Abstract

A general formalism is developed for optimizing homodyne detection of quadrature-noise squeezing by selection of the local-oscillator (LO) field. The optimum LO is the minimum-eigenvalue eigenfunction of a particular Fredholm integral equation whose kernel depends on the signal field's normally ordered and phase-sensitive covariance functions. The squeezing that results from use of the optimum LO equals one plus twice its associated eigenvalue. A continuous-wave (cw) simplification of the general formalism is presented for the case of stationary signal-field covariances when the homodyne photocurrent is spectrum analyzed. Another simplified special case is exhibited for single-spatial-mode operation, such as is encountered in fiber-based quantum-noise experiments. The cw-source–spectrum-analysis approach is used to determine the optimum LO field and its squeezing performance for cw squeezed-state generation in a bulk Kerr medium with a Gaussian spatial-response function. The single-spatial-mode framework is employed to find the optimum LO field and its squeezing performance for pulsed squeezed-state generation in a single-mode optical fiber whose Kerr nonlinearity has a noninstantaneous response function. Comparison of the cw limit of this pulsed analysis with previous cw fiber-squeezing theory reveals a new regime for quadrature-noise reduction: Raman squeezing in fiber four-wave mixing.

© 1997 Optical Society of America

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References

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  1. R. H. Kingston, Detection of Optical and Infrared Radiation (Springer-Verlag, Berlin, 1978), Chap. 3.
  2. R. M. Gagliardi and S. Karp, Optical Communications (Wiley, New York, 1976), Chap. 6.
  3. H. P. Yuen and J. H. Shapiro, “Optical communication with two-photon coherent states—Part III: Quantum measurements realizable with photoemissive detectors,” IEEE Trans. Inf. Theory IT-26, 78–92 (1980).
    [CrossRef]
  4. J. H. Shapiro, “Quantum noise and excess noise in optical homodyne and heterodyne receivers,” IEEE J. Quantum Electron. QE-21, 237–250 (1985).
    [CrossRef]
  5. H. J. Kimble and D. F. Walls, eds., feature on squeezed states of the electromagnetic field, J. Opt. Soc. Am. B 4, 1450–1741 (1987).
    [CrossRef]
  6. D. Fink, “Coherent detection signal-to-noise ratio,” Appl. Opt. 14, 689–690 (1975).
    [CrossRef] [PubMed]
  7. B. J. Rye, “Antenna parameters for incoherent backscatter heterodyne lidar,” Appl. Opt. 18, 1390–1398 (1979).
    [CrossRef] [PubMed]
  8. J. H. Shapiro, “Heterodyne mixing efficiency for detector arrays,” Appl. Opt. 26, 3600–3606 (1987).
    [CrossRef] [PubMed]
  9. O. Aytür and P. Kumar, “Squeezed-light generation with a mode-locked Q-switched laser and detection by using a matched local oscillator,” Opt. Lett. 17, 529–531 (1992).
    [CrossRef]
  10. J. H. Shapiro and L. G. Joneckis, “Enhanced fiber squeezing via local-oscillator pulse compression,” in Proceedings of 1994 IEEE Nonlinear Optics (Institute of Electrical and Electronics Engineers, New York, 1994), pp. 347–349.
  11. L. Boivin, F. X. Kärtner, and H. A. Haus, “Analytical solution to the quantum field theory of self-phase modulation with a finite response time,” Phys. Rev. Lett. 73, 240–243 (1994).
    [CrossRef] [PubMed]
  12. J. H. Shapiro and L. Boivin, “Raman-noise limit on squeezing in continuous-wave four-wave mixing,” Opt. Lett. 20, 925–927 (1995).
    [CrossRef] [PubMed]
  13. H. L. Van Trees, Detection, Estimation, and Modulation Theory (Wiley, New York, 1968), Part I, Chap. 3.
  14. Strictly speaking, we should require that K(x, t, x, t) be square integrable over Ad×[−T/2, T/2] to vouchsafe the eigenfunction–eigenvalue properties that we shall assert. For finite area photodetectors and finite measurement intervals, however, the principal pathological cases that can occur in Eq. (25) arise from covariance functions that are delta correlated in space, time, or both. In Section 3 we shall see how such a case can be handled directly from Eq. (18), with a solution that can be understood from Eq. (25).
  15. Note that the cos(ωt) term in Eq. (47) can be phase shifted by an arbitrary amount without invalidating Eq. (48). Physically, this is because the spectrum analysis measurement is insensitive to radio-frequency (frequency ω) phase shifts.
  16. R. M. Shelby, M. D. Levenson, R. G. DeVoe, S. H. Perlmutter, and D. F. Walls, “Broad-band parametric deamplification of quantum noise in an optical fiber,” Phys. Rev. Lett. 57, 691–694 (1986).
    [CrossRef] [PubMed]
  17. K. Bergman and H. A. Haus, “Squeezing in fibers with optical pulses,” Opt. Lett. 16, 663–665 (1991).
    [CrossRef] [PubMed]
  18. H. P. Yuen, “Two-photon coherent states of the radiation field,” Phys. Rev. A 13, 2226–2243 (1976).
    [CrossRef]
  19. Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984), Chap. 17.
  20. In anticipation of the photon-units formulation that we shall employ for the quantum theory to come, we are expressing the classical fields in these units too; see Ref. 4.
  21. See L. G. Joneckis and J. H. Shapiro, “Quantum propagation in a Kerr medium: lossless, dispersionless fiber,” J. Opt. Soc. Am. B 10, 1102–1120 (1993), for an analogous reduction of full SPM to FWM in a single-spatial-mode, multiple-temporal-mode quantum setting. Note, however, that this reference also indicates the necessity of the Kerr nonlinearity’s having a noninstantaneous temporal response to ensure that quantum SPM—the quantized version of Eq. (72)—properly includes the classical limit. In this regard, Eq. (73) should include a causal, noninstantaneous temporal response, with its attendant Raman noise.12 We shall eschew the inclusion of these temporal effects in our bulk Kerr medium example because our principal aim, in this section, is to develop quantum-mechanically consistent signal-field covariances with which to explore the issue of LO optimization. Our fiber FWM treatment, in Section 4, will specifically address the noninstantaneous Kerr nonlinearity.
    [CrossRef]
  22. I. S. Gradshteyn and I. M. Rhyzhik, Table of Integrals, Series, and Products (Academic, New York, 1980).
  23. M. Shirasaki and H. A. Haus, “Squeezing of pulses in a nonlinear interferometer,” J. Opt. Soc. Am. B 7, 30–34 (1990).
    [CrossRef]
  24. K. J. Blow, R. Loudon, and S. J. D. Phoenix, “Exact solution for quantum self-phase modulation,” J. Opt. Soc. Am. B 8, 1750–1756 (1991).
    [CrossRef]
  25. K. J. Blow, R. Loudon, and S. J. D. Phoenix, “Quantum theory of nonlinear loop mirrors,” Phys. Rev. A 45, 8064–8073 (1992).
    [CrossRef]
  26. L. G. Joneckis and J. H. Shapiro, “Quantum propagation in a Kerr medium: lossless, dispersionless fiber,” J. Opt. Soc. Am. B 10, 1102–1120 (1993).
    [CrossRef]
  27. W. M. Siebert, Circuits, Signals, and Systems (McGraw-Hill, New York, 1986), Chap. 15.
  28. L. Boivin, “Sagnac-loop squeezer at zero dispersion with a response time for the Kerr nonlinearity,” Phys. Rev. A 52, 754–766 (1995).
    [CrossRef] [PubMed]
  29. The peak Raman gain in fused silica fiber occurs at ω0/2π= 13.2 THz, so our Ω0 choice implies that γ=4.15 1013 s−1. With this γ value our Γ choice dictates that tP=0.19 ps. Computing the peak Raman gain from these parameter values, along the lines laid out in Ref. 12, then leads to a value ~10 times larger than the actual 1.2 10−13 m/W value found in fused-silica fiber. Likewise, the parameter values that we are using predict a low-frequency Raman limit, kBTRγ/ħω02, that is ~10 times larger at TR=300 K than the 0.03 value of a typical fiber; cf. Ref. 12.
  30. If the maximum singular value is degenerate, this procedure will still yield a λH1 value that we can use in our upper and lower bounds. With the Gaussian-pulse seed, convergence was complete at n=20. The same singular value and eigenfunctions resulted when we used the second-order Gauss–Hermite seed: ϕH1(0)(t)=[1−2(t/tP)2]exp[−(t/ tP)2]/(9πtP2/32)1/4. When we used the first-order Gauss– Hermite function as the seed, i.e., ϕH1(0)(t)(t/tP) exp[−(t/tP)2]/(πtP2/32)1/4, our iteration procedure converged to essentially the same λH1 value but had a somewhat different set of eigenfunctions. Unlike the Fourier transforms of the eigenfunctions obtained by use of the zeroth-order and second-order seed functions, those found with the first-order seed were zero at zero frequency. Inasmuch as any odd symmetry seed function can be shown, analytically, to converge to eigenfunction estimates with no zero-frequency content, we believe that the results obtained from the zeroth-order and second-order seeds correctly capture the values that we need; viz., they provide accurate estimates of {λH1, ϕH1(t), ΦH1(t)}.
  31. A. Shakeel, “Enhanced squeezing in homodyne detection via local-oscillator optimization,” Master’s thesis (Massachusetts Institute of Technology, Cambridge, Mass., 1995).
  32. Pulse compression without pump-phase rotation was previously suggested as a practical LO choice, in which case compression factor much than the r=6 value that suffices with pump-phase rotation is apt to be needed.10
  33. It should be remembered that Fig. 12 presumes a single-resonance H(ω) with Ω0≡2ω0/γ=4 and h̄Γ=ħγ/4kBTR=0.28. These parameters do not accurately represent fused-silica fiber. Specifically, in conjunction with ω0/2π=13.2 THz, the preceding parameters make the peak Raman gain too large. Reducing the peak Raman gain, in our calculations, to a more realistic value would have the effect of increasing σ20) in Fig. 12 relative to Smin(0).
  34. D. Lee and N. C. Wong, “Tunable optical frequency division using a phase-locked optical parametric oscillator,” Opt. Lett. 17, 13–15 (1992).
    [CrossRef] [PubMed]
  35. L. R. Brothers, D. Lee, and N. C. Wong, “Terahertz optical frequency comb generation and phase locking of an optical parametric oscillator at 665 GHz,” Opt. Lett. 19, 245–247 (1994).
    [CrossRef] [PubMed]

