Abstract

The transverse distribution of intensity noise in the far field of semiconductor lasers has been experimentally studied. For a single-mode edge-emitting laser, it has been found that a large amount of noise is present in higher-order nonlasing transverse modes parallel to the diode junction. In the case of a spatially multimode vertical-cavity surface-emitting laser, each mode exhibits a large noise, but these noises show strong anticorrelations.

© 1999 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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1999 (3)

S. Lathi and Y. Yamamoto, “Influence of nonlinear gain and loss on the intensity noise of a multimode semiconductor laser,” Phys. Rev. A 59, 819–825 (1999).
[CrossRef]

F. Marin and G. Giacomelli, “Polarization and transverse mode dynamics of VCSELs,” Quantum Semiclassic. Opt. 1, 128–132 (1999).
[CrossRef]

A. Bramati, J.-P. Hermier, A. Z. Khoury, E. Giacobino, P. Schnitzer, R. Michalzik, K. J. Ebeling, J.-Ph. Poizat, and Ph. Grangier, “Spatial distribution of the intensity noise of a vertical-cavity surface-emitting laser,” Opt. Lett. 24, 893–895 (1999).
[CrossRef]

1998 (3)

J.-Ph. Poizat, T.-J. Chang, O. Ripoll, and Ph. Grangier, “Spatial quantum noise of laser diodes,” J. Opt. Soc. Am. B 15, 1757–1761 (1998).
[CrossRef]

D. Wiedenmann, P. Schnitzer, C. Jung, M. Grabherr, R. Jäger, R. Michalzik, and K. J. Ebeling, “Noise characteristics of 850 nm single-mode vertical cavity surface emitting lasers,” Appl. Phys. Lett. 73, 717–719 (1998).
[CrossRef]

C. Becher, E. Gehrig, and K.-J. Boller, “Spectrally asymmetric mode correlation and intensity noise in pump-noise-suppressed laser diodes,” Phys. Rev. A 57, 3952–3960 (1998).
[CrossRef]

1997 (2)

1996 (1)

1995 (2)

E. Goobar, J. W. Scott, B. Thibeault, G. Robinson, Y. Akulova, and L. A. Coldren, “Noise in homodyne and heterodyne detection,” Appl. Phys. Lett. 67, 3697–3699 (1995).
[CrossRef]

F. Marin, A. Bramati, E. Giacobino, T.-C. Zhang, J.-Ph. Poizat, J.-F. Roch, and P. Grangier, “Squeezing and intermode correlations in laser diodes,” Phys. Rev. Lett. 75, 4606–4609 (1995).
[CrossRef] [PubMed]

1994 (2)

D. Pieroux and P. Mandel, “Transient dynamics of a multimode laser: oscillation frequencies and decay rates,” Opt. Commun. 107, 245–248 (1994).
[CrossRef]

K. Otsuka, D. Pieroux, and P. Mandel, “Modulation dynamics in a multimode laser with feedback,” Opt. Commun. 108, 273–277 (1994).
[CrossRef]

1993 (2)

1992 (1)

S. Inoue, H. Ohzu, S. Machida, and Y. Yamamoto, “Quantum correlation between longitudinal-mode intensities in a multimode squeezed semiconductor laser,” Phys. Rev. A 46, 2757–2765 (1992).
[CrossRef] [PubMed]

1987 (1)

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]

1986 (1)

Y. Yamamoto, S. Machida, and O. Nilsson, “Amplitude squeezing in a pump-noise-suppressed laser oscillator,” Phys. Rev. A 34, 4025–4042 (1986).
[CrossRef] [PubMed]

1985 (1)

H. A. Haus and S. Kawakami, “On the ‘excess spontaneous emission factor’ in gain-guided laser amplifiers,” IEEE J. Quantum Electron. QE-21, 63–69 (1985).
[CrossRef]

1984 (1)

Yu. M. Golubev and I. V. Sokolov, “Photon antibunching in a coherent light source and suppression of the photorecording noise,” Sov. Phys. JETP 60, 234–238 (1984).

