Abstract

We consider the problem of measurement of optical transverse profile parameters and their conjugate variable. Using multimode analysis, we introduce the concept of detection noise modes. For Gaussian beams, displacement and tilt are a pair of transverse-profile conjugate variables. We experimentally demonstrate the optimal encoding and detection of these variables with a spatial homodyning scheme. Using higher-order spatial mode squeezing, we show the sub-shot-noise measurements for the displacement and tilt of a Gaussian beam.

© 2006 Optical Society of America

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  1. W. P. Bowen, R. Schnabel, H.-A. Bachor, and P. K. Lam, Phys. Rev. Lett. 88, 093601 (2002).
    [CrossRef] [PubMed]
  2. N. Korolkova, G. Leuchs, R. Loudon, T. C. Ralph, and C. Silberhorn, Phys. Rev. A 65, 052306 (2002).
    [CrossRef]
  3. A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, Nature 412, 3123 (2001).
    [CrossRef]
  4. N. K. Langford, R. B. Dalton, M. D. Harvey, J. L. O'Brien, G. J. Pryde, A. Gilchrist, S. D. Bartlett, and A. G. White, Phys. Rev. Lett. 93, 053601 (2004).
    [CrossRef] [PubMed]
  5. N. Treps, V. Delaubert, A. Maître, J. M. Courty, and C. Fabre, Phys. Rev. A 71, 013820 (2005).
    [CrossRef]
  6. N. Treps, N. Grosse, W. P. Bowen, M. T. L. Hsu, A. Maître, C. Fabre, H.-A. Bachor, and P. K. Lam, J. Opt. B: Quantum Semiclassical Opt. 6, S664 (2004).
    [CrossRef]
  7. M. T. L. Hsu, W. P. Bowen, N. Treps, and P. K. Lam, Phys. Rev. A 72, 013802 (2005).
    [CrossRef]
  8. M. T. L. Hsu, V. Delaubert, P. K. Lam, and W. P. Bowen, J. Opt. B: Quantum Semiclassical Opt. 6, 495 (2004).
    [CrossRef]
  9. V. Delaubert, D. A. Shaddock, P. K. Lam, B. C. Buchler, H.-A. Bachor, and D. E. McClelland, J. Opt. A, Pure Appl. Opt. 4, 393 (2002).
    [CrossRef]
  10. N. Treps, N. Grosse, W. P. Bowen, A. Maître, C. Fabre, H.-A. Bachor, and P. K. Lam, Science 301, 940 (2003).
    [CrossRef] [PubMed]

2005 (2)

N. Treps, V. Delaubert, A. Maître, J. M. Courty, and C. Fabre, Phys. Rev. A 71, 013820 (2005).
[CrossRef]

M. T. L. Hsu, W. P. Bowen, N. Treps, and P. K. Lam, Phys. Rev. A 72, 013802 (2005).
[CrossRef]

2004 (3)

M. T. L. Hsu, V. Delaubert, P. K. Lam, and W. P. Bowen, J. Opt. B: Quantum Semiclassical Opt. 6, 495 (2004).
[CrossRef]

N. K. Langford, R. B. Dalton, M. D. Harvey, J. L. O'Brien, G. J. Pryde, A. Gilchrist, S. D. Bartlett, and A. G. White, Phys. Rev. Lett. 93, 053601 (2004).
[CrossRef] [PubMed]

N. Treps, N. Grosse, W. P. Bowen, M. T. L. Hsu, A. Maître, C. Fabre, H.-A. Bachor, and P. K. Lam, J. Opt. B: Quantum Semiclassical Opt. 6, S664 (2004).
[CrossRef]

2003 (1)

N. Treps, N. Grosse, W. P. Bowen, A. Maître, C. Fabre, H.-A. Bachor, and P. K. Lam, Science 301, 940 (2003).
[CrossRef] [PubMed]

2002 (3)

V. Delaubert, D. A. Shaddock, P. K. Lam, B. C. Buchler, H.-A. Bachor, and D. E. McClelland, J. Opt. A, Pure Appl. Opt. 4, 393 (2002).
[CrossRef]

W. P. Bowen, R. Schnabel, H.-A. Bachor, and P. K. Lam, Phys. Rev. Lett. 88, 093601 (2002).
[CrossRef] [PubMed]

N. Korolkova, G. Leuchs, R. Loudon, T. C. Ralph, and C. Silberhorn, Phys. Rev. A 65, 052306 (2002).
[CrossRef]

2001 (1)

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, Nature 412, 3123 (2001).
[CrossRef]

Bachor, H.-A.