1995 (2)

J. H. Shapiro and L. Boivin, “Raman-noise limit on squeezing in continuous-wave four-wave mixing,” Opt. Lett. 20, 925–927 (1995).
[CrossRef] [PubMed]

L. Boivin, “Sagnac-loop squeezer at zero dispersion with a response time for the Kerr nonlinearity,” Phys. Rev. A 52, 754–766 (1995).
[CrossRef] [PubMed]

1994 (2)

L. R. Brothers, D. Lee, and N. C. Wong, “Terahertz optical frequency comb generation and phase locking of an optical parametric oscillator at 665 GHz,” Opt. Lett. 19, 245–247 (1994).
[CrossRef] [PubMed]

L. Boivin, F. X. Kärtner, and H. A. Haus, “Analytical solution to the quantum field theory of self-phase modulation with a finite response time,” Phys. Rev. Lett. 73, 240–243 (1994).
[CrossRef] [PubMed]

1993 (2)

1992 (3)

1991 (2)

1990 (1)

1987 (2)

J. H. Shapiro, “Heterodyne mixing efficiency for detector arrays,” Appl. Opt. 26, 3600–3606 (1987).
[CrossRef] [PubMed]

H. J. Kimble and D. F. Walls, eds., feature on squeezed states of the electromagnetic field, J. Opt. Soc. Am. B 4, 1450–1741 (1987).
[CrossRef]

1986 (1)

