## Abstract

We consider polarization changes of randomly fluctuating electromagnetic pulsed light in temporal imaging. The polarization properties of pulses formed by the time lens are formulated in terms of the Stokes parameters. For Gaussian Schell-model pulses we show that the degree and state of polarization of the time-imaged pulse can be tailored in versatile ways, depending on the temporal polarization and coherence of the input pulse and the system parameters. In particular, weakly polarized central region of the pulse may become fully polarized without energy absorption. The results have potential applications in optical communication, micromachining, and light–matter interactions.

© 2013 OSA

## 1. Introduction

Partial polarization and electromagnetic coherence of light beams play a significant role in many areas of optics, e.g., imaging, microscopy, interferometry, and astronomy [1–4]. For stationary fields the theory of partial polarization in time domain has a long history, originating from the 1950s [5, 6]. Thirty years later, the theory was extended to frequency domain [7], implying wider applicability of the formalism. The polarization of non-stationary (pulsed) light has received much less attention, even though the coherence properties of scalar pulses has been investigated for some time [8–11]. A systematic treatment for both temporal and spectral partial polarization of electromagnetic pulses was presented very recently [12]. The formalism brings extra richness to the theory of partial polarization, and it can be applied to short pulses obtained, for instance, from Q-switched or mode-locked laser sources.

In this work, we study the changes in partial polarization experienced by a pulsed, partially polarized and partially coherent light beam in temporal imaging. The imaging system is composed of a time lens [13–15] (temporal quadratic phase modulator) surrounded by two sections of linearly dispersive waveguides. Time lenses have been essential in the development for single-shot measurements of short pulses via temporal microscopy [16], chronocyclic tomography [17], or optical Fourier transform [18]. Other applications have included production of picosecond pulses from continuous-wave or mode-locked lasers [19]. However, so far only the case of a scalar or linearly polarized field has been considered. Obviously there are numerous phenomena to be observed in the partial polarization and coherence of temporally manipulated fields, both in time and frequency domains. We present a general theory for the imaging and apply it to Gaussian Schell-model (GSM) pulses in the cases of symmetric, antisymmetric, and asymmetric lens magnifications. Depending on the temporal coherence of the input pulse, these situations allow versatile manipulation of the Stokes parameters and hence the intensity and the degree and state of polarization within the output pulse. For example, the Stokes parameters of the image pulse can be scaled and magnified/demagnified replicas of the object pulse, or their structure within the pulse can be otherwise significantly altered. In particular, a weakly polarized central region of a pulse may be made polarized, without absorption of energy.

This paper is organized as follows: in Secs. 2 and 3 we briefly recall the polarization theory of random electromagnetic pulses in time domain and the principle of temporal imaging, respectively. In Sec. 4 the general expressions are derived for the Stokes parameters of the image pulse. Polarization changes of GSM pulsed beams in various temporal imaging systems are considered in Sec. 5. The main conclusions are summarized in Sec. 6.

## 2. Polarization of an electromagnetic pulse

The second-order coherence properties of a non-stationary beam-like field, described by the complex analytic signal **E**(**r**, *t*), are encoded in the 2 × 2 electric coherence matrix **Γ**(**r**_{1}, **r**_{2}, *t*_{1}, *t*_{2}) [12], whose elements are

*i*,

*j*=

*x*,

*y*label the orthogonal electric field components, the asterisk stands for complex conjugation, and the angle brackets denote ensemble average over many pulses. The origin of time for each pulse in the train is determined by the trigger that generates the pulse. Hence the ensemble average is general and accounts, e.g., for jitter, chirp, and phase and amplitude fluctuations of the pulses. The coherence matrix includes information on both temporal and spatial coherence, but in the context of this article only temporal coherence is of interest. The information on the polarization state of the field is contained in the polarization matrix

**J**(

**r**,

*t*), defined as

**J**(

**r**,

*t*) =

**Γ**(

**r**,

**r**,

*t*,

*t*). This indicates that the state of polarization is specified by the intensities of the field components and the correlations between them. The diagonal elements

*J*(

_{xx}**r**,

*t*) and

*J*(

_{yy}**r**,

*t*) are the intensities

*I*(

_{x}**r**,

*t*) and

*I*(

_{y}**r**,

*t*) of the orthogonal field components, and the trace tr

**J**(

**r**,

*t*) gives the total intensity

*I*(

**r**,

*t*) =

*I*(

_{x}**r**,

*t*) +

*I*(

_{y}**r**,

*t*) of the field. The off-diagonal elements

*J*(

_{xy}**r**,

*t*) and ${J}_{yx}\left(\mathbf{r},t\right)={J}_{xy}^{*}\left(\mathbf{r},t\right)$ describe the correlations between the field components. The correlations can also be expressed using the correlation coefficient ${\gamma}_{xy}\left(\mathbf{r},t\right)={J}_{xy}\left(\mathbf{r},t\right)/{\left[{J}_{xx}\left(\mathbf{r},t\right){J}_{yy}\left(\mathbf{r},t\right)\right]}^{1/2}={\gamma}_{yx}^{*}\left(\mathbf{r},t\right)$. The coefficient is bounded in absolute value by |

*γ*(

_{xy}**r**,

*t*)| ≤ 1.

