Abstract

Matrix optics is applied to a class of random, in time and space, electromagnetic pulsed beam-like (REMPB) radiation interacting with linear optical elements. A 6×6 order matrix describing transformation of a six-dimensional state vector including four spatial and two temporal positions within the field is used to derive conditions for spatio-temporal coupling. An example is included which deals with a spatio-temporal coupling in a typical REMPB on reflection from a reflecting grating. Electromagnetic nature of such interaction is explored via considering dependence of the degree of polarization of the reflected REMPB on its source and on the structure of the grating.

©2010 Optical Society of America

1. Introduction

Matrix optics techniques serve as an extremely convenient tool in dealing with transformation of electromagnetic radiation by linear optical elements [1]. Matrices of various dimensions and structures have been proposed for description of local transformation of beams and pulses of deterministic and random nature. Perhaps the most important classic theories belong to Jones [2] and Muller [3], which deal with modulation of electromagnetic fields and Stokes vectors, respectively, of stationary waves, providing with single-position transformations. The Jones and Mueller calculi were generalized to two-positions and unified by Korotkova and Wolf [4], [5] taking into account transformation of field correlations. The very convenient tensor method was used by Lin and Cai [6] for description of a vide class of electromagnetic Gaussian Schell-model sources. This tensor method has proven to be a very convenient tool for solving problems relating to random light interaction with resonators [7] as well as LIDAR systems operating in turbulent atmosphere [8]. All the aforementioned references, however, dealt with wide-sense stationary fields and did not include temporal variations. Matrix description of pulsed fields via 3x3 matrices was first introduced by Martinez [9], [10] and extended to a more complete approach by Kostenbauder [11] which employed 4x4 matrices, and also to a more convenient ABCD (2x2) matrices by Dijaili et al. [12]. Other extensions were proposed in Refs [13], [14].

Recently a class of electromagnetic pulses that exhibit non-stationary variations in time and are also spatially random was introduced by Paakkonen et al. [15] (see also [16], where only temporally random fields are discussed). Matrix treatment involving 6×6 tensors which are constructed from ABCD cells was suggested for such fields and their free-space evolution was treated [17]. Propagation, generation, coherent mode-expansion and ghost interference of such electromagnetic pulses were investigated widely [1828].

Spatio-temporal coupling of electromagnetic pulsed waves on interaction with dispersive systems was known for long time (e.g. in [29] on focusing by a lens). The matrix approach to such coupling was introduced by Lin et al. [30], where the formalism of 6x6 matrices was used for reflection-induced coupling by a grating. That analysis was limited, however, to initially deterministic pulses, both in time and space. In a somewhat different perspective reflection of electromagnetic random (stationary) wave from a polarization grating was discussed by Piquero et al. [31], highlighting the aspect of polarization modulation.

We generalize the results of [30] to the class of REMPB and derive general conditions to be satisfied by the transformation matrix in order to describe interaction of REMPB with dispersive linear media, with and without spatio-temporal coupling. We also provide an example which provides coupling conditions for REMPB interacting with a reflecting grating and explore the effect of interaction and subsequent free-space propagation of the pulse from the grating on pulse’s degree of polarization.

2. Theory

We begin by reviewing matrix formalism of Ref [30]. applied for REMPB, in which we employ the Gaussian Schell-model for temporal and spatial fluctuations of Refs [27]. and [28]. The notations are explained in Fig. 1 . We assume that the initial field does not carry spatiotemporal coupling. Hence, the elements of the mutual correlation matrix (MCM) of the pulsed beams are given by the expressions [27,28]

