## Polarization changes in temporal imaging with pulses of random light |

Optics Express, Vol. 21, Issue 7, pp. 8987-9004 (2013)

http://dx.doi.org/10.1364/OE.21.008987

Acrobat PDF (2391 KB)

### 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

4. R. Martínez-Herrero, P. M. Mejías, and G. Piquero, *Characterization of Partially Polarized Light Fields* (Springer, 2009) [CrossRef] .

5. E. Wolf, “Coherence properties of partially polarized electromagnetic radiation,” Nuovo Cimento **13**, 1165–1181 (1959) [CrossRef] .

7. R. Barakat, “*n*-fold polarization measures and associated thermodynamic entropy of *N* partially coherent pencils of radiation,” Optica Acta **30**, 1171–1182 (1983) [CrossRef] .

8. P. Pääkkönen, J. Turunen, P. Vahimaa, A. T. Friberg, and F. Wyrowski, “Partially coherent Gaussian pulses,” Opt. Commun. **204**, 53–58 (2002) [CrossRef] .

11. K. Saastamoinen, J. Turunen, P. Vahimaa, and A. T. Friberg, “Spectrally partially coherent propagation-invariant fields,” Phys. Rev. A **80**, 053804 (2009) [CrossRef] .

12. T. Voipio, T. Setälä, and A. T. Friberg, “Partial polarization theory of optical pulses,” J. Opt. Soc. Am. A **30**, 71–81 (2013) [CrossRef] .

13. B. Kolner and M. Nazarathy, “Temporal imaging with a time lens,” Opt. Lett. **14**, 630–632 (1989) [CrossRef] [PubMed] .

15. V. Torres-Company, J. Lancis, and P. Andrés, “Space-time analogies in optics,” in *Progress in Optics*, ed. E. Wolf, vol. 56 (Elsevier, 2011), pp. 1–80 [CrossRef] .

16. C. V. Bennett and B. H. Kolner, “Upconversion time microscope demonstrating 103× magnification of femtosecond waveforms,” Opt. Lett. **24**, 783–785 (1999) [CrossRef] .

17. Ch. Dorrer and I. Kang, “Complete temporal characterization of short optical pulses by simplified chronocyclic tomography,” Opt. Lett. **28**, 1481–1483 (2003) [CrossRef] [PubMed] .

18. M. A. Foster, R. Salem, D. F. Geraghty, A. C. Turner-Foster, M. Lipson, and A. L. Gaeta, “Silicon-chip-based ultrafast optical oscilloscope,” Nature **456**, 81–85 (2008) [CrossRef] [PubMed] .

19. A. A. Godil, B. A. Auld, and D. M. Bloom, “Time-lens producing 1.9 ps optical pulses,” Appl. Phys. Lett. **62**, 1047–1049 (1992) [CrossRef] .

## 2. Polarization of an electromagnetic pulse

**E**(

**r**,

*t*), are encoded in the 2 × 2 electric coherence matrix

**Γ**(

**r**

_{1},

**r**

_{2},

*t*

_{1},

*t*

_{2}) [12

12. T. Voipio, T. Setälä, and A. T. Friberg, “Partial polarization theory of optical pulses,” J. Opt. Soc. Am. A **30**, 71–81 (2013) [CrossRef] .

*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

*γ*(

_{xy}**r**,

*t*)| ≤ 1.

## 3. Temporal imaging using a time lens

14. I. A. Walmsley and C. Dorrer, “Characterization of ultrashort electromagnetic pulses,” Adv. Opt. Photon. **1**, 308–437 (2009) [CrossRef] .

15. V. Torres-Company, J. Lancis, and P. Andrés, “Space-time analogies in optics,” in *Progress in Optics*, ed. E. Wolf, vol. 56 (Elsevier, 2011), pp. 1–80 [CrossRef] .

*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

*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] with the propagation kernel Technically, taking the upper integration limit in Eq. (10) as infinity instead of

*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.

*t*̃ =

*t*−

*β*

_{1}

*z*, Eq. (10) becomes together with where we have introduced the group delay dispersion (GDD) parameter Φ

*=*

_{i}*β*

_{2i}

*z*.

_{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*and

_{x}*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.