R. M. Shelby, M. D. Levenson, R. G. DeVoe, S. H. Perlmutter, and D. F. Walls, “Broad-band parametric deamplification of quantum noise in an optical fiber,” Phys. Rev. Lett. 57, 691–694 (1986).
[CrossRef] [PubMed]

1985 (1)

J. H. Shapiro, “Quantum noise and excess noise in optical homodyne and heterodyne receivers,” IEEE J. Quantum Electron. QE-21, 237–250 (1985).
[CrossRef]

1980 (1)

H. P. Yuen and J. H. Shapiro, “Optical communication with two-photon coherent states—Part III: Quantum measurements realizable with photoemissive detectors,” IEEE Trans. Inf. Theory IT-26, 78–92 (1980).
[CrossRef]

1979 (1)

1976 (1)

H. P. Yuen, “Two-photon coherent states of the radiation field,” Phys. Rev. A 13, 2226–2243 (1976).
[CrossRef]

1975 (1)

Aytür, O.

Bergman, K.

Blow, K. J.

K. J. Blow, R. Loudon, and S. J. D. Phoenix, “Quantum theory of nonlinear loop mirrors,” Phys. Rev. A 45, 8064–8073 (1992).
[CrossRef]

K. J. Blow, R. Loudon, and S. J. D. Phoenix, “Exact solution for quantum self-phase modulation,” J. Opt. Soc. Am. B 8, 1750–1756 (1991).
[CrossRef]

Boivin, L.

L. Boivin, “Sagnac-loop squeezer at zero dispersion with a response time for the Kerr nonlinearity,” Phys. Rev. A 52, 754–766 (1995).
[CrossRef] [PubMed]

J. H. Shapiro and L. Boivin, “Raman-noise limit on squeezing in continuous-wave four-wave mixing,” Opt. Lett. 20, 925–927 (1995).
[CrossRef] [PubMed]

L. Boivin, F. X. Kärtner, and H. A. Haus, “Analytical solution to the quantum field theory of self-phase modulation with a finite response time,” Phys. Rev. Lett. 73, 240–243 (1994).
[CrossRef] [PubMed]

Brothers, L. R.

DeVoe, R. G.

R. M. Shelby, M. D. Levenson, R. G. DeVoe, S. H. Perlmutter, and D. F. Walls, “Broad-band parametric deamplification of quantum noise in an optical fiber,” Phys. Rev. Lett. 57, 691–694 (1986).
[CrossRef] [PubMed]

Fink, D.

Haus, H. A.

Joneckis, L. G.

Kärtner, F. X.

L. Boivin, F. X. Kärtner, and H. A. Haus, “Analytical solution to the quantum field theory of self-phase modulation with a finite response time,” Phys. Rev. Lett. 73, 240–243 (1994).
[CrossRef] [PubMed]

Kimble, H. J.

H. J. Kimble and D. F. Walls, eds., feature on squeezed states of the electromagnetic field, J. Opt. Soc. Am. B 4, 1450–1741 (1987).
[CrossRef]

Kumar, P.

Lee, D.

Levenson, M. D.

R. M. Shelby, M. D. Levenson, R. G. DeVoe, S. H. Perlmutter, and D. F. Walls, “Broad-band parametric deamplification of quantum noise in an optical fiber,” Phys. Rev. Lett. 57, 691–694 (1986).
[CrossRef] [PubMed]

Loudon, R.

K. J. Blow, R. Loudon, and S. J. D. Phoenix, “Quantum theory of nonlinear loop mirrors,” Phys. Rev. A 45, 8064–8073 (1992).
[CrossRef]

K. J. Blow, R. Loudon, and S. J. D. Phoenix, “Exact solution for quantum self-phase modulation,” J. Opt. Soc. Am. B 8, 1750–1756 (1991).
[CrossRef]

Perlmutter, S. H.

R. M. Shelby, M. D. Levenson, R. G. DeVoe, S. H. Perlmutter, and D. F. Walls, “Broad-band parametric deamplification of quantum noise in an optical fiber,” Phys. Rev. Lett. 57, 691–694 (1986).
[CrossRef] [PubMed]

Phoenix, S. J. D.

K. J. Blow, R. Loudon, and S. J. D. Phoenix, “Quantum theory of nonlinear loop mirrors,” Phys. Rev. A 45, 8064–8073 (1992).
[CrossRef]

K. J. Blow, R. Loudon, and S. J. D. Phoenix, “Exact solution for quantum self-phase modulation,” J. Opt. Soc. Am. B 8, 1750–1756 (1991).
[CrossRef]

Rye, B. J.

Shapiro, J. H.

J. H. Shapiro and L. Boivin, “Raman-noise limit on squeezing in continuous-wave four-wave mixing,” Opt. Lett. 20, 925–927 (1995).
[CrossRef] [PubMed]

See L. G. Joneckis and J. H. Shapiro, “Quantum propagation in a Kerr medium: lossless, dispersionless fiber,” J. Opt. Soc. Am. B 10, 1102–1120 (1993), for an analogous reduction of full SPM to FWM in a single-spatial-mode, multiple-temporal-mode quantum setting. Note, however, that this reference also indicates the necessity of the Kerr nonlinearity’s having a noninstantaneous temporal response to ensure that quantum SPM—the quantized version of Eq. (72)—properly includes the classical limit. In this regard, Eq. (73) should include a causal, noninstantaneous temporal response, with its attendant Raman noise.12 We shall eschew the inclusion of these temporal effects in our bulk Kerr medium example because our principal aim, in this section, is to develop quantum-mechanically consistent signal-field covariances with which to explore the issue of LO optimization. Our fiber FWM treatment, in Section 4, will specifically address the noninstantaneous Kerr nonlinearity.
[CrossRef]

L. G. Joneckis and J. H. Shapiro, “Quantum propagation in a Kerr medium: lossless, dispersionless fiber,” J. Opt. Soc. Am. B 10, 1102–1120 (1993).
[CrossRef]

J. H. Shapiro, “Heterodyne mixing efficiency for detector arrays,” Appl. Opt. 26, 3600–3606 (1987).
[CrossRef] [PubMed]

J. H. Shapiro, “Quantum noise and excess noise in optical homodyne and heterodyne receivers,” IEEE J. Quantum Electron. QE-21, 237–250 (1985).
[CrossRef]

H. P. Yuen and J. H. Shapiro, “Optical communication with two-photon coherent states—Part III: Quantum measurements realizable with photoemissive detectors,” IEEE Trans. Inf. Theory IT-26, 78–92 (1980).
[CrossRef]

Shelby, R. M.

