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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 7 — Apr. 8, 2013
  • pp: 8987–9004
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Polarization changes in temporal imaging with pulses of random light

Timo Voipio, Tero Setälä, and Ari T. Friberg  »View Author Affiliations


Optics Express, Vol. 21, Issue 7, pp. 8987-9004 (2013)
http://dx.doi.org/10.1364/OE.21.008987


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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

1. J. W. Goodman, Statistical Optics (Wiley, 1985).

4

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

]. For stationary fields the theory of partial polarization in time domain has a long history, originating from the 1950s [5

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

, 6

6. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 2001).

]. Thirty years later, the theory was extended to frequency domain [7

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] .

], 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

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

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

]. A systematic treatment for both temporal and spectral partial polarization of electromagnetic pulses was presented very recently [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] .

]. 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

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

15

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] .

] (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

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

], chronocyclic tomography [17

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] .

], or optical Fourier transform [18

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] .

]. Other applications have included production of picosecond pulses from continuous-wave or mode-locked lasers [19

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] .

]. 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(r1, r2, t1, t2) [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] .

], whose elements are
Γij(r1,r2,t1,t2)=Ei*(r1,t1)Ej(r2,t2),
(1)
where 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 Jxx(r, t) and Jyy(r, t) are the intensities Ix(r, t) and Iy(r, t) of the orthogonal field components, and the trace tr J(r, t) gives the total intensity I(r, t) = Ix(r, t) + Iy(r, t) of the field. The off-diagonal elements Jxy(r, t) and Jyx(r,t)=Jxy*(r,t) describe the correlations between the field components. The correlations can also be expressed using the correlation coefficient γxy(r,t)=Jxy(r,t)/[Jxx(r,t)Jyy(r,t)]1/2=γyx*(r,t). The coefficient is bounded in absolute value by |γxy(r, t)| ≤ 1.

Fig. 1 Illustration of the Poincaré sphere. The dashed circle shows the intersection of the s3(t) = 0 plane with the sphere. Dots indicate intersections of the sphere with the coordinate axes, with the respective fully polarized states indicated (beam propagates towards the viewer).

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

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

, 15

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] .

].

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 βi(ω), i = x, y, of the waveguide may be reasonably approximated with the Taylor polynomials βi(ω) = β0i +β1i(ωω0)+β2i(ωω0)2/2, where the Taylor coefficients are βni = dnβi(ω)/dωn evaluated at ω = ω0. Physically, ω0/β0i are the phase velocities at the center frequency and β1i1 are the group velocities at which the envelopes of the orthogonal components propagate. The parameters β2i 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 β0x = β0y = β0 and β1x = β1y = β1, but β2xβ2y, allowing anisotropic pulse broadening on propagation. The complex envelope after propagation over a distance z in such a waveguide is obtained via [21

21. B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 2nd ed. (Wiley, 2007).

]
Ai(z,t)=Ai(0,t)Ki(f)(tt;z)dt,i=x,y,
(10)
with the propagation kernel
Ki(f)(tt;z)=eiβ0zi2πβ2izexp{i[(tt)+β1z]22β2iz},i=x,y.
(11)
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.

We see that by examining the pulse in the local time frame, which moves at the group velocity, i.e., t̃ = tβ1z, Eq. (10) becomes
Ai(z,t˜)=Ai(0,t)Ki(f)(t˜t;z)dt,i=x,y,
(12)
together with
Ki(f)(t˜t;z)=eiβ0zi2πΦiexp[i(t˜t)22Φi],i=x,y,
(13)
where we have introduced the group delay dispersion (GDD) parameter Φi = β2iz.

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 Φax and Φay 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 γx and γy, 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 Φbx and Φby, retains the modified spectral profile but changes the relative phases of different frequency components of the pulse, altering the temporal profile.