An important quantity characterizing the partial polarization of the field is the degree of polarization, which for nonstationary fields depends on time and is given by [12]

*P*(

**r**,

*t*) ≤ 1, where the lower bound describes an unpolarized field and the upper a fully polarized field at (

**r**,

*t*). In addition, it has the property that if the intensities

*I*(

_{x}**r**,

*t*) and

*I*(

_{y}**r**,

*t*) of the orthogonal field components are equal,

*I*(

_{x}**r**,

*t*) =

*I*(

_{y}**r**,

*t*), then

*P*(

**r**,

*t*) = |

*γ*(

_{xy}**r**,

*t*)|. Conventionally, the polarization state is represented by the Stokes parameters

*S*(

_{i}**r**,

*t*),

*i*= 0 ...3, which are related to the elements of the polarization matrix via

*α*and

*β*in Eq. (5) refer to the coordinate axes of a frame rotated by +45° with respect to the

*x*axis. The subscripts

*r*and

*l*in Eq. (6) refer to right and left circularly polarized states, respectively. The polarization state and the total intensity of the field can be separated by defining the normalized Stokes parameters

**s**(

**r**,

*t*) =

*s*

_{1}(

**r**,

*t*)

*û*

_{1}+

*s*

_{2}(

**r**,

*t*)

*û*

_{2}+

*s*

_{3}(

**r**,

*t*)

*û*

_{3}, where

*û*

_{1},

*û*

_{2}, and

*û*

_{3}constitute an orthonormal right-handed basis. Geometrically, the locus of the possible points given by the Poincaré vector is a unit sphere called the Poincaré sphere, which is illustrated in Fig. 1. Points on the surface of the sphere correspond to fully polarized fields, the origin to an unpolarized field, and intermediate points to partially polarized fields. The ‘north’ and ‘south’ poles of the Poincaré sphere refer to fully right- and left-circularly polarized states, respectively, and points on the equator correspond to fully linearly polarized states. Points which are off the equatorial plane and not on the poles represent elliptically polarized states.

## 3. Temporal imaging using a time lens

The temporal profile of an optical pulse can be changed by propagation through a cascade consisting of a linearly dispersive medium (e.g., an optical fiber), a temporal quadratic phase modulator (QPM), and another linearly dispersive medium. Such a system is mathematically fully analogous to a conventional spatial imaging system consisting of Fresnel propagation, a thin lens, and Fresnel propagation, which forms a magnified or demagnified image. Due to the spatial analogy, quadratic phase modulators are also called time lenses [14, 15].

Consider an optical pulse propagating in the *z* direction with the electric field in the *xy* plane. The complex envelopes of the field components are *A _{i}*(

*z*,

*t*) =

*E*(

_{i}*z*,

*t*)exp(i

*ω*

_{0}

*t*),

*i*=

*x*,

*y*, where

*E*(

_{i}*z*,

*t*) are the related complex analytic signals and

*ω*

_{0}is the center frequency of the pulse. A QPM affects the envelope as

*γ*is the ‘temporal focal length’ specifying the magnitude and sign of the modulation. Clearly, the intensity profile of the pulse remains unchanged on propagation through the time lens, but the spectral profile is affected by the temporal phase change. Such a modulator can be realized using, e.g., active electro-optic modulation [17, 19], cross phase modulation [20], sum frequency generation, or four wave mixing [16].

