrossRef] [PubMed]

1979 (1)

A. E. Siegman, “Orthogonal Properties of Optical Resonator Eigenmodes,” Opt. Commun. 31, 369–000 (1979).
[CrossRef]

1964 (1)

1960 (1)

Anderson, D. Z.

Bilger, H. R.

Billing, H.

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

Collins, S. A.

Drever, R. W. P.

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. B 31, 97 (1983).
[CrossRef]

Drever, W. R. P.

W. R. P. Drever, R. Spero, California Institute of Technology; private communication (1984).

Ezekiel, S.

Ford, G. M.

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. B 31, 97 (1983).
[CrossRef]

Grafstrom, S.

S. Grafstrom, U. Harbarth, J. Kowalski, R. Neumann, S. Noehte, “Fast Laser Beam Position Control with Sub-Microradian Precision,” Opt. Commun. 65, 121 (1988).
[CrossRef]

Habib, T.

Hall, J. L.

J. L. Hall, L-S. Ma, G. Kramer, “Principles of Optical Phase Locking: With Application to Internal Mirror He–Ne Lasers Phase Locked via Fast Control of the Discharge Current,” IEEE J. Quantum Electron. (Special issue on Applications of Q.E. to Frequency Standards, Clocks, and Inertial Rotation Sensors) QE-23, 427 (1987).
[CrossRef]

J. Hough, D. Hils, M. D. Rayman, L.-S. Ma, L. Hollberg, J. L. Hall, “Dye Laser Stabilization Using Optical Resonators,” Appl. Phys. B 33, 179 (1984).
[CrossRef]

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. B 31, 97 (1983).
[CrossRef]

Harbarth, U.

S. Grafstrom, U. Harbarth, J. Kowalski, R. Neumann, S. Noehte, “Fast Laser Beam Position Control with Sub-Microradian Precision,” Opt. Commun. 65, 121 (1988).
[CrossRef]

Hils, D.

J. Hough, D. Hils, M. D. Rayman, L.-S. Ma, L. Hollberg, J. L. Hall, “Dye Laser Stabilization Using Optical Resonators,” Appl. Phys. B 33, 179 (1984).
[CrossRef]

Hollberg, L.

J. Hough, D. Hils, M. D. Rayman, L.-S. Ma, L. Hollberg, J. L. Hall, “Dye Laser Stabilization Using Optical Resonators,” Appl. Phys. B 33, 179 (1984).
[CrossRef]

Hough, J.

J. Hough, D. Hils, M. D. Rayman, L.-S. Ma, L. Hollberg, J. L. Hall, “Dye Laser Stabilization Using Optical Resonators,” Appl. Phys. B 33, 179 (1984).
[CrossRef]

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. B 31, 97 (1983).
[CrossRef]

Kogelnik, H.

Kowalski, F. V.

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. B 31, 97 (1983).
[CrossRef]

Kowalski, J.

S. Grafstrom, U. Harbarth, J. Kowalski, R. Neumann, S. Noehte, “Fast Laser Beam Position Control with Sub-Microradian Precision,” Opt. Commun. 65, 121 (1988).
[CrossRef]

Kramer, G.

J. L. Hall, L-S. Ma, G. Kramer, “Principles of Optical Phase Locking: With Application to Internal Mirror He–Ne Lasers Phase Locked via Fast Control of the Discharge Current,” IEEE J. Quantum Electron. (Special issue on Applications of Q.E. to Frequency Standards, Clocks, and Inertial Rotation Sensors) QE-23, 427 (1987).
[CrossRef]

Li, T.

Ma, L.-S.

J. Hough, D. Hils, M. D. Rayman, L.-S. Ma, L. Hollberg, J. L. Hall, “Dye Laser Stabilization Using Optical Resonators,” Appl. Phys. B 33, 179 (1984).
[CrossRef]

Ma, L-S.

J. L. Hall, L-S. Ma, G. Kramer, “Principles of Optical Phase Locking: With Application to Internal Mirror He–Ne Lasers Phase Locked via Fast Control of the Discharge Current,” IEEE J. Quantum Electron. (Special issue on Applications of Q.E. to Frequency Standards, Clocks, and Inertial Rotation Sensors) QE-23, 427 (1987).
[CrossRef]

Maischberger, K.