R. M. Shelby, M. D. Levenson, R. G. DeVoe, S. H. Perlmutter, and D. F. Walls, “Broad-band parametric deamplification of quantum noise in an optical fiber,” Phys. Rev. Lett. 57, 691–694 (1986).
[CrossRef] [PubMed]

Shirasaki, M.

Walls, D. F.

H. J. Kimble and D. F. Walls, eds., feature on squeezed states of the electromagnetic field, J. Opt. Soc. Am. B 4, 1450–1741 (1987).
[CrossRef]

R. M. Shelby, M. D. Levenson, R. G. DeVoe, S. H. Perlmutter, and D. F. Walls, “Broad-band parametric deamplification of quantum noise in an optical fiber,” Phys. Rev. Lett. 57, 691–694 (1986).
[CrossRef] [PubMed]

Wong, N. C.

Yuen, H. P.

H. P. Yuen and J. H. Shapiro, “Optical communication with two-photon coherent states—Part III: Quantum measurements realizable with photoemissive detectors,” IEEE Trans. Inf. Theory IT-26, 78–92 (1980).
[CrossRef]

H. P. Yuen, “Two-photon coherent states of the radiation field,” Phys. Rev. A 13, 2226–2243 (1976).
[CrossRef]

Appl. Opt. (3)

IEEE J. Quantum Electron. (1)

J. H. Shapiro, “Quantum noise and excess noise in optical homodyne and heterodyne receivers,” IEEE J. Quantum Electron. QE-21, 237–250 (1985).
[CrossRef]

IEEE Trans. Inf. Theory (1)

H. P. Yuen and J. H. Shapiro, “Optical communication with two-photon coherent states—Part III: Quantum measurements realizable with photoemissive detectors,” IEEE Trans. Inf. Theory IT-26, 78–92 (1980).
[CrossRef]

J. Opt. Soc. Am. B (5)

See L. G. Joneckis and J. H. Shapiro, “Quantum propagation in a Kerr medium: lossless, dispersionless fiber,” J. Opt. Soc. Am. B 10, 1102–1120 (1993), for an analogous reduction of full SPM to FWM in a single-spatial-mode, multiple-temporal-mode quantum setting. Note, however, that this reference also indicates the necessity of the Kerr nonlinearity’s having a noninstantaneous temporal response to ensure that quantum SPM—the quantized version of Eq. (72)—properly includes the classical limit. In this regard, Eq. (73) should include a causal, noninstantaneous temporal response, with its attendant Raman noise.12 We shall eschew the inclusion of these temporal effects in our bulk Kerr medium example because our principal aim, in this section, is to develop quantum-mechanically consistent signal-field covariances with which to explore the issue of LO optimization. Our fiber FWM treatment, in Section 4, will specifically address the noninstantaneous Kerr nonlinearity.
[CrossRef]

M. Shirasaki and H. A. Haus, “Squeezing of pulses in a nonlinear interferometer,” J. Opt. Soc. Am. B 7, 30–34 (1990).
[CrossRef]

K. J. Blow, R. Loudon, and S. J. D. Phoenix, “Exact solution for quantum self-phase modulation,” J. Opt. Soc. Am. B 8, 1750–1756 (1991).
[CrossRef]

L. G. Joneckis and J. H. Shapiro, “Quantum propagation in a Kerr medium: lossless, dispersionless fiber,” J. Opt. Soc. Am. B 10, 1102–1120 (1993).
[CrossRef]

H. J. Kimble and D. F. Walls, eds., feature on squeezed states of the electromagnetic field, J. Opt. Soc. Am. B 4, 1450–1741 (1987).
[CrossRef]

Opt. Lett. (5)

Phys. Rev. A (3)

L. Boivin, “Sagnac-loop squeezer at zero dispersion with a response time for the Kerr nonlinearity,” Phys. Rev. A 52, 754–766 (1995).
[CrossRef] [PubMed]

K. J. Blow, R. Loudon, and S. J. D. Phoenix, “Quantum theory of nonlinear loop mirrors,” Phys. Rev. A 45, 8064–8073 (1992).
[CrossRef]

H. P. Yuen, “Two-photon coherent states of the radiation field,” Phys. Rev. A 13, 2226–2243 (1976).
[CrossRef]

Phys. Rev. Lett. (2)

L. Boivin, F. X. Kärtner, and H. A. Haus, “Analytical solution to the quantum field theory of self-phase modulation with a finite response time,” Phys. Rev. Lett. 73, 240–243 (1994).
[CrossRef] [PubMed]

R. M. Shelby, M. D. Levenson, R. G. DeVoe, S. H. Perlmutter, and D. F. Walls, “Broad-band parametric deamplification of quantum noise in an optical fiber,” Phys. Rev. Lett. 57, 691–694 (1986).
[CrossRef] [PubMed]

Other (15)

The peak Raman gain in fused silica fiber occurs at ω0/2π= 13.2 THz, so our Ω0 choice implies that γ=4.15 1013 s−1. With this γ value our Γ choice dictates that tP=0.19 ps. Computing the peak Raman gain from these parameter values, along the lines laid out in Ref. 12, then leads to a value ~10 times larger than the actual 1.2 10−13 m/W value found in fused-silica fiber. Likewise, the parameter values that we are using predict a low-frequency Raman limit, kBTRγ/ħω02, that is ~10 times larger at TR=300 K than the 0.03 value of a typical fiber; cf. Ref. 12.