_{i}Propagation of a pulse through a linearly dispersive waveguide (e.g., an optical fiber) is mathematically analogous to Fresnel diffraction in free space. More precisely, throughout the bandwidth of the pulse of interest the propagation constants *β _{i}*(

*ω*),

*i*=

*x*,

*y*, of the waveguide may be reasonably approximated with the Taylor polynomials

*β*(

_{i}*ω*) =

*β*

_{0i}+

*β*

_{1i}(

*ω*−

*ω*

_{0})+

*β*

_{2i}(

*ω*−

*ω*

_{0})

^{2}/2, where the Taylor coefficients are

*β*= d

_{ni}*(*

^{n}β_{i}*ω*)/d

*ω*evaluated at

^{n}*ω*=

*ω*

_{0}. Physically,

*ω*

_{0}/

*β*

_{0i}are the phase velocities at the center frequency and ${\beta}_{1i}^{-1}$ are the group velocities at which the envelopes of the orthogonal components propagate. The parameters

*β*

_{2i}are the dispersion coefficients, which are responsible for pulse broadening. The propagation constants are, in general, different for the orthogonal polarization components, e.g., due to waveguide shape and material anisotropy. However, we take the propagation media to be such that

*β*

_{0x}=

*β*

_{0y}=

*β*

_{0}and

*β*

_{1x}=

*β*

_{1y}=

*β*

_{1}, but

*β*

_{2x}≠

*β*

_{2y}, allowing anisotropic pulse broadening on propagation. The complex envelope after propagation over a distance

*z*in such a waveguide is obtained via [21]

*t*violates the principle of causality; however, the approximation is valid if the input signal is sufficiently narrow-band. It should be borne in mind that using the second-order Taylor polynomial for the propagation constant also necessitates a narrow-band input signal.

We see that by examining the pulse in the local time frame, which moves at the group velocity, i.e., *t*̃ = *t* − *β*_{1}*z*, Eq. (10) becomes

*=*

_{i}*β*

_{2i}

*z*.

Using the equations presented above we are able to analyze a temporal imaging system consisting of two single-mode fibers (SMF) and a QPM placed between them. The fibers before and after the lens are referred to with subscripts a and b, respectively. The imaging system is shown in Fig. 2, where the middle row illustrates the geometry of the cascade, the top row shows the qualitative behavior of the pulse’s electric field at different points in the system, and the bottom row shows the spectrum of the pulse before and after the temporal modulator. The first SMF with group delay dispersion parameters Φ_{ax} and Φ_{ay} acts as a spectral phase filter for the originally unchirped pulse, introducing phase differences between different spectral components, thus changing the temporal profile of the pulse and introducing chirp while retaining the spectrum of the pulse. The QPM, with modulation parameters *γ _{x}* and

*γ*, leaves the temporal profile of the pulse unchanged, but changes the phases of the orthogonal components by different amounts at different instants of the pulse, modifying the frequency spectrum of the pulse. The second SMF, with GDD parameters Φ

_{y}_{bx}and Φ

_{by}, retains the modified spectral profile but changes the relative phases of different frequency components of the pulse, altering the temporal profile.

The relation between the input and output complex envelopes ${A}_{i}^{\left(\text{in}\right)}\left({t}^{\prime}\right)$ and ${A}_{i}^{\left(\text{out}\right)}\left(t\right)$ of the system in Fig. 2 is obtained by consecutive applications of Eq. (9) and Eq. (12) [21], which gives

*t*′ and

*t*, respectively. The kernel

*K*(

_{i}*t*,

*t*′) is

*ϕ*(

_{i}*t*,

*t*′) =

*β*

_{0a}

*z*

_{a}+

*β*

_{0b}

*z*

_{b}− (

*t*′

^{2}/Φ

_{ai}+

*t*

^{2}/Φ

_{bi})/2,

*i*=

*x*,

*y*. The quantities

*β*

_{0a}and

*β*

_{0b}are the propagation constants of fibers a and b at

*ω*

_{0}, respectively, and

*z*

_{a}and

*z*

_{b}are the lengths of the fibers. We observe that the quantity

*L*= 1/Φ

_{i}_{bi}+ 1/Φ

_{ai}− 1/

*γ*bears resemblance to the lens law both in its form and in the role it plays. If the temporal focal parameter

_{i}*γ*and the GDD parameters Φ

_{i}_{ai}and Φ

_{bi}are chosen such that

*L*= 0, i.e.,

_{i}*M*|

_{i}^{−1/2}), magnified (|

*M*| >1) or demagnified (|

_{i}*M*| < 1), and inverted (

_{i}*M*< 0) or erect (

_{i}*M*> 0) image of the input. If the envelope of the pulse is inverted, the interactions within the system have caused the envelope of the leading edge of the pulse to attain the form of the original trailing edge, and vice versa [22]. Equation (18) also indicates that propagation through the system imparts a constant phase factor, determined by the propagation constants, lengths and group dispersion delays of the fibers, and a quadratically time-dependent phase factor. The temporal phase factor does not have any effect on the intensity profile of the pulse, but affects the spectrum when

_{i}*M*≠ 1.