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

Meyer, R. E.

Munley, A. J.

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. B 31, 97 (1983).
[CrossRef]

Neumann, R.

S. Grafstrom, U. Harbarth, J. Kowalski, R. Neumann, S. Noehte, “Fast Laser Beam Position Control with Sub-Microradian Precision,” Opt. Commun. 65, 121 (1988).
[CrossRef]

Noehte, S.

S. Grafstrom, U. Harbarth, J. Kowalski, R. Neumann, S. Noehte, “Fast Laser Beam Position Control with Sub-Microradian Precision,” Opt. Commun. 65, 121 (1988).
[CrossRef]

Oughstun, K. E.

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

Prentiss, M. G.

Rayman, M. D.

J. Hough, D. Hils, M. D. Rayman, L.-S. Ma, L. Hollberg, J. L. Hall, “Dye Laser Stabilization Using Optical Resonators,” Appl. Phys. B 33, 179 (1984).
[CrossRef]

Rudiger, A.

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

Sanders, G. A.

Sayeh, M. R.

Schnupp, L.

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

Schulling, R.

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

Siegman, A. E.

A. E. Siegman, “Orthogonal Properties of Optical Resonator Eigenmodes,” Opt. Commun. 31, 369–000 (1979).
[CrossRef]

A. E. Siegman, Lasers (University Science Books, Mill Valley, Ca, 1986), pp. 642–652.

Spero, R.

W. R. P. Drever, R. Spero, California Institute of Technology; private communication (1984).

Ward, H.

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. B 31, 97 (1983).
[CrossRef]

Winkler, W.

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

Appl. Opt. (4)

Appl. Phys. B (2)

J. Hough, D. Hils, M. D. Rayman, L.-S. Ma, L. Hollberg, J. L. Hall, “Dye Laser Stabilization Using Optical Resonators,” Appl. Phys. B 33, 179 (1984).
[CrossRef]

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. B 31, 97 (1983).
[CrossRef]

IEEE J. Quantum Electron. (1)

J. L. Hall, L-S. Ma, G. Kramer, “Principles of Optical Phase Locking: With Application to Internal Mirror He–Ne Lasers Phase Locked via Fast Control of the Discharge Current,” IEEE J. Quantum Electron. (Special issue on Applications of Q.E. to Frequency Standards, Clocks, and Inertial Rotation Sensors) QE-23, 427 (1987).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Acta (1)

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

Opt. Commun. (3)

S. Grafstrom, U. Harbarth, J. Kowalski, R. Neumann, S. Noehte, “Fast Laser Beam Position Control with Sub-Microradian Precision,” Opt. Commun. 65, 121 (1988).
[CrossRef]

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

A. E. Siegman, “Orthogonal Properties of Optical Resonator Eigenmodes,” Opt. Commun. 31, 369–000 (1979).
[CrossRef]

Opt. Lett. (1)

Other (2)

A. E. Siegman, Lasers (University Science Books, Mill Valley, Ca, 1986), pp. 642–652.

W. R. P. Drever, R. Spero, California Institute of Technology; private communication (1984).

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Figures (7)

Fig. 1
Fig. 1

Profile of laser beam injected into an optical resonator for each of the four possible alignment error types. The waist of the input beam is indicated by the arrows and the waist of the cavity by the vertical dotted line. Error types can by classified by the relative sizes and positions of the waists alone: (a) transverse beam displacement; (b) angular displacement of input beam; (c) longitudinal waist displacement; (d) waist size mismatch.

Fig. 2
Fig. 2

Frequency spectra of electric fields and cavity transmission. In normal conditions the input beam carrier frequency is locked to the cavity fundamental mode frequency ν = ν00. The beat of the two transmitted frequencies produces the rf alignment error signals: (a) electric field of phase modulated input beam; (b) transmission of resonator; (c) electric field of transmitted beam.

