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Optics Express

  • Editor: J. H. Eberly
  • Vol. 8, Iss. 12 — Jun. 2, 2001
  • pp: 664–669
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Action on pulse position and momentum using dispersion and phase modulation

M. E. Grein, H. A. Haus, L. A. Jiang, and E. P. Ippen  »View Author Affiliations


Optics Express, Vol. 8, Issue 12, pp. 664-669 (2001)
http://dx.doi.org/10.1364/OE.8.000664


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Abstract

The timing jitter and frequency jitter of quantized optical pulses obey Heisenberg’s uncertainty principle. We show how one jitter may be reduced at the expense of the other, using dispersion and phase modulation.

© Optical Society of America

1 Introduction

In this paper, we study linear Hamiltonian processes, such as dispersion in an optical fiber, and time-dependent processes, such as phase modulation, which affect the position (timing) and momentum (carrier frequency) of an optical pulse. These processes are closely analogous to the quantum propagation of a free particle[1

1. L. S. Brown, Quantum Field Theory (Cambridge University Press, 1992).

] and its focusing by a lens. While the problem of quantizing a propagating electric field in both linear dispersionless inhomogeneous dielectric media and in linear homogeneous dielectric media with dispersion have been studied[2

2. L. Knöll, W. Vogel, and D.-G. Welsch, “Action of passive, lossless optical systems in quantum optics,” Phys. Rev. A 36, 3803–3818 (1987). [CrossRef] [PubMed]

7

7. P. W. Milonni, “Field quantization and radiative processes in dispersive dielectric media,” J. Mod. Opt. 42, 1991 (1995).

], the effects of phase modulation have not. The combination of dispersion and phase modulation can reduce the position fluctuations (timing jitter) at the expense of momentum fluctuations (carrier frequency jitter). This can be contrasted with generation of squeezed states–using parametric processes utilizing either second order (χ (2)) or third order (χ (3)) nonlinearities[8

8. R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, and J. F. Valley, “Observation of squeezed states generated by 4-wave mixing in an optical cavity,” Phys. Rev. Lett. 55, 2409–2412 (1985). [CrossRef] [PubMed]

14

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

]–in which the projections of zero-point fluctuations are manipulated through their phase-sensitive nature. Classical analogues to squeezing have also been demonstrated[15

15. F. Hong-Yi and J. VanderLinde, “Squeezed-state wave functions and their relation to classical phase-space maps,” Phys. Rev. A 40, 4785 (1989). [CrossRef]

17

17. F. DiFilippo, V. Natarajan, K. R. Boyce, and D. E. Pritchard, “Classical amplitude squeezing for precisionm easurements,” Phys. Rev. Lett. 68, 2859 (1992). [CrossRef] [PubMed]

]. The approach here achieves a reduction of position uncertainty by renormalization of the excitations and is entirely classical. Aside from its theoretical interest, this finding is of practical importance: the reduction of the timing jitter of a pulse can improve the accuracy of optical sampling for analog-to-digital conversion[18

18. A. E. Siegman and D. J. Kuizenga, “Proposed method for measuring picosecond pulse widths and pulse shapes in CW mode-locked lasers,” IEEE J. Quantum Electron. 6, 212–215 (1970). [CrossRef]

21

21. J. C. Twichell and R. Helkey, “Phase-encoded optical sampling for analog-to-digital converters,” Phot. Tech. Lett. 12, 1237–1239 (2000). [CrossRef]

] and for retiming pulses at the end of a transmission link. A process that reduces the timing jitter at the expense of frequency jitter leads to improvement of the timing signal, since the optical bandwidth of detectors is large enough to be insensitive to frequency jitter. We consider Gaussian pulses, such as are produced by active mode-locking, and quantize them. We develop equations of motion for the position and momentum operators as affected by dispersion and phase modulation. Then we show that this system conserves commutator brackets and thus the area of the uncertainty ellipse. Reduction of pulse position fluctuations (timing jitter) is demonstrated.