N. Treps, N. Grosse, W. P. Bowen, M. T. L. Hsu, A. Maître, C. Fabre, H.-A. Bachor, and P. K. Lam, J. Opt. B: Quantum Semiclassical Opt. 6, S664 (2004).
[CrossRef]

N. Treps, N. Grosse, W. P. Bowen, A. Maître, C. Fabre, H.-A. Bachor, and P. K. Lam, Science 301, 940 (2003).
[CrossRef] [PubMed]

V. Delaubert, D. A. Shaddock, P. K. Lam, B. C. Buchler, H.-A. Bachor, and D. E. McClelland, J. Opt. A, Pure Appl. Opt. 4, 393 (2002).
[CrossRef]

W. P. Bowen, R. Schnabel, H.-A. Bachor, and P. K. Lam, Phys. Rev. Lett. 88, 093601 (2002).
[CrossRef] [PubMed]

Bartlett, S. D.

N. K. Langford, R. B. Dalton, M. D. Harvey, J. L. O'Brien, G. J. Pryde, A. Gilchrist, S. D. Bartlett, and A. G. White, Phys. Rev. Lett. 93, 053601 (2004).
[CrossRef] [PubMed]

Bowen, W. P.

M. T. L. Hsu, W. P. Bowen, N. Treps, and P. K. Lam, Phys. Rev. A 72, 013802 (2005).
[CrossRef]

M. T. L. Hsu, V. Delaubert, P. K. Lam, and W. P. Bowen, J. Opt. B: Quantum Semiclassical Opt. 6, 495 (2004).
[CrossRef]

N. Treps, N. Grosse, W. P. Bowen, M. T. L. Hsu, A. Maître, C. Fabre, H.-A. Bachor, and P. K. Lam, J. Opt. B: Quantum Semiclassical Opt. 6, S664 (2004).
[CrossRef]

N. Treps, N. Grosse, W. P. Bowen, A. Maître, C. Fabre, H.-A. Bachor, and P. K. Lam, Science 301, 940 (2003).
[CrossRef] [PubMed]

W. P. Bowen, R. Schnabel, H.-A. Bachor, and P. K. Lam, Phys. Rev. Lett. 88, 093601 (2002).
[CrossRef] [PubMed]

Buchler, B. C.

V. Delaubert, D. A. Shaddock, P. K. Lam, B. C. Buchler, H.-A. Bachor, and D. E. McClelland, J. Opt. A, Pure Appl. Opt. 4, 393 (2002).
[CrossRef]

Courty, J. M.

N. Treps, V. Delaubert, A. Maître, J. M. Courty, and C. Fabre, Phys. Rev. A 71, 013820 (2005).
[CrossRef]

Dalton, R. B.

N. K. Langford, R. B. Dalton, M. D. Harvey, J. L. O'Brien, G. J. Pryde, A. Gilchrist, S. D. Bartlett, and A. G. White, Phys. Rev. Lett. 93, 053601 (2004).
[CrossRef] [PubMed]

Delaubert, V.

N. Treps, V. Delaubert, A. Maître, J. M. Courty, and C. Fabre, Phys. Rev. A 71, 013820 (2005).
[CrossRef]

M. T. L. Hsu, V. Delaubert, P. K. Lam, and W. P. Bowen, J. Opt. B: Quantum Semiclassical Opt. 6, 495 (2004).
[CrossRef]

V. Delaubert, D. A. Shaddock, P. K. Lam, B. C. Buchler, H.-A. Bachor, and D. E. McClelland, J. Opt. A, Pure Appl. Opt. 4, 393 (2002).
[CrossRef]

Fabre, C.