If the maximum singular value is degenerate, this procedure will still yield a λH1 value that we can use in our upper and lower bounds. With the Gaussian-pulse seed, convergence was complete at n=20. The same singular value and eigenfunctions resulted when we used the second-order Gauss–Hermite seed: ϕH1(0)(t)=[1−2(t/tP)2]exp[−(t/ tP)2]/(9πtP2/32)1/4. When we used the first-order Gauss– Hermite function as the seed, i.e., ϕH1(0)(t)(t/tP) exp[−(t/tP)2]/(πtP2/32)1/4, our iteration procedure converged to essentially the same λH1 value but had a somewhat different set of eigenfunctions. Unlike the Fourier transforms of the eigenfunctions obtained by use of the zeroth-order and second-order seed functions, those found with the first-order seed were zero at zero frequency. Inasmuch as any odd symmetry seed function can be shown, analytically, to converge to eigenfunction estimates with no zero-frequency content, we believe that the results obtained from the zeroth-order and second-order seeds correctly capture the values that we need; viz., they provide accurate estimates of {λH1, ϕH1(t), ΦH1(t)}.

A. Shakeel, “Enhanced squeezing in homodyne detection via local-oscillator optimization,” Master’s thesis (Massachusetts Institute of Technology, Cambridge, Mass., 1995).

Pulse compression without pump-phase rotation was previously suggested as a practical LO choice, in which case compression factor much than the r=6 value that suffices with pump-phase rotation is apt to be needed.10

It should be remembered that Fig. 12 presumes a single-resonance H(ω) with Ω0≡2ω0/γ=4 and h̄Γ=ħγ/4kBTR=0.28. These parameters do not accurately represent fused-silica fiber. Specifically, in conjunction with ω0/2π=13.2 THz, the preceding parameters make the peak Raman gain too large. Reducing the peak Raman gain, in our calculations, to a more realistic value would have the effect of increasing σ20) in Fig. 12 relative to Smin(0).

I. S. Gradshteyn and I. M. Rhyzhik, Table of Integrals, Series, and Products (Academic, New York, 1980).

W. M. Siebert, Circuits, Signals, and Systems (McGraw-Hill, New York, 1986), Chap. 15.

Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984), Chap. 17.

In anticipation of the photon-units formulation that we shall employ for the quantum theory to come, we are expressing the classical fields in these units too; see Ref. 4.

H. L. Van Trees, Detection, Estimation, and Modulation Theory (Wiley, New York, 1968), Part I, Chap. 3.

Strictly speaking, we should require that K(x, t, x, t) be square integrable over Ad×[−T/2, T/2] to vouchsafe the eigenfunction–eigenvalue properties that we shall assert. For finite area photodetectors and finite measurement intervals, however, the principal pathological cases that can occur in Eq. (25) arise from covariance functions that are delta correlated in space, time, or both. In Section 3 we shall see how such a case can be handled directly from Eq. (18), with a solution that can be understood from Eq. (25).

Note that the cos(ωt) term in Eq. (47) can be phase shifted by an arbitrary amount without invalidating Eq. (48). Physically, this is because the spectrum analysis measurement is insensitive to radio-frequency (frequency ω) phase shifts.

J. H. Shapiro and L. G. Joneckis, “Enhanced fiber squeezing via local-oscillator pulse compression,” in Proceedings of 1994 IEEE Nonlinear Optics (Institute of Electrical and Electronics Engineers, New York, 1994), pp. 347–349.

R. H. Kingston, Detection of Optical and Infrared Radiation (Springer-Verlag, Berlin, 1978), Chap. 3.

R. M. Gagliardi and S. Karp, Optical Communications (Wiley, New York, 1976), Chap. 6.

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Figures (12)

Fig. 1
Fig. 1

Balanced homodyne detection. Quantum signal and LO fields, with operator-valued complex envelopes ÊS(x, t) and ÊLO(x, t), are combined through a lossless 50/50 beam splitter on the surfaces of two photodetectors, D1 and D2. The resulting photocurrents are then subtracted.

Fig. 2
Fig. 2

Schematic of FWM in a bulk Kerr medium with a Gaussian spatial-response function.

Fig. 3
Fig. 3

Optimum squeezing performance for the bulk Kerr-medium example as a function of the effective nonlinear phase shift.

Fig. 4
Fig. 4

Ratio of the optimum LO's effective nonlinear phase shift to the pump's peak nonlinear phase shift, Φeff/ΦNL, versus normalized pump-beam beam waist, ρrP/rG, for the bulk Kerr-medium example.

Fig. 5
Fig. 5

Ratio of the optimum LO's normalized beam waist, rGP/rG, to the pump beam's normalized beam waist, ρ rP/rG, for the bulk Kerr-medium example.

Fig. 6
Fig. 6

Schematic of FWM in a lossless, dispersionless, single-mode fiber with a noninstantaneous Kerr nonlinearity.

Fig. 7
Fig. 7

Normalized-time plots of the normalized Kerr-nonlinearity response function, (tP/ΛH)1/2h(t), and the normalized pump pulse, eP(t).

Fig. 8
Fig. 8

Normalized-time plot of the maximum-singular-value input eigenfunction, tP1/2ϕH1(t).

Fig. 9
Fig. 9

Normalized-time plot of the maximum-singular-value output eigenfunction, tP1/2ΦH1(t).

Fig. 10
Fig. 10

Normalized-frequency plots of the frequency content of the maximum-eigenvalue eigenfunctions, tP-1/2M(ω)tP-1/2×|ϕ¯H1(ω)|=tP-1/2|Φ˜H1(ω)|; the imaginary part of the Kerr nonlinearity's frequency response, Hi(ω); and the Raman noise's spectral density, Hi(ω)coth(ω/2kBTR).

Fig. 11
Fig. 11

Normalized charge variances (in decibels) sK 10 log10(σK2), versus nonlinear phase shift, ΦNL: bright-fringe (BF) LO, K=BF; weak FWM LO, K=FWM; pulse-compression (PC) LO, K=PC; upper bound on optimum-LO performance, K=UB; lower bound on optimum-LO performance, K=LB.