1983 (1)

1979 (1)

K. Petermann, “Calculated spontaneous emission factor for double-heterostructure injection lasers with gain-induced waveguiding,” IEEE J. Quantum Electron. QE-15, 566–570 (1979).
[CrossRef]

1966 (1)

A. W. Smith and J. A. Armstrong, “Intensity noise in multimode GaAs laser emission,” IBM Syst. J. 10, 225–232 (1966).
[CrossRef]

Akulova, Y.

E. Goobar, J. W. Scott, B. Thibeault, G. Robinson, Y. Akulova, and L. A. Coldren, “Noise in homodyne and heterodyne detection,” Appl. Phys. Lett. 67, 3697–3699 (1995).
[CrossRef]

Armstrong, J. A.

A. W. Smith and J. A. Armstrong, “Intensity noise in multimode GaAs laser emission,” IBM Syst. J. 10, 225–232 (1966).
[CrossRef]

Bachor, H.-A.

Becher, C.

C. Becher, E. Gehrig, and K.-J. Boller, “Spectrally asymmetric mode correlation and intensity noise in pump-noise-suppressed laser diodes,” Phys. Rev. A 57, 3952–3960 (1998).
[CrossRef]

Boller, K.-J.

C. Becher, E. Gehrig, and K.-J. Boller, “Spectrally asymmetric mode correlation and intensity noise in pump-noise-suppressed laser diodes,” Phys. Rev. A 57, 3952–3960 (1998).
[CrossRef]

Bramati, A.

A. Bramati, J.-P. Hermier, A. Z. Khoury, E. Giacobino, P. Schnitzer, R. Michalzik, K. J. Ebeling, J.-Ph. Poizat, and Ph. Grangier, “Spatial distribution of the intensity noise of a vertical-cavity surface-emitting laser,” Opt. Lett. 24, 893–895 (1999).
[CrossRef]

F. Marin, A. Bramati, E. Giacobino, T.-C. Zhang, J.-Ph. Poizat, J.-F. Roch, and P. Grangier, “Squeezing and intermode correlations in laser diodes,” Phys. Rev. Lett. 75, 4606–4609 (1995).
[CrossRef] [PubMed]

Carlsten, J. L.

D. C. Kilper, P. A. Roos, J. L. Carlsten, and K. L. Lear, “Squeezed light generated by a microcavity laser,” Phys. Rev. A 55, R3323–R3326 (1997).
[CrossRef]

Chan, V. W. S.

Chang, T.-J.

Coldren, L. A.

E. Goobar, J. W. Scott, B. Thibeault, G. Robinson, Y. Akulova, and L. A. Coldren, “Noise in homodyne and heterodyne detection,” Appl. Phys. Lett. 67, 3697–3699 (1995).
[CrossRef]

Craig, R.

Ebeling, K. J.

A. Bramati, J.-P. Hermier, A. Z. Khoury, E. Giacobino, P. Schnitzer, R. Michalzik, K. J. Ebeling, J.-Ph. Poizat, and Ph. Grangier, “Spatial distribution of the intensity noise of a vertical-cavity surface-emitting laser,” Opt. Lett. 24, 893–895 (1999).
[CrossRef]

D. Wiedenmann, P. Schnitzer, C. Jung, M. Grabherr, R. Jäger, R. Michalzik, and K. J. Ebeling, “Noise characteristics of 850 nm single-mode vertical cavity surface emitting lasers,” Appl. Phys. Lett. 73, 717–719 (1998).
[CrossRef]

Freeman, M. J.

Freitag, I.

Gehrig, E.

C. Becher, E. Gehrig, and K.-J. Boller, “Spectrally asymmetric mode correlation and intensity noise in pump-noise-suppressed laser diodes,” Phys. Rev. A 57, 3952–3960 (1998).
[CrossRef]

Giacobino, E.

A. Bramati, J.-P. Hermier, A. Z. Khoury, E. Giacobino, P. Schnitzer, R. Michalzik, K. J. Ebeling, J.-Ph. Poizat, and Ph. Grangier, “Spatial distribution of the intensity noise of a vertical-cavity surface-emitting laser,” Opt. Lett. 24, 893–895 (1999).
[CrossRef]

F. Marin, A. Bramati, E. Giacobino, T.-C. Zhang, J.-Ph. Poizat, J.-F. Roch, and P. Grangier, “Squeezing and intermode correlations in laser diodes,” Phys. Rev. Lett. 75, 4606–4609 (1995).
[CrossRef] [PubMed]

Giacomelli, G.