Fig. 3
Fig. 3

Block diagram of 1-D two degrees-of-freedom alignment system.

Fig. 4
Fig. 4

Block diagram of 2-D detector used to derive the rf error signals for the four-degrees-of-freedom alignment system.

Fig. 5
Fig. 5

Block diagram of mirror control amplifiers.

Fig. 6
Fig. 6

X-axis translational error signal noise measurements from 0.1 Hz to 10 kHz. The uppermost trace indicates the extent of the open loop alignment fluctuations. The next lower trace indicates the noise level caused by both the instrument noise and shot noise that results when white light producing the same photocurrent as the transmitted beam is incident on the detector. This represents the actual performance limit. The closed loop response is given by the lowest curve.

Fig. 7
Fig. 7

X-axis angular error signal noise measurements from 0.1 Hz to 10 kHz.

Tables (1)

Tables Icon

Table I Coupling of Off-Axis Modes Due to Cavity Misalignments

Equations (16)

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Ψ ( x , y ) = m = 0 n = 0 C m n U m n ( x , y ) ,
U m n ( x , y ) = ( 2 π 2 m 2 n m ! n ! ) 1 / 2 H m ( 2 x w 0 x ) × H n ( 2 y w 0 y ) exp ( - x 2 w 0 x - y 2 w 0 y ) .
Ψ ( x , y ) C 00 U 00 ( x , y ) + C 10 U 10 ( x , y ) + C 01 U 01 ( x , y ) + C 20 U 20 ( x , y ) + C 02 U 02 ( x , y ) ,
Δ ν m n ν s π { m arccos [ ( 1 - L 2 f 1 x ) ( 1 - L 2 f 2 x ) ] 1 / 2 + n arccos [ ( 1 - L 2 f 1 y ) ( 1 - L 2 f 2 y ) ] 1 / 2 } ,
w 0 μ 4 = ( λ π ) 2 ( 2 f 1 μ - L ) ( 2 f 2 μ - L ) [ 2 ( f 1 μ + f 2 μ ) - L ] 4 ( f 1 μ + f 2 μ - L ) 2 .
α 0 μ = λ π w 0 μ .
E p m = E o { k = 0 J k ( m ) exp [ i ( ω + k ω m ) t ] + k = 1 ( - 1 ) k J k ( m ) exp [ i ( ω + k ω m ) t ] } ,
E t E 0 t c { C 00 J 0 ( m ) exp ( i ω t U 0 ) ( x ) + C 10 J 1 ( m ) exp [ i ( ω + ω m ) t U 1 ] ( x ) } U 0 ( y ) ,
I t ( x , y ) = E t * · E t = T c E 0 2 { C 00 2 J 0 2 ( m ) U 0 2 ( x ) + C 01 2 J 1 2 ( m ) U 1 2 ( x ) + 2 J 0 ( m ) J 1 ( m ) U 0 ( x ) U 1 ( x ) × [ a x 0 cos ω m t - α α x 0 sin ω m t ] } U 0 2 ( y ) .
i det e η h ν - d y { - I ( x , y ) d x - - o I ( x , y ) d x } ( 2 π ) 1 / 2 ( 2 e η λ h c ) P 0 T c J 0 ( m ) J 1 ( m ) × [ a x 0 cos ω m t - α α x 0 sin ω m t ] ,
i d c η e λ h c T c P 0 J 0 2 ( m ) ,
i s n [ 2 η e 2 λ h c T c P 0 J 0 2 ( m ) Δ f ] 1 / 2 ,
( S N ) max = i ω m 2 i s n 2 η λ π h c P o T c J 1 ( m ) Δ f [ a 2 x 0 2 + α 2 α x 0 2 ] .
( P 10 P 00 ) min a 2 x 0 2 + α 2 α x 0 2 π h c Δ f η λ P 0 T c J 1 2 ( m ) .
( P 10 P 00 ) min m 1 4 π h c Δ f η λ P 0 T c m 2 .
in - phase : a x 0 cos ϕ - α α 0 x sin ϕ , in - quadrature : a x 0 sin ϕ - α α 0 x cos ϕ ,

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