N. Treps, V. Delaubert, A. Maître, J. M. Courty, and C. Fabre, Phys. Rev. A 71, 013820 (2005).
[CrossRef]

N. Treps, N. Grosse, W. P. Bowen, M. T. L. Hsu, A. Maître, C. Fabre, H.-A. Bachor, and P. K. Lam, J. Opt. B: Quantum Semiclassical Opt. 6, S664 (2004).
[CrossRef]

N. Treps, N. Grosse, W. P. Bowen, A. Maître, C. Fabre, H.-A. Bachor, and P. K. Lam, Science 301, 940 (2003).
[CrossRef] [PubMed]

Gilchrist, A.

N. K. Langford, R. B. Dalton, M. D. Harvey, J. L. O'Brien, G. J. Pryde, A. Gilchrist, S. D. Bartlett, and A. G. White, Phys. Rev. Lett. 93, 053601 (2004).
[CrossRef] [PubMed]

Grosse, N.

N. Treps, N. Grosse, W. P. Bowen, M. T. L. Hsu, A. Maître, C. Fabre, H.-A. Bachor, and P. K. Lam, J. Opt. B: Quantum Semiclassical Opt. 6, S664 (2004).
[CrossRef]

N. Treps, N. Grosse, W. P. Bowen, A. Maître, C. Fabre, H.-A. Bachor, and P. K. Lam, Science 301, 940 (2003).
[CrossRef] [PubMed]

Harvey, M. D.

N. K. Langford, R. B. Dalton, M. D. Harvey, J. L. O'Brien, G. J. Pryde, A. Gilchrist, S. D. Bartlett, and A. G. White, Phys. Rev. Lett. 93, 053601 (2004).
[CrossRef] [PubMed]

Hsu, M. T. L.

M. T. L. Hsu, W. P. Bowen, N. Treps, and P. K. Lam, Phys. Rev. A 72, 013802 (2005).
[CrossRef]

N. Treps, N. Grosse, W. P. Bowen, M. T. L. Hsu, A. Maître, C. Fabre, H.-A. Bachor, and P. K. Lam, J. Opt. B: Quantum Semiclassical Opt. 6, S664 (2004).
[CrossRef]

M. T. L. Hsu, V. Delaubert, P. K. Lam, and W. P. Bowen, J. Opt. B: Quantum Semiclassical Opt. 6, 495 (2004).
[CrossRef]

Korolkova, N.

N. Korolkova, G. Leuchs, R. Loudon, T. C. Ralph, and C. Silberhorn, Phys. Rev. A 65, 052306 (2002).
[CrossRef]

Lam, P. K.

M. T. L. Hsu, W. P. Bowen, N. Treps, and P. K. Lam, Phys. Rev. A 72, 013802 (2005).
[CrossRef]

M. T. L. Hsu, V. Delaubert, P. K. Lam, and W. P. Bowen, J. Opt. B: Quantum Semiclassical Opt. 6, 495 (2004).
[CrossRef]

N. Treps, N. Grosse, W. P. Bowen, M. T. L. Hsu, A. Maître, C. Fabre, H.-A. Bachor, and P. K. Lam, J. Opt. B: Quantum Semiclassical Opt. 6, S664 (2004).
[CrossRef]

N. Treps, N. Grosse, W. P. Bowen, A. Maître, C. Fabre, H.-A. Bachor, and P. K. Lam, Science 301, 940 (2003).
[CrossRef] [PubMed]

V. Delaubert, D. A. Shaddock, P. K. Lam, B. C. Buchler, H.-A. Bachor, and D. E. McClelland, J. Opt. A, Pure Appl. Opt. 4, 393 (2002).
[CrossRef]

W. P. Bowen, R. Schnabel, H.-A. Bachor, and P. K. Lam, Phys. Rev. Lett. 88, 093601 (2002).
[CrossRef] [PubMed]

Langford, N. K.

N. K. Langford, R. B. Dalton, M. D. Harvey, J. L. O'Brien, G. J. Pryde, A. Gilchrist, S. D. Bartlett, and A. G. White, Phys. Rev. Lett. 93, 053601 (2004).
[CrossRef] [PubMed]

Leuchs, G.

N. Korolkova, G. Leuchs, R. Loudon, T. C. Ralph, and C. Silberhorn, Phys. Rev. A 65, 052306 (2002).
[CrossRef]

Loudon, R.