*t*′ and

*t*, respectively. The kernel

*K*(

_{i}*t*,

*t*′) is where

*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., the quadratic term in the integrand in Eq. (15) vanishes and the kernel simplifies to Using this kernel, the complex envelope at the output of the system is where we have introduced the temporal magnification We see that the output is possibly scaled (with a factor of |

_{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

_{i}22. C. V. Bennett, R. P. Scott, and B. H. Kolner, “Temporal magnification and reversal of 100 Gb/s optical data with an up conversion time microscope,” Appl. Phys. Lett. **65**, 2513–2515 (1994) [CrossRef] .

*M*≠ 1.

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

**Γ**

^{(e)}(

*t*

_{1},

*t*

_{2}) with the elements where Γ

*(*

_{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

**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.

*x*and

*y*components, the coherence matrix becomes The polarization matrix

**J**(

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

*t*

_{1}=

*t*

_{2}=

*t*: with the quadratic phase where

*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.

*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.

## 5. Examples

12. T. Voipio, T. Setälä, and A. T. Friberg, “Partial polarization theory of optical pulses,” J. Opt. Soc. Am. A **30**, 71–81 (2013) [CrossRef] .

23. A. Shevchenko, T. Setälä, M. Kaivola, and A. T. Friberg, “Characterization of polarization fluctuations in random electromagnetic beams,” New J. Phys. **11**, 073004 (2009), http://iopscience.iop.org/1367-2630/11/7/073004 [CrossRef] .

24. T. Voipio, T. Setälä, A. Shevchenko, and A. T. Friberg, “Polarization dynamics and polarization time of random three-dimensional electromagnetic fields,” Phys. Rev. A **82**, 063807 (2010) [CrossRef] .

25. K. Lindfors, A. Priimägi, T. Setälä, A. Shevchenko, A. T. Friberg, and M. Kaivola, “Local polarization of tightly focused unpolarized light,” Nature Phot. **1**, 228–231 (2007) [CrossRef] .

26. Ph. Réfrégier, T. Setälä, and A. T. Friberg, “Maximal polarization order of random optical beams: reversible and irreversible polarization variations,” Opt. Lett. **37**, 3750–3752 (2012) [CrossRef] [PubMed] .

27. Ph. Réfrégier, T. Setälä, and A. T. Friberg, “Temporal and spectral degrees of polarization of light,” Proc. SPIE **8171**, 817102 (2011) [CrossRef] .

*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 is defined by the amplitude

_{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 where

*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 where is the coherence function of the original linearly polarized field.

*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.

*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

*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 and

*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

_{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.

*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.

*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

**30**, 71–81 (2013) [CrossRef] .

*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.

_{a}

*≠ Φ*

_{x}_{ay}and Φ

_{bx}≠ Φ

_{by}, leading to where we have introduced the quadratic phase constant to simplify notation. The phase does not affect the intensities

*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 Correspondingly, the Stokes parameters

*S*

_{0}(

*t*) and

*S*

_{1}(

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

*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*).

*M*= 0.6,

*τ*=

_{d}*T*=

_{c}*T*

_{0}. The chosen GDD parameters lead to

*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

*M*= 0.6 and

_{x}*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

*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

*ϕ*(

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

*M*= 2.0 and

_{x}*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.

*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

## Acknowledgments

## References and links

1. | J. W. Goodman, |

2. | L. Mandel and E. Wolf, |

3. | E. Collett, |

4. | R. Martínez-Herrero, P. M. Mejías, and G. Piquero, |

5. | E. Wolf, “Coherence properties of partially polarized electromagnetic radiation,” Nuovo Cimento |

6. | M. Born and E. Wolf, |

7. | R. Barakat, “ |

8. | P. Pääkkönen, J. Turunen, P. Vahimaa, A. T. Friberg, and F. Wyrowski, “Partially coherent Gaussian pulses,” Opt. Commun. |

9. | H. Lajunen, J. Tervo, and P. Vahimaa, “Overall coherence and coherent-mode expansion of spectrally partially coherent plane-wave pulses,” J. Opt. Soc. Am. A |

10. | H. Lajunen, P. Vahimaa, and J. Tervo, “Theory of spatially and spectrally partially coherent pulses,” J. Opt. Soc. Am. A |

11. | K. Saastamoinen, J. Turunen, P. Vahimaa, and A. T. Friberg, “Spectrally partially coherent propagation-invariant fields,” Phys. Rev. A |