F. Marin and G. Giacomelli, “Polarization and transverse mode dynamics of VCSELs,” Quantum Semiclassic. Opt. 1, 128–132 (1999).
[CrossRef]

Golubev, Yu. M.

Yu. M. Golubev and I. V. Sokolov, “Photon antibunching in a coherent light source and suppression of the photorecording noise,” Sov. Phys. JETP 60, 234–238 (1984).

Goobar, E.

E. Goobar, J. W. Scott, B. Thibeault, G. Robinson, Y. Akulova, and L. A. Coldren, “Noise in homodyne and heterodyne detection,” Appl. Phys. Lett. 67, 3697–3699 (1995).
[CrossRef]

Grabherr, M.

D. Wiedenmann, P. Schnitzer, C. Jung, M. Grabherr, R. Jäger, R. Michalzik, and K. J. Ebeling, “Noise characteristics of 850 nm single-mode vertical cavity surface emitting lasers,” Appl. Phys. Lett. 73, 717–719 (1998).
[CrossRef]

Grangier, P.

F. Marin, A. Bramati, E. Giacobino, T.-C. Zhang, J.-Ph. Poizat, J.-F. Roch, and P. Grangier, “Squeezing and intermode correlations in laser diodes,” Phys. Rev. Lett. 75, 4606–4609 (1995).
[CrossRef] [PubMed]

Grangier, Ph.

Harb, C. C.

Haus, H. A.

H. A. Haus and S. Kawakami, “On the ‘excess spontaneous emission factor’ in gain-guided laser amplifiers,” IEEE J. Quantum Electron. QE-21, 63–69 (1985).
[CrossRef]

Hermier, J.-P.

Huntington, E. H.

Inoue, S.

S. Inoue, H. Ohzu, S. Machida, and Y. Yamamoto, “Quantum correlation between longitudinal-mode intensities in a multimode squeezed semiconductor laser,” Phys. Rev. A 46, 2757–2765 (1992).
[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]

Jäger, R.

D. Wiedenmann, P. Schnitzer, C. Jung, M. Grabherr, R. Jäger, R. Michalzik, and K. J. Ebeling, “Noise characteristics of 850 nm single-mode vertical cavity surface emitting lasers,” Appl. Phys. Lett. 73, 717–719 (1998).
[CrossRef]

Jung, C.

D. Wiedenmann, P. Schnitzer, C. Jung, M. Grabherr, R. Jäger, R. Michalzik, and K. J. Ebeling, “Noise characteristics of 850 nm single-mode vertical cavity surface emitting lasers,” Appl. Phys. Lett. 73, 717–719 (1998).
[CrossRef]

Kawakami, S.

H. A. Haus and S. Kawakami, “On the ‘excess spontaneous emission factor’ in gain-guided laser amplifiers,” IEEE J. Quantum Electron. QE-21, 63–69 (1985).
[CrossRef]

Khoury, A. Z.

Kilper, D. C.

D. C. Kilper, P. A. Roos, J. L. Carlsten, and K. L. Lear, “Squeezed light generated by a microcavity laser,” Phys. Rev. A 55, R3323–R3326 (1997).
[CrossRef]

D. C. Kilper, D. G. Steel, R. Craig, and D. R. Scifres, “Polarization-dependent noise in a photon-number squeezed light generated by quantum-well lasers,” Opt. Lett. 21, 1283–1285 (1996).
[CrossRef] [PubMed]

Lathi, S.

S. Lathi and Y. Yamamoto, “Influence of nonlinear gain and loss on the intensity noise of a multimode semiconductor laser,” Phys. Rev. A 59, 819–825 (1999).
[CrossRef]

Lear, K. L.

D. C. Kilper, P. A. Roos, J. L. Carlsten, and K. L. Lear, “Squeezed light generated by a microcavity laser,” Phys. Rev. A 55, R3323–R3326 (1997).
[CrossRef]

Machida, S.

