1. Introduction

Longitudinal modulation of optical waveguides is integral to photonic components such as filters, gratings, mode converters, and periodic polling in nonlinear optics [1]. The propagation of light in modulated waveguides is also of fundamental importance. Optical waveguides provide a test bed to construct optical analogs of diverse quantum effects such as coherent population transfer, Zeno dynamics, Bloch oscillations, and Anderson localization (see [2] for a review). In particular, the effect of harmonic modulation of optical waveguides on light propagation and localization has been extensively studied, and its connection to quantum mechanics elaborated [3,4].

With few exceptions [5,6], most of the works on modulated waveguides remain concerned with spatially coherent beams. The sources in nature, however, are partially coherent, and the subject of partial coherence is a field of its own [7]. Partial coherence is not always undesirable, as it has been proven to be robust against turbulent media [8]. It is therefore natural to extend the previous work to partially coherent beams. We can also ask what, if any, is the corresponding quantum mechanical analog?

We address these issues for the particular case of a parametrically modulated graded-index (GRIN) waveguide. We first show that the corresponding quantum mechanical analog for parametric modulation of such a waveguide is the degenerate parametric amplifier and the corresponding generation of squeezed states [7]. This connection, to the best of our knowledge, has not been made in the literature. We then show how partially coherent Gaussian Schell-model (GSM) beams evolve in parametrically modulated media. Two interesting features are shown: (1) The widths of the GSM beam’s intensity (also referred to as spectral density) and coherence are parametrically amplified. (2) The beam undergoes self-imaging of intensity alongside the parametric amplification. Note that the energy of the beams is constant in our analysis, in contrast to the conventional temporal parametric amplification. It is only the width parameters of the beam that are amplified. Furthermore, we consider temporally coherent monochromatic beams in this work, and are concerned only with spatial coherence.

We consider GRIN media with a parabolic refractive index profile given by

The paraxial wave equation for an optical field of amplitude $A(x,z)=\langle x|A(z)\rangle$ and wavelength $\lambda _{0}$ can be written as

To set the stage, we first revisit important results for $\epsilon =0,~a(z)=a_{0}$. In this case, $H$ in Eq. (2) is independent of $z$. Firstly, the eigenmodes are the Fock states i.e. the Hermite–Gauss (HG) beams. These beams remain invariant, upto a Gouy phase [12], upon propagation. Secondly, an initially coherent state $|A(0)\rangle =|\alpha _{0}\rangle$ evolves as $|A(z)\rangle =|\alpha _{0}e^{-i\kappa _{0} z}\rangle$, i.e., the state remains coherent upon evolution. The coherent states correspond to a set of complex Gaussian beams of width $\sigma =\sqrt {1/2m\kappa _{0}}$, with each beam characterized by an arbitrary position displacement and tilt phase, and the beam centroid exhibiting a sinusoidal motion with frequency $\kappa _{0}=\sqrt {a_{0}}/n_{0}$. Finally, any arbitrary input field and coherence function will self-image with a period $\Lambda _{0}=2\pi /\kappa _{0}$ [11,13].

The general treatment for the case of time-dependent quadratic Hamiltonians such as given by Eq. (2) is a well-known problem, and has been treated in, for example, [14–17]. These works showed that time-dependent quadratic Hamiltonians lead to squeezing of input fields. In general, any Fock state evolves into a squeezed Fock state, and any coherent state evolves into a squeezed coherent state, also known as the two-photon coherent state [16]. The particular details of the squeezing are determined by the form of $\kappa (z)$. There are only a handful of systems for which there are closed form solutions. Among these, the most well-known is the parametric amplifier [7], the spatial analog of which we consider in this work.

The solution $|A(z)\rangle$ to Eq. (2) is found as follows. Using the harmonic oscillator ladder operators $a,a^{\dagger }$ such that $a=(ip+m\kappa _{0} x)/\sqrt {2m\kappa _{0}}$, and writing $\kappa ^2(z)=\kappa ^2_{0}\beta (z)$, the Hamiltonian in Eq. (2) can be recast into the form

We now explain the transverse structure of the field $A(x,z)=\langle x|A(z)\rangle =\langle x|U(z)|A(0)\rangle .$ The squeeze operator $S(\chi )=\textrm {exp}(\chi K_{+}-\chi ^{*} K_{-})$ can be decomposed as $S(\chi )=e^{i\frac {\theta _{\chi }}{2} a^{\dagger }a}S(|\chi |)e^{-i\frac {\theta _{\chi }}{2} a^{\dagger }a}$ [19]. The operator $U(z)$ can then be written as

