Abstract

The transverse properties of squeezed light are studied for a number-squeezed edge diode laser working in an external-cavity configuration. Light noise is analyzed as a function of the transverse wave vector transmitted by a single slit at the focal plane of a short focal lens. Noise measurements obtained by opening the slit centered at the kx=0 position show a minimum squeezing at the slit center that increases up to the full-beam noise-compression. Noise is also measured by displacing a single slit of fixed width at the focal plane. The results of this wave-vector dependence are compared with a theory in which the slit is treated as a beam splitter for the incoming Hermite–Gaussian laser beam. A summation of the noise contributions at each wave vector cannot give the obtained result with the slit being open, which indicates a possible noise correlation of light with distinct wave vectors.

© 2001 Optical Society of America

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References

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  1. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, New York, 1995).
  2. S. Machida, Y. Yamamoto, and Y. Itaya, “Observation of amplitude squeezing in a constant-current-driven semiconductor laser,” Phys. Rev. Lett. 58, 1000–1003 (1987).
    [CrossRef] [PubMed]
  3. P. R. Tapster, J. G. Rarity, and J. S. Satchell, “Generation of sub-Poissonian light by high-efficiency light-emitting-diodes,” Europhys. Lett. 4, 293–299 (1987).
    [CrossRef]
  4. H. Wang, M. J. Freeman, and D. G. Steel, “Squeezed light from injection-locked quantum well lasers,” Phys. Rev. Lett. 71, 3951–3954 (1993).
    [CrossRef] [PubMed]
  5. R.-D. Li, S.-K. Choi, C. Kim, and P. Kumar, “Generation of sub-Poissonian pulses of light,” Phys. Rev. A 51, R3429–R3432 (1995).
    [CrossRef] [PubMed]
  6. Yong-qing Li and Min Xiao, “Generation and applications of amplitude-squeezed states of light from semiconductor diode lasers,” Opt. Express 2, 110–117 (1998), and references therein.
    [CrossRef] [PubMed]
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    [CrossRef]
  8. J. Ph. Poizat, T. Chang, O. Ripoll, and Ph. Grangier, “Spatial quantum noise of laser diodes,” J. Opt. Soc. Am. B 15, 1757–1761 (1998).
    [CrossRef]
  9. G. Giacomelli, F. Marin, M. Gabrysch, K. H. Gulden, and M. Moser, “Polarization competition and noise properties of VCSELs,” Opt. Commun. 146, 136–140 (1998).
    [CrossRef]
  10. M. D. Levenson, W. H. Richardson, and S. H. Perlmutter, “Stochastic noise in TEM00 laser beam position,” Opt. Lett. 14, 779–781 (1989).
    [CrossRef] [PubMed]
  11. The high-impedance power supply was designed at the Laboratoire Kastler Brossell, Université Pierre et Marie Curie, Paris.
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    [CrossRef] [PubMed]
  13. R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, and J. F. Valley, “Observation of squeezed states generated by four-wave mixing in an optical cavity,” Phys. Rev. Lett. 55, 2409–2412 (1985).
    [CrossRef] [PubMed]
  14. J. W. Goodman Introduction to Fourier Optics (McGraw-Hill, San Francisco, Calif., 1968).
  15. G. Grynberg, A. Aspect, and C. Fabre, Introduction aux Lasers et à L’Optique Quantique (Ellipses, Paris, 1997).

1999 (1)

1998 (3)

1995 (1)

R.-D. Li, S.-K. Choi, C. Kim, and P. Kumar, “Generation of sub-Poissonian pulses of light,” Phys. Rev. A 51, R3429–R3432 (1995).
[CrossRef] [PubMed]

1993 (1)

H. Wang, M. J. Freeman, and D. G. Steel, “Squeezed light from injection-locked quantum well lasers,” Phys. Rev. Lett. 71, 3951–3954 (1993).
[CrossRef] [PubMed]

1989 (1)

1987 (2)

S. Machida, Y. Yamamoto, and Y. Itaya, “Observation of amplitude squeezing in a constant-current-driven semiconductor laser,” Phys. Rev. Lett. 58, 1000–1003 (1987).
[CrossRef] [PubMed]

P. R. Tapster, J. G. Rarity, and J. S. Satchell, “Generation of sub-Poissonian light by high-efficiency light-emitting-diodes,” Europhys. Lett. 4, 293–299 (1987).
[CrossRef]

1985 (1)

R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, and J. F. Valley, “Observation of squeezed states generated by four-wave mixing in an optical cavity,” Phys. Rev. Lett. 55, 2409–2412 (1985).
[CrossRef] [PubMed]

1983 (1)

Bramati, A.

