Abstract

We theoretically investigate the quantum uncertainty in the beam width of transverse optical modes and, for this purpose, define a corresponding quantum operator. Single mode states are studied as well as multimode states with small quantum noise. General relations are derived, and specific examples of different modes and quantum states are examined. For the multimode case, we show that the quantum uncertainty in the beam width can be completely attributed to the amplitude quadrature uncertainty of one specific mode, which is uniquely determined by the field under investigation. This discovery provides us with a strategy for the reduction of the beam width noise by an appropriate choice of the quantum state.

© 2015 Optical Society of America

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References

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  9. P. H. SoutoRibeiro, C. Schwob, A. Maître, and C. Fabre, “Sub-shot-noise high-sensitivity spectroscopy with optical parametric oscillator twin beams,” Opt. Lett. 22, 1893–1895 (1997).
    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref] [PubMed]
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  24. N. Treps, V. Delaubert, A. Maître, J. M. Courty, and C. Fabre, “Quantum noise in multipixel image processing,” Phys. Rev. A 71, 013820 (2005).
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref]

2013 (3)

The LIGO Scientific Collaboration, “Enhanced sensitivity of the LIGO gravitational wave detector by using squeezed states of light,” Nature Photon. 7, 613–619 (2013).
[Crossref]

M. A. Taylor, J. Janousek, V. Daria, J. Kittel, B. Hage, H.-A. Bachor, and W. P. Bowen, “Biological measurement beyond the quantum limit,” Nature Photon. 7, 229–233 (2013).
[Crossref]

R. W. Bowman and M. J. Padgett, “Optical trapping and binding,” Rep. Prog. Phys. 76, 026401 (2013).
[Crossref] [PubMed]

2012 (1)

M. Mazilu, A. Mourka, T. Vettenburg, E. M. Wright, and K. Dholakia, “Simultaneous determination of the constituent azimuthal and radial mode indices for light fields possessing orbital angular momentum,” Appl. Phys. Lett. 100, 231115 (2012).
[Crossref]

2011 (2)

H. Shin, K. W. C. Chan, H. J. Chang, and R. W. Boyd, “Quantum spatial superresolution by optical centroid measurements,” Phys. Rev. Lett. 107, 083603 (2011).
[Crossref] [PubMed]

C. Gabriel, A. Aiello, W. Zhong, T. G. Euser, N. Y. Joly, P. Banzer, M. Förtsch, D. Elser, U. L. Andersen, Ch. Marquardt, P. St. J. Russell, and G. Leuchs, “Entangling different degrees of freedom by quadrature squeezing cylindrically polarized modes,” Phys. Rev. Lett. 106, 060502 (2011).
[Crossref] [PubMed]

2007 (1)

J. Alda, “Laser and Gaussian beam propagation and transformation,” Encyclopedia on Optical Engineering 98, 999–1013 (2007).

2005 (1)

N. Treps, V. Delaubert, A. Maître, J. M. Courty, and C. Fabre, “Quantum noise in multipixel image processing,” Phys. Rev. A 71, 013820 (2005).
[Crossref]

2004 (1)

S. Franke-Arnold, S. M. Barnett, E. Yao, J. Leach, J. Courtial, and M. Padgett, “Uncertainty principle for angular position and angular momentum,” New J. Phys. 6, 103 (2004).
[Crossref]

2003 (2)

N. Treps, N. Grosse, W. P. Bowen, C. Fabre, H.-A. Bachor, and P. K. Lam, “A quantum laser pointer,” Science 301, 940–943 (2003).
[Crossref] [PubMed]

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901 (2003).
[Crossref] [PubMed]

2002 (1)

N. Treps, U. Andersen, B. Buchler, P. K. Lam, A. Maître, H.-A. Bachor, and C. Fabre, “Surpassing the standard quantum limit for optical imaging using nonclassical multimode light,” Phys. Rev. Lett. 88, 203601 (2002).
[Crossref] [PubMed]

2000 (1)

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Optics Commun. 179, 1–7 (2000).
[Crossref]

1999 (2)

M. Meron, P. J. Viccaro, and B. Lin, “Geometrical and wave optics of paraxial beams,” Phys. Rev. E 59, 7152–7165 (1999).
[Crossref]

M. I. Kolobov, “The spatial behavior of nonclassical light,” Rev. Mod. Phys. 71, 1539–1589 (1999).
[Crossref]

1998 (2)

S. R. J. Brueck, S. H. Zaidi, X. Chen, and Z. Zhang, “Interferometric lithography - from periodic arrays to arbitrary patterns,” Microelectro. Eng. 41-42, 145–148 (1998).
[Crossref]

J. Gao, F. Cui, C. Xue, C. Xie, and P. Kunchi, “Generation and application of twin beams from an optical parametric oscillator including an α-cut KTP crystal,” Opt. Lett. 23, 870–872 (1998).
[Crossref]

1997 (2)

P. H. SoutoRibeiro, C. Schwob, A. Maître, and C. Fabre, “Sub-shot-noise high-sensitivity spectroscopy with optical parametric oscillator twin beams,” Opt. Lett. 22, 1893–1895 (1997).
[Crossref]

F. Marin, A. Bramati, V. Jost, and E. Giacobino, “Demonstration of high sensitivity spectroscopy with squeezed semiconductor lasers,” Optics Commun. 140, 146–157 (1997).
[Crossref]

1996 (2)