2. Coherent field propagation

To propagate any arbitrary field, it is sufficient to have knowledge of propagation of modes forming a complete set. We therefore explicitly write out the propagated fields for the Fock states , i.e., the HG beams. Let an HG mode of order $n$ be denoted by $\phi _{n}(x)=\langle x|n\rangle =\left(\frac {\sqrt {m\kappa _{0}/\pi }}{2^{n}n!}\right)^{1/2}H_{n}(\sqrt {m\kappa _{0}}x)e^{-m\kappa _{0}x^2/2}$, where $H_{n}$ denotes Hermite polynomials of order $n$, and $A(x,z=0)=\phi _{n}(x)$. Then

Equation (9) has been previously derived in the context of squeezed Fock states [19,22]. In the context of GRIN media specifically, Eq. (9) shows that, like free space propagation, HG beams retain their general structure upon propagation in parametrically modulated GRIN media. The only quantities affected by any longitudinal refractive index modulation are the beam width and curvature. We stress the fact that such self-evolution of HG beams is true for arbitrary longitudinal modulation. The fact that Gaussian wavepackets retain their Gaussian structure in time-dependent quadratic potentials has been known since the beginning of Quantum Mechanics (see [23] and references therein), but has not been pointed out in the context of modulated parabolic GRIN waveguides to the best of our knowledge.

Figure 1 shows the Gaussian beam corresponding to the ground state wavefunction $\phi _{0}(x)$ evolving in a parametrically driven GRIN medium. We use the well known beam propagation method to propagate beams in GRIN media [24]. The algorithm consists of a cascade of free space propagation and lens focusing operations. The simulation parameters are : $\epsilon =0.04,~\theta _{\chi }=0$, Total Z array size=1000, Total X array size=512, dx=0.25 $\mu$m, dz=0.01 $\Lambda _{0}$ where $\Lambda _{0}=2.4$ mm and $\lambda =1~\mu$m. For the GRIN parameters we choose $n_{0}=1.47, a_{0}=2n^2_{0}\Delta /R^2$, where $\Delta =0.0088$ is the index step and $R=50~\mu$m is the fiber core radius. Even though the exact numbers are proprietary, these parameters are close to those of commercially available fibers [25]. Note that the self-imaging period $\Lambda _{0}=2.4$ mm implies a parametric modulation on a scale of $\Lambda _{0}/2=1.2$ mm. While they do not occur naturally in GRIN fibers, longitudinal refractive index modulations on a mm scale can be achieved, among other means, by i) modulating an applied stress on the fiber, ii) modulating the doping of the fiber core, and iii) using ultraviolet absorption techniques employed in the design of fiber gratings [26].

 

Fig. 1. Simulated amplitude of an input Gaussian mode ($A(x,0)=\langle x|0\rangle$) propagating through the parametrically modulated GRIN medium. The cyan dashed line indicates a location where the beam has no transverse quadratic phase, and also where the beam flips from being divergent to being convergent. The red dashed and dotted lines indicate locations where the beam intensity is equal to the input intensity, i.e., locations where the beam is self-imaged. The width of the beam is parametrically amplified.

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3. Gaussian Schell-model beam propagation

The beams considered so far were spatially coherent. We now extend our results to propagation of partially coherent beams. The evolution of partially coherent beams in unmodulated GRIN media ($\epsilon =0$) is well known. We refer the reader to [27] for the treatment of GSM beams and to [11,28] for propagation of beams with arbitrary coherence properties through unmodulated GRIN media. In this section, we consider the specific class of extensively studied GSM beams propagating through parametrically modulated GRIN media.

Any partially coherent field is characterized by the Cross-Spectral Density (CSD) $W(x_{1},x_{2},z)=\langle A^{*}(x_{1},z)A(x_{2},z)\rangle$. The CSD for a GSM beam is given by

For mode-matched GRIN beams, each coherent mode $\phi _{n}=\psi _{n}$ evolves according to Eqs. (7) and (9). In general, each mode acquires a quadratic phase $\Pi$ as given in Eq. (9). By direct inspection of Eqs. (9) and (11), we see that the mode-matched GSM CSD evolves as

In certain conditions the quadratic phase can vanish, and then the CSD is an exact scaled version of the input CSD. Firstly, if we restrict ourselves to symmetric points about the GRIN axis such that $x_{1}=-x_{2}$, the quadratic phase vanishes and we obtain the simple result $W(x,-x,z)=(D/s)W(Dx/s,-Dx/s,z=0)$, i.e., the input CSD simply scales by the $z$-dependent factor $D/s$, and the factor of $(D/s)$ upfront preserves normalization of the modes $\phi_n$. Note that symmetric coherence functions are simpler to measure experimentally than the more general two-point CSD [31–33]. Secondly, there are locations $z_{0}$ where $\Pi$ itself vanishes. For distances $z_{0}$ and an integer $j$ such that $s=1,~2B=(2\kappa _{0}z_{0}-\theta _{\chi })=j\pi$, the quadratic phase vanishes. Upto a global phase, the mode profile at $z_{0}$ is then given as $\phi _{n}(D_{0}x)$, with $D_{0}=\exp {(-\kappa _{0}\epsilon z_{0}/4)}$. The CSD at $z_{0}$ is then given as