Cahng, T. J.

Chan, V. W. S.

Chang, T.

Choi, S.-K.

R.-D. Li, S.-K. Choi, C. Kim, and P. Kumar, “Generation of sub-Poissonian pulses of light,” Phys. Rev. A 51, R3429–R3432 (1995).
[CrossRef] [PubMed]

Freeman, M. J.

H. Wang, M. J. Freeman, and D. G. Steel, “Squeezed light from injection-locked quantum well lasers,” Phys. Rev. Lett. 71, 3951–3954 (1993).
[CrossRef] [PubMed]

Gabrysch, M.

G. Giacomelli, F. Marin, M. Gabrysch, K. H. Gulden, and M. Moser, “Polarization competition and noise properties of VCSELs,” Opt. Commun. 146, 136–140 (1998).
[CrossRef]

Giacobino, E.

Giacomelli, G.

G. Giacomelli, F. Marin, M. Gabrysch, K. H. Gulden, and M. Moser, “Polarization competition and noise properties of VCSELs,” Opt. Commun. 146, 136–140 (1998).
[CrossRef]

Grangier, Ph.

Gulden, K. H.

G. Giacomelli, F. Marin, M. Gabrysch, K. H. Gulden, and M. Moser, “Polarization competition and noise properties of VCSELs,” Opt. Commun. 146, 136–140 (1998).
[CrossRef]

Hermier, J. P.

Hollberg, L. W.

R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, and J. F. Valley, “Observation of squeezed states generated by four-wave mixing in an optical cavity,” Phys. Rev. Lett. 55, 2409–2412 (1985).
[CrossRef] [PubMed]

Itaya, Y.

S. Machida, Y. Yamamoto, and Y. Itaya, “Observation of amplitude squeezing in a constant-current-driven semiconductor laser,” Phys. Rev. Lett. 58, 1000–1003 (1987).
[CrossRef] [PubMed]

Khoury, A. Z.

Kim, C.

R.-D. Li, S.-K. Choi, C. Kim, and P. Kumar, “Generation of sub-Poissonian pulses of light,” Phys. Rev. A 51, R3429–R3432 (1995).
[CrossRef] [PubMed]

Kumar, P.

R.-D. Li, S.-K. Choi, C. Kim, and P. Kumar, “Generation of sub-Poissonian pulses of light,” Phys. Rev. A 51, R3429–R3432 (1995).
[CrossRef] [PubMed]

Levenson, M. D.

Li, R.-D.

R.-D. Li, S.-K. Choi, C. Kim, and P. Kumar, “Generation of sub-Poissonian pulses of light,” Phys. Rev. A 51, R3429–R3432 (1995).
[CrossRef] [PubMed]

Li, Yong-qing

Machida, S.

S. Machida, Y. Yamamoto, and Y. Itaya, “Observation of amplitude squeezing in a constant-current-driven semiconductor laser,” Phys. Rev. Lett. 58, 1000–1003 (1987).
[CrossRef] [PubMed]

Marin, F.

G. Giacomelli, F. Marin, M. Gabrysch, K. H. Gulden, and M. Moser, “Polarization competition and noise properties of VCSELs,” Opt. Commun. 146, 136–140 (1998).
[CrossRef]

Mertz, J. C.

R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, and J. F. Valley, “Observation of squeezed states generated by four-wave mixing in an optical cavity,” Phys. Rev. Lett. 55, 2409–2412 (1985).
[CrossRef] [PubMed]

Moser, M.

G. Giacomelli, F. Marin, M. Gabrysch, K. H. Gulden, and M. Moser, “Polarization competition and noise properties of VCSELs,” Opt. Commun. 146, 136–140 (1998).
[CrossRef]

Perlmutter, S. H.

Poizat, J. Ph.

Rarity, J. G.

P. R. Tapster, J. G. Rarity, and J. S. Satchell, “Generation of sub-Poissonian light by high-efficiency light-emitting-diodes,” Europhys. Lett. 4, 293–299 (1987).
[CrossRef]

Richardson, W. H.