2 Quantization of the Optical Pulse

We consider a periodic train of Gaussian pulses. The envelope of the electric field amplitude is quantized if described in terms of the annihilation operator â(T, x), where T and x represent two time scales: the slow time scale T of pulse evolution, and the fast time scale (expressed as a spatial scale) x=υgt, where υg is the group velocity. The operator obeys the commutation relation[22

22. H. A. Haus, “Steady-state quantum analysis of linear systems,” Proc. of the IEEE 58, 1599–1611 (1970). [CrossRef]

23

23. H. A. Haus and J. A. Mullen, “Quantum noise in linear amplifiers,” Phys. Rev. 128, 2407 (1962). [CrossRef]

]

[â(T,x),â(T,x)]=δ(xx)
(1)

We separate the excitation into a classical c-number part a0 in terms of which the pulses are described, and a perturbation operator Δâ that characterizes the noise

â=a0+Δâ
(2)

The commutator associated with the perturbation operator is:

[Δâ(T,x),Δâ(T,x)]=δ(xx)
(3)

Focusing on one of the pulses, the c-number part of the excitation is of the form

a0=A0ψ0(xξ)
(4)

where ψ 0 is the zeroth-order Hermite-Gaussian of the set:

ψn(xξ)=12nn!ξπHn(xξ)eix22ω(Tib)ei(n+1)ϕ
(5)

where b=ξ02/ω , ω =d 2 ω/d β 2 is the group-velocity dispersion, and ϕ=tan-1(z/b). This set expresses, in general, chirped pulses after a time T starting with minimum width ξ 0. The pulsewidth changes when the pulse propagates in a dispersive medium according to the law

ξ2=ξ02[1+(Tb)2]
(6)

The perturbation is expanded in a complete set

Δâ=ΣnΔÂnψn(xξ)
(7)

The annihilation operators ΔÂ n can be expressed as sums of Hermitian operators

ΔÂn=ΔÂn(1)+iΔÂn(2)
(8)

where ΔÂn(1) and ΔÂn(2) are in quadrature. A displacement and a carrier frequency change are represented by the first-order Hermite-Gaussian. The coefficient ΔA 1 is obtained from the expansion of a pulse that has been displaced by ΔX and frequency shifted by Δω. The pulse is described by

A0ψ0(x,T,ΔX,Δω)=A0ξπexp[i(xΔX+ωΔωTνg)22ω(Tib)]exp[iΔωxνg]exp[iϕ]
(9)

The expansion is carried out to first order in ΔX and Δω, the perturbations are replaced by operators, and the result is equated to ΔÂ1ψ̂1(x, T) with the result

ΔÂ1=12(ΔX̂ξ0iΔω̂νgξ0)=12(ΔX̂ξ0+iΔP̂ξ0)
(10)

where we have introduced the momentum operator

Δω̂νg=ΔP̂
(11)

ΔX̂=2A0ξ0ΔÂ1(1)andΔP̂=2A01ξ0ΔÂ1(2)
(12)

The operators ΔÂ1(1) and ΔÂ1(2) obey a commutation relation that can be gleaned from (3) and (8)

[ΔÂ1(1),ΔÂ1(2)]=i2
(13)

We find the commutation relation for position and momentum

[ΔX̂,ΔP̂]=in
(14)

where 〈n〉=A02 is the average photon number.

A carrier frequency shift changes the group velocity of the pulse propagation. A pulse propagating over a time T changes its position in a manner proportional to the carrier frequency change and the dispersion ω =d 2 ω/ 2. This leads to an equation relating the position after propagation over a time T to its initial value Δ(0)

ΔX̂(T)=ΔX̂(0)+ωTΔP̂(0)
(15)

A phase modulator multiplies the pulses of equation [7

7. P. W. Milonni, “Field quantization and radiative processes in dispersive dielectric media,” J. Mod. Opt. 42, 1991 (1995).