Γ0αβ(r01,τ01;r02,τ02)=AαAβBαβexp[r012+r0224σI0αβ2(r01r02)22δI0αβ2]                                   ×exp[τ012+τ0222στ0αβ2(τ01τ02)22δτ0αβ2]exp[iω0(τ01τ02)],
where τ0=c(tt0) and r0 represent longitudinal and transverse positions in the pulsed beams, respectively; c=λ0ν0 is light velocity in vacuum, and t0 is an arrival time at a certain transverse plane; σI0αβ and δI0αβ are the r.m.s. widths of the intensity and of the degree of coherence in transverse plane, respectively; στ0αβ denotes the r.m.s. longitudinal width, being the averaged pulse duration; δτ0αβ stands for the r.m.s. width of longitudinal degree of coherence, describing the state of temporal coherence. According to the tensor method [6,17,30], Eq. (1) can be expressed in following tensor form
Γ0αβ(r¯0)=AαAβBαβexp[ik02r¯0TQ¯0αβ1r¯0]exp[iω0(τ01τ02)],  (α=x,y;β=x,y)
with
Q¯0αβ1=(ik0(12σI0αβ2+δI0αβ2)I021ik0δI0αβ2I021012ik0(στ0αβ2+δτ0αβ2)012ik0δτ0αβ2ik0δI0αβ2I021ik0(12σI0αβ2+δI0αβ2)I021012ik0δτ0αβ2012ik0(στ0αβ2+δτ0αβ2))
Here I is the 2×2 identity matrix. 021 is 2×1 zero matrix, and 012 is 1×2 zero matrix. r¯0T=(r˜01Tr˜02T) and r˜0iT=(x0iy0iτ0i) (i=1,2); k0=ω0/c is the wave number and the corresponding ω0 is the central angular frequency; AxandAy are the amplitudes of x and y components of the electric field, respectively; Bαβ=|Bαβ|exp(iϕαβ)=Bβα* is the correlation coefficient between Ex and Ey field components.

 

Fig. 1 Propagation scheme for a pulse beam through dispersive optical system and free space.

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Just after interacting with a linear dispersive medium the mutual correlation matrix of the Gaussian Schell-model REMPB can be expressed by the formula (see Ref [17]. for details of derivation):

Γ1αβ(r¯1)=AαAβBαβ[det(A¯1+B¯1Q¯0αβ1)]1/2exp[ik02r¯1TQ¯1αβ1r¯1],
where r¯1T=(r˜11Tr˜12T), det stands for the determinant of a matrix and
Q¯1αβ1=(C¯1+D¯1Q¯0αβ1)(A¯1+B¯1Q¯0αβ1)1.
Q¯0αβ1 and Q¯1αβ1 denote spatiotemporal complex curvature tensor of input and output plane. Q¯1αβ1 is a 6×6 matrix of the form
Q¯1αβ1=(q11q12q13q14q15q16q21q22q23q24q25q26q31q32q33q34q35q36q41q42q43q44q45q46q51q52q53q54q55q56q61q62q63q64q65q66).
In Eqs. (4) and (5) A¯1, B¯1, C¯1 and D¯1 are also the 6×6 matrices of the form
A¯1=(A˜11033033A˜12), B¯1=(B˜11033033B˜12*), C¯1=(C˜11033033C˜12*), D1=(D˜11033033D˜12),
where “*” denotes complex conjugate. A˜11,B˜11,C˜11,D˜11 and A˜12,B˜12,C˜12,D˜12 are the 3×3 matrices of the dispersive optical elements (see Appendix A of Ref [30]. for matrices of typical dispersive elements. We will confine our analysis to the case when the reference plane is perpendicular to the direction of propagation of the field. Thus the optical elements must satisfy the relationsA˜11=A˜12=A˜1, B˜11=B˜12=B˜1, C˜11=C˜12=C˜1 and D˜11=D˜12=D˜1 [17,30].

Matrix (6) may provide the information about the spatio-temporal coupling in pulsed beams on interacting with dispersive linear systems. If elements q13,q16,q23,q26, q31,q32,q34,q35, q43,q46,q53,q56,q61,q62,q64,q65of this matrix remain non-zero [compare with corresponding elements of matrix (3)], then no coupling takes place, and it does otherwise.