S. Inoue, H. Ohzu, S. Machida, and Y. Yamamoto, “Quantum correlation between longitudinal-mode intensities in a multimode squeezed semiconductor laser,” Phys. Rev. A 46, 2757–2765 (1992).
[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]

Y. Yamamoto, S. Machida, and O. Nilsson, “Amplitude squeezing in a pump-noise-suppressed laser oscillator,” Phys. Rev. A 34, 4025–4042 (1986).
[CrossRef] [PubMed]

Mandel, P.

D. Pieroux and P. Mandel, “Transient dynamics of a multimode laser: oscillation frequencies and decay rates,” Opt. Commun. 107, 245–248 (1994).
[CrossRef]

K. Otsuka, D. Pieroux, and P. Mandel, “Modulation dynamics in a multimode laser with feedback,” Opt. Commun. 108, 273–277 (1994).
[CrossRef]

Marin, F.

F. Marin and G. Giacomelli, “Polarization and transverse mode dynamics of VCSELs,” Quantum Semiclassic. Opt. 1, 128–132 (1999).
[CrossRef]

F. Marin, A. Bramati, E. Giacobino, T.-C. Zhang, J.-Ph. Poizat, J.-F. Roch, and P. Grangier, “Squeezing and intermode correlations in laser diodes,” Phys. Rev. Lett. 75, 4606–4609 (1995).
[CrossRef] [PubMed]

McClelland, D. E.

Michalzik, R.

A. Bramati, J.-P. Hermier, A. Z. Khoury, E. Giacobino, P. Schnitzer, R. Michalzik, K. J. Ebeling, J.-Ph. Poizat, and Ph. Grangier, “Spatial distribution of the intensity noise of a vertical-cavity surface-emitting laser,” Opt. Lett. 24, 893–895 (1999).
[CrossRef]

D. Wiedenmann, P. Schnitzer, C. Jung, M. Grabherr, R. Jäger, R. Michalzik, and K. J. Ebeling, “Noise characteristics of 850 nm single-mode vertical cavity surface emitting lasers,” Appl. Phys. Lett. 73, 717–719 (1998).
[CrossRef]

Nilsson, O.

Y. Yamamoto, S. Machida, and O. Nilsson, “Amplitude squeezing in a pump-noise-suppressed laser oscillator,” Phys. Rev. A 34, 4025–4042 (1986).
[CrossRef] [PubMed]

Ohzu, H.

S. Inoue, H. Ohzu, S. Machida, and Y. Yamamoto, “Quantum correlation between longitudinal-mode intensities in a multimode squeezed semiconductor laser,” Phys. Rev. A 46, 2757–2765 (1992).
[CrossRef] [PubMed]

Otsuka, K.

K. Otsuka, D. Pieroux, and P. Mandel, “Modulation dynamics in a multimode laser with feedback,” Opt. Commun. 108, 273–277 (1994).
[CrossRef]

Petermann, K.

K. Petermann, “Calculated spontaneous emission factor for double-heterostructure injection lasers with gain-induced waveguiding,” IEEE J. Quantum Electron. QE-15, 566–570 (1979).
[CrossRef]

Pieroux, D.

K. Otsuka, D. Pieroux, and P. Mandel, “Modulation dynamics in a multimode laser with feedback,” Opt. Commun. 108, 273–277 (1994).
[CrossRef]

D. Pieroux and P. Mandel, “Transient dynamics of a multimode laser: oscillation frequencies and decay rates,” Opt. Commun. 107, 245–248 (1994).
[CrossRef]

Poizat, J.-Ph.

Ralph, T. C.

Ripoll, O.

Robinson, G.

E. Goobar, J. W. Scott, B. Thibeault, G. Robinson, Y. Akulova, and L. A. Coldren, “Noise in homodyne and heterodyne detection,” Appl. Phys. Lett. 67, 3697–3699 (1995).
[CrossRef]

Roch, J.-F.

F. Marin, A. Bramati, E. Giacobino, T.-C. Zhang, J.-Ph. Poizat, J.-F. Roch, and P. Grangier, “Squeezing and intermode correlations in laser diodes,” Phys. Rev. Lett. 75, 4606–4609 (1995).
[CrossRef] [PubMed]

Roos, P. A.