] by iM cosΩ M (tX/υg ), where M is the depth of modulation, and ΩM is the modualation frequency. When expanded to first order in ΔX, we find a first-order Hermite Gaussian in quadrature. This term produces a momentum perturbation

ΔP̂out=ΔP̂inMΩM2vg2ΔX̂
(16)

We have found that the operators Δ and Δ between input and output experience transformations that can be described by ABCD matrices

[ΔX̂outΔP̂out]=[ABCD][ΔX̂inΔP̂in]
(17)

The matrices are, respectively, for propagation through dispersion:

[ABCD]=[1ωT01]
(18)

and through a phase modulator

[ABCD]=[10MΩM2vg21]
(19)

The determinants of these matrices are unity, and cascading of the components leads to ABCD matrices that also have a unity determinant as required by Heisenberg’s uncertainty principle.

An example of a position fluctuation reduction arrangement is shown in Fig. 1. A section of dispersive fiber ω1 and delay T 1 is followed by a phase modulator, and then followed by another dispersive fiber ω2 with delay T 2. The computed position fluctuation reduction is shown in Figs. 2a and 2b for the cases where the input position and momentum fluctuations are uncorrelated. The uncertainty of the pulse position at the output has been reduced at the expense of the momentum uncertainty in regions of phase space in which S<1. We find that the best position fluctuation reduction occurs for the case where the modulation depth and modulation frequency are large, and the dispersion ωT is large and can be either anomalous or normal. Experimentally, the measurement of position fluctuations is classical and can be achieved using conventional rf phase noise detection techniques employing direct detection with a fast photodetector, a double-balanced mixer, and a local oscillator [24

24. T. R. Clark, T. F. Carruthers, P. J. Matthews, and I. N. Duling III, “Phase noise measurements of ultrastable 10 GHz harmonically modelocked fibre laser,” Electron. Lett. 35, 720–721 (1999). [CrossRef]

], and the position fluctuations before and after the setup of Fig. 1 can be compared.

Fig. 1. Schematic of system analyzed for Fig. 2 where ω is the group-velocity dispersion, and T the propagation delay.
Fig. 2. R=(MΩM2 /υg2 )2〈|Δ|〉2/〈|Δ|〉2 for the cases (a) Rin =2 an d (b) Rin =1 where SRout /Rin , X≡(MΩM2 /υg2 )ω T 1, and Y≡(MΩM2 /υg2 )ω2 T 2. The pulse position fluctuations are reduced in the regions where S<1. Regions for S>1 are not shown.

3 Conclusions

We have shown that the perturbation operators of pulse position and momentum, when acted upon by dispersive propagation and/or phase modulation, obey an ABCD matrix transformation. The uncertainty ellipse of position and momentum fluctuations can be transformed using a system containing phase modulators and dispersive propagation segments. The reduction of position fluctuations can be used to improve timing signals obtained from detected pulse trains. The system need not operate at the minimum uncertainty limit to be of use.

4 Acknowledgments

References and links

1.

L. S. Brown, Quantum Field Theory (Cambridge University Press, 1992).

2.

L. Knöll, W. Vogel, and D.-G. Welsch, “Action of passive, lossless optical systems in quantum optics,” Phys. Rev. A 36, 3803–3818 (1987). [CrossRef] [PubMed]

3.

R. J. Glauber and M. Lewenstein, “Quantum optics of dielectic media,” Phys Rev. A 43, 467–491 (1991). [CrossRef] [PubMed]

4.

H. Khosravi and R. Loudon, “Vacuum field fluctuations and spontaneous emission in a dielectric slab,” Proc. R. Soc. Lon don Ser. A 436, 373–389 (1992). [CrossRef]

5.

P. D. Drummond, “Electromagnetic quantization in dispersive inhomogeneous nonlinear dielectrics,” Phys. Rev. A 42, 6845–6857 (1990). [CrossRef] [PubMed]

6.

B. Huttner, J. J. Baumberg, and S. M. Barnett, “Canonical quantization of light in a linear dielectric,” Europhys. Lett. 16, 177 (1991). [CrossRef]

7.

P. W. Milonni, “Field quantization and radiative processes in dispersive dielectric media,” J. Mod. Opt. 42, 1991 (1995).

8.

R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, and J. F. Valley, “Observation of squeezed states generated by 4-wave mixing in an optical cavity,” Phys. Rev. Lett. 55, 2409–2412 (1985). [CrossRef] [PubMed]

9.