Fig. 2 Schematic of a temporal imaging system. The middle row illustrates the arrangement with two single-mode fibers (SMF) whose GDD parameters are Φa and Φb, and the quadratic phase modulator (QPM). The phase ϕ(t) imparted by the QPM is determined by the parameter γ. The three figures in the top row illustrate the behavior of the electric field E(t) of an originally transform-limited pulse with center frequency ω0. The first plot shows the electric field at the input of the system. The second plot shows that propagation through the first SMF has broadened the pulse and introduced chirp. Finally, propagation through the second SMF compresses the pulse. The bottom row shows the spectral densities S(ω) of the pulse before and after propagation through the QPM. In the case of an originally transform-limited pulse, the spectrum is broadened by the phase modulation.

The relation between the input and output complex envelopes Ai(in)(t) and Ai(out)(t) of the system in Fig. 2 is obtained by consecutive applications of Eq. (9) and Eq. (12) [21

21. B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 2nd ed. (Wiley, 2007).

], which gives
Ai(out)(t)=Ai(in)(t)Ki(t,t)dt,i=x,y.
(14)
The explicit spatial dependence has been omitted as the pulses are considered at the beginning and the end of the system in their local time frames, denoted by t′ and t, respectively. The kernel Ki(t, t′) is
Ki(t,t)=eiϕi(t,t)i2πΦaii2πΦbiexp[i2Lit2+i(tΦai+tΦbi)t]dt,
(15)
where ϕi(t, t′) = β0aza + β0bzb − (t2ai + t2bi)/2, i = x, y. The quantities β0a and β0b are the propagation constants of fibers a and b at ω0, respectively, and za and zb are the lengths of the fibers. We observe that the quantity Li = 1/Φbi + 1/Φai − 1/γi bears resemblance to the lens law both in its form and in the role it plays. If the temporal focal parameter γi and the GDD parameters Φai and Φbi are chosen such that Li = 0, i.e.,
1γi=1Φai+1Φbi,i=x,y,
(16)
the quadratic term in the integrand in Eq. (15) vanishes and the kernel simplifies to
Ki(t,t)=eiϕi(t,t)iΦaiiΦbi|Φai|δ(t+ΦaiΦbit),i=x,y.
(17)
Using this kernel, the complex envelope at the output of the system is
Ai(out)(t)=exp{i[β0aza+β0bzb+π4(sgnΦai+sgnΦbi)+Mi112Φbit2]}1|Mi|1/2Ai(in)(tMi),
(18)
where we have introduced the temporal magnification
Mi=Φbi/Φai,i=x,y.
(19)
We see that the output is possibly scaled (with a factor of |Mi|−1/2), magnified (|Mi| >1) or demagnified (|Mi| < 1), and inverted (Mi < 0) or erect (Mi > 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

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] .

]. 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 Mi ≠ 1.

4. Polarization of a temporally imaged pulse

The elements of the envelope coherence matrix at the output are
Γij(e)(t1,t2)=Γ0ij(e)(t1,t2)Kij(t1,t2;t1,t2)dt1dt2,i,j=x,y,
(21)
where
Kij(t1,t2;t1,t2)=Ki*(t1,t1)Kj(t2,t2),i,j=x,y,
(22)
as obtained from Eq. (15). If the imaging condition (16) holds for both x and y components, the coherence matrix becomes
Γij(e)(t1,t2)=exp[iϕij(t1,t2)]|MiMj|1/2Γ0ij(e)(t1Mi,t2Mj),i,j=x,y.
(23)
The polarization matrix J(t) at the output of the imaging system is obtained from the coherence matrix by setting t1 = t2 = t:
Jij(t)=exp[iϕij(t)]|MiMj|1/2Γ0ij(tMi,tMj)exp[iω0(Mi1Mj1)t],i,j=x,y,
(24)
with the quadratic phase
ϕij(t)=π4[sgn(Φaj)+sgn(Φbj)sgn(Φai)sgn(Φbi)]+(Mj112ΦbjMi112Φbi)t2,
(25)
where i, j = x, y. We observe that ϕxx(t) = ϕyy(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 ϕxy(t) and ϕyx(t) are generally non-zero and obey the equation ϕyx(t) = −ϕxy(t). Recalling the definition of the temporal magnification presented in Eq. (19), we see that if both Mx and My are positive, the time-independent term in ϕxy(t) is zero. If both Mx and My are negative, then the term is either −π, zero, or π, depending on the signs of the GDD parameters. When Mx and My have different signs, the time-independent phase is π/2 if My < 0, Φay > 0 or Mx < 0, Φax < 0 and −π/2 in other cases.