On propagation from a dispersive element in free space the correlation matrix of the pulse undergoes the following change:

Γ2αβ(r¯2)=AαAβBαβ[det(A¯1+B¯1Q¯0αβ1)]1/2[det(A¯2+B¯2Q¯1αβ1)]1/2exp[ik02r¯2TQ¯2αβ1r¯2],
wherer¯2T=(r˜21Tr˜22T)and Q¯2αβ1=(C¯2+D¯2Q¯1αβ1)(A¯2+B¯2Q¯1αβ1)1.The submatrices of a matrix describing free space propagation have the form [19]
A˜21=D˜21=(100010001), B˜21=(z000z0001), C˜21=(000000000).
In the most cases, there is no spatiotemporal coupling in pulsed beams on free-space propagation [1]. However, if the coupling has already occurred it may be further modified by free space propagation.

Using the Fourier-transform, the elements of the correlation matrices Γ0, Γ1 and Γ2 can be replaced by their counterparts in space-frequency domain:

Wjαβ(rj1,ω1;rj2,ω2)=12π++Γjαβ(rj1,τj1;rj2,τj2)exp[(ω1τj1ω2τj2)]dτj1dτj2,
wherej=0, 1, 2. Even though matrices in Eq. (10) can be used for calculation of all the second-order statistical properties of pulsed beams in what follows we will only consider the changes in the degree of polarization [32,33]:
Pjω(rj,ω)=14Det[Wj(rj,ω,rj,ω)]{Tr[Wj(rj,ω,rj,ω)]}2,
wherej=0, 1, 2. We believe the evolution of the degree of polarization on interaction with dispersive systems was not previously studied.

3. Spatio-temporal coupling of REMPB reflected from a reflecting grating

To illustrate the general theory of Section 2 we will now provide an example dealing with spatio-temporal coupling and the evolution of the degree of polarization of a REMPB reflected from a reflecting grating.

In order to analyze the spatiotemporal coupling and its effects on transmission characteristics, we consider a REMPB reflected by the reflecting grating and propagating in the free space, as shown in Fig. 2 . The grooves of the grating are assumed to coincide with y-direction.

 

Fig. 2 Illustrating notation for reflection of a pulse from a reflecting grating.

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The submatrices of 6x6 matrix describing reflecting grating have the form [12,30]

A˜11=(sinϕsinψ00010cosψcosϕsinψ01),  B˜11=C˜11=0,  D˜11=(sinψsinϕ0cosψcosϕsinϕ010001),
where ψ and ϕ are the incident and the reflective angles, respectively (see Fig. 2), which satisfy the grating equation
dG(cosϕcosψ)=mλ0,
with dG being the groove spacing and m being the diffraction order. The matrix describing reflecting grating (Eq. (12)) was first derived in [12] by only considering the effect of interference and diffraction of reflecting grating on light, and the Fresnel reflection coefficient and the absorption of reflecting grating were not taken into consideration. This assumption has been adopted in many previous literatures. For the most general case, the Fresnel reflection coefficient and absorption of reflecting grating should be taken into consideration, while up to now, no such general matrix was developed, and we leave this for future study.

Using Eqs. (4)-(7) and (12) we can easily obtain the elements of the correlation matrix just after reflection:

Γ1αβ(r¯1)=AαAβBαβsinψsinϕexp[ik02r¯1TQ¯1αβ1r¯1],
where Q¯1αβ1 has the form
Q¯1αβ1=(q110q13q140q160q2200q250q310q33q340q36q410q43q440q460q5200q550q610q63q640q66).
with
q11=q44=ig122k0(σI0αβ2+2δI0αβ2)ig22k0(στ0αβ2+δτ0αβ2),q13=q31=q46=q64=ig2k0(στ0αβ2+δτ0αβ2),q14=q41=ik0(g12δI0αβ2+g22δτ0αβ2),q16=q34=q43=q61=ig2k0δτ0αβ2,q22=q55=i2k0(σI0αβ2+2δI0αβ2),q25=q52=ik0δI0αβ2,q33=q66=ik0(στ0αβ2+δτ0αβ2),q36=q63=ik0δτ0αβ2.
Here g1=sinψ/sinϕ and g2=(cosψcosϕ)/sinϕ. It can be seen from Eq. (16) that since q13 and q16 are not zero, there exist spatiotemporal coupling. It means that only the pulsed beam of zero diffraction order m ([given values of ψ and φ]) has no spatiotemporal coupling. Also it is seen from Eqs. (15) and (16) that the spatial-temporal coupling of the pulsed beam at output plane depends on the diffraction order of grating, i.e. reflective angle of the pulsed beam. For the sake of convenience we confine our analysis to values of degree of polarization when r¯2=0, i.e. on the axis of the beam in space domain, and at the pulse center in time domain. For this position, on using Eqs. (8) and (9), we find that the correlation matrix takes the form
Γ2αβ=AαAβBαβg1[1+z24k02σI0αβ2(4δI0αβ2+σI0αβ2)]1/2          ×{1+z24k02(g12σI0αβ2+2g22στ0αβ2)[g12(4δI0αβ2+σI0αβ2)+2g22(2δτ0αβ2+στ0αβ2)]}1/2,
and the degree of polarization becomes
Pτ(z)=14Det(Γ2)Tr(Γ2)=|Γ2xxΓ2yyΓ2xx+Γ2yy|=|Ax2ΔxxAy2ΔyyAx2Δxx+Ay2Δyy|,
with
Δαβ=[1+z24k02σI0αβ2(4δI0αβ2+σI0αβ2)]1/2      ×{1+z24k02(g12σI0αβ2+2g22στ0αβ2)[g12(4δI0αβ2+σI0αβ2)+2g22(2δτ0αβ2+στ0αβ2)]}1/2.
It is seen from Eqs. (17)-(19) that if the incident angle and the reflected angle coincide, i.e. if ψ=ϕ, then g1=1 and g2=0 and, hence is not spatial-temporal coupling. Equation (19) then reduces to
Δαβ=[1+z24k02σI0αβ2(4δI0αβ2+σI0αβ2)]1.
On substituting Eq. (20) into Eq. (18), we find that the result for the degree of polarization of the pulsed beam agrees with that of reference [27] by Ding et al.

If ψϕ then the spatial-temporal coupling occurs and, as it can be seen from the Eq. (18), it affects the evolution of degree of polarization. In particular in the limiting case of the far-field the degree of polarization is given by the expression

limzPτ(z)=|Ax2ΩxxAy2ΩyyAx2Ωxx+Ay2Ωyy|,
where Ωαβ is a parameter independent of the propagation distance z, and given by the formula
Ωαβ={σI0αβ2(4δI0αβ2+σI0αβ2)(g12σI0αβ2+2g22στ0αβ2)}1/2         ×{[g12(4δI0αβ2+σI0αβ2)+2g22(2δτ0αβ2+στ0αβ2)]}1/2.
Thus the degree of polarization tends to a constant value after the pulsed beam travels in free space at a sufficiently large distance. It is well known that the degree of polarization of a coherent electromagnetic beam remains invariant on propagation in free space, while the degree of polarization of a stochastic electromagnetic beam varies on propagation in the near field and in the intermediate propagation distances [32]. The degree of coherence γ(r¯1,r¯2)=Tr[Γ(r¯1,r¯2)]/Tr[Γ(r¯1,r¯1)]Tr[Γ(r¯2,r¯2)] of a REMPB increases on propagation, and the REMPB approaches to a coherent electromagnetic pulse beam in the far field, thus its degree of polarization tends to a constant value in the far field.