Fig. 12
Fig. 12

Cw squeezing curves versus nonlinear phase shift, ΦNL: Raman-peak frequency, sR10 log10[σ-2(ω0)], and zero frequency, s010 log10[SN(0)]=10 log10[σ-2(0)].

Equations (173)

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[EˆK(x, t), EˆK(x, t)]=δ(x-x)δ(t-t)
forK=S, LO.
iˆ(t)2q ReAddxEˆS(x, t)ELO*(x, t),
-T/2tT/2,
iˆ(t)2q ReAddxEˆS(x, t)·ELO*(x, t),
-T/2tT/2,
KS(n)(x, t, x, t)ΔEˆS(x, t)ΔEˆS(x, t),
KS(p)(x, t, x, t)ΔEˆS(x, t)ΔEˆS(x, t),
ΔEˆS(x, t)EˆS(x, t)-EˆS(x, t)
Ki(t, t)Δiˆ(t)Δiˆ(t)
=q2Addx|ELO(x, t)|2δ(t-t)+2q2 ReAddxAddxELO(x, t)×KS(n)(x, t, x, t)ELO*(x, t)+2q2 ReAddxAddxELO*(x, t)×KS(p)(x, t, x, t)ELO*(x, t).
KS(n)(x, t, x, t)=KS(p)(x, t, x, t)=0,
KiCS(t, t)Δiˆ(t)Δiˆ(t)=q2 Addx|ELO(x, t)|2δ(t-t),
Qˆ-T/2T/2 dtiˆ(t).
σQ2var(Qˆ)=-T/2T/2 dt -T/2T/2 dtKi(t, t).
σCS2=-T/2T/2dt -T/2T/2dtKiCS(t, t)=q2NLO,
NLO-T/2T/2dt Addx|ELO(x, t)|2
Qˆ2q Im-T/2T/2dt AddxEˆS(x, t)ELO*(x, t),
ξLO(x, t)ELO(x, t)/NLO,
xAd,t[-T/2, T/2],
σN2σQ2/σCS2
=1+2 -T/2T/2 dt -T/2T/2 dt Ad dx Addx×KS(n)(x, t, x, t)ξLO(x, t)ξLO*(x, t)+2 Re-T/2T/2 dt -T/2T/2 dt Ad dx Ad×dxKS(p)(x, t, x, t)ξLO*(x, t)ξLO*(x, t).
ξ(x, t)Re[ξLO(x, t)]Im[ξLO(x, t)],
K(x, t, x, t)KR(n)(x, t, x, t)+KR(p)(x, t, x, t)KI(p)(x, t, x, t)-KI(n)(x, t, x, t)KI(n)(x, t, x, t)+KI(p)(x, t, x, t)KR(n)(x, t, x, t)-KR(p)(x, t, x, t),
σN2=1+2 -T/2T/2 dt -T/2T/2 dt Ad dx Ad×dxξT(x, t)K(x, t, x, t)ξ(x, t),
KS(n)(x, t, x, t)=KS(n)*(x, t, x, t),
KS(p)(x, t, x, t)=KS(p)(x, t, x, t),
K(x, t, x, t)=KT(x, t, x, t).
-T/2T/2 dt Ad dxK(x, t, x, t)ϕn(x, t)=λnϕn(x, t),xAd,
t[-T/2, T/2],n=1, 2, 3,.
-T/2T/2 dt Ad dxϕnT(x, t)ϕm(x, t)=δnm,
n, m=1, 2, 3,.
n=1 ϕn(x, t)ϕnT(x, t)=δ(x-x)δ(t-t),
x, xAd,t, t[-T/2, T/2].
ξ(x, t)=n=1 ξnϕn(x, t),
ξn-T/2T/2 dt Ad dxϕnT(x, t)ξ(x, t),
n=1 ξn2=-T/2T/2 dt Ad dxξT(x, t)ξ(x, t)=1.
σN2=1+2 n=1 ξn2λn.
σmin2minξ(σN2)=1+2 minn(λn),
KS(n)(x, t+τ, x, t)=-dω2πSS(n)(x, x, ω)×exp(-iωτ),
KS(p)(x, t+τ, x, t)=-dω2πSS(p)(x,x, ω)×exp(-iωτ),
Si(ω)=- dτKi(τ)exp(iωτ)
=q2PLO+2q2PLO Ad dx Ad dx×SS(n)(x, x, ω)ξLO(x)ξLO*(x)+2q2PLO ReAd dx Ad dx×SS(p)(x, x, ω)ξLO*(x)ξLO*(x),
ξLO(x)ELO(x)/PLOforxAd,
PLOAd dx|ELO(x)|2
SiCS(ω)=q2PLO.
ξ(x)Re[ξLO(x)]Im[ξLO(x)],
S(x, x, ω)SR(n)(x, x, ω)+SR(p)(x, x, ω)SI(p)(x, x, ω)-SI(n)(x, x, ω)SI(n)(x, x, ω)+SI(p)(x, x, ω)SR(n)(x, x, ω)-SR(p)(x, x, ω),
SN(ω)=1+2 Ad dx Ad dxξT(x)S(x, x, ω)ξ(x).
S(x, x, ω)=n=1λn(ω)ϕn(x, ω)ϕnT(x, ω),
x, xAd,
AddxS(x, x, ω)ϕn(x, ω)=λn(ω)ϕn(x, ω),
xAd,n=1, 2, 3,.
minξ[SN(ω)]=1+2 minn[λn(ω)],
Sminminξ,ω SN(ω)=1+2 minn,ω[λn(ω)],