According to Eq. (24), the intensities at the output are
Ii(t)=1|Mi|I0i(tMi),i=x,y,
(26)
where I0i(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 |Mi| ensures that the total energy in the i component is conserved in propagation through the imaging system.

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

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] .

]. Systems of this type have been studied in the stationary case as polarization state controllers [23

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

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] .

] and they are, in particular, the method of choice in experiments to create unpolarized light beams [25

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] .

]. 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

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

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

]. 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 Ex(t) = E (t) and Ey(t) = E (tτd), where τd is the time delay. The signal
E(t)=A0a(t)exp[t2/(2T02)]exp(iω0t)
(35)
is defined by the amplitude A0, the normalized amplitude function a(t) with 〈|a(t)|2〉 = 1, the pulse length T0, and the central frequency ω0. The amplitude function a(t) describes the random fluctuations of the field, with the temporal correlations of the Gaussian form
a*(t1)a(t2)=exp[(t2t1)2/Tc2],
(36)
where Tc is the coherence time. The intensity of the signal E (t) is likewise Gaussian with
I(t)=A02exp(t2/T02).
(37)
The coherence matrix of the GSM beam thus becomes
Γ0(t1,t2)=[Γ(t1,t2)Γ(t1,t2τd)Γ(t1τd,t2)Γ(t1τd,t2τd)],
(38)
where
Γ(t1,t2)=[I(t1)I(t2)]1/2exp[(t2t1)2/Tc2]exp[iω0(t2t1)]
(39)
is the coherence function of the original linearly polarized field.

The Stokes parameters are obtained by using Eqs. (3)(6) in Eqs. (40)(42):
S0(t)=1|Mx|I(tMx)+1|My|I(tMyτd),
(44)
S1(t)=1|Mx|I(tMx)1|My|I(tMyτd),
(45)
S2(t)=2|MxMy|1/2[I(tMx)I(tMyτd)]1/2exp[Δ2(t)/Tc2]cos[ϕxy(t)+ω0τd],
(46)
S3(t)=2|MxMy|1/2[I(tMx)I(tMyτd)]1/2exp[Δ2(t)/Tc2]sin[ϕxy(t)+ω0τd].
(47)
We observe, based on the expressions for S1(t), S2(t), and S3(t), that the polarization state of the output beam can be altered by changing the time delay τd, the GDD parameters and the ‘focal length’ of the QPM such that the imaging condition in Eq. (16) holds for both orthogonal components.

