## Abstract

Correlation dependent, propagation-induced shifts in the generalized spectra of cyclostationary, random fields are predicted. This result generalizes the Wolf shift for stationary fields and is applicable to periodic trains of fast pulses such as might be generated in comb spectroscopy or other mode-locked pulsed systems. Examples illustrate these shifts for intrinsically stationary fields and the fields generated by a mode-locked laser.

©2009 Optical Society of America

## 1. Cyclostationary fields

The theory of statistically stationary optical fields, coherence theory [1, 2], provides a means to relate time-averaged measurements of optical intensities to the statistical properties of a scatterer or source. However, as ultrafast optical devices become more common and applications of short pulses become more wide-spread, it is clear that the theory of stationary fields alone is not sufficient to interpret measurements of optical fields or intensities [3]. A framework for analyzing physically realizable measurements from statistically periodic sequences of fast pulses based on the theory of cyclostationary processes [4] has recently been presented [5]. Cyclostationary signals differ from stationary signals in that the field may be correlated across distinct temporal frequencies. Thus the Wiener-Khintchine theorem [2] that defines the power spectrum must be generalized for cyclostationary processes, resulting in a sequence of generalized spectra. It has been shown that, for stationary optical sources, the power spectrum may undergo correlation-induced changes on propagation, the so-called Wolf shift [6, 7]. In this article, changes in the generalized power spectra and the two-frequency cross-spectral density are predicted for fields generated by cyclostationary sources.

Consider a stochastic, cyclostationary source σ(**r**,*t*) with mutual coherence function

where the brackets denote an ensemble average. Cycloergodicity is assumed, so the appropriate ensemble averages and long time averages are equivalent. Cyclostationary processes exhibit a discrete time translation symmetry rather than the continuous symmetry of stationary processes, so Γ(**r**
_{1} ,**r**
_{2},*t* − *τ*,*t*) = Γ(**r**
_{1}, **r**
_{2}, *t* − *τ* + *T*
_{0}, *t* + *T*
_{0}),*i.e*. the absolute time is only important modulo *T*
_{0}. Because of this statistical periodicity, the mutual coherence function admits the Fourier series representation

where ω_{0} = 2π/*T*
_{0}.

The Fourier transform of the mutual coherence function is the two-frequency cross-spectral density, viz.,

From Eq. (2) and Eq. (3), one finds that

where *C*̃* _{n}* is the Fourier transform of

*C*. Equation (4) is a generalization of the Wiener-Khintchine theorem as may be seen by taking the

_{n}*T*

_{0}→ ∞ (ω

_{0}→ 0) limit. The

*C*̃

*(*

_{n}**r**,

**r**, ω) are generalized spectra and the

*C*̃

*(*

_{n}**r**

_{1},

**r**

_{2}, ω) are generalized cross-spectra. In the usual form of the Wiener-Khintchine theorem, the delta function

*δ*(Ω) indicates that the field is uncorrelated across frequencies. In this generalized form, it is clear that the field is correlated only at discretely-spaced frequencies.

The measurement of the standard spectrum, where Ω = 0, may be accomplished by means such as grating spectroscopy or Fourier transform spectroscopy. The spectral density of a source, secondary source, or field is given by the expression *S*(**r**, ω) = *W*(**r**,**r**,ω, ω) [8], which implies that for cyclostationary fields the spectrum as would be measured in a standard spectrometer is given by the zeroth-order generalized spectra *S*(**r**, ω) = *C*̃_{0}(**r**,**r**, ω) [5]. In Sec. 2, the propagation laws for the cross-spectra are examined and the generalized Wolf shifts are identified. Examples are used to illustrate the Wolf shift for cyclostationary fields in Sec. 3. A scheme to measure the generalized spectra and cross-spectra by heterodyne techniques is proposed in Sec. 4.