_{i}## 4. Polarization of a temporally imaged pulse

As a precursor to the polarization studies, we introduce the envelope coherence matrix **Γ**^{(e)}(*t*_{1}, *t*_{2}) with the elements

*(*

_{ij}*t*

_{1},

*t*

_{2}) are the elements of the electric coherence matrix of the complex analytic signals

*E*(

_{i}*t*). When a pulse traverses a temporal imaging system, the coherence matrix ${\mathbf{\Gamma}}_{0}^{\left(\text{e}\right)}\left({t}_{1}^{\prime},{t}_{2}^{\prime}\right)$ at the input of the system is transformed into

**Γ**

^{(e)}(

*t*

_{1},

*t*

_{2}) at the output. The primed time variables serve to remind that the origin of time is different at input and output, as discussed in the previous section, with

*t*=

_{m}*t*′

*−*

_{m}*β*

_{1a}

*z*

_{a}−

*β*

_{1b}

*z*

_{b},

*m*= 1, 2, where

*β*

_{1a}and

*β*

_{1b}are the reciprocal group velocities in fibers a and b, respectively.

The elements of the envelope coherence matrix at the output are

*x*and

*y*components, the coherence matrix becomes

**J**(

*t*) at the output of the imaging system is obtained from the coherence matrix by setting

*t*

_{1}=

*t*

_{2}=

*t*:

*i*,

*j*=

*x*,

*y*. We observe that

*ϕ*(

_{xx}*t*) =

*ϕ*(

_{yy}*t*) = 0, consistently with the fact that the diagonal elements of

**J**(

*t*) are intensities and thus real and non-negative. The off-diagonal phases

*ϕ*(

_{xy}*t*) and

*ϕ*(

_{yx}*t*) are generally non-zero and obey the equation

*ϕ*(

_{yx}*t*) = −

*ϕ*(

_{xy}*t*). Recalling the definition of the temporal magnification presented in Eq. (19), we see that if both

*M*and

_{x}*M*are positive, the time-independent term in

_{y}*ϕ*(

_{xy}*t*) is zero. If both

*M*and

_{x}*M*are negative, then the term is either −

_{y}*π*, zero, or

*π*, depending on the signs of the GDD parameters. When

*M*and

_{x}*M*have different signs, the time-independent phase is

_{y}*π*/2 if

*M*< 0, Φ

_{y}_{ay}> 0 or

*M*< 0, Φ

_{x}_{a}

*< 0 and −*

_{x}*π*/2 in other cases.

According to Eq. (24), the intensities at the output are

*I*

_{0i}(

*t*) is the input intensity. Thus the intensity at the output of the temporal imaging system is a scaled, magnified/demagnified, and direct/inverted copy of the input intensity. Scaling the peak intensity by |

*M*| ensures that the total energy in the

_{i}*i*component is conserved in propagation through the imaging system.

The relations between the input and output degree and state of polarization are more complicated. In particular, we note from Eq. (24) that in the case of *M _{x}* ≠

*M*, the off-diagonal terms of the polarization matrix at the output depend not only on the polarization properties of the input, but also on the temporal coherence properties, as

_{y}*t*/

*M*and

_{x}*t*/

*M*are not equal. In an equal-magnification system (

_{y}*M*=

_{x}*M*=

_{y}*M*) the output polarization matrix is fully specified by the input polarization matrix. The input and output degrees of polarization are, in this case, related via

*P*

_{0}(

*t*) is the degree of polarization of the input pulse. Clearly, the output degree of polarization is simply magnified or demagnified with respect to the input degree of polarization. The Stokes parameters at the input and at the output are connected via

*S*

_{0}

*(*

_{n}*t*),

*n*= 0 ... 3, refer to the Stokes parameters and

*s*

_{0}

*(*

_{n}*t*),

*n*= 1, 2, 3, to the normalized Stokes parameters of the input pulse. We observe that the polarization state, as described by the Stokes parameters, is not necessarily only a magnified/demagnified copy of the original pulse’s polarization state. This stems from the additional phase difference

*ϕ*(

_{xy}*t*), which generally depends on both the GDD parameters of the fibers and time. Equations (33) and (34) indicate that for

*ϕ*(

_{xy}*t*) ≠ 0 the polarization state is necessarily modulated by propagation through the time lens system. Recalling Eq. (25), we see that the polarization modulation can be controlled by properly choosing the GDD parameters of the fibers.