C. H. Henry and R. F. Kazarinov, “Quantum noise in photonics,” Rev. Mod. Phys. 68, 801–853 (1996).
[Crossref]

L. Davidovich, “Sub-Poissonian processes in quantum optics,” Rev. Mod. Phys. 68, 127–173 (1996).
[Crossref]

1994 (1)

F. Gori, “Flattened gaussian beams,” Opt. Commun. 107, 335–341 (1994).
[Crossref]

1992 (1)

E. Polzik, J. Carri, and H. J. Kimble, “Atomic spectroscopy with squeezed light for sensitivity beyond the vacuum-state limit,” Appl. Phys. B 55, 279–290 (1992).
[Crossref]

1987 (2)

M. Xiao, L.-A. Wu, and H. J. Kimble, “Precision measurement beyond the shot-noise limit,” Phys. Rev. Lett. 59, 278–281 (1987).
[Crossref] [PubMed]

P. Grangier, R. E. Slusher, B. Yurke, and A. LaPorta, “Squeezed-lightenhanced polarization interferometer,” Phys. Rev. Lett. 59, 2153–2156 (1987).
[Crossref] [PubMed]

1979 (1)

Aiello, A.

C. Gabriel, A. Aiello, W. Zhong, T. G. Euser, N. Y. Joly, P. Banzer, M. Förtsch, D. Elser, U. L. Andersen, Ch. Marquardt, P. St. J. Russell, and G. Leuchs, “Entangling different degrees of freedom by quadrature squeezing cylindrically polarized modes,” Phys. Rev. Lett. 106, 060502 (2011).
[Crossref] [PubMed]

Alda, J.

J. Alda, “Laser and Gaussian beam propagation and transformation,” Encyclopedia on Optical Engineering 98, 999–1013 (2007).

Andersen, U.

N. Treps, U. Andersen, B. Buchler, P. K. Lam, A. Maître, H.-A. Bachor, and C. Fabre, “Surpassing the standard quantum limit for optical imaging using nonclassical multimode light,” Phys. Rev. Lett. 88, 203601 (2002).
[Crossref] [PubMed]

Andersen, U. L.

C. Gabriel, A. Aiello, W. Zhong, T. G. Euser, N. Y. Joly, P. Banzer, M. Förtsch, D. Elser, U. L. Andersen, Ch. Marquardt, P. St. J. Russell, and G. Leuchs, “Entangling different degrees of freedom by quadrature squeezing cylindrically polarized modes,” Phys. Rev. Lett. 106, 060502 (2011).
[Crossref] [PubMed]

Aspect, A.

G. Grynberg, A. Aspect, and C. Fabre, Introduction to Quantum Optics: From the Semi-classical Approach to Quantized Light (Cambridge Press, 2010), for the linearization of the annihilation operator see p. 366, for the basis change see p. 335.
[Crossref]

Bachor, H.-A.

M. A. Taylor, J. Janousek, V. Daria, J. Kittel, B. Hage, H.-A. Bachor, and W. P. Bowen, “Biological measurement beyond the quantum limit,” Nature Photon. 7, 229–233 (2013).
[Crossref]

N. Treps, N. Grosse, W. P. Bowen, C. Fabre, H.-A. Bachor, and P. K. Lam, “A quantum laser pointer,” Science 301, 940–943 (2003).
[Crossref] [PubMed]

N. Treps, U. Andersen, B. Buchler, P. K. Lam, A. Maître, H.-A. Bachor, and C. Fabre, “Surpassing the standard quantum limit for optical imaging using nonclassical multimode light,” Phys. Rev. Lett. 88, 203601 (2002).
[Crossref] [PubMed]

H.-A. Bachor and T. C. Ralph, A Guide to Experiments in Quantum Optics (Wiley-VCH, 2004), for the linearization of the annihilation operator see p. 84/85.

Banzer, P.

C. Gabriel, A. Aiello, W. Zhong, T. G. Euser, N. Y. Joly, P. Banzer, M. Förtsch, D. Elser, U. L. Andersen, Ch. Marquardt, P. St. J. Russell, and G. Leuchs, “Entangling different degrees of freedom by quadrature squeezing cylindrically polarized modes,” Phys. Rev. Lett. 106, 060502 (2011).
[Crossref] [PubMed]

Barnett, S. M.

S. Franke-Arnold, S. M. Barnett, E. Yao, J. Leach, J. Courtial, and M. Padgett, “Uncertainty principle for angular position and angular momentum,” New J. Phys. 6, 103 (2004).
[Crossref]

S. M. Barnett and P. M. Radmore, Methods in Theoretical Quantum Optics (Oxford Science Publications, 1997).

Bowen, W. P.

M. A. Taylor, J. Janousek, V. Daria, J. Kittel, B. Hage, H.-A. Bachor, and W. P. Bowen, “Biological measurement beyond the quantum limit,” Nature Photon. 7, 229–233 (2013).
[Crossref]

N. Treps, N. Grosse, W. P. Bowen, C. Fabre, H.-A. Bachor, and P. K. Lam, “A quantum laser pointer,” Science 301, 940–943 (2003).
[Crossref] [PubMed]

Bowman, R. W.

R. W. Bowman and M. J. Padgett, “Optical trapping and binding,” Rep. Prog. Phys. 76, 026401 (2013).
[Crossref] [PubMed]

Boyd, R. W.

H. Shin, K. W. C. Chan, H. J. Chang, and R. W. Boyd, “Quantum spatial superresolution by optical centroid measurements,” Phys. Rev. Lett. 107, 083603 (2011).
[Crossref] [PubMed]

Bramati, A.