5.1. Symmetric magnification

The first case we consider is where the x and y components of the pulse undergo similar magnification, Mx = My = M. The simplest way to ensure symmetric magnification is to choose the system parameters such that Φax = Φay and Φbx = Φby. As evident from Eqs. (43) and (25), the effective time delay Δ(t) becomes time-independent and the phase term ϕxy(t) is zero at all times. The polarization matrix elements at the output are
Jxx(t)=1|M|I(tM),
(48)
Jyy(t)=1|M|I(tMτd),
(49)
Jxy(t)=1|M|[I(tM)I(tMτd)]1/2exp(iω0τdτd2/Tc2),
(50)
and Jyx(t)=Jxy*(t). These result in the Stokes parameters
S0(t)=1|M|[I(tM)+I(tMτd)],
(51)
S1(t)=1|M|[I(tM)I(tMτd)],
(52)
S2(t)=2|M|[I(tM)I(tMτd)]1/2exp(τd2/Tc2)cos(ω0τd),
(53)
S3(t)=2|M|[I(tM)I(tMτd)]1/2exp(τd2/Tc2)sin(ω0τd),
(54)
and the normalized Stokes parameters
s1(t)=tanh[(t/Mτd/2)τd/T02],
(55)
s2(t)=exp(τd2/Tc2)cosh[(t/Mτd/2)τd/T02]cos(ω0τd),
(56)
s3(t)=exp(τd2/Tc2)cosh[(t/Mτd/2)τd/T02]sin(ω0τd).
(57)
The degree of polarization becomes
P(t)={tanh2[(τd2t/M)τd2T02]+sech2[(τd2t/M)τd2T02]exp(2τd2/Tc2)}1/2,
(58)
where Eq. (8) was used. We see that in the high-coherence limit (τdTc), the exponential term tends to unity and thus P(t) = 1 at all times, i.e., the beam is fully polarized throughout. In the opposite case, τdTc, the exponential term goes to zero and P(t)=|tanh(τd2t/M)τd/2T02|, which leads to P(d/2) = 0. In all cases the pulse edges are fully polarized, as seen from the fact that limt→±∞P(t) = 1.

The effect of symmetric magnification is investigated numerically in the case of τd = Tc = T0, ω0 = 2000/T0, 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 d = 0.6τd. Temporal development of the degree of polarization has similarly been demagnified, but the shape of the 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.

Fig. 3 (a) Intensities of x (blue dash-dotted) and y (green dashed) components and the degree of polarization (red solid) before and after temporal imaging, shown with thin and thick curves, respectively. The vertical axis represents intensity normalized such that the peak intensity of the components at the input of the temporal imaging system is 1.0. (b) Behavior of the normalized Stokes parameters as a function of time. Solid blue line, dashed green line, and dash-dotted red line indicate values of s1(t), s2(t), and s3(t), respectively. (c) Evolution of the polarization state (solid curve) represented on the Poincaré sphere. The dashed circles show the intersection of the s1(t) = 0, s2(t) = 0, and s3(t) = 0 planes with the surface of the sphere. Correspondence between the intensities, the Stokes parameters, and the respective locations on the Poincaré sphere is further illustrated in Media 1.

Figure 3(b) presents the normalized Stokes parameters of the temporally demagnified pulse: s1(t) (dash-dotted blue curve), s2(t) (dashed green curve), and s3(t) (solid red curve). At small t the pulse is linearly x polarized, as indicated by s1(t) = 1. As t increases, s1(t) starts to decrease, and both s2(t) and s3(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

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] .

]. Since s3(t) and both s1(t) and s2(t) are non-zero, we see that the fully polarized part of the field is elliptically polarized. As t increases further, both s2(t) and s3(t) tend to zero and s1(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 s1(t) = 1 to s1(t) = −1, as seen from Fig. 3 ( Media 1), with most of the curve being inside the sphere corresponding to partially polarized state.

Fig. 4 Polarization state of a pulse after propagation through a temporal imaging system which imparts a quadratic phase difference between the orthogonal components ( Media 2). (a) Behavior of the normalized Stokes parameters as a function of time. Solid blue line, dashed green line, and dash-dotted red line indicate values of s1(t), s2(t), and s3(t), respectively. (b) Evolution of the polarization state (solid curve) represented on the Poincaré sphere. The dashed circles show the intersection of the s1(t) = 0, s2(t) = 0, and s3(t) = 0 planes with the surface of the sphere.