## 2. Propagation of generalized cross-spectra

In general, the spectrum, or for cyclostationary processes, the generalized spectra, cannot be determined at arbitrary points from the known spectra of the source or even the spectra on some plane. Instead, it is necessary to propagate the two-point, and for nonstationary processes, two-frequency, cross-spectral density. It has been shown that the two-point, two-frequency cross-spectral density satisfies the Wolf equation [2, 9],

where *W _{σ}* is the source cross-spectral density and

*W*is the cross-spectral density for the radiated field,

_{R}*k*

_{1}= ω/

*c*and

*k*

_{2}= (ω + Ω)/

*c*. The radiated field can thus be found by the method of Green functions, and the cross-spectral density for the radiated field in the far zone takes the form

where **r**
* _{i}* =

*r*and

_{i}r̂_{i}*W*is the six-dimensional spatial Fourier transform of

_{σ}*W*. For stationary fields [6], the peak of the spectrum of the radiated field generally shifts compared to the source spectrum as a direct consequence of the correlations in the source. From Eqs. (4) and (6) and using the sifting property of the

_{σ}*δ*-function, one obtains

$$\phantom{\rule{9.2em}{0ex}}\times \delta \left(\Omega -n{\omega}_{0}\right)$$

where *k*
_{2,n} = (ω + *n*ω_{0})/*c* and *$\tilde{C}$ _{n}* is the six-dimensional spatial Fourier transform of

*C̃*. Note that Eq. (7) is of the same form as Eq. (4), with

_{n}*C̃*(

_{n}**r**

_{1},

**r**

_{2},ω) → exp(

*ik*

_{2,n}

*r*

_{2}−

*ik*

_{1}

*r*

_{1})

*$\tilde{C}$*(−

_{n}*k*

_{1}

*r*̂

_{1},

*k*

_{2,n}

*r*̂, ω)/(

*r*

_{1}

*r*

_{2}). Because

*$\tilde{C}$*and

_{n}*k*

_{2,n}differ for each

*n*, each generalized spectrum may experience a different spectral shift. Thus the standard Wolf shift may be seen in

*C*̃

_{0}, and a sequence of generalized Wolf shifts may be seen in each of these generalized spectra. This is the main result of this article, which is illustrated below through example.

The measured spectrum in the far zone is given by

Equation (8) is of the same form as the propagation rule for the spectrum in the stationary case [2], where *$\tilde{C}$*
_{0} takes the place of the one-frequency cross-spectral density. There is no equivalent expression for the propagation rule for the generalized spectra in the theory of stationary fields. It is also important to note that all of the generalized spectra contribute to the optical intensity *I*(**r**,*t*) = Γ(**r**,**r**,*t*,*t*), and thus the propagated pulse shape.

## 3. Examples

As a model for a cyclostationary field, consider a stationary source with mutual coherence function $\overline{\Gamma}$
(**r**
_{1}, **r**
_{2}, *τ*) that is modulated periodically in time with period *T*
_{0}. Such a field is called intrinsically stationary [10, 11]. Let the periodic modulation function be given by *h*(*t*) with period *T*
_{0}. The Fourier transform of the underlying stationary mutual coherence function is the single frequency cross-spectral density,

The modulation function has a Fourier representation *h*(*t*) = ∑* _{n}h_{n}* exp(−

*i*ω

_{0}

*nt*), where ω

_{0}= 2

*π*/

*T*

_{0}. After modulation, the source has a two-time mutual coherence function, Γ(

**r**

_{1},

**r**

_{2},

*t*−

*τ*,

*t*) = $\overline{\Gamma}$ (

**r**

_{1},

**r**

_{2},

*τ*)

*h*(

*t*)

*h*

^{*}(

*t*−

*τ*). Inserting this expression for the mutual coherence function into Eq. (3) and using Eq. (8), the measured spectrum in the far zone is found to be

where *W*̄ is the six-dimensional Fourier transform of the stationary cross-spectral density of the source *W*̄ and *k* =ω/*c*. The modulation and the propagation both affect the spectrum in the far zone in a complicated manner for a general source. When the source can be factorized *W*̄ (**r**
_{1},**r**
_{2},ω) = *A*(ω)*D*(**r**
_{1},**r**
_{2}), the effects
of modulation and propagation can be separated:

The original spectrum, *A*, is broadened by the modulation, and then shifted in the course of propagation as a result of the factor *D*.