## 5. Examples

We explore the effect of temporal imaging using a pulsed beam constructed by superposing a linearly polarized Gaussian Schell-model (GSM) pulse and its delayed, orthogonally polarized replica [12]. Systems of this type have been studied in the stationary case as polarization state controllers [23, 24] and they are, in particular, the method of choice in experiments to create unpolarized light beams [25]. We note, though, that such unpolarized light differs from natural unpolarized light in that it is completely polarized at each frequency. The full spectral polarization indicates that the depolarization can be fully or partially reversed without energy absorption [26, 27]. Here, we first derive the general expressions for the Stokes parameters at the output of the imaging system, which are then considered for the specific cases of symmetric, antisymmetric, and asymmetric magnifications in the following subsections.

The complex analytic signals of the *x* and *y* components of the pulse are *E _{x}*(

*t*) =

*E*(

*t*) and

*E*(

_{y}*t*) =

*E*(

*t*−

*τ*), where

_{d}*τ*is the time delay. The signal

_{d}*A*

_{0}, the normalized amplitude function

*a*(

*t*) with 〈|

*a*(

*t*)|

^{2}〉 = 1, the pulse length

*T*

_{0}, and the central frequency

*ω*

_{0}. The amplitude function

*a*(

*t*) describes the random fluctuations of the field, with the temporal correlations of the Gaussian form

*T*is the coherence time. The intensity of the signal

_{c}*E*(

*t*) is likewise Gaussian with The coherence matrix of the GSM beam thus becomes

When the GSM pulse described above propagates through a time lens system presented in the previous section, the polarization matrix at the output is obtained by inserting Eq. (38) into Eq. (24), which gives

*I*(

*t*) in Eqs. (40) and (41) is the Gaussian intensity function defined in Eq. (37), and

*ϕ*(

_{xy}*t*) in Eq. (42) is the quadratic phase function given in Eq. (25). The physical meaning of the effective time delay Δ(

*t*) is obtained as follows: the

*x*and

*y*components of the output pulse at time

*t*are derived from the field at times

*t*

_{0x}and

*t*

_{0y}, respectively, of the original linearly polarized pulse. For the imaged

*x*component we have

*t*

_{0x}=

*t*/

*M*, while for the delayed and imaged

_{x}*y*component we have

*t*

_{0y}=

*t*/

*M*−

_{y}*τ*. Thus at time

_{d}*t*, the correlation between the

*x*and

*y*components of the output pulse is dictated by the time difference

*t*

_{0y}−

*t*

_{0x}= Δ(

*t*) in the original pulse.

The Stokes parameters are obtained by using Eqs. (3)–(6) in Eqs. (40)–(42):

*S*

_{1}(

*t*),

*S*

_{2}(

*t*), and

*S*

_{3}(

*t*), that the polarization state of the output beam can be altered by changing the time delay

*τ*, the GDD parameters and the ‘focal length’ of the QPM such that the imaging condition in Eq. (16) holds for both orthogonal components.

_{d}#### 5.1. Symmetric magnification

The first case we consider is where the *x* and *y* components of the pulse undergo similar magnification, *M _{x}* =

*M*=

_{y}*M*. The simplest way to ensure symmetric magnification is to choose the system parameters such that Φ

_{a}

*= Φ*

_{x}_{ay}and Φ

_{bx}= Φ

_{by}. As evident from Eqs. (43) and (25), the effective time delay Δ(

*t*) becomes time-independent and the phase term

*ϕ*(

_{xy}*t*) is zero at all times. The polarization matrix elements at the output are

*τ*≪

_{d}*T*), the exponential term tends to unity and thus

_{c}*P*(

*t*) = 1 at all times, i.e., the beam is fully polarized throughout. In the opposite case,

*τ*≫

_{d}*T*, the exponential term goes to zero and $P\left(t\right)=\left|\text{tanh}\left({\tau}_{d}-2t/M\right){\tau}_{d}/2{T}_{0}^{2}\right|$, which leads to

_{c}*P*(

*Mτ*/2) = 0. In all cases the pulse edges are fully polarized, as seen from the fact that lim

_{d}

_{t→}_{±∞}

*P*(

*t*) = 1.

The effect of symmetric magnification is investigated numerically in the case of *τ _{d}* =

*T*=

_{c}*T*

_{0},

*ω*

_{0}= 2000/

*T*

_{0}, and

*M*= 0.6. These parameters correspond to a pulse whose components are themselves partially coherent and which are markedly shifted in time with respect to each other, but still have significant amount of overlap. The intensities of the

*x*and

*y*components, shown with blue solid and green dashed lines, respectively, and the degree of polarization (red dash-dotted curve) before (thin) and after (thick curves) imaging are shown in Fig. 3(a). The intensity profiles show that the pulse has been demagnified in time, and on the other hand the peak intensity of both components has increased by the reciprocal of the temporal magnification factor. The delay between the components is changed by the demagnification to

*Mτ*= 0.6

_{d}*τ*. Temporal development of the degree of polarization has similarly been demagnified, but the shape of the

_{d}*P*(

*t*) curve remains otherwise unchanged, as predicted by Eq. (27). During the pulse, the degree of polarization alters between values 0.37 and unity, indicating that the pulse varies between partially and fully polarized states.