Ripoll, O.

Satchell, J. S.

P. R. Tapster, J. G. Rarity, and J. S. Satchell, “Generation of sub-Poissonian light by high-efficiency light-emitting-diodes,” Europhys. Lett. 4, 293–299 (1987).
[CrossRef]

Slusher, R. E.

R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, and J. F. Valley, “Observation of squeezed states generated by four-wave mixing in an optical cavity,” Phys. Rev. Lett. 55, 2409–2412 (1985).
[CrossRef] [PubMed]

Steel, D. G.

H. Wang, M. J. Freeman, and D. G. Steel, “Squeezed light from injection-locked quantum well lasers,” Phys. Rev. Lett. 71, 3951–3954 (1993).
[CrossRef] [PubMed]

Tapster, P. R.

P. R. Tapster, J. G. Rarity, and J. S. Satchell, “Generation of sub-Poissonian light by high-efficiency light-emitting-diodes,” Europhys. Lett. 4, 293–299 (1987).
[CrossRef]

Valley, J. F.

R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, and J. F. Valley, “Observation of squeezed states generated by four-wave mixing in an optical cavity,” Phys. Rev. Lett. 55, 2409–2412 (1985).
[CrossRef] [PubMed]

Wang, H.

H. Wang, M. J. Freeman, and D. G. Steel, “Squeezed light from injection-locked quantum well lasers,” Phys. Rev. Lett. 71, 3951–3954 (1993).
[CrossRef] [PubMed]

Xiao, Min

Yamamoto, Y.

S. Machida, Y. Yamamoto, and Y. Itaya, “Observation of amplitude squeezing in a constant-current-driven semiconductor laser,” Phys. Rev. Lett. 58, 1000–1003 (1987).
[CrossRef] [PubMed]

Yuen, H. P.

Yurke, B.

R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, and J. F. Valley, “Observation of squeezed states generated by four-wave mixing in an optical cavity,” Phys. Rev. Lett. 55, 2409–2412 (1985).
[CrossRef] [PubMed]

Europhys. Lett. (1)

P. R. Tapster, J. G. Rarity, and J. S. Satchell, “Generation of sub-Poissonian light by high-efficiency light-emitting-diodes,” Europhys. Lett. 4, 293–299 (1987).
[CrossRef]

J. Opt. Soc. Am. B (2)

Opt. Commun. (1)

G. Giacomelli, F. Marin, M. Gabrysch, K. H. Gulden, and M. Moser, “Polarization competition and noise properties of VCSELs,” Opt. Commun. 146, 136–140 (1998).
[CrossRef]

Opt. Express (1)

Opt. Lett. (2)

Phys. Rev. A (1)

R.-D. Li, S.-K. Choi, C. Kim, and P. Kumar, “Generation of sub-Poissonian pulses of light,” Phys. Rev. A 51, R3429–R3432 (1995).
[CrossRef] [PubMed]

Phys. Rev. Lett. (3)

R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, and J. F. Valley, “Observation of squeezed states generated by four-wave mixing in an optical cavity,” Phys. Rev. Lett. 55, 2409–2412 (1985).
[CrossRef] [PubMed]

H. Wang, M. J. Freeman, and D. G. Steel, “Squeezed light from injection-locked quantum well lasers,” Phys. Rev. Lett. 71, 3951–3954 (1993).
[CrossRef] [PubMed]

S. Machida, Y. Yamamoto, and Y. Itaya, “Observation of amplitude squeezing in a constant-current-driven semiconductor laser,” Phys. Rev. Lett. 58, 1000–1003 (1987).
[CrossRef] [PubMed]

Other (4)

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, New York, 1995).

The high-impedance power supply was designed at the Laboratoire Kastler Brossell, Université Pierre et Marie Curie, Paris.

J. W. Goodman Introduction to Fourier Optics (McGraw-Hill, San Francisco, Calif., 1968).

G. Grynberg, A. Aspect, and C. Fabre, Introduction aux Lasers et à L’Optique Quantique (Ellipses, Paris, 1997).

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Figures (7)

Fig. 1
Fig. 1

Experimental setup. See text for abbreviations.

Fig. 2
Fig. 2

Squeezing as a function of frequency.