ξLO(x, t)=(2/T)1/2 cos(ωt)ξLO(x),t[-T/2, T/2],
limTσN2=SN(ω),
EˆK(x, t)=EˆK(t)ζ(x)+EˆKvac(x, t),K=S, LO,
[EˆK(t), EˆK(t)]=δ(t-t),K=S, LO,
iˆ(t)2q Re[EˆS(t)ELO*(t)],-T/2tT/2
σN2=1+2 -T/2T/2 dt -T/2T/2 dtξT(t)K(t, t)ξ(t),
K(t, t)KR(n)(t, t)+KR(p)(t, t)KI(p)(t, t)-KI(n)(t, t)KI(n)(t, t)+KI(p)(t, t)KR(n)(t, t)-KR(p)(t, t),
-T/2T/2dtK(t, t)ϕn(t)=λnϕn(t),t[-T/2, T/2],
n=1, 2, 3, ,
σmin2minξ(σN2)=1+2 minn(λn).
EˆS(x, t)=μ(x, t)EˆP(x, t)+ν(x, t)EˆP(x, t),
|μ(x, t)|2-|ν(x, t)|2=1,xAd,
t[-T/2, T/2]
KS(n)(x, t, x, t)=|ν(x, t)|2δ(x-x)δ(t-t),
KS(p)(x, t, x, t)=μ(x, t)ν(x, t)δ(x-x)δ(t-t),
σN2=-T/2T/2dtAddx|μ(x, t)ξLO*(x, t)+ν*(x, t)ξLO(x, t)|2.
σN2-T/2T/2dtAddx[|μ(x, t)|-|ν(x, t)|]2|ξLO(x, t)|2
minx, t[|μ(x, t)|-|ν(x, t)|]2.
|ξLO(x, t)|2δ(x-x0)δ(t-t0),
[|μ(x0, t0)|-|ν(x0, t0)|]2=minx, t[|μ(x, t)|-|ν(x, t)|]2.
Λ(x, t)ξ(x, t)=λnξ(x, t),xAd,
t[-T/2, T/2],
Λ(x, t)|ν(x, t)|2+Re[μ(x, t)ν(x, t)]Im[μ(x, t)ν(x, t)]Im[μ(x, t)ν(x, t)]|ν(x, t)|2-Re[μ(x, t)ν(x, t)].
minx, t[σN2(x, t)]1+2 minx, t[λn(x, t)]
=[|μ(x0, t0)|-|ν(x0, t0)|]2.
T2E(x, z)+2ikE(x, z)z
=-2kκ  dxg(x-x)|E(x, z)|2E(x, z),
0zL
g(x)exp(-x·x/rG2)/πrG2
E(x, z)z=iκ  dxg(x-x)|E(x, z)|2E(x, z),
0zL.
ES(x)=exp[iκL  dxg(x-x)|EP(x)|2]EP(x).
ΔEˆS(x, t)=expiκL  dxg(x-x)|EP(x)|2×ΔEˆP(x, t)+iκL  dxEP(x)×g(x-x)EP*(x)ΔEˆP(x, t)+iκL  dxEP(x)×g(x-x)EP(x)ΔEˆP(x, t).
EP(x)=IP exp(-x·x/rP2),
KS(n)(x, t, x, t)=ΦNL2eP(x)eP(x)δ(t-t) dx×g(x-x)eP2(x)g(x-x),
KS(p)(x, t, x, t)=expiΦNL  dxg(x-x)eP2(x)×expiΦNL dxg(x-x)eP2(x)×δ(t-t)iΦNLeP(x)×g(x-x)eP(x)-ΦNL2 dx×g(x-x)eP2(x)×g(x-x),
ΦNLκIPL
eP(x)exp(-x·x/rP2)
ξLO(x)ξLO(x)exp-iΦNL  dxg(x-x)eP2(x);
 dxS(x, x)ϕn(x)=λnϕn(x),
S(x, x)eP(x)eP(x)0ΦNLg(x-x)ΦNLg(x-x)2ΦNL2  dxg(x-x)eP2(x)g(x-x).
KG(2)(x, x) dxKG(x, x)KG(x, x),
-<x,x<
KG(x, x)exp{-(x/rP)2-[(x-x)/rG]2-(x/rP)2}/πrG2,
-<x,x<.
ξnm(+)(x)sin(θnm)cos(θnm)ϕGn(x)ϕGm(y),
n, m=0, 1, 2, ,
ξnm(-)(x)cos(θnm)-sin(θnm)ϕGn(x)ϕGm(y),
n, m=0, 1, 2, 
λnm(±)±|νnm|(|μnm|±|νnm|),
SN(±)=1+2λnm(±)=(|μnm|±|νnm|)2.
ξLO(x)=ξopt(x)expiΦNL  dxg(x-x)eP2(x),
ξopt(x)cos(θ00)-sin(θ00)ϕG0(x)ϕG0(y).
Smin=1+2Φeff2-2Φeff1+Φeff2,
ϕGn(x)=exp[-(x/rGP)2]Hn(2x/rGP)(πrGP2/2)1/4(2nn!)1/2,
λGn=ρ(1+ρ2-1+2ρ2)n/2(1+ρ2+1+2ρ2)(n+1)/2,
ξopt(x)=exp[iΦNL(rGP/rP)2 exp(-x·xrGP2/rP4)]×exp(-iθ00)exp(-x·x/rGP2)/πrGP2/2,
Φeff=ρ2ΦNL/(1+ρ2+1+2ρ2)
ϕGnm(x)expiΦNL  dxg(x-x)×eP2(x)ϕGn(x)ϕGm(y).
ΔEˆSnm(t)=μnmΔEˆPnm(t)+νnmΔEˆPnm(t).
smin10 log10(1+2Φeff2-2Φeff1+Φeff2),
ES(t)=expiκL  dth(t-t)|EP(t)|2EP(t)
ΔEˆS(t)=expiκL  dth(t-t)|EP(t)|2ΔEˆP(t)+iκL  dtEP(t)h(t-t)EP*(t)ΔEˆP(t)+iκL  dtEP(t)h(t-t)EP(t)ΔEˆP(t)+iӨˆ(t)EP(t),