4. Numerical results

We will now discuss the results of the previous section with the help of numerical calculations of the degree of polarization of the REMPB after its interaction with a reflection grating. Unless it is specified otherwise in figure captions the following parameters of the source of the pulse and the grating were selected: Ax=3/2,  Ay=1, Bxx=Byy=1, Bxy=Byx=0, σI0xx=σI0yy=σI0, δI0yy=2δI0xx, στ0xx=στ0yy=στ0, δτ0xx=δτ0yy=δτ0, ω0=2.36rad fs1, dG=1/600mm, ψ=300. Also, we set σt0=στ0/c and δt0=δτ0/c, where σt0 and δt0 represent the pulsed duration and the temporal degree of coherence respectively.

In Figs. 3 -6 we demonstrate with the help of three-dimensional plots of the degree of polarization varying with propagation distance z from the reflecting grating and several parameters of the pulse, for several chosen diffraction orders of the grating (m = 0, −1, −7). In particular, Fig. 3 shows the effect of pulse durationσt0, Fig. 4 that of r.m.s. width σI0of the intensity, Fig. 5 that of r.m.s. width of temporal degree of coherence δt0 and Fig. 6 that of r.m.s. width of spatial degree of coherence δI0xx (while the other correlations are kept fixed). One sees from these plots that only in case when m = 0 no spatio-temporal coupling occurs in the reflected pulsed beam. In order to provide quantitatively better results we show in Figs. 7 -11 two dimensional plots the degree of polarization depending on various source and system parameters, for several fixed values of diffraction order m. In particular, in Fig. 7 the dependence of the degree of polarization on propagation distance from the source is given, in the rest of the figures the distance was kept fixed while the parameters of the source were varied. One notices that in some cases, such as in Fig. 7 and 9 the degree of polarization does not depend on propagation distance and r.m.s intensity in a monotonic way.

 

Fig. 3 Dependence of the degree of polarization of a REMPB on the propagation distance z and the pulse duration σt0 for different diffraction orders m with σI0=1mm, δI0xx=1mm, δt0=2ps.

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Fig. 6 Dependence of the degree of polarization P on the propagation distance z and the r.m.s. width of spatial degree of coherence δI0xx for different diffraction orders m with σI0=1mm, σt0=2ps, δt0=2ps.

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Fig. 4 Dependence of the degree of polarization P of a REMPB on the propagation distance z and the r.m.s. width σI0of the intensity for different diffraction orders m with δI0xx=1mm, σt0=2ps, δt0=2ps.

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Fig. 5 Dependence of the degree of polarization P on the propagation distance z and the r.m.s. width of temporal degree of coherence δt0 for different diffraction orders m with σI0=1mm, δI0xx=1mm, σt0=2ps.

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Fig. 7 Dependence of the degree of polarization P on the propagation distance z for different diffraction orders m with σI0=1mm, δI0xx=1mm, σt0=2ps, δt0=2ps.

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Fig. 11 Dependence of the degree of polarization P on the r.m.s. width of spatial degree of coherence δI0xx for different diffraction orders m at z=200mwith σI0=1mm, σt0=2ps, δt0=2ps.

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Fig. 9 Dependence of the degree of polarization P on the r.m.s. width σI0of the intensity for different diffraction orders m at z = 200m with δI0xx=1mm, σt0=2ps, δt0=2ps.

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Fig. 8 Dependence of the degree of polarization P on the pulse duration σt0for different diffraction orders m at z = 200m with σI0=1mm, δI0xx=1mm, δt0=2ps.

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Fig. 10 Dependence of the degree of polarization P on the r.m.s. width of temporal degree of coherence δt0 for different diffraction orders m at z = 200m with σI0=1mm, δI0xx=1mm, δt0=2ps.

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5. Conclusion

With the help of matrix optics, we have analyzed the interaction of a REMPB with linear optical systems. As an application example, the degree of polarization of a typical REMPB reflected from a reflecting grating was explored. It is found that spatio-temporal coupling occurs in the reflected REMPB of nonzero diffraction order, and the statistics properties of the reflected REMPB depend closely on the parameters of the source beam and the reflecting grating.