F. Marin, A. Bramati, V. Jost, and E. Giacobino, “Demonstration of high sensitivity spectroscopy with squeezed semiconductor lasers,” Optics Commun. 140, 146–157 (1997).
[Crossref]

Brueck, S. R. J.

S. R. J. Brueck, S. H. Zaidi, X. Chen, and Z. Zhang, “Interferometric lithography - from periodic arrays to arbitrary patterns,” Microelectro. Eng. 41-42, 145–148 (1998).
[Crossref]

Buchler, B.

N. Treps, U. Andersen, B. Buchler, P. K. Lam, A. Maître, H.-A. Bachor, and C. Fabre, “Surpassing the standard quantum limit for optical imaging using nonclassical multimode light,” Phys. Rev. Lett. 88, 203601 (2002).
[Crossref] [PubMed]

Carri, J.

E. Polzik, J. Carri, and H. J. Kimble, “Atomic spectroscopy with squeezed light for sensitivity beyond the vacuum-state limit,” Appl. Phys. B 55, 279–290 (1992).
[Crossref]

Chan, K. W. C.

H. Shin, K. W. C. Chan, H. J. Chang, and R. W. Boyd, “Quantum spatial superresolution by optical centroid measurements,” Phys. Rev. Lett. 107, 083603 (2011).
[Crossref] [PubMed]

Chang, H. J.

H. Shin, K. W. C. Chan, H. J. Chang, and R. W. Boyd, “Quantum spatial superresolution by optical centroid measurements,” Phys. Rev. Lett. 107, 083603 (2011).
[Crossref] [PubMed]

Chen, X.

S. R. J. Brueck, S. H. Zaidi, X. Chen, and Z. Zhang, “Interferometric lithography - from periodic arrays to arbitrary patterns,” Microelectro. Eng. 41-42, 145–148 (1998).
[Crossref]

Courtial, J.

S. Franke-Arnold, S. M. Barnett, E. Yao, J. Leach, J. Courtial, and M. Padgett, “Uncertainty principle for angular position and angular momentum,” New J. Phys. 6, 103 (2004).
[Crossref]

Courty, J. M.

N. Treps, V. Delaubert, A. Maître, J. M. Courty, and C. Fabre, “Quantum noise in multipixel image processing,” Phys. Rev. A 71, 013820 (2005).
[Crossref]

Cui, F.

Daria, V.

M. A. Taylor, J. Janousek, V. Daria, J. Kittel, B. Hage, H.-A. Bachor, and W. P. Bowen, “Biological measurement beyond the quantum limit,” Nature Photon. 7, 229–233 (2013).
[Crossref]

Davidovich, L.

L. Davidovich, “Sub-Poissonian processes in quantum optics,” Rev. Mod. Phys. 68, 127–173 (1996).
[Crossref]

Delaubert, V.

N. Treps, V. Delaubert, A. Maître, J. M. Courty, and C. Fabre, “Quantum noise in multipixel image processing,” Phys. Rev. A 71, 013820 (2005).
[Crossref]

Dholakia, K.

M. Mazilu, A. Mourka, T. Vettenburg, E. M. Wright, and K. Dholakia, “Simultaneous determination of the constituent azimuthal and radial mode indices for light fields possessing orbital angular momentum,” Appl. Phys. Lett. 100, 231115 (2012).
[Crossref]

Dorn, R.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901 (2003).
[Crossref] [PubMed]

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Optics Commun. 179, 1–7 (2000).
[Crossref]

Eberler, M.

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Optics Commun. 179, 1–7 (2000).
[Crossref]

Elser, D.

C. Gabriel, A. Aiello, W. Zhong, T. G. Euser, N. Y. Joly, P. Banzer, M. Förtsch, D. Elser, U. L. Andersen, Ch. Marquardt, P. St. J. Russell, and G. Leuchs, “Entangling different degrees of freedom by quadrature squeezing cylindrically polarized modes,” Phys. Rev. Lett. 106, 060502 (2011).
[Crossref] [PubMed]

Euser, T. G.

C. Gabriel, A. Aiello, W. Zhong, T. G. Euser, N. Y. Joly, P. Banzer, M. Förtsch, D. Elser, U. L. Andersen, Ch. Marquardt, P. St. J. Russell, and G. Leuchs, “Entangling different degrees of freedom by quadrature squeezing cylindrically polarized modes,” Phys. Rev. Lett. 106, 060502 (2011).
[Crossref] [PubMed]

Fabre, C.

N. Treps, V. Delaubert, A. Maître, J. M. Courty, and C. Fabre, “Quantum noise in multipixel image processing,” Phys. Rev. A 71, 013820 (2005).
[Crossref]

N. Treps, N. Grosse, W. P. Bowen, C. Fabre, H.-A. Bachor, and P. K. Lam, “A quantum laser pointer,” Science 301, 940–943 (2003).
[Crossref] [PubMed]

N. Treps, U. Andersen, B. Buchler, P. K. Lam, A. Maître, H.-A. Bachor, and C. Fabre, “Surpassing the standard quantum limit for optical imaging using nonclassical multimode light,” Phys. Rev. Lett. 88, 203601 (2002).
[Crossref] [PubMed]

P. H. SoutoRibeiro, C. Schwob, A. Maître, and C. Fabre, “Sub-shot-noise high-sensitivity spectroscopy with optical parametric oscillator twin beams,” Opt. Lett. 22, 1893–1895 (1997).
[Crossref]

G. Grynberg, A. Aspect, and C. Fabre, Introduction to Quantum Optics: From the Semi-classical Approach to Quantized Light (Cambridge Press, 2010), for the linearization of the annihilation operator see p. 366, for the basis change see p. 335.
[Crossref]

Förtsch, M.