Figure 3(b) presents the normalized Stokes parameters of the temporally demagnified pulse: *s*_{1}(*t*) (dash-dotted blue curve), *s*_{2}(*t*) (dashed green curve), and *s*_{3}(*t*) (solid red curve). At small *t* the pulse is linearly *x* polarized, as indicated by *s*_{1}(*t*) = 1. As *t* increases, *s*_{1}(*t*) starts to decrease, and both *s*_{2}(*t*) and *s*_{3}(*t*) become non-zero. From Fig. 3(a) we see that this coincides with a decrease in the degree of polarization, which indicates that the originally fully polarized field becomes partially polarized. The polarization state of the fully polarized part of the partially polarized field is specified by the normalized Stokes parameters [12]. Since *s*_{3}(*t*) and both *s*_{1}(*t*) and *s*_{2}(*t*) are non-zero, we see that the fully polarized part of the field is elliptically polarized. As *t* increases further, both *s*_{2}(*t*) and *s*_{3}(*t*) tend to zero and *s*_{1}(*t*) approaches −1, demonstrating that the tail of the pulse is fully polarized in the *y* direction. Figure 3(c) illustrates the evolution of the polarization state in terms of the Poincaré sphere. The Poincaré vector traces a flattened arc from *s*_{1}(*t*) = 1 to *s*_{1}(*t*) = −1, as seen from Fig. 3 (
Media 1), with most of the curve being inside the sphere corresponding to partially polarized state.

Next we consider symmetric magnification in the case that the quadratic phase given in Eq. (25) is not eliminated, i.e., Φ_{a}* _{x}* ≠ Φ

_{ay}and Φ

_{bx}≠ Φ

_{by}, leading to

*J*(

_{xx}*t*) and

*J*(

_{yy}*t*) of the components, and they retain the forms given in Eqs. (48) and (49), but the off-diagonal element

*J*(

_{xy}*t*) becomes

*S*

_{0}(

*t*) and

*S*

_{1}(

*t*) remain as given in Eqs. (51) and (52), but the last two parameters become

*s*

_{1}(

*t*) remains as given in Eq. (55). Based on Eq. (8) we observe that the degree of polarization is exactly the same as given in Eq. (58). The effect of

*ϕ*is, however, manifested in the polarization state of the polarized part of the field, as indicated by the harmonic factors in

_{q}*s*

_{2}(

*t*) and

*s*

_{3}(

*t*).

The effect of the quadratic phase is shown in Fig. 4 with *M* = 0.6,
${\mathrm{\Phi}}_{\text{a}x}=0.20{T}_{0}^{2}$,
${\mathrm{\Phi}}_{\text{b}x}=-0.12{T}_{0}^{2}$,
${\mathrm{\Phi}}_{\text{a}y}=2.0{T}_{0}^{2}$,
${\mathrm{\Phi}}_{\text{b}y}=-1.2{T}_{0}^{2}$, *τ _{d}* =

*T*=

_{c}*T*

_{0}. The chosen GDD parameters lead to ${\varphi}_{q}=2.5{T}_{0}^{-2}$. These values are of realistic magnitude, which can be verified by calculating the corresponding propagation length

*z*. A typical single-mode fiber might have the parameter

*β*

_{2}= 20,000 fs

^{2}m

^{−1}[28]. Intensities

*I*(

_{x}*t*) and

*I*(

_{y}*t*) and the degree of polarization

*P*(

*t*) within the pulse are exactly as shown in Fig. 3(a) corresponding to the previous case of Φ

_{a}

*= Φ*

_{x}_{ay}and Φ

_{bx}= Φ

_{by}. The behavior of the polarization state, expressed using the normalized Stokes parameters, is demonstrated in Fig. 4(a), where

*s*

_{1}(

*t*),

*s*

_{2}(

*t*), and

*s*

_{3}(

*t*) as a function of

*t*are shown with blue solid curve, green dashed curve, and red dash-dotted curve, respectively. As in the previous case, the leading edge of the pulse is fully polarized in the

*x*direction and the trailing edge in the

*y*direction. The notable difference in the polarization behavior is observed in the middle of the pulse, where the polarization state of the fully polarized part of the field oscillates due to the quadratic time dependence of

*s*

_{2}(

*t*) and

*s*

_{3}(

*t*) induced by

*ϕ*. The polarization state is also shown in terms of the Poincaré sphere with a solid line in Fig. 4(b), where the Poincaré vector traces out a spiralling curve inside the sphere from

_{q}*s*

_{1}(

*t*) = 1 to

*s*

_{1}(

*t*) = −1 [see Fig. 4 ( Media 2)], corresponding to fully

*x*-polarized state and fully

*y*-polarized state, respectively.