Another interesting result is the phenomenon of self-imaging of intensity $I(x,z)=W(x,x,z)$. By self-imaging we mean the condition $I(x,0)=I(x,z_{i})$ for a particular $z_{i}$. This occurs whenever $D=s$, which corresponds to the solution of the transcendental equation $\exp {(-2|\chi |)}=1+\sin ^2{[(h-\theta _{\chi })/2]}[\exp {(-4|\chi |)}-1]$. For the Gaussian field propagation in Fig. 1, the different $z_{i}$ are marked by the blue circles in Fig. 2. Figure 2 shows the overlap integrals of the propagated complex field (amplitude) with the input complex field (amplitude). The field overlap gradually decreases, but there are revivals in the (absolute) amplitude overlap at each $z_{i}$. The small ripples in the field overlap simulations are the contributions from the non-resonant terms that were ignored in writing Eq. (4).

 

Fig. 2. (Dimensionless) Overlap integral for the input field $\left [|\int dx A^{*}(x,z)A(x,0)|\right ]$ and amplitude $\left [\int dx |A(x,z)||A(x,0)|\right ]$ as a function of $z/\Lambda _{0}$ where $\Lambda _{0}=2.4~$mm is the spatial period associated with the GRIN oscillator. The simulated results are derived from the propagated Gaussian field shown in Fig. 1. The dashed black line shows the theoretically expected overlap integral for the field, while the solid blue line shows the corresponding results for the simulation. The blue circles show theoretically expected points of self-imaging of intensity, while the solid red line shows the intensity overlap calculated for the simulated Gaussian beam in Fig. 1. Note also that by choice we use a field amplitude overlap metric instead of the intensity overlap metric because the former is bounded above by unity while the latter has an upper bound greater than unity, i.e., it is possible to have $\int dx~I^2(x,z) > 1$. For the simulated results, the overlap integral has been normalized to the maximum value, which also occurs at the input plane $z=0$.

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A mode-matched GSM beam can be characterized by the factor $\eta =\sigma _{\mu }/\sigma _{I}$, which characterizes the ‘degree of global coherence’ of a beam [29]. The limits $\eta \xrightarrow []{}0,\infty$ correspond to completely spatially incoherent and coherent fields respectively. Note that for a mode-matched beam each $\eta$ maps to a unique $\sigma _{I}$, and the two are related as

 

Fig. 3. Spectral Density propagation for GSM beams with different $\eta$ parameters. The input beam is assumed to be mode matched and therefore the more incoherent the beam (smaller $\eta$), the wider the beam width. All beams exhibit the exponential amplification of beam width and self imaging of intensity as depicted in Fig. 2. For these figures, $\epsilon =0.05$ and X array size=1024. The rest of the parameters are the same as used in Fig. 1.

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Figure 4 shows the evolution of the second moment $\langle \Delta x^2 (z) \rangle =\int dx x^2I(x,z)$ of a GSM beam with $\eta =0.5$. The solid red line plots the numerically computed ratio $\langle \Delta x^2 (z) \rangle /\langle \Delta x^2 (0) \rangle$. The blue lines are the envelopes given by $D^{\pm 2}=\exp {(\mp \kappa _{0}\epsilon z_{0}/2)}$. The exponential growth is due to parametric modulation of the refractive index. The oscillation of the beam width is due to the focusing and expansion of the beam in the GRIN medium, and is given simply by the factor of $(s/D)^2$. Note that $s/D$ has a quasi-spatial frequency of $2\kappa _{0}$, which is also the frequency of the parametric drive $a(z)$. As shown in Fig. 4, this means that, apart from the width amplification and suppression, the beam expands and focuses every $\Lambda _{0}/2$. In the quantum mechanical analogy, it would be the variance of the quadrature $(a+a^{\dagger })$ that would be parametrically amplified [7, Chap. 22], and the dashed blue line is the analog of the effective ‘vacuum’ noise of the field mode. The focusing of the beam width below the dashed blue line is the analog of generating a ‘temporal’ squeezed state [7, Chap. 21]. Care must be taken, however, when making these analogies. The analogy is mathematically exact because the Hamiltonian in Eq. (4) has the same form as a temporal parametric amplifier. However, the mechanism to generate the parametric modulation can correspond to different physical processes in each case. While in the GRIN medium the refractive index is modulated longitudinally in space, the temporal parametric amplifier uses nonlinear interaction between electric fields to generate a time-varying polarization in the medium. The former is a linear optics problem, while the latter falls in the domain of nonlinear optics. Another feature expressed in Fig. 4 is the self-imaging of the beam. When the solid red line crosses the dashed blue line, $\langle \Delta x^2 (z) \rangle =\langle \Delta x^2 (0) \rangle$ and the beam self images. These are precisely the locations $z_{i}$, also discussed in Fig. 2. Notice that the lower envelope $\exp {(-\kappa _{0}\epsilon z_{0}/2)}$ shows that the beam focus also becomes narrower with propagation, a feature that might find application in fiber-based imaging systems. A similar result was also reported in [5], although using a slightly different approach based on Green’s functions. Recall, however, that our results are valid under the paraxial approximation. At some point the paraxial approximation will break down because the diffraction angles required by the beam to expand and focus within $\Lambda _{0}/2$ increase with propagation.

