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] 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[2–7], 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[8–14]–in which the projections of zero-point fluctuations are manipulated through their phase-sensitive nature. Classical analogues to squeezing have also been demonstrated[15–17]. 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[18–21] 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=υ_gt, where υ_g is the group velocity. The operator obeys the commutation relation[22–23]

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

The commutator associated with the perturbation operator is:

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

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

where b=${\xi }_0^2$/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

The perturbation is expanded in a complete set

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

where Δ${Â}_n^{\left(1\right)}$ and Δ${Â}_n^{\left(2\right)}$ 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

The expansion is carried out to first order in ΔX and Δω, the perturbations are replaced by operators, and the result is equated to ΔÂ_1ψ̂1(x, T) with the result

where we have introduced the momentum operator

The position and momentum operators can be related to the in-phase and quadrature components of the expansion coefficient ΔÂ₁

The operators Δ${Â}_1^{\left(1\right)}$ and Δ${Â}_1^{\left(2\right)}$ obey a commutation relation that can be gleaned from (3) and (8)

We find the commutation relation for position and momentum

where 〈n〉=$A_0^2$ 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/dβ ². This leads to an equation relating the position after propagation over a time T to its initial value ΔX̂(0)

A phase modulator multiplies the pulses of equation [7] by iM cosΩ _M (t -ΔX/υ_g ), where M is the depth of modulation, and Ω_M 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

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

The matrices are, respectively, for propagation through dispersion:

and through a phase modulator

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 ${\omega }_1^{″}$ and delay T ₁ is followed by a phase modulator, and then followed by another dispersive fiber ${\omega }_2^{″}$ 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 ${\omega }^{″}T$ 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], 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.

Download Full Size | PDF

 

Fig. 2. R=(M${\mathrm{\Omega }}_M^2$ /${\upsilon }_g^2$ )²〈|ΔX̂|〉²/〈|ΔP̂|〉² for the cases (a) R_in =2 an d (b) R_in =1 where S≡R_out /R_in , X≡(M${\mathrm{\Omega }}_M^2$ /${\upsilon }_g^2$ )ω ^″ T ₁, and Y≡(M${\mathrm{\Omega }}_M^2$ /${\upsilon }_g^2$ )${\omega }_2^{″}$ T ₂. The pulse position fluctuations are reduced in the regions where S<1. Regions for S>1 are not shown.

Download Full Size | PDF

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.