#### 5.2. Antisymmetric magnification

Next we investigate the case where *M _{x}* = −

*M*, i.e., the temporal magnifications experienced by the

_{y}*x*and

*y*components are antisymmetric. The other parameters are as before, i.e.,

*τ*=

_{d}*T*=

_{c}*T*

_{0}and

*ω*

_{0}= 2000/

*T*

_{0}. Let us define

*M*=

_{x}*M*and

*M*= −

_{y}*M*. We then obtain the polarization matrix elements

*ϕ*(

_{xy}*t*) is ±

*π*/2 as discussed previously. The quadratic term

*ϕ*is eliminated if the dispersion parameters are chosen such that Φ

_{q}_{by}= Φ

_{bx}(

*M*+ 1)/(

*M*− 1); we assume that such choice has been made, and denote this by

*ϕ*(

_{xy}*t*) =

*ϕ*. The Stokes parameters are

_{xy}*t*of the numerator in expressions for

*s*

_{2}(

*t*) and

*s*

_{3}(

*t*). As discussed previously, the effective time delay Δ(

*t*) = −2

*t*/

*M*−

*τ*, together with the coherence time

_{d}*T*of the original pulse, dictates the correlation between the

_{c}*x*and

*y*components of the pulse. The orthogonal components are correlated only when Δ(

*t*) is of the order of the coherence time. The effective time delay is zero when

*t*= −

*Mτ*/2; this coincides with the midpoint of the intensity peaks,

_{d}*s*

_{1}(−

*Mτ*/2) = 0. We recall from Sec. 2 that when

_{d}*I*(

_{x}*t*) =

*I*(

_{y}*t*) =

*I*(

*t*), the degree of polarization is given by

*P*(

*t*) = |

*γ*(

_{xy}*t*)|. Since in this case Δ(

*t*) = 0 when the component intensities are equal, the field is fully polarized at that time, independently of the degree of polarization at other points of the pulse.

The component intensities and the degree of polarization of a pulse after passing through a system with *M _{x}* = 0.6 and

*M*= −0.6 are shown in Fig. 5(a). The

_{y}*x*and

*y*component intensities and the degree of polarization are shown with blue dash-dotted, green dashed, and red solid lines, respectively, with thin lines corresponding to the pulse prior to imaging and thick lines to after imaging. Due to the negative

*y*magnification, the

*y*component precedes the

*x*component at the output of the system, with the delay −

*Mτ*= −0.6

_{d}*τ*. The degree of polarization no longer has its minima at the midpoint between the intensity maxima, but on the contrary, the minima are on either side of the midpoint and at the midpoint there is a local maximum with

_{d}*P*(

*t*) = 1. The complete polarization is due to the full correlation between the

*x*and

*y*components, caused by zero effective time delay at the midpoint, as described above. The depth of the degree of polarization modulation is smaller than in the previous cases, with

*P*(

*t*) ranging from 0.55 to unity. The leading edge of the pulse is

*y*-polarized, in contrast to the case with

*M*=

_{x}*M*. As

_{y}*t*becomes larger, the

*x*component begins to affect the polarization of the pulse. We observe from the values of

*s*

_{2}(

*t*) and

*s*

_{3}(

*t*) in Fig. 5(b) that the correlations between the components are negligible outside the interval −

*T*

_{0}<

*t*< 0.3

*T*

_{0}, leading to partial polarization for

*t*< −

*T*

_{0}. The path traced by the polarization state on the Poincaré sphere from

*s*

_{1}(

*t*) = −1 to

*s*

_{1}(

*t*) = 1 is shown in Fig. 5(c) ( Media 3).

#### 5.3. Asymmetric magnification

The final example in this work is the case of asymmetric magnification, i.e., |*M _{x}*| ≠ |

*M*|. As before, the parameters characterizing the input beam are

_{y}*τ*=

_{d}*T*=

_{c}*T*

_{0}and

*ω*

_{0}= 2000/

*T*

_{0}. We restrict our consideration to the situation where the quadratically time-dependent phase term

*ϕ*(

_{xy}*t*) in Eq. (25) is eliminated. This choice enables us to observe the effect of the asymmetric magnification alone, without the additional phase modulation effect. We see that the quadratic phase term is eliminated if the GDD parameters are such that $\left({M}_{x}^{-1}-1\right)/{\mathrm{\Phi}}_{\text{b}x}=\left({M}_{y}^{-1}-1\right)/{\mathrm{\Phi}}_{\text{b}y}$. The Stokes parameters are the general ones put forward in Eqs. (44)–(47) with the simplification that

*ϕ*(

_{xy}*t*) is independent of time.