M. J. Potasek and B. Yurke, “Dissipative effects on squeezed light generated in systems governed by the nonlinear Schrödinger equation,” Phys. Rev. A 38, 1335–1348 (1988). [CrossRef] [PubMed]

10.

L. Wu, H. J. Kimble, J. L. Hall, and H. Wu, “Generation of squeezed states by parametric down conversion,” Phys. Rev. Lett. 57, 2520 (1986). [CrossRef] [PubMed]

11.

M. Xiao, L. Wu, and H. J. Kimble, “Precision measurement beyond the shot-noise limit,” Phys. Rev. Lett. 53, 278 (1987). [CrossRef]

12.

M. W. Maeda, P. Kumar, and J. H. Shapiro, “Observation of squeezed noise produced by forward four-wave mixing in sodium vapor,” Op. Lett. 12, 161 (1987). [CrossRef]

13.

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

14.

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

15.

F. Hong-Yi and J. VanderLinde, “Squeezed-state wave functions and their relation to classical phase-space maps,” Phys. Rev. A 40, 4785 (1989). [CrossRef]

16.

D. Rugar and P. Grutter, “Mechanical parametric amplification and thermomechanical noise squeezing,” Phys. Rev. Lett. 67, 699 (1991). [CrossRef] [PubMed]

17.

F. DiFilippo, V. Natarajan, K. R. Boyce, and D. E. Pritchard, “Classical amplitude squeezing for precisionm easurements,” Phys. Rev. Lett. 68, 2859 (1992). [CrossRef] [PubMed]

18.

A. E. Siegman and D. J. Kuizenga, “Proposed method for measuring picosecond pulse widths and pulse shapes in CW mode-locked lasers,” IEEE J. Quantum Electron. 6, 212–215 (1970). [CrossRef]

19.

D. H. Auston, “Picosecond optoelectronic switching and gating in silicon,” Appl. Phys. Lett. 26, 101–103 (1975). [CrossRef]

20.

H. F. Taylor, “An electrooptic analog-to-digital converter-design and analysis,” IEEE J. Quantum. Electron. 15, 210–216 (1979). [CrossRef]

21.

J. C. Twichell and R. Helkey, “Phase-encoded optical sampling for analog-to-digital converters,” Phot. Tech. Lett. 12, 1237–1239 (2000). [CrossRef]

22.

H. A. Haus, “Steady-state quantum analysis of linear systems,” Proc. of the IEEE 58, 1599–1611 (1970). [CrossRef]

23.

H. A. Haus and J. A. Mullen, “Quantum noise in linear amplifiers,” Phys. Rev. 128, 2407 (1962). [CrossRef]

24.

T. R. Clark, T. F. Carruthers, P. J. Matthews, and I. N. Duling III, “Phase noise measurements of ultrastable 10 GHz harmonically modelocked fibre laser,” Electron. Lett. 35, 720–721 (1999). [CrossRef]

OCIS Codes
(140.3460) Lasers and laser optics : Lasers
(270.0270) Quantum optics : Quantum optics
(270.6570) Quantum optics : Squeezed states

ToC Category:
Research Papers

History
Original Manuscript: April 24, 2001
Published: June 2, 2001

Citation
Matthew Grein, Hermann Haus, Leaf Jiang, and Erich Ippen, "Action on pulse position and momentum using dispersion and phase modulation," Opt. Express 8, 664-669 (2001)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-8-12-664