Acknowledgments

Min Yao acknowledges the support by Scientific Research Fund of Zhejiang Provincial Education Department, China (Grant No. Y200908631). Yangjian Cai acknowledges the support by the National Natural Science Foundation of China under Grant No. 10904102, the Foundation for the Author of National Excellent Doctoral Dissertation of PR China under Grant No. 200928, the Natural Science of Jiangsu Province under Grant No. BK2009114, the Huo Ying Dong Education Foundation of China under Grant No. 121009 and the Key Project of Chinese Ministry of Education under Grant No. 210081. O. Korotkova's research is funded by the US AFOSR (grant FA 95500810102) and US ONR (Grant N0018909P1903). Qiang Lin acknowledges the support by the National Natural Science Foundation of China under Grant No. 60925022. Zhaoying Wang acknowledges the support by the Zhejiang Provincial Qian-Jiang-Ren-Cai Project of China (2009R10034) and the Fundamental Research Funds for the Central Universities (2010QNA3024).

References and links

1. S. M. Wang and D. M. Zhao, Matrix Optics, (Springer, 2000).

2. For the collection of original papers by R. C. Jones see W. Swindel, Polarized light, (Dowden, Hutchinson & Ross, (Stroudsburg, Pennsylvania, 1975).

3. H. Mueller, “The foundations of optics,” J. Opt. Soc. Am. 38, 661–661 (1948); for account of the Mueller’s theory see also E. Collett, Polarized Light: Fundamentals and Applications (Marcel Dekker, Inc., New York, 1993). Chap. 5.

4. O. Korotkova and E. Wolf, “Effects of linear non-image forming devices on coherence and polarization properties of random electromagnetic beams. Part I. General theory,” J. Mod. Opt. 52, 2659–2671 (2005). [CrossRef]  

5. O. Korotkova and E. Wolf, “Effects of linear non-image forming devices on spectra and on coherence and polarization properties ofstochastic electromagnetic beams. Part II. Examples,” J. Mod. Opt. 52, 2673–2685 (2005). [CrossRef]  

6. Q. Lin and Y. Cai, “Tensor ABCD law for partially coherent twisted anisotropic Gaussian-Schell model beams,” Opt. Lett. 27(4), 216–218 (2002). [CrossRef]  

7. M. Yao, Y. Cai, H. T. Eyyuboğlu, Y. Baykal, and O. Korotkova, “Evolution of the degree of polarization of an electromagnetic Gaussian Schell-model beam in a Gaussian cavity,” Opt. Lett. 33(19), 2266–2268 (2008). [CrossRef]   [PubMed]  

8. O. Korotkova, Y. Cai, and E. Watson, “Stochastic electromagnetic beams for LIDAR systems operating through turbulent atmosphere,” Appl. Phys. B 94(4), 681–690 (2009). [CrossRef]  

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14. P. A. Bélanger, “Beam propagation and the ABCD ray matrices,” Opt. Lett. 16(4), 196–198 (1991). [CrossRef]   [PubMed]  

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17. L. G. Wang, Q. Lin, H. Chen, and S. Y. Zhu, “Propagation of partially coherent pulsed beams in the spatiotemporal domain,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 67(5), 056613 (2003). [CrossRef]   [PubMed]  

18. 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(11), 2117–2123 (2004). [CrossRef]  

19. Y. Cai and Q. Lin, “The fractional Fourier transform for a partially coherent pulse,” J. Opt. A, Pure Appl. Opt. 6(4), 307–311 (2004). [CrossRef]  

20. J. Lancis, V. Torres-Company, E. Silvestre, and P. Andrés, “Space-time analogy for partially coherent plane-wave-type pulses,” Opt. Lett. 30(22), 2973–2975 (2005). [CrossRef]   [PubMed]  

21. H. Lajunen, J. Turunen, P. Vahimaa, J. Tervo, and F. Wyrowski, “Spectrally partially coherent pulse trains in dispersive media,” Opt. Commun. 255(1-3), 12–22 (2005). [CrossRef]  