12. | T. Voipio, T. Setälä, and A. T. Friberg, “Partial polarization theory of optical pulses,” J. Opt. Soc. Am. A |

13. | B. Kolner and M. Nazarathy, “Temporal imaging with a time lens,” Opt. Lett. |

14. | I. A. Walmsley and C. Dorrer, “Characterization of ultrashort electromagnetic pulses,” Adv. Opt. Photon. |

15. | V. Torres-Company, J. Lancis, and P. Andrés, “Space-time analogies in optics,” in |

16. | C. V. Bennett and B. H. Kolner, “Upconversion time microscope demonstrating 103× magnification of femtosecond waveforms,” Opt. Lett. |

17. | Ch. Dorrer and I. Kang, “Complete temporal characterization of short optical pulses by simplified chronocyclic tomography,” Opt. Lett. |

18. | M. A. Foster, R. Salem, D. F. Geraghty, A. C. Turner-Foster, M. Lipson, and A. L. Gaeta, “Silicon-chip-based ultrafast optical oscilloscope,” Nature |

19. | A. A. Godil, B. A. Auld, and D. M. Bloom, “Time-lens producing 1.9 ps optical pulses,” Appl. Phys. Lett. |

20. | L. Kh. Mouradian, F. Louradour, V. Messager, A. Barthélémy, and C. Froehly, “Spectro-temporal imaging of femtosecond events,” IEEE J. Quantum Electron. |

21. | B. E. A. Saleh and M. C. Teich, |

22. | C. V. Bennett, R. P. Scott, and B. H. Kolner, “Temporal magnification and reversal of 100 Gb/s optical data with an up conversion time microscope,” Appl. Phys. Lett. |

23. | A. Shevchenko, T. Setälä, M. Kaivola, and A. T. Friberg, “Characterization of polarization fluctuations in random electromagnetic beams,” New J. Phys. |

24. | T. Voipio, T. Setälä, A. Shevchenko, and A. T. Friberg, “Polarization dynamics and polarization time of random three-dimensional electromagnetic fields,” Phys. Rev. A |

25. | K. Lindfors, A. Priimägi, T. Setälä, A. Shevchenko, A. T. Friberg, and M. Kaivola, “Local polarization of tightly focused unpolarized light,” Nature Phot. |

26. | Ph. Réfrégier, T. Setälä, and A. T. Friberg, “Maximal polarization order of random optical beams: reversible and irreversible polarization variations,” Opt. Lett. |

27. | Ph. Réfrégier, T. Setälä, and A. T. Friberg, “Temporal and spectral degrees of polarization of light,” Proc. SPIE |

28. | A. Yariv and P. Yeh, |

**OCIS Codes**

(260.2110) Physical optics : Electromagnetic optics

(260.5430) Physical optics : Polarization

(320.5550) Ultrafast optics : Pulses

**ToC Category:**

Physical Optics

**History**

Original Manuscript: February 11, 2013

Revised Manuscript: March 13, 2013

Manuscript Accepted: March 14, 2013

Published: April 4, 2013

**Citation**

Timo Voipio, Tero Setälä, and Ari T. Friberg, "Polarization changes in temporal imaging with pulses of random light," Opt. Express **21**, 8987-9004 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-7-8987