N. Korolkova, G. Leuchs, R. Loudon, T. C. Ralph, and C. Silberhorn, Phys. Rev. A 65, 052306 (2002).
[CrossRef]

Mair, A.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, Nature 412, 3123 (2001).
[CrossRef]

Maître, A.

N. Treps, V. Delaubert, A. Maître, J. M. Courty, and C. Fabre, Phys. Rev. A 71, 013820 (2005).
[CrossRef]

N. Treps, N. Grosse, W. P. Bowen, M. T. L. Hsu, A. Maître, C. Fabre, H.-A. Bachor, and P. K. Lam, J. Opt. B: Quantum Semiclassical Opt. 6, S664 (2004).
[CrossRef]

N. Treps, N. Grosse, W. P. Bowen, A. Maître, C. Fabre, H.-A. Bachor, and P. K. Lam, Science 301, 940 (2003).
[CrossRef] [PubMed]

McClelland, D. E.

V. Delaubert, D. A. Shaddock, P. K. Lam, B. C. Buchler, H.-A. Bachor, and D. E. McClelland, J. Opt. A, Pure Appl. Opt. 4, 393 (2002).
[CrossRef]

O'Brien, J. L.

N. K. Langford, R. B. Dalton, M. D. Harvey, J. L. O'Brien, G. J. Pryde, A. Gilchrist, S. D. Bartlett, and A. G. White, Phys. Rev. Lett. 93, 053601 (2004).
[CrossRef] [PubMed]

Pryde, G. J.

N. K. Langford, R. B. Dalton, M. D. Harvey, J. L. O'Brien, G. J. Pryde, A. Gilchrist, S. D. Bartlett, and A. G. White, Phys. Rev. Lett. 93, 053601 (2004).
[CrossRef] [PubMed]

Ralph, T. C.

N. Korolkova, G. Leuchs, R. Loudon, T. C. Ralph, and C. Silberhorn, Phys. Rev. A 65, 052306 (2002).
[CrossRef]

Schnabel, R.

W. P. Bowen, R. Schnabel, H.-A. Bachor, and P. K. Lam, Phys. Rev. Lett. 88, 093601 (2002).
[CrossRef] [PubMed]

Shaddock, D. A.

V. Delaubert, D. A. Shaddock, P. K. Lam, B. C. Buchler, H.-A. Bachor, and D. E. McClelland, J. Opt. A, Pure Appl. Opt. 4, 393 (2002).
[CrossRef]

Silberhorn, C.

N. Korolkova, G. Leuchs, R. Loudon, T. C. Ralph, and C. Silberhorn, Phys. Rev. A 65, 052306 (2002).
[CrossRef]

Treps, N.

N. Treps, V. Delaubert, A. Maître, J. M. Courty, and C. Fabre, Phys. Rev. A 71, 013820 (2005).
[CrossRef]

M. T. L. Hsu, W. P. Bowen, N. Treps, and P. K. Lam, Phys. Rev. A 72, 013802 (2005).
[CrossRef]

N. Treps, N. Grosse, W. P. Bowen, M. T. L. Hsu, A. Maître, C. Fabre, H.-A. Bachor, and P. K. Lam, J. Opt. B: Quantum Semiclassical Opt. 6, S664 (2004).
[CrossRef]

N. Treps, N. Grosse, W. P. Bowen, A. Maître, C. Fabre, H.-A. Bachor, and P. K. Lam, Science 301, 940 (2003).
[CrossRef] [PubMed]

Vaziri, A.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, Nature 412, 3123 (2001).
[CrossRef]

Weihs, G.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, Nature 412, 3123 (2001).
[CrossRef]

White, A. G.