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References

  1. L. S. Brown, Quantum Field Theory (Cambridge University Press, 1992).
  2. L. Knöll, W. Vogel, and D.-G. Welsch, "Action of passive, lossless optical systems in quantum optics," Phys. Rev. A 36, 3803-3818 (1987). [CrossRef] [PubMed]
  3. R. J. Glauber and M. Lewenstein, "Quantum optics of dielectic media," Phys Rev. A 43, 467-491 (1991). [CrossRef] [PubMed]
  4. H. Khosravi and R. Loudon, "Vacuum field fluctuations and spontaneous emission in a dielectric slab," Proc. R. Soc. London Ser. A 436, 373-389 (1992). [CrossRef]
  5. P. D. Drummond, "Electromagnetic quantization in dispersive inhomogeneous nonlinear dielectrics," Phys. Rev. A 42, 6845-6857 (1990). [CrossRef] [PubMed]
  6. B. Huttner, J. J. Baumberg, and S. M. Barnett, "Canonical quantization of light in a linear dielectric," Europhys. Lett. 16, 177 (1991). [CrossRef]
  7. P. W. Milonni, "Field quantization and radiative processes in dispersive dielectric media," J. Mod. Opt. 42, 1991 (1995).
  8. R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, and J. F. Valley, "Observation of squeezed states generated by 4-wave mixing in an optical cavity," Phys. Rev. Lett. 55, 2409-2412 (1985). [CrossRef] [PubMed]
  9. M. J. Potasek and B. Yurke, "Dissipative effects on squeezed light generated in systems governed by the nonlinear Schrödinger equation," Phys. Rev. A 38, 1335-1348 (1988). [CrossRef] [PubMed]
  10. L. Wu, H. J. Kimble, J. L. Hall, and H. Wu, "Generation of squeezed states by parametric down conversion," Phys. Rev. Lett. 57, 2520 (1986). [CrossRef] [PubMed]
  11. M. Xiao, L. Wu, and H. J. Kimble, "Precision measurement beyond the shot-noise limit," Phys. Rev. Lett. 53, 278 (1987). [CrossRef]
  12. M. W. Maeda, P. Kumar, and J. H. Shapiro, "Observation of squeezed noise produced by forward four-wave mixing in sodium vapor," Opt. Lett. 12, 161 (1987). [CrossRef]
  13. R. M. Shelby, M. D. Levenson, S. H. Perlmutter, R. G. DeVoe, and D. F. Walls, "Broad-band parametric deamplification of quantum noise in an optical fiber," Phys. Rev. B 57, 691 (1986). [CrossRef]
  14. M. Shirasaki and H. A. Haus, "Squeezing of pulses in a nonlinear interferometer," J. Opt. Soc. Am. B 7, 30 (1990). [CrossRef]
  15. F. Hong-Yi and J. VanderLinde, "Squeezed-state wave functions and their relation to classical phase-space maps," Phys. Rev. A 40, 4785 (1989). [CrossRef]
  16. D. Rugar and P. Grutter, "Mechanical parametric amplification and thermomechanical noise squeezing," Phys. Rev. Lett. 67, 699 (1991). [CrossRef] [PubMed]
  17. F. DiFilippo, V. Natarajan, K. R. Boyce, and D. E. Pritchard, "Classical amplitude squeezing for precision measurements," Phys. Rev. Lett. 68, 2859 (1992). [CrossRef] [PubMed]
  18. A. E. Siegman and D. J. Kuizenga, "Proposed method for measuring picosecond pulse widths and pulse shapes in CW mode-locked lasers," IEEE J. Quantum Electron. 6, 212-215 (1970). [CrossRef]
  19. D. H. Auston, "Picosecond optoelectronic switching and gating in silicon," Appl. Phys. Lett. 26, 101-103 (1975). [CrossRef]
  20. H. F. Taylor, "An electrooptic analog-to-digital converter-design and analysis," IEEE J. Quantum. Electron. 15, 210-216 (1979). [CrossRef]
  21. J. C. Twichell and R. Helkey, "Phase-encoded optical sampling for analog-to-digital converters," Phot. Tech. Lett. 12, 1237-1239 (2000). [CrossRef]
  22. H. A. Haus, "Steady-state quantum analysis of linear systems," Proc. of the IEEE 58, 1599-1611 (1970). [CrossRef]
  23. H. A. Haus and J. A. Mullen, "Quantum noise in linear amplifiers," Phys. Rev. 128, 2407 (1962). [CrossRef]
  24. T. R. Clark, T. F. Carruthers, P. J. Matthews, and I. N. Duling III, "Phase noise measurements of ultrastable 10 GHz harmonically modelocked fibre laser," Electron. Lett. 35, 720-721 (1999). [CrossRef]

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