[Өˆ(t), Өˆ(t)]=-iκL[h(t-t)-h(t-t)]
=-i(κL/π) dωHi(ω)sin[ω(t-t)],
H(ω) dth(t)exp(iωt).
KӨ(τ)Өˆ(t+τ)Өˆ(t)+Өˆ(t)Өˆ(t+τ)/2=(κL/2π) dωHi(ω)coth(ω/2kBTR)cos(ωτ),
h(t)=ω0 exp(-γt/2)sin(ω02-γ2/4t)ω02-γ2/4,t0,
H(ω)=ω02ω02-ω2-iωγ,-<ω<,
ΦNLκPPL,
KS(n)(t, t)=eP(t)eP(t)iΦNL[h(t-t)-h(t-t)]/2+ΦNL2  dth(t-t)eP2(t)h(t-t)+ΦNLKӨ(t-t),
KS(p)(t, t)=expiΦNL dth(t-t)eP2(t)+ dth(t-t)eP2(t)eP(t)eP(t)×iΦNL[h(t-t)+h(t-t)]/2-ΦNL2  dth(t-t)eP2(t)h(t-t)-ΦNLKӨ(t-t),
ξLO(t)ξLO(t)expiΦNL  dth(t-t)eP2(t)
 dtK(t, t)ϕn(t)=λnϕn(t),
K(t, t)0ΦNLKH(t, t)ΦNLKH(t, t)2ΦNL2KH(2)(t, t)+2ΦNLeP(t)KӨ(t-t)eP(t),
KH(2)(t, t) dtKH(t, t)KH(t, t),
-<t,t<.
KH(t, t)=n=1λHnΦHn(t)ϕHn(t),
 dtKH(2)(t, t)ΦHn(t)=λHnΦHn(t);
 dtKH(t, t)ΦHn(t)=λHnϕHn(t).
ξopt(t)cos(θ)ψ(t)-sin(θ)Ψ(t),
σmin2=1-2 sin(2θ)ΦNLn=1ψnΨnλHn+2[1-cos2θ]ΦNL2n=1Ψn2λHn+ΦNLλRopt,
λRopt dt  dtΨ(t)eP(t)KӨ(t-t)eP(t)Ψ(t).
σmin21-2 sin(2θ)ΦNLn=1Ψn2λHn1/2
+2[1-cos(2θ)]
×ΦNL2 n=1 Ψn2λHn+ΦNLλRopt.
σmin21+2ΦNL2 n=1 Ψn2λHn+2ΦNLλRopt-2ΦNL2 n=1 Ψn2λHn+ΦNL2 n=1 Ψn2λHn+ΦNLλRopt21/2.
λ˜Ropt= dt  dtΨ(t)eP(t)KӨ(t-t)eP(t)Ψ(t) dt  dtΨ(t)eP(t)KH(2)(t-t)eP(t)Ψ(t)
 dt  dtΨ(t)eP(t)KӨ(t-t)eP(t)Ψ(t) dt  dtΨ(t)eP(t)[ dth(t-t)h(t-t)]eP(t)Ψ(t)
= dω|Ψ˜(ω)|2Hi(ω)coth(ω/2kBTR) dω|Ψ˜(ω)|2|H(ω)|2
minω[Hi(ω)coth(ω/2kBTR)/|H(ω)|2]
=minω[ωγ coth(ω/2kBTR)/ω02]=λ˜RL2kBTRγ/ω02,
σmin2σLB21+2ΦNL2λH1+2ΦNLλH1λ˜RL
-2ΦNL2λH1+(ΦNL2λH1+ΦNLλH1λ˜RL)2.
ξ(t)=cos(θ˜)ϕH1(t)-sin(θ˜)ΦH1(t),
σmin2σUB21+2ΦNL2λH1+2ΦNLλRU-2ΦNL2λH1+(ΦNL2λH1+ΦNLλRU)2,
λRU dt  dtΦH1(t)eP(t)×KӨ(t-t)eP(t)ΦH1(t),
σmin2=1+2Φeff2-2Φeff2+Φeff4,
ξBF(t)=CBF expiΦNL  dth(t-t)eP2(t)eP(t),
ξPC(t)=CPC expiΦNL  dth(t-t)eP2(t)eP(rt),
ξFWM(t)=CFWM expiΦNL  dth(t-t)eP2(t)×exp(-iϕ)+2i cos(ϕ)ΦNL  dt×h(t-t)eP2(t)eP(t),
ΛHn=1λHn= dtKH(2)(t, t),
kH(t, t)KH(t, t)/ΛH
=n=1λHnΦHn(t)ϕHn(t),
 dtkH(t, t)ϕH1(n)(t)=λH1(n+1)ΦH1(n+1)(t),
n=0, 2, 4, ,
 dtkH(t, t)ΦH1(n)(t)=λH1(n+1)ϕH1(n+1)(t),
n=1, 3, 5,.
SN(ω)=1+2ΦNL2|H(ω)|2+2ΦNLHi(ω)coth(ω/2kBTR)-2{ΦNL2Hr2(ω)+[ΦNL2|H(ω)|2+ΦNLHi(ω)coth(ω/2kBTR)]2}1/2,
K(τ)=0ΦNLh(τ)ΦNLh(-τ)2ΦNL2  dth(t+τ)h(t)+2ΦNLKӨ(τ).
ξω(+)(t)2T sin(θω)cos(ωt)cos(θω)cos(ωt-αω),
t[-T/2, T/2],
ξω(-)(t)2T cos(θω)cos(ωt)-sin(θω)cos(ωt-αω),
t[-T/2, T/2]
tan(2θω)=ΦNL|H(ω)|ΦNL2|H(ω)|2+ΦNLHi(ω)coth(ω/2kBTR).
σ±2(ω)=1+2ΦNL2|H(ω)|2+2ΦNLHi(ω)coth(ω/2kBTR)±2{ΦNL2|H(ω)|2+[ΦNL2|H(ω)|2+ΦNLHi(ω)coth(ω/2kBTR)]2}1/2,
σ-2(ω)Hi(ω)coth(ω/2kBTR)/|H(ω)|2ΦNL,
ΔE˜ˆS(ω) dtΔEˆS(t)exp(iωt)=exp(iΦNL){[1+iΦNLH(ω)]ΔE˜ˆP(ω)+iΦNLH(ω)ΔE˜ˆP(-ω)+iΦNLӨ˜ˆ(ω)},

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