C. Gabriel, A. Aiello, W. Zhong, T. G. Euser, N. Y. Joly, P. Banzer, M. Förtsch, D. Elser, U. L. Andersen, Ch. Marquardt, P. St. J. Russell, and G. Leuchs, “Entangling different degrees of freedom by quadrature squeezing cylindrically polarized modes,” Phys. Rev. Lett. 106, 060502 (2011).
[Crossref] [PubMed]

Franke-Arnold, S.

S. Franke-Arnold, S. M. Barnett, E. Yao, J. Leach, J. Courtial, and M. Padgett, “Uncertainty principle for angular position and angular momentum,” New J. Phys. 6, 103 (2004).
[Crossref]

Gabriel, C.

C. Gabriel, A. Aiello, W. Zhong, T. G. Euser, N. Y. Joly, P. Banzer, M. Förtsch, D. Elser, U. L. Andersen, Ch. Marquardt, P. St. J. Russell, and G. Leuchs, “Entangling different degrees of freedom by quadrature squeezing cylindrically polarized modes,” Phys. Rev. Lett. 106, 060502 (2011).
[Crossref] [PubMed]

Gao, J.

Giacobino, E.

F. Marin, A. Bramati, V. Jost, and E. Giacobino, “Demonstration of high sensitivity spectroscopy with squeezed semiconductor lasers,” Optics Commun. 140, 146–157 (1997).
[Crossref]

Glöckl, O.

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Optics Commun. 179, 1–7 (2000).
[Crossref]

Gori, F.

F. Gori, “Flattened gaussian beams,” Opt. Commun. 107, 335–341 (1994).
[Crossref]

Grangier, P.

P. Grangier, R. E. Slusher, B. Yurke, and A. LaPorta, “Squeezed-lightenhanced polarization interferometer,” Phys. Rev. Lett. 59, 2153–2156 (1987).
[Crossref] [PubMed]

Grosse, N.

N. Treps, N. Grosse, W. P. Bowen, C. Fabre, H.-A. Bachor, and P. K. Lam, “A quantum laser pointer,” Science 301, 940–943 (2003).
[Crossref] [PubMed]

Grynberg, G.

G. Grynberg, A. Aspect, and C. Fabre, Introduction to Quantum Optics: From the Semi-classical Approach to Quantized Light (Cambridge Press, 2010), for the linearization of the annihilation operator see p. 366, for the basis change see p. 335.
[Crossref]

Hage, B.

M. A. Taylor, J. Janousek, V. Daria, J. Kittel, B. Hage, H.-A. Bachor, and W. P. Bowen, “Biological measurement beyond the quantum limit,” Nature Photon. 7, 229–233 (2013).
[Crossref]

Henry, C. H.

C. H. Henry and R. F. Kazarinov, “Quantum noise in photonics,” Rev. Mod. Phys. 68, 801–853 (1996).
[Crossref]

Janousek, J.

M. A. Taylor, J. Janousek, V. Daria, J. Kittel, B. Hage, H.-A. Bachor, and W. P. Bowen, “Biological measurement beyond the quantum limit,” Nature Photon. 7, 229–233 (2013).
[Crossref]

Joly, N. Y.

C. Gabriel, A. Aiello, W. Zhong, T. G. Euser, N. Y. Joly, P. Banzer, M. Förtsch, D. Elser, U. L. Andersen, Ch. Marquardt, P. St. J. Russell, and G. Leuchs, “Entangling different degrees of freedom by quadrature squeezing cylindrically polarized modes,” Phys. Rev. Lett. 106, 060502 (2011).
[Crossref] [PubMed]

Jost, V.

F. Marin, A. Bramati, V. Jost, and E. Giacobino, “Demonstration of high sensitivity spectroscopy with squeezed semiconductor lasers,” Optics Commun. 140, 146–157 (1997).
[Crossref]

Kazarinov, R. F.

C. H. Henry and R. F. Kazarinov, “Quantum noise in photonics,” Rev. Mod. Phys. 68, 801–853 (1996).
[Crossref]

Kimble, H. J.

E. Polzik, J. Carri, and H. J. Kimble, “Atomic spectroscopy with squeezed light for sensitivity beyond the vacuum-state limit,” Appl. Phys. B 55, 279–290 (1992).
[Crossref]

M. Xiao, L.-A. Wu, and H. J. Kimble, “Precision measurement beyond the shot-noise limit,” Phys. Rev. Lett. 59, 278–281 (1987).
[Crossref] [PubMed]

Kittel, J.

M. A. Taylor, J. Janousek, V. Daria, J. Kittel, B. Hage, H.-A. Bachor, and W. P. Bowen, “Biological measurement beyond the quantum limit,” Nature Photon. 7, 229–233 (2013).
[Crossref]

Kolobov, M. I.

M. I. Kolobov, “The spatial behavior of nonclassical light,” Rev. Mod. Phys. 71, 1539–1589 (1999).
[Crossref]

Kunchi, P.

Lam, P. K.