 

Fig. 4. Longitudinal evolution of beam width for a mode-matched input GSM beam with $\eta =0.5$. The solid red line denotes the simulation results for the ratio $\langle \Delta x^2 (z) \rangle /\langle \Delta x^2 (0) \rangle$, where $\langle \Delta x^2 (z) \rangle =\int dx x^2I(x,z)$ is the transverse beam width at $z$, and $\langle \Delta x^2 (0) \rangle$ is the input beam width. The dashed blue line denotes the input beam width level. The solid blue lines are theoretical envelopes $\exp [\pm \kappa _{0}\epsilon z_{0}/2]$. The beam width is parametrically amplified as shown by the exponential envelopes. As explained in the text, this parametric amplification is mathematically analogous to the quantum mechanical amplification of the photon number in a conventional parametric amplifier. The phase of the beam width oscillations is controlled by $\theta _{m}$, the phase of the longitudinal refractive index modulation. For this figure, we set $\theta _{m}=0$. The beam is self-imaged at the locations where the solid red line crosses the dashed blue line, see Fig. 2. All simulation parameters are the same as used for Fig. 3.

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4. Beam displacement and tilt: centroid amplification

The previous results of coherence width amplification and self-imaging assumed the input beam to be normally incident and centered on the input face of the GRIN medium. If the input GSM beam is displaced from the GRIN core or strikes at an angle, the beam centroid is parametrically amplified upon propagation. Beam centroid amplification has been observed in spatial solitons [35]. A coherent mode $\phi _{n}(x)$ that is displaced in position by $x_{0}$ and in momentum by $p_{0}$ is characterized by a complex parameter $\alpha =(ip_{0}+m\kappa _{0}x_{0})/(2m\kappa _{0})^{(1/2)}$ such that $|\alpha ,n\rangle =D(\alpha )|n\rangle$, where $D(\alpha )=\textrm {exp}(\alpha a^{\dagger }-\alpha ^{*}a)$ is the displacement operator. The states $|\alpha ,n\rangle$ are also referred to as displaced Fock states [22]. Similar to the derivation leading to Eq. (9), we find the expression for $\langle x|U(z)|\alpha \rangle$, given the initial state $A(x,z=0)=\langle x|\alpha ,n\rangle$ as

Many features of our work such as mode matching and beam focusing share similarities with the work done in [5,6]. Our results could, in principle, be derived from the analysis in these papers, which solve the general problem of propagation of partially coherent beams in longitudinally modulated GRIN media. In our work we examine in detail the special case of a parametric modulation, which was not the focus of these previous works. To the best of our knowledge, we are not aware of any other works that show the i) the connection to the squeezing Hamiltonian of quantum mechanics to parametrically modulated GRIN media, and the corresponding parametric amplification of the intensity and coherence widths, ii) the use of the CMD for the GSM beam propagation in longitudinally modulated GRIN media.

 

Fig. 5. (a) Centroid amplification of an input Gaussian mode which is displaced from the optical axis by $0.27\sigma$. The centroid amplification is sensitive to the pump phase $\theta _{m}$. (b) Simulated centroid (solid blue line) of the beam in (a), and the envelope (dashed red line) predicted by Eq. (19) for $\theta _{m}=0$. (c) Centroid evolution for $\theta _{m}=\pi$. All other simulation parameters are same as in Fig. 3.

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5. Conclusion

In summary, using results from quantum theory of parametric amplifiers we have shown the amplification of intensity width and coherence width of GSM beams in parametrically modulated GRIN media. We also show that self-imaging of GSM beams still occurs alongside amplification. Moreover, the degree of global coherence is invariant upon propagation, a feature shared by GSM beams in free space. Our results can be used in fiber-based imaging systems, and also to study quantum statistics of squeezed states in an optical analog setting. Our analysis can be extended to the time domain, and employed in temporal imaging with partially coherent pulses. Specifically, our work can be extended to partially coherent temporal GSM pulses propagating in ‘temporal’ GRIN media, i.e., nonlinear dispersive fibers [36]. There, the temporal coherence function would be amplified and the pulse would also show self-imaging.