As an example, consider a collection of *M* point-sources that are identically modulated. The source density for this system is given by the expression

where *σ _{p}*(

*t*) is the source density at point

**r**

^{(p)}. The underlying sources,

*σ*, are assumed to be partially correlated, stationary, random processes. The frequency-domain correlation function for the {

_{p}*σ*} is written

_{p}where 〈·〉 denotes an ensemble average and *μ _{pq}* is the spectral degree of coherence between the point sources at

**r**

^{(P)}and

**r**

^{(q)}, respectively. Each point source is taken to have the same spectrum,

*S*(ω). The field radiated from the collection of modulated sources has a two-frequency cross-spectral density

$$\phantom{\rule{9.2em}{0ex}}\times S\left(\omega -n{\omega}_{0}\right){\mu}_{\mathrm{pq}}\left(\omega -n{\omega}_{0}\right)\delta \left(\Omega -n{\omega}_{0}\right).$$

In Fig. 1 the normalized spectra, *C*̃_{0} and |*C*̃_{4}|, are shown at both the sources and at a point far away for a collection of three sources, located along the *x*-axis at *x*= 100 mm, *x* = 0 mm, and *x* = −100 mm, respectively. The {*h _{n}*} represent a square wave with a 10% duty cycle,

*S*(ω) is taken to be a Gaussian function with 20% bandwidth and center frequency ω

*= 5 × 10*

_{c}^{15}rad/s, and the spectral degree of coherence is taken to be

where *χ* is the bandwidth of the coherence function, assumed to be the bandwidth of the spectrum for this example. In the top two panels, the normalized generalized spectra for *C*̃_{0} and |*C*
_{4}| are shown for sources that are highly correlated (*μ* = 0.8). The middle two panels show simulations of the same system as in the top two panels, except that the sources are less correlated (*μ* = 0.1). The peak frequency for both *C*̃_{0} (ω) and |*C*
_{4}(ω)| are each less red-shifted in the lower-coherence case. In the bottom panels of Fig. 1, *C*̃_{0} and |*C*
_{4}| are shown for the same system as in the middle two panels, except that the repetition rate of the system has been changed from 1% of the center frequency to 5% of the center frequency. In this case, the length of the pulse is two optical cycles. Note that while the plot for *C*̃_{0} is only slightly changed on propagation at the higher repetition rate, the plot for *C*̃_{4} is much changed, with the secondary peak red-shifted instead of blue-shifted from the primary peak.

The example above may be realized in experiments in which a partially spatially coherent, cyclosta-tionary, optical field illuminates small, point-like objects, for example in imaging of nanoparticles and quantum dots [12, 13, 14]. The objects are then secondary sources, and the field scattered from those objects can exhibit shifts in the generalized spectra.

As another model for the generation of cyclostationary light, consider a mode-locked laser in which the carrier envelope has phase fluctuations. The electric field at the output of a mode-locked laser can be modeled by *U*(**r**,*t*) =*f*(**r**)∑_{n}*A _{n}* exp[

*i*(ω

*+*

_{c}*n*ω

_{0})

*t*] [15], where {

*A*} describe the periodic envelope around the carrier wave centered at ω

_{n}*. While in the idealized case the phase between the carrier wave and the envelope can be perfectly matched [16], in reality, there is some random phase between the output comb lines. The {*

_{c}*A*} are random coefficients with correlation matrix

_{n}*α*= 〈

_{nm}*A*

_{n}^{*}

*A*〉. The two-frequency cross-spectral density at the output plane can be put in the form of Eq. (4) with

_{m}and *F*(**r**
_{1},**r**
_{2}) = *f*
^{*}(**r**
_{1}/**r**
_{2}). If the phase fluctuations at the output plane are depend on position, the function *F*(**r**
_{1} ,**r**
_{2}) will not be separable and the source will be spatially partially coherent. The field that propagates away from the output plane will then exhibit the correlation-induced shifts presented in this article.