N. Treps, N. Grosse, W. P. Bowen, C. Fabre, H.-A. Bachor, and P. K. Lam, “A quantum laser pointer,” Science 301, 940–943 (2003).
[Crossref] [PubMed]

N. Treps, U. Andersen, B. Buchler, P. K. Lam, A. Maître, H.-A. Bachor, and C. Fabre, “Surpassing the standard quantum limit for optical imaging using nonclassical multimode light,” Phys. Rev. Lett. 88, 203601 (2002).
[Crossref] [PubMed]

LaPorta, A.

P. Grangier, R. E. Slusher, B. Yurke, and A. LaPorta, “Squeezed-lightenhanced polarization interferometer,” Phys. Rev. Lett. 59, 2153–2156 (1987).
[Crossref] [PubMed]

Leach, J.

S. Franke-Arnold, S. M. Barnett, E. Yao, J. Leach, J. Courtial, and M. Padgett, “Uncertainty principle for angular position and angular momentum,” New J. Phys. 6, 103 (2004).
[Crossref]

Leuchs, G.

C. Gabriel, A. Aiello, W. Zhong, T. G. Euser, N. Y. Joly, P. Banzer, M. Förtsch, D. Elser, U. L. Andersen, Ch. Marquardt, P. St. J. Russell, and G. Leuchs, “Entangling different degrees of freedom by quadrature squeezing cylindrically polarized modes,” Phys. Rev. Lett. 106, 060502 (2011).
[Crossref] [PubMed]

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901 (2003).
[Crossref] [PubMed]

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Optics Commun. 179, 1–7 (2000).
[Crossref]

Lin, B.

M. Meron, P. J. Viccaro, and B. Lin, “Geometrical and wave optics of paraxial beams,” Phys. Rev. E 59, 7152–7165 (1999).
[Crossref]

Loudon, R.

R. Loudon, The Quantum Theory of Light (Oxford Science Publications, 2000).

Maître, A.

N. Treps, V. Delaubert, A. Maître, J. M. Courty, and C. Fabre, “Quantum noise in multipixel image processing,” Phys. Rev. A 71, 013820 (2005).
[Crossref]

N. Treps, U. Andersen, B. Buchler, P. K. Lam, A. Maître, H.-A. Bachor, and C. Fabre, “Surpassing the standard quantum limit for optical imaging using nonclassical multimode light,” Phys. Rev. Lett. 88, 203601 (2002).
[Crossref] [PubMed]

P. H. SoutoRibeiro, C. Schwob, A. Maître, and C. Fabre, “Sub-shot-noise high-sensitivity spectroscopy with optical parametric oscillator twin beams,” Opt. Lett. 22, 1893–1895 (1997).
[Crossref]

Mandel, L.

Marin, F.

F. Marin, A. Bramati, V. Jost, and E. Giacobino, “Demonstration of high sensitivity spectroscopy with squeezed semiconductor lasers,” Optics Commun. 140, 146–157 (1997).
[Crossref]

Marquardt, Ch.

C. Gabriel, A. Aiello, W. Zhong, T. G. Euser, N. Y. Joly, P. Banzer, M. Förtsch, D. Elser, U. L. Andersen, Ch. Marquardt, P. St. J. Russell, and G. Leuchs, “Entangling different degrees of freedom by quadrature squeezing cylindrically polarized modes,” Phys. Rev. Lett. 106, 060502 (2011).
[Crossref] [PubMed]

Mazilu, M.

M. Mazilu, A. Mourka, T. Vettenburg, E. M. Wright, and K. Dholakia, “Simultaneous determination of the constituent azimuthal and radial mode indices for light fields possessing orbital angular momentum,” Appl. Phys. Lett. 100, 231115 (2012).
[Crossref]

Meron, M.

M. Meron, P. J. Viccaro, and B. Lin, “Geometrical and wave optics of paraxial beams,” Phys. Rev. E 59, 7152–7165 (1999).
[Crossref]

Mourka, A.

M. Mazilu, A. Mourka, T. Vettenburg, E. M. Wright, and K. Dholakia, “Simultaneous determination of the constituent azimuthal and radial mode indices for light fields possessing orbital angular momentum,” Appl. Phys. Lett. 100, 231115 (2012).
[Crossref]

Padgett, M.

S. Franke-Arnold, S. M. Barnett, E. Yao, J. Leach, J. Courtial, and M. Padgett, “Uncertainty principle for angular position and angular momentum,” New J. Phys. 6, 103 (2004).
[Crossref]

Padgett, M. J.

R. W. Bowman and M. J. Padgett, “Optical trapping and binding,” Rep. Prog. Phys. 76, 026401 (2013).
[Crossref] [PubMed]

Polzik, E.

E. Polzik, J. Carri, and H. J. Kimble, “Atomic spectroscopy with squeezed light for sensitivity beyond the vacuum-state limit,” Appl. Phys. B 55, 279–290 (1992).
[Crossref]

Quabis, S.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901 (2003).
[Crossref] [PubMed]

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Optics Commun. 179, 1–7 (2000).
[Crossref]

Radmore, P. M.

S. M. Barnett and P. M. Radmore, Methods in Theoretical Quantum Optics (Oxford Science Publications, 1997).

Ralph, T. C.

H.-A. Bachor and T. C. Ralph, A Guide to Experiments in Quantum Optics (Wiley-VCH, 2004), for the linearization of the annihilation operator see p. 84/85.

Russell, P. St. J.