5.2. Antisymmetric magnification

Fig. 5 (a) Component intensities and the degree of polarization before (thin lines) and after (thick lines) temporal imaging with opposite magnifications for x and y components. Intensities of the x and y components and the degree of polarization are shown with blue dash-dotted, green dashed, and red solid curves, respectively. Intensities before and after imaging are represented on the same scale to show the temporal compression-induced change in peak intensity. The main difference to the case Mx = My in Fig. 3 is the short period of full polarization midway between the intensity peaks. (b) Normalized Stokes si(t) parameters of a temporally imaged pulse with opposite magnifications for the orthogonally polarized components. (c) Polarization state represented using a Poincaré sphere, solid curve, and the intersections of the planes s1(t) = 0, s2(t) = 0, and s3(t) = 0 with the surface of the Poincaré sphere, dashed circles. The locus of the points representing the polarization states of the beam at different instants of time forms a closed curve. The connection between the Poincaré sphere, the component intensities, and the Stokes parameters is illustrated in Media 3.

5.3. Asymmetric magnification

The final example in this work is the case of asymmetric magnification, i.e., |Mx| ≠ |My|. As before, the parameters characterizing the input beam are τd = Tc = T0 and ω0 = 2000/T0. We restrict our consideration to the situation where the quadratically time-dependent phase term ϕxy(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 (Mx11)/Φbx=(My11)/Φby. The Stokes parameters are the general ones put forward in Eqs. (44)(47) with the simplification that ϕxy(t) is independent of time.

Fig. 6 (a) Intensities of the x and y components (blue dash-dotted and green dashed lines, respectively), and the degree of polarization (red solid lines) before and after imaging (thin and thick lines, respectively). (b) Normalized Stokes parameters s1(t), s2(t), and s3(t) drawn with blue solid, green dashed, and red dash-dotted lines, respectively. (c) Polarization state (red solid curve) in terms of the Poincaré sphere. The dashed circles depict the intersections of the planes s1(t) = 0, s2(t) = 0, and s3(t) = 0 with the surface of the Poincaré sphere. The time evolution of the polarization state is demonstrated in 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.

References and links

1.

J. W. Goodman, Statistical Optics (Wiley, 1985).

2.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

3.

E. Collett, Polarized Light in Fiber Optics (SPIE, 2003).

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] .

6.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 2001).

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] .

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 21, 2117–2123 (2004) [CrossRef] .

10.

H. Lajunen, P. Vahimaa, and J. Tervo, “Theory of spatially and spectrally partially coherent pulses,” J. Opt. Soc. Am. A 22, 1536–1545 (2005) [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] .

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] .

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] .

20.

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] .

21.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 2nd ed. (Wiley, 2007).

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] .

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] .

28.

A. Yariv and P. Yeh, Photonics, 6th ed. (Oxford University, 2007).

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


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References

  1. J. W. Goodman, Statistical Optics (Wiley, 1985).
  2. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).
  3. E. Collett, Polarized Light in Fiber Optics (SPIE, 2003).
  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 Cimento13, 1165–1181 (1959). [CrossRef]
  6. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 2001).
  7. R. Barakat, “n-fold polarization measures and associated thermodynamic entropy of N partially coherent pencils of radiation,” Optica Acta30, 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]
  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. A21, 2117–2123 (2004). [CrossRef]
  10. H. Lajunen, P. Vahimaa, and J. Tervo, “Theory of spatially and spectrally partially coherent pulses,” J. Opt. Soc. Am. A22, 1536–1545 (2005). [CrossRef]
  11. K. Saastamoinen, J. Turunen, P. Vahimaa, and A. T. Friberg, “Spectrally partially coherent propagation-invariant fields,” Phys. Rev. A80, 053804 (2009). [CrossRef]
  12. T. Voipio, T. Setälä, and A. T. Friberg, “Partial polarization theory of optical pulses,” J. Opt. Soc. Am. A30, 71–81 (2013). [CrossRef]
  13. B. Kolner and M. Nazarathy, “Temporal imaging with a time lens,” Opt. Lett.14, 630–632 (1989). [CrossRef] [PubMed]
  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]
  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,” Nature456, 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]
  20. 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]
  21. B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 2nd ed. (Wiley, 2007).
  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]
  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. A82, 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. SPIE8171, 817102 (2011). [CrossRef]
  28. A. Yariv and P. Yeh, Photonics, 6th ed. (Oxford University, 2007).

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