## 4. Detection of the generalized cross-correlation functions

Interferometric measurements are often made to determine the statistical properties of stationary fields. Because of the increase in the dimensionality in the case of cyclostationary fields (two time variables instead of one), conventional interferometric measurements must be augmented in some manner to make possible the inference of the statistical properties of the field [5]. For a cyclostationary field with correlation function Γ(**ρ**
_{1},**ρ**
_{2},*t* − *τ*,*t*) = ∑* _{m}C_{m}*(

**ρ**

_{1},

**ρ**

_{2},

*τ*)exp(−

*im*ω

_{0}

*t*) at the plane

*z*= 0 of a Sagnac or Mach-Zehnder interferometer [17, see Ch. 2], the intensity at the output of the interferometer is given by the expression

$$+2\sum _{m=1}^{\infty}\{\mid {C}_{m}(\rho ,\rho ,0)\mid \mathrm{cos}\left(\frac{m{\omega}_{0}\tau}{2}\right)\mathrm{cos}\left[m{\omega}_{0}\left(t-\tau /2\right)-{\alpha}_{m}(\rho ,0)\right]$$

$$+\mid {C}_{m}(\rho ,\rho ,\tau )\mid \mathrm{cos}\left[m{\omega}_{0}t-{\alpha}_{m}(\rho ,\tau )\right]+\mid {C}_{-m}(\rho ,\rho ,\tau )\mid \mathrm{cos}\left[m{\omega}_{0}t+{\alpha}_{-m}(\rho ,\tau )\right]\},$$

where **ρ** is the planar coordinate on the detector, *α _{m}*(

**ρ**,

*τ*) = arg[

*C*(

_{m}**ρ**,

**ρ**,

*τ*)] and

*τ*is the delay in one of the interferometer arms. Since the time dependence is purely sinusoidal, a heterodyne detection scheme [18, 19] at the output of the interferometer allows each term in the sum in Eq. (17) to be determined individually. In heterodyne detection, the electrical signal from a detector is mixed with a sinusoidal electrical signal (the local oscillator). The output of a heterodyne detector is the amplitude of the original signal at the frequency of the local oscillator. The local oscillator frequency may be tuned to values

*m*ω

_{0}= 2

*πm*/

*T*

_{0}to obtain each term in Eq. (17) separately.

At the observation plane of a Young’s two-pinhole interferometer [20], the measured intensity of an incident cyclostationary field is given by the expression

$$+2\sum _{m=1}^{\infty}\{\mid {C}_{m}^{\left(11\right)}\left(0\right)\mid \mathrm{cos}\left[m{\omega}_{0}\left(t-{R}_{1}/c\right)+{\alpha}_{m}^{\left(11\right)}\left(0\right)\right]$$

$$+\mid {C}_{m}^{\left(22\right)}\left(0\right)\mid \mathrm{cos}\left[m{\omega}_{0}\left(t-{R}_{2}/c\right)+{\alpha}_{m}^{\left(22\right)}\left(0\right)\right]$$

$$+\mid {C}_{m}^{\left(12\right)}\left(\tau \right)\mid \mathrm{cos}\left[m{\omega}_{0}\left(t-{R}_{2}/c\right)+{\alpha}_{m}^{\left(12\right)}\left(\tau \right)\right]$$