C. Gabriel, A. Aiello, W. Zhong, T. G. Euser, N. Y. Joly, P. Banzer, M. Förtsch, D. Elser, U. L. Andersen, Ch. Marquardt, P. St. J. Russell, and G. Leuchs, “Entangling different degrees of freedom by quadrature squeezing cylindrically polarized modes,” Phys. Rev. Lett. 106, 060502 (2011).
[Crossref] [PubMed]

Schwob, C.

Shin, H.

H. Shin, K. W. C. Chan, H. J. Chang, and R. W. Boyd, “Quantum spatial superresolution by optical centroid measurements,” Phys. Rev. Lett. 107, 083603 (2011).
[Crossref] [PubMed]

Siegman, A. E.

A. E. Siegman, “How to (maybe) measure laser beam quality,” OSA Annual Meeting (1997).

Slusher, R. E.

P. Grangier, R. E. Slusher, B. Yurke, and A. LaPorta, “Squeezed-lightenhanced polarization interferometer,” Phys. Rev. Lett. 59, 2153–2156 (1987).
[Crossref] [PubMed]

SoutoRibeiro, P. H.

Taylor, M. A.

M. A. Taylor, J. Janousek, V. Daria, J. Kittel, B. Hage, H.-A. Bachor, and W. P. Bowen, “Biological measurement beyond the quantum limit,” Nature Photon. 7, 229–233 (2013).
[Crossref]

Treps, N.

N. Treps, V. Delaubert, A. Maître, J. M. Courty, and C. Fabre, “Quantum noise in multipixel image processing,” Phys. Rev. A 71, 013820 (2005).
[Crossref]

N. Treps, N. Grosse, W. P. Bowen, C. Fabre, H.-A. Bachor, and P. K. Lam, “A quantum laser pointer,” Science 301, 940–943 (2003).
[Crossref] [PubMed]

N. Treps, U. Andersen, B. Buchler, P. K. Lam, A. Maître, H.-A. Bachor, and C. Fabre, “Surpassing the standard quantum limit for optical imaging using nonclassical multimode light,” Phys. Rev. Lett. 88, 203601 (2002).
[Crossref] [PubMed]

Vettenburg, T.

M. Mazilu, A. Mourka, T. Vettenburg, E. M. Wright, and K. Dholakia, “Simultaneous determination of the constituent azimuthal and radial mode indices for light fields possessing orbital angular momentum,” Appl. Phys. Lett. 100, 231115 (2012).
[Crossref]

Viccaro, P. J.

M. Meron, P. J. Viccaro, and B. Lin, “Geometrical and wave optics of paraxial beams,” Phys. Rev. E 59, 7152–7165 (1999).
[Crossref]

Vogel, W.

W. Vogel and D.-G. Welsch, Quantum Optics (Wiley-VCH, 2006).
[Crossref]

Welsch, D.-G.

W. Vogel and D.-G. Welsch, Quantum Optics (Wiley-VCH, 2006).
[Crossref]

Wright, E. M.

M. Mazilu, A. Mourka, T. Vettenburg, E. M. Wright, and K. Dholakia, “Simultaneous determination of the constituent azimuthal and radial mode indices for light fields possessing orbital angular momentum,” Appl. Phys. Lett. 100, 231115 (2012).
[Crossref]

Wu, L.-A.

M. Xiao, L.-A. Wu, and H. J. Kimble, “Precision measurement beyond the shot-noise limit,” Phys. Rev. Lett. 59, 278–281 (1987).
[Crossref] [PubMed]

Xiao, M.

M. Xiao, L.-A. Wu, and H. J. Kimble, “Precision measurement beyond the shot-noise limit,” Phys. Rev. Lett. 59, 278–281 (1987).
[Crossref] [PubMed]

Xie, C.

Xue, C.

Yao, E.

S. Franke-Arnold, S. M. Barnett, E. Yao, J. Leach, J. Courtial, and M. Padgett, “Uncertainty principle for angular position and angular momentum,” New J. Phys. 6, 103 (2004).
[Crossref]

Yurke, B.

P. Grangier, R. E. Slusher, B. Yurke, and A. LaPorta, “Squeezed-lightenhanced polarization interferometer,” Phys. Rev. Lett. 59, 2153–2156 (1987).
[Crossref] [PubMed]

Zaidi, S. H.

S. R. J. Brueck, S. H. Zaidi, X. Chen, and Z. Zhang, “Interferometric lithography - from periodic arrays to arbitrary patterns,” Microelectro. Eng. 41-42, 145–148 (1998).
[Crossref]

Zhang, Z.

S. R. J. Brueck, S. H. Zaidi, X. Chen, and Z. Zhang, “Interferometric lithography - from periodic arrays to arbitrary patterns,” Microelectro. Eng. 41-42, 145–148 (1998).
[Crossref]

Zhong, W.

C. Gabriel, A. Aiello, W. Zhong, T. G. Euser, N. Y. Joly, P. Banzer, M. Förtsch, D. Elser, U. L. Andersen, Ch. Marquardt, P. St. J. Russell, and G. Leuchs, “Entangling different degrees of freedom by quadrature squeezing cylindrically polarized modes,” Phys. Rev. Lett. 106, 060502 (2011).
[Crossref] [PubMed]

Appl. Phys. B (1)

E. Polzik, J. Carri, and H. J. Kimble, “Atomic spectroscopy with squeezed light for sensitivity beyond the vacuum-state limit,” Appl. Phys. B 55, 279–290 (1992).
[Crossref]

Appl. Phys. Lett. (1)

M. Mazilu, A. Mourka, T. Vettenburg, E. M. Wright, and K. Dholakia, “Simultaneous determination of the constituent azimuthal and radial mode indices for light fields possessing orbital angular momentum,” Appl. Phys. Lett. 100, 231115 (2012).
[Crossref]

Encyclopedia on Optical Engineering (1)

J. Alda, “Laser and Gaussian beam propagation and transformation,” Encyclopedia on Optical Engineering 98, 999–1013 (2007).