N. K. Langford, R. B. Dalton, M. D. Harvey, J. L. O'Brien, G. J. Pryde, A. Gilchrist, S. D. Bartlett, and A. G. White, Phys. Rev. Lett. 93, 053601 (2004).
[CrossRef] [PubMed]

Zeilinger, A.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, Nature 412, 3123 (2001).
[CrossRef]

J. Opt. A, Pure Appl. Opt. (1)

V. Delaubert, D. A. Shaddock, P. K. Lam, B. C. Buchler, H.-A. Bachor, and D. E. McClelland, J. Opt. A, Pure Appl. Opt. 4, 393 (2002).
[CrossRef]

J. Opt. B: Quantum Semiclassical Opt. (2)

N. Treps, N. Grosse, W. P. Bowen, M. T. L. Hsu, A. Maître, C. Fabre, H.-A. Bachor, and P. K. Lam, J. Opt. B: Quantum Semiclassical Opt. 6, S664 (2004).
[CrossRef]

M. T. L. Hsu, V. Delaubert, P. K. Lam, and W. P. Bowen, J. Opt. B: Quantum Semiclassical Opt. 6, 495 (2004).
[CrossRef]

Nature (1)

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, Nature 412, 3123 (2001).
[CrossRef]

Phys. Rev. A (3)

N. Korolkova, G. Leuchs, R. Loudon, T. C. Ralph, and C. Silberhorn, Phys. Rev. A 65, 052306 (2002).
[CrossRef]

N. Treps, V. Delaubert, A. Maître, J. M. Courty, and C. Fabre, Phys. Rev. A 71, 013820 (2005).
[CrossRef]

M. T. L. Hsu, W. P. Bowen, N. Treps, and P. K. Lam, Phys. Rev. A 72, 013802 (2005).
[CrossRef]

Phys. Rev. Lett. (2)

W. P. Bowen, R. Schnabel, H.-A. Bachor, and P. K. Lam, Phys. Rev. Lett. 88, 093601 (2002).
[CrossRef] [PubMed]

N. K. Langford, R. B. Dalton, M. D. Harvey, J. L. O'Brien, G. J. Pryde, A. Gilchrist, S. D. Bartlett, and A. G. White, Phys. Rev. Lett. 93, 053601 (2004).
[CrossRef] [PubMed]

Science (1)

N. Treps, N. Grosse, W. P. Bowen, A. Maître, C. Fabre, H.-A. Bachor, and P. K. Lam, Science 301, 940 (2003).
[CrossRef] [PubMed]

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Figures (2)

Fig. 1
Fig. 1

Schematic diagram of the experiment for optimal displacement and tilt measurements with a spatial homodyne detector. A TEM 00 mode, which is displaced and tilted using a PZT actuator (i), is mode matched to the TEM 10 local oscillator (ii) of a balanced homodyne detector (iii). The TEM 10 local oscillator selects the quadratures amplitude of the TEM 10 component of the input beam that contains the small displacement and tilt information of the incident TEM 00 beam (iv).

Fig. 2
Fig. 2

Demonstration of sub-shot-noise measurements of (a) displacement and (b) tilt modulations using the spatial homodyne detector. The figures show an example where there was 90% of tilt and only 10% of displacement modulations. The left-hand side of each part shows the scanning of local oscillator phase ϕ LO that successively accesses the pure displacement (at ϕ LO = 0 and π) to pure tilt (at ϕ LO = π 2 and 3 π 2 ) information of the beam. SQZ, quadrature noise of squeezed light with 2 dB of squeezing and 8 dB of antisqueezing on the TEM 10 mode but without any modulation signal. MOD, measured modulation with coherent light. MOD-SQZ, measured modulation with squeezed light. The right-hand side of each figure part shows the locked local oscillator phase corresponding to the (a) displacement or (b) tilt measurement. SQZ, at ϕ LO = 0 is the squeezed noise level 2 dB below the shot noise and at ϕ LO = π 2 is 8 dB of antisqueezing noise. DISP, MOD-SQZ curve locked to ϕ LO = 0 for displacement measurement. TL, MOD-SQZ curve locked to ϕ LO = π 2 for tilt measurement. Displacement measurement is improved by the 2 dB of squeezing, while the tilt measurement is degraded by the 8 dB of antisqueezing.

Equations (3)

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u 00 ( p ) u 00 ( p 0 ) + ( p p 0 ) ( u 00 ) p ,
u 00 ( d ) = u 00 + d ( u 00 ) x = u 00 + d w u 10 ,
u 00 ( θ ) = u 00 + θ ( u 00 ) θ = u 00 + i π θ w λ u 10 ,

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