Fig. 3
Fig. 3

Slit S is placed at the focal plane of the lens L1 (z=-f). Elm(x, y, z) is the incoming field amplitude associated with mode (l, m). At the focal plane of the lens the electric field amplitude in wave-vector space is Ef(l,m)(kxf, kyf), which is written in terms of xf as Ef(l,m)(xf, yf).

Fig. 4
Fig. 4

Slit system seen as a beam splitter, followed by the homodyne detection system. The transmittance and the reflectance of the slit are Tj and Rj, respectively, where Rj+Tj=1, and t and r(=t) are the transmissivity and the reflectivity of the beam splitter BS. The annihilation operator aˆj specifies photons from the laser beam in a jth light mode, aˆlj is the light that does not reach the detectors through the slit, and aˆvj is the vacuum input to the idealized beam splitter. bˆj is the annihilation operator for photons transmitted by the slit and diverted to bˆ1j and bˆ2j by the polarizing beam splitter PBS. The short focal lens L1 of focus f is at a distance z=-f from the slit.

Fig. 5
Fig. 5

Fano factor as a function of the slit aperture in a linear scale. The solid curve is the fitting from Eqs. (15) and (3). The dashed curve is the Fano factor obtained from a normalized integration on the variable kx of the Fano factor F(kx) and represents the case with a fixed aperture being displaced along kx. The dotted curve shows the integration of the variances normalized by the beam intensity.

Fig. 6
Fig. 6

Fano factor as a function of the kx transverse wave-vector position in a linear scale. The fixed slit width opening is 7 µm and corresponds to a resolution in kx of Δkx=2π7 µm/(λf)=6.510-3 µm-1.

Fig. 7
Fig. 7

(a) Isolated single modes TEM00 and TEM10 are compared with the curve that considers both modes simultaneously. (b) The two modes are shown simultaneously but with different degrees of correlation between c=0.002, c=0.23, and c=-0.23.

Equations (15)

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Elm(x, y, z)=ω0xω0yωx(z)ωy(z) exp(-ikz)×exp{i[(l+1/2)arctan(z/zRx)+(m+1/2)arctan(z/zRy)]}×exp{-ikx2/[2Rx(z)]}×exp{-iky2/[2Ry(z)]}exp[-x2/ωx2(x)]×exp[-y2/ωy2(z)]Hlx2ωx(z)Hmy2ωy(z),
Ef(l,m)(kxf, kyf)Ef(l,m)(xf, yf)=exp[ik(xf2+yf2)/(2f)]iλf-+-+dxdy×exp[-i2π(xxf+yyf)/(λf)]×Ein(l,m)(x, y),
Tlm=-a/2+a/2dxf-+dyf|Ef(l,m)(xf, yf)|2-+dx-+dy|Elm(x, y, z=-f)|2,
Tlm=xs-d/2xs+d/2dxf-+dyf|Ef(l,m)(xf, yf)|2-dx-dy|Elm(x, y, z=-f)|2.
bˆj=Tjaˆj-Rjaˆvj,aˆlj=Rjaˆj+Tjaˆvj.
bˆ1j=tbˆj+rbˆvj,bˆ2j=-rbˆj+tbˆvj,
Nˆ-=0,Nˆ+=jbˆjbˆj,
(ΔNˆ+)2=j,l(bˆjbˆlbˆjbˆl-bˆjbˆjbˆlbˆl)+jbˆjbˆj,
(ΔNˆ-)2=jbˆjbˆj.
F=(ΔNˆ+)2(ΔNˆ-)2=j,l(bˆjbˆlbˆjbˆl-bˆjbˆjbˆlbˆl)jbˆjbˆj+1.
F=j,lTjTl(aˆjaˆlaˆjaˆl-aˆjaˆjaˆlaˆl)jTjaˆjaˆj+1.
(Δnˆj)2aˆjaˆjaˆjaˆj-aˆjaˆj2,
F=T02(Δnˆ0)2+T12(Δnˆ1)2+2T0T1(nˆ0nˆ1-nˆ0nˆ1)T0nˆ0+T1nˆ1+1.
qnˆ1nˆ0,vj(Δnˆj)2nˆj,cnˆ0nˆ1-nˆ0nˆ1(Δnˆ0)2(Δnˆ1)2,
F=T02v0+qT12v1+2T0T1cq(1+v0)(1+v1)T0+qT1+1.

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