Microelectro. Eng. (1)

S. R. J. Brueck, S. H. Zaidi, X. Chen, and Z. Zhang, “Interferometric lithography - from periodic arrays to arbitrary patterns,” Microelectro. Eng. 41-42, 145–148 (1998).
[Crossref]

Nature Photon. (2)

M. A. Taylor, J. Janousek, V. Daria, J. Kittel, B. Hage, H.-A. Bachor, and W. P. Bowen, “Biological measurement beyond the quantum limit,” Nature Photon. 7, 229–233 (2013).
[Crossref]

The LIGO Scientific Collaboration, “Enhanced sensitivity of the LIGO gravitational wave detector by using squeezed states of light,” Nature Photon. 7, 613–619 (2013).
[Crossref]

New J. Phys. (1)

S. Franke-Arnold, S. M. Barnett, E. Yao, J. Leach, J. Courtial, and M. Padgett, “Uncertainty principle for angular position and angular momentum,” New J. Phys. 6, 103 (2004).
[Crossref]

Opt. Commun. (1)

F. Gori, “Flattened gaussian beams,” Opt. Commun. 107, 335–341 (1994).
[Crossref]

Opt. Lett. (3)

Optics Commun. (2)

F. Marin, A. Bramati, V. Jost, and E. Giacobino, “Demonstration of high sensitivity spectroscopy with squeezed semiconductor lasers,” Optics Commun. 140, 146–157 (1997).
[Crossref]

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Optics Commun. 179, 1–7 (2000).
[Crossref]

Phys. Rev. A (1)

N. Treps, V. Delaubert, A. Maître, J. M. Courty, and C. Fabre, “Quantum noise in multipixel image processing,” Phys. Rev. A 71, 013820 (2005).
[Crossref]

Phys. Rev. E (1)

M. Meron, P. J. Viccaro, and B. Lin, “Geometrical and wave optics of paraxial beams,” Phys. Rev. E 59, 7152–7165 (1999).
[Crossref]

Phys. Rev. Lett. (6)

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901 (2003).
[Crossref] [PubMed]

C. Gabriel, A. Aiello, W. Zhong, T. G. Euser, N. Y. Joly, P. Banzer, M. Förtsch, D. Elser, U. L. Andersen, Ch. Marquardt, P. St. J. Russell, and G. Leuchs, “Entangling different degrees of freedom by quadrature squeezing cylindrically polarized modes,” Phys. Rev. Lett. 106, 060502 (2011).
[Crossref] [PubMed]

M. Xiao, L.-A. Wu, and H. J. Kimble, “Precision measurement beyond the shot-noise limit,” Phys. Rev. Lett. 59, 278–281 (1987).
[Crossref] [PubMed]

P. Grangier, R. E. Slusher, B. Yurke, and A. LaPorta, “Squeezed-lightenhanced polarization interferometer,” Phys. Rev. Lett. 59, 2153–2156 (1987).
[Crossref] [PubMed]

H. Shin, K. W. C. Chan, H. J. Chang, and R. W. Boyd, “Quantum spatial superresolution by optical centroid measurements,” Phys. Rev. Lett. 107, 083603 (2011).
[Crossref] [PubMed]

N. Treps, U. Andersen, B. Buchler, P. K. Lam, A. Maître, H.-A. Bachor, and C. Fabre, “Surpassing the standard quantum limit for optical imaging using nonclassical multimode light,” Phys. Rev. Lett. 88, 203601 (2002).
[Crossref] [PubMed]

Rep. Prog. Phys. (1)

R. W. Bowman and M. J. Padgett, “Optical trapping and binding,” Rep. Prog. Phys. 76, 026401 (2013).
[Crossref] [PubMed]

Rev. Mod. Phys. (3)

M. I. Kolobov, “The spatial behavior of nonclassical light,” Rev. Mod. Phys. 71, 1539–1589 (1999).
[Crossref]

C. H. Henry and R. F. Kazarinov, “Quantum noise in photonics,” Rev. Mod. Phys. 68, 801–853 (1996).
[Crossref]

L. Davidovich, “Sub-Poissonian processes in quantum optics,” Rev. Mod. Phys. 68, 127–173 (1996).
[Crossref]

Science (1)

N. Treps, N. Grosse, W. P. Bowen, C. Fabre, H.-A. Bachor, and P. K. Lam, “A quantum laser pointer,” Science 301, 940–943 (2003).
[Crossref] [PubMed]

Other (6)

H.-A. Bachor and T. C. Ralph, A Guide to Experiments in Quantum Optics (Wiley-VCH, 2004), for the linearization of the annihilation operator see p. 84/85.