Sort: Year | Journal | Reset

### References

- J. W. Goodman, Statistical Optics (Wiley, 1985).
- L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).
- E. Collett, Polarized Light in Fiber Optics (SPIE, 2003).
- R. Martínez-Herrero, P. M. Mejías, and G. Piquero, Characterization of Partially Polarized Light Fields (Springer, 2009). [CrossRef]
- E. Wolf, “Coherence properties of partially polarized electromagnetic radiation,” Nuovo Cimento13, 1165–1181 (1959). [CrossRef]
- M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 2001).
- R. Barakat, “n-fold polarization measures and associated thermodynamic entropy of N partially coherent pencils of radiation,” Optica Acta30, 1171–1182 (1983). [CrossRef]
- P. Pääkkönen, J. Turunen, P. Vahimaa, A. T. Friberg, and F. Wyrowski, “Partially coherent Gaussian pulses,” Opt. Commun.204, 53–58 (2002). [CrossRef]
- H. Lajunen, J. Tervo, and P. Vahimaa, “Overall coherence and coherent-mode expansion of spectrally partially coherent plane-wave pulses,” J. Opt. Soc. Am. A21, 2117–2123 (2004). [CrossRef]
- H. Lajunen, P. Vahimaa, and J. Tervo, “Theory of spatially and spectrally partially coherent pulses,” J. Opt. Soc. Am. A22, 1536–1545 (2005). [CrossRef]
- K. Saastamoinen, J. Turunen, P. Vahimaa, and A. T. Friberg, “Spectrally partially coherent propagation-invariant fields,” Phys. Rev. A80, 053804 (2009). [CrossRef]
- T. Voipio, T. Setälä, and A. T. Friberg, “Partial polarization theory of optical pulses,” J. Opt. Soc. Am. A30, 71–81 (2013). [CrossRef]
- B. Kolner and M. Nazarathy, “Temporal imaging with a time lens,” Opt. Lett.14, 630–632 (1989). [CrossRef] [PubMed]
- I. A. Walmsley and C. Dorrer, “Characterization of ultrashort electromagnetic pulses,” Adv. Opt. Photon.1, 308–437 (2009). [CrossRef]
- V. Torres-Company, J. Lancis, and P. Andrés, “Space-time analogies in optics,” in Progress in Optics, ed. E. Wolf, vol. 56 (Elsevier, 2011), pp. 1–80. [CrossRef]
- C. V. Bennett and B. H. Kolner, “Upconversion time microscope demonstrating 103× magnification of femtosecond waveforms,” Opt. Lett.24, 783–785 (1999). [CrossRef]
- Ch. Dorrer and I. Kang, “Complete temporal characterization of short optical pulses by simplified chronocyclic tomography,” Opt. Lett.28, 1481–1483 (2003). [CrossRef] [PubMed]
- M. A. Foster, R. Salem, D. F. Geraghty, A. C. Turner-Foster, M. Lipson, and A. L. Gaeta, “Silicon-chip-based ultrafast optical oscilloscope,” Nature456, 81–85 (2008). [CrossRef] [PubMed]
- A. A. Godil, B. A. Auld, and D. M. Bloom, “Time-lens producing 1.9 ps optical pulses,” Appl. Phys. Lett.62, 1047–1049 (1992). [CrossRef]
- L. Kh. Mouradian, F. Louradour, V. Messager, A. Barthélémy, and C. Froehly, “Spectro-temporal imaging of femtosecond events,” IEEE J. Quantum Electron.36, 795–801 (2000). [CrossRef]
- B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 2nd ed. (Wiley, 2007).
- C. V. Bennett, R. P. Scott, and B. H. Kolner, “Temporal magnification and reversal of 100 Gb/s optical data with an up conversion time microscope,” Appl. Phys. Lett.65, 2513–2515 (1994). [CrossRef]
- A. Shevchenko, T. Setälä, M. Kaivola, and A. T. Friberg, “Characterization of polarization fluctuations in random electromagnetic beams,” New J. Phys.11, 073004 (2009), http://iopscience.iop.org/1367-2630/11/7/073004 . [CrossRef]
- T. Voipio, T. Setälä, A. Shevchenko, and A. T. Friberg, “Polarization dynamics and polarization time of random three-dimensional electromagnetic fields,” Phys. Rev. A82, 063807 (2010). [CrossRef]
- K. Lindfors, A. Priimägi, T. Setälä, A. Shevchenko, A. T. Friberg, and M. Kaivola, “Local polarization of tightly focused unpolarized light,” Nature Phot.1, 228–231 (2007). [CrossRef]
- Ph. Réfrégier, T. Setälä, and A. T. Friberg, “Maximal polarization order of random optical beams: reversible and irreversible polarization variations,” Opt. Lett.37, 3750–3752 (2012). [CrossRef] [PubMed]
- Ph. Réfrégier, T. Setälä, and A. T. Friberg, “Temporal and spectral degrees of polarization of light,” Proc. SPIE8171, 817102 (2011). [CrossRef]
- A. Yariv and P. Yeh, Photonics, 6th ed. (Oxford University, 2007).

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

### Supplementary Material

» Media 1: MOV (1853 KB)

» Media 2: MOV (2351 KB)

» Media 3: MOV (2356 KB)

» Media 4: MOV (2356 KB)

« Previous Article | Next Article »

OSA is a member of CrossRef.