D. C. Kilper, P. A. Roos, J. L. Carlsten, and K. L. Lear, “Squeezed light generated by a microcavity laser,” Phys. Rev. A 55, R3323–R3326 (1997).
[CrossRef]

Schnitzer, P.

A. Bramati, J.-P. Hermier, A. Z. Khoury, E. Giacobino, P. Schnitzer, R. Michalzik, K. J. Ebeling, J.-Ph. Poizat, and Ph. Grangier, “Spatial distribution of the intensity noise of a vertical-cavity surface-emitting laser,” Opt. Lett. 24, 893–895 (1999).
[CrossRef]

D. Wiedenmann, P. Schnitzer, C. Jung, M. Grabherr, R. Jäger, R. Michalzik, and K. J. Ebeling, “Noise characteristics of 850 nm single-mode vertical cavity surface emitting lasers,” Appl. Phys. Lett. 73, 717–719 (1998).
[CrossRef]

Scifres, D. R.

Scott, J. W.

E. Goobar, J. W. Scott, B. Thibeault, G. Robinson, Y. Akulova, and L. A. Coldren, “Noise in homodyne and heterodyne detection,” Appl. Phys. Lett. 67, 3697–3699 (1995).
[CrossRef]

Smith, A. W.

A. W. Smith and J. A. Armstrong, “Intensity noise in multimode GaAs laser emission,” IBM Syst. J. 10, 225–232 (1966).
[CrossRef]

Sokolov, I. V.

Yu. M. Golubev and I. V. Sokolov, “Photon antibunching in a coherent light source and suppression of the photorecording noise,” Sov. Phys. JETP 60, 234–238 (1984).

Steel, D. G.

Thibeault, B.

E. Goobar, J. W. Scott, B. Thibeault, G. Robinson, Y. Akulova, and L. A. Coldren, “Noise in homodyne and heterodyne detection,” Appl. Phys. Lett. 67, 3697–3699 (1995).
[CrossRef]

Wang, H.

Wiedenmann, D.

D. Wiedenmann, P. Schnitzer, C. Jung, M. Grabherr, R. Jäger, R. Michalzik, and K. J. Ebeling, “Noise characteristics of 850 nm single-mode vertical cavity surface emitting lasers,” Appl. Phys. Lett. 73, 717–719 (1998).
[CrossRef]

Yamamoto, Y.

S. Lathi and Y. Yamamoto, “Influence of nonlinear gain and loss on the intensity noise of a multimode semiconductor laser,” Phys. Rev. A 59, 819–825 (1999).
[CrossRef]

S. Inoue, H. Ohzu, S. Machida, and Y. Yamamoto, “Quantum correlation between longitudinal-mode intensities in a multimode squeezed semiconductor laser,” Phys. Rev. A 46, 2757–2765 (1992).
[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]

Y. Yamamoto, S. Machida, and O. Nilsson, “Amplitude squeezing in a pump-noise-suppressed laser oscillator,” Phys. Rev. A 34, 4025–4042 (1986).
[CrossRef] [PubMed]

Yuen, H. P.

Zhang, T.-C.

F. Marin, A. Bramati, E. Giacobino, T.-C. Zhang, J.-Ph. Poizat, J.-F. Roch, and P. Grangier, “Squeezing and intermode correlations in laser diodes,” Phys. Rev. Lett. 75, 4606–4609 (1995).
[CrossRef] [PubMed]

Appl. Phys. Lett. (2)

E. Goobar, J. W. Scott, B. Thibeault, G. Robinson, Y. Akulova, and L. A. Coldren, “Noise in homodyne and heterodyne detection,” Appl. Phys. Lett. 67, 3697–3699 (1995).
[CrossRef]

D. Wiedenmann, P. Schnitzer, C. Jung, M. Grabherr, R. Jäger, R. Michalzik, and K. J. Ebeling, “Noise characteristics of 850 nm single-mode vertical cavity surface emitting lasers,” Appl. Phys. Lett. 73, 717–719 (1998).
[CrossRef]