A. E. Siegman, “How to (maybe) measure laser beam quality,” OSA Annual Meeting (1997).

G. Grynberg, A. Aspect, and C. Fabre, Introduction to Quantum Optics: From the Semi-classical Approach to Quantized Light (Cambridge Press, 2010), for the linearization of the annihilation operator see p. 366, for the basis change see p. 335.
[Crossref]

R. Loudon, The Quantum Theory of Light (Oxford Science Publications, 2000).

S. M. Barnett and P. M. Radmore, Methods in Theoretical Quantum Optics (Oxford Science Publications, 1997).

W. Vogel and D.-G. Welsch, Quantum Optics (Wiley-VCH, 2006).
[Crossref]

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Figures (3)

Fig. 1
Fig. 1

The intensity distribution in the cross section of a fundamental Gaussian beam is depicted. 〈Ŵ〉 represents the variance of the transverse distribution (see Eq. (1)), the indicated quantity W ^ is thus its standard deviation. The uncertainty in the width of the distribution can be quantified by 〈δŴ2〉 (see Eq. (4)). In the figure, δ W ^ 2 4 is indicated, possessing the same dimension as W ^ .

Fig. 2
Fig. 2

(a) Noise in the beam width of a fundamental Gaussian beam for different single mode quantum states, normalized with the noise of the coherent state. The uncertainty in the beam width depends strongly on the quantum state. The lowest noise is achieved for a Fock state, for which the photon number fluctuations are equal to 0. (b) Noise in the beam width of a single mode coherent state in a Laguerre-Gauss mode LGlp with p = 0 and different values for the azimuthal values of l, normalized with the squared mean value of the beam width. The influence of the quantum noise is getting less significant for larger total beam sizes. The uncertainty of the beam width decreases with increasing azimuthal parameter l.

Fig. 3
Fig. 3

The plots show two examples of mean field modes and their respective detection mode evaluated at y = 0: In (a), the mean field mode (blue) is a fundamental Gaussian mode, and its detection mode (red) a superposition of Hermite-Gauss modes of the zeroth and the second order. (b) shows an example for a flattened Gaussian beam as the mean field mode and its detection mode consisting of two peaks at the position of the edges of the flattened Gaussian beam. The horizontal axis gives the transverse length x in terms of the standard deviation w0 of the fundamental Gaussian beam and the equivalent parameter for the flattened Gaussian beam (see main text). The vertical axis is scaled such that the maximum of the mean field modes is 1.

Equations (21)

Equations on this page are rendered with MathJax. Learn more.

W ^ = 1 E ^ ( ) ( x , y ) E ^ ( + ) ( x , y ) d x d y ( x 2 + y 2 ) E ^ ( ) ( x , y ) E ^ ( + ) ( x , y ) d x d y = 1 N all i , j D i j a ^ i a ^ j ,
D i j = ( x 2 + y 2 ) u i * ( x , y ) u j ( x , y ) d x d y
N all = i N i = i a ^ i a ^ i .
δ W ^ 2 = 1 N all 2 [ i j k l D i j D k l ( a ^ i a ^ k a ^ j a ^ l a ^ i a ^ j a ^ k a ^ l ) + i l F i l a ^ i a ^ l ] ,
F i l = ( x 2 + y 2 ) 2 u i * ( x , y ) u l ( x , y ) d x d y .
δ W ^ 2 = 1 n ^ 0 [ D 00 2 ( δ n ^ 0 2 n ^ 0 1 ) + F 00 ] ,
δ W ^ 2 Fock δ W ^ 2 Coh = 1 D 00 2 F 00 .
δ W ^ 2 SqVac δ W ^ 2 Coh = D 00 2 F 00 ( 2 n ¯ 0 + 1 ) + 1 ,
δ W ^ 2 DisplSq δ W ^ 2 Coh = [ sinh 2 ( s ) e 2 s + 2 sinh 2 ( s ) ( sinh 2 ( s ) + 1 ) ] D 00 2 F 00 n ¯ 0 + [ 1 + D 00 2 F 00 ( e 2 s 1 ) ] ,
δ W ^ 2 Thermal δ W ^ 2 Coh = D 00 2 F 00 n ¯ 0 + 1.
δ W ^ 2 DisplThermal δ W ^ 2 Coh = D 00 2 F 00 ( 2 n ¯ th n ¯ 0 ) n ¯ t h + 1 ,
δ W ^ = 1 N all i j D i j ( a ^ i δ a ^ j + δ a ^ i a ^ j ) .
δ W ^ 2 = a ^ 0 2 F 00 N all 2 ( δ A ^ + δ A ^ ) 2 ,
δ W ^ 2 = a ^ 0 2 F 00 N all 2 δ 2 X ^ v 0 ( + )
v 0 ( x , y ) = 1 F 00 ( x 2 + y 2 ) u 0 ( x , y ) .
u 0 ( x , y ) = D 00 F 00 v 0 ( x , y ) + 1 D 00 2 F 00 v 1 ( x , y ) .
v 0 ( x , y ) = 1 3 u HG 00 ( x , y ) + 2 3 u HG 20 ( x , y )
U N ( x , y ) = A exp ( N ( x 2 + y 2 ) w 0 2 ) m = 0 N 1 2 m m = 0 N ( 1 ) n ( m n ) L n ( 2 N ( x 2 + y 2 ) w 0 2 ) ,
Θ ^ = 1 N all i j D ˜ i j a ^ i a ^ j ,
D ˜ i j = 1 k 2 u i * ( x , y ) ( x 2 + y 2 ) u j ( x , y ) d x d y .
m 0 ( x , y ) = 1 k 2 F ˜ 00 ( x 2 + y 2 ) u 0 ( x , y ) .

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