$$+\mid {C}_{m}^{\left(21\right)}(-\tau )\mid \mathrm{cos}\left[m{\omega}_{0}\left(t-{R}_{1}/c\right)+{\alpha}_{m}^{\left(21\right)}(-\tau )\right]\},$$

where *C _{m}*

^{(ij)}(

*τ*) =

*C*(

_{m}*P*,

_{i}*P*,

_{j}*τ*),

*P*is the location of the pinhole

_{i}*i*,

*α*

_{m}^{(ij)}(

*τ*) = arg[C

_{m}

^{(ij)}(

*τ*)] and

*τ*= (

*R*

_{1}−

*R*

_{2})/

*c*, with

*R*being the distance between the pinhole located at

_{i}*P*. Again, a heterodyne detection scheme will yield each term in the sum in Eq. (18) individually. By varying the locations of the pinholes, the generalized correlation functions

_{i}*C*may be recovered.

_{m}The correlation induced changes in the field upon propagation are observable in the detection scheme described above. Suppose the field with correlation function Γ(**ρ**,**ρ**,*t* − *τ*,*t*) = ∑_{m}*C _{m}*(

**ρ**,

**ρ**,

*τ*)exp(−

*im*ω

_{0}

*t*) on a plane

*z*= 0 is allowed to propagate to a plane

*z*=

*z*′,

*z*′ ≫ 0 where another Mach-Zahnder interferometer is present. The intensity at the output of the interferometer is given by

$$+2\sum _{m=1}^{\infty}\{\mid {F}_{m}(\hat{r},0)\mid \mathrm{cos}\left(\frac{m{\omega}_{0}\tau}{2}\right)\mathrm{cos}\left[m{\omega}_{0}\left(t-\tau /2-r/c\right)-{\varphi}_{m}(\hat{r},0)\right]$$

$$+\mid {F}_{m}(\hat{r},\tau )\mid \mathrm{cos}\left[m{\omega}_{0}\left(t-r/c\right)-{\varphi}_{m}(\hat{r},\tau )\right]$$

$$+\mid {F}_{-m}(\hat{r},\tau )\mid \mathrm{cos}\left[m{\omega}_{0}\left(t-r/c\right)-{\varphi}_{-m}(\hat{r},\tau )\right]\}$$

where ${F}_{m}(\hat{r},\tau )=\frac{1}{2\pi}\int {\tilde{C}}_{m}\left(-{k}_{1}\hat{r},{k}_{2,m}\hat{r},\omega \right)\mathrm{exp}\left(-\mathrm{i\omega \tau}\right)\mathrm{d\omega},$
*ϕ _{m}*(

*r*̂,

*τ*) = arg[

*F*(

_{m}*r*̂,

*τ*)],

*r*is the distance between the detector and the source, and the mapping between

*r*̂ and

**ρ**is

**ρ**/

*r*=

*r*̂

_{⊥}. Heterodyne detection can then be used to find each term in the sum. The coherence-induced changes in the field can then be determined by comparison of the terms at frequency m(O0 at the planes

*z*= 0 and

*z*=

*z*′.

In this article, it has been shown that the Wolf shift from the theory of stationary, partially-coherent fields, can be seen as well in fields governed by the theory of cyclostationary, partially coherent random processes. In this more general setting, a new set of phenomena are predicted: distinct generalized Wolf shifts for each of the generalized spectra. The spectral correlation functions for cyclostationary fields, each representing a correlation between distinct frequencies, undergo separate shifts based on the repetition frequency of the system and the spatial properties of the source. These correlation-dependent, propagation-induced shifts will each contribute separately to changes in the measured intensity of the field far from the source, and may be seen individually in a combined interferometric-heterodyne detection scheme.

## Acknowledgments

RWS, BJD and PSC acknowledge support by a grant from the Grainger Program in Emerging Technologies, the National Science Foundation under CAREER Grant No. 0239265 and the Air Force Office of Scientific Research (Grant No. MURI F-49620-03-1-0379).

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