The polarization of the image beam shows several features which distinguish it from the previous cases, as we see from Fig. 6 illustrating the output of a system with *M _{x}* = 2.0 and

*M*= 0.5. Figure 6(a) shows the intensities of the

_{y}*x*and

*y*components with dash-dotted blue and dashed green curves, respectively, and the degree of polarization with red solid curves. The thin and thick curves illustrate the properties of the pulse before and after imaging, respectively. We observe that the

*x*component is stretched in time with a corresponding decrease in the peak intensity, and the

*y*component is compressed in time with increased peak intensity. The time delay between the components has also changed to

*M*= 0.5

_{y}τ_{d}*τ*. The degree of polarization exhibits two local minima, not only one as in the previous cases. We observe that the variations in the degree of polarization within the pulse are large, with

_{d}*P*(

*t*) between 0.25 and unity. Thus parts of the pulse are weakly polarized, while others are fully polarized. The full polarization at

*t*= 0.67

*T*

_{0}, between the periods of weak polarization, occurs due to the effective time delay Δ(

*t*) being zero at that time, corresponding to complete correlation between the orthogonal components.

The temporal behavior of the Stokes parameters *s*_{1}(*t*), *s*_{2}(*t*), and *s*_{3}(*t*) is illustrated in Fig. 6(b) with solid blue, dashed green, and dash-dotted red lines, respectively. The polarization states of the leading and trailing edges of the pulse are, in contrast to the previous examples, the same, i.e., *x*-polarized, as seen from the curve of *s*_{1}(*t*) as *t* → ±∞. This is due to the fact that the shifted and demagnified *y* component is contained well within the interval where the *x* component has appreciable intensity [see Fig. 6(a)]. Inside the pulse the polarized part of the partially polarized field is elliptically polarized, as evident from the non-zero values of *s*_{3}(*t*) in Fig. 6(b). The evolution of the polarization state within the pulse is shown in Fig. 6(c) using the Poincaré sphere. We see that the Poincaré vector traces out a closed path (red curve), which demonstrates that the polarization states at the leading and trailing edges are the same. Most of the curve is inside the sphere, representing partially polarized states, but at one point the curve touches the surface of the sphere near the ‘north’ pole, indicating an almost right-handed circularly polarized state. The evolution of the Poincaré vector is further illustrated in Fig. 6 (
Media 4).

## 6. Conclusion

We have shown how a temporal imaging system, consisting of two dispersive waveguides and a temporal phase modulator, can be used to tailor the degree and state of polarization within a light pulse. In general, the polarization of the output pulse depends on the temporal coherence of the input, but for symmetric magnification the output polarization is specified by the input polarization and the system parameters only.

We illustrated the temporal imaging by considering the propagation of an electromagnetic Gaussian Schell-model (GSM) pulsed beam through imaging systems with symmetric, antisymmetric, and asymmetric magnifications. In the symmetric case, the profiles of the intensity and the degree of polarization of the output pulse are scaled and magnified/demagnified copies of those of the input pulse. However, the polarization state within the pulse can differ from that of the input, depending on the induced quadratic phase difference between the components. For the antisymmetric magnification the degrees of polarization of the input and output pulses can have different profiles. More precisely, even though the input pulse is weakly polarized at its central region, the output pulse exhibits a duration of full polarization near the center, surrounded by regions of weak polarization. In the asymmetric case, the distribution of the degree and state of polarization can be further altered in versatile ways. We also showed that in the case of symmetric and antisymmetric magnifications the leading and trailing edges have orthogonal polarizations, while in the asymmetric case the edges have the same polarization state.

Here we considered partially polarized GSM pulses created by introducing a time delay between the orthogonal polarization components. Pulses with more elaborate temporal and spectral polarization properties could be produced by, for instance, making use of wave plates, controllable spatial light modulators (SLMs), or other elements in one or both of the arms. The results of this work can find applications in various fields of optics, e.g., optical communications and information processing, laser-assisted micromachining, polarization microscopy, and generation and characterization of polarization-dependent phenomena.

## Acknowledgments

This work was partially funded by the Academy of Finland (grant 272414). A major part of this work was performed while the authors were with Aalto University, Espoo, Finland.

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