IBM Syst. J. (1)

A. W. Smith and J. A. Armstrong, “Intensity noise in multimode GaAs laser emission,” IBM Syst. J. 10, 225–232 (1966).
[CrossRef]

IEEE J. Quantum Electron. (2)

K. Petermann, “Calculated spontaneous emission factor for double-heterostructure injection lasers with gain-induced waveguiding,” IEEE J. Quantum Electron. QE-15, 566–570 (1979).
[CrossRef]

H. A. Haus and S. Kawakami, “On the ‘excess spontaneous emission factor’ in gain-guided laser amplifiers,” IEEE J. Quantum Electron. QE-21, 63–69 (1985).
[CrossRef]

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

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We assume that operation involves the first two transverse modes with Hermite–Gauss profiles, which corresponds to the experimental situation, both for VCSEL’s and for edge-emitting lasers.

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Figures (4)

Fig. 1
Fig. 1

Experimental setup for the measurement of the spatial intensity noise distribution.

Fig. 2
Fig. 2

Normalized intensity noise of a VCSEL versus the position of the razorblade, normalized to the beam size. The smooth (solid) trace is the theoretical prediction, and the noisy trace corresponds to the experimental results. Trace (S) is the SNL (proportional to the intensity). The dashed curve shows the theoretical intensity noise corresponding to an overall beam attenuation equal to the transmission of the razorblade. The fitting parameters are as follows: (a) For VCSEL 1: C=-0.75, v0=25.82, v1=330.9, and q=6.13; (b) for VCSEL 2: C=-0.98, v0=80.2, v1=3648, q=34.8.

Fig. 3
Fig. 3

Normalized intensity noise of an edge-emitting laser versus the position of the razorblade, normalized to the beam size. Trace (S) is the SNL (proportional to the intensity). Trace (L) [(R)] corresponds to the razorblade entering the beam from the left (right). For each trace the solid curve is the best fit. Trace (R) has been flipped over for convenient comparison with trace (L). The fitting parameters are v0=-0.10±0.02, v1=14.9±0.2, v2=1±0.5, C=0.23±0.02. The inset, a zoom of the SNL-normalized noise of trace (L), shows that the maximum squeezing is obtained when part of the beam is screened. For each trace, the solid curve is the best fit.

Fig. 4
Fig. 4

Normalized intensity noise of the total beam (points), of mode 0 (squares), and of mode 1 (triangles) versus the driving current (a) for the VCSEL and (b) for the edge-emitting laser.

Equations (15)

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u0(x)=(2/π)1/41/w exp(-x2/w2),
u1(x)=(2/π)1/41/w2(x/w)exp(-x2/w2),
T0(X)=Xu02(x)dx,T1(X)=Xu12(x)dx,
Q01(X)=Xu0(x)u1(x)dx.
n(X)=T0(X)n0+T1(X)n1.
δn2(X)=n2(X)-n(X)2=T0(X)n0+T1(X)×n1+T02(X):δn02:+T12(X):δn12:+2T0(X)T1(X)δn0δn1,
C=δn0δn1(δn02δn12)1/2.
δn2(X)=T0(X)n0+T1(X)n1+T02(X):δn02:+T12(X):δn12:+2T0(X)T1(X)×C(δn02δn12)1/2.
S(X)=1q+1{qT0(X)+T1(X)+T02(X)qv0+T12(X)v1+2T0(X)T1(X)C[q(1+v0)(1+v1)]1/2}.
SSNL(X)=qT0(X)+T1(X)q+1.
E(+)(x)=E0u0(x)+δE0(+)u0(x)+δE1(+)u1(x),
n(X)=T0(X)E02+E0[T0(X)δP0+Q01(X)δP1],
δn2(X)=T0(X)E02+E02[T02(X):δP02:+Q012(X)×:δP12:+2T0(X)Q01(X)δP0δP1]=T0(X)n0+T02(X):δn02:+E02[Q012(X)×:δP12:+2T0(X)Q01(X)δP0δP1],
C=δP0δP1[(1+v0)(1+v1)]1/2.
Q01(X)X+u0(x)u1(x)dx=---Xu0(x)u1(x)dx.

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