22. H. Lajunen, P. Vahimaa, and J. Tervo, “Theory of spatially and spectrally partially coherent pulses,” J. Opt. Soc. Am. A 22(8), 1536–1545 (2005). [CrossRef]  

23. V. Torres-Company, G. Mínguez-Vega, J. Lancis, and A. T. Friberg, “Controllable generation of partially coherent light pulses with direct space-to-time pulse shaper,” Opt. Lett. 32(12), 1608–1610 (2007). [CrossRef]   [PubMed]  

24. A. T. Friberg, H. Lajunen, and V. Torres-Company, “Spectral elementary-coherence-function representation for partially coherent light pulses,” Opt. Express 15(8), 5160–5165 (2007). [CrossRef]   [PubMed]  

25. V. Torres-Company, H. Lajunen, J. Lancis, and A. T. Friberg, “Ghost interference with classical partially coherent light pulses,” Phys. Rev. A 77(4), 043811 (2008). [CrossRef]  

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27. C. Ding, L. Pan, and B. Lu, “Characterization of stochastic spatially and spectrally partially coherent electromagnetic pulsed beams,” N. J. Phys. 11(8), 083001 (2009). [CrossRef]  

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References

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  1. S. M. Wang and D. M. Zhao, Matrix Optics, (Springer, 2000).
  2. For the collection of original papers by R. C. Jones see W. Swindel, Polarized light, (Dowden, Hutchinson & Ross, (Stroudsburg, Pennsylvania, 1975).
  3. H. Mueller, “The foundations of optics,” J. Opt. Soc. Am. 38, 661–661 (1948); for account of the Mueller’s theory see also E. Collett, Polarized Light: Fundamentals and Applications (Marcel Dekker, Inc., New York, 1993). Chap. 5.
  4. O. Korotkova and E. Wolf, “Effects of linear non-image forming devices on coherence and polarization properties of random electromagnetic beams. Part I. General theory,” J. Mod. Opt. 52, 2659–2671 (2005).
    [Crossref]
  5. O. Korotkova and E. Wolf, “Effects of linear non-image forming devices on spectra and on coherence and polarization properties ofstochastic electromagnetic beams. Part II. Examples,” J. Mod. Opt. 52, 2673–2685 (2005).
    [Crossref]
  6. Q. Lin and Y. Cai, “Tensor ABCD law for partially coherent twisted anisotropic Gaussian-Schell model beams,” Opt. Lett. 27(4), 216–218 (2002).
    [Crossref]
  7. M. Yao, Y. Cai, H. T. Eyyuboğlu, Y. Baykal, and O. Korotkova, “Evolution of the degree of polarization of an electromagnetic Gaussian Schell-model beam in a Gaussian cavity,” Opt. Lett. 33(19), 2266–2268 (2008).
    [Crossref] [PubMed]
  8. O. Korotkova, Y. Cai, and E. Watson, “Stochastic electromagnetic beams for LIDAR systems operating through turbulent atmosphere,” Appl. Phys. B 94(4), 681–690 (2009).
    [Crossref]
  9. O. E. Martinez, “Matrix formalism for pulse compressors,” IEEE J. Quantum Electron. 24(12), 2530–2536 (1988).
    [Crossref]
  10. O. E. Martinez, “Matrix formalism for dispersive laser cavities,” IEEE J. Quantum Electron. 25(3), 296–300 (1989).
    [Crossref]
  11. A. G. Kostenbauder, “Ray-pulse matrices: a rational treatment for dispersive optical systems,” IEEE J. Quantum Electron. 26(6), 1148–1157 (1990).
    [Crossref]
  12. S. P. Dijaili, A. Dienes, and J. S. Smith, “ABCD matrices for dispersive pulse propagation,” IEEE J. Quantum Electron. 26(6), 1158–1164 (1990).
    [Crossref]
  13. Q. Lin, S. Wang, J. Alda, and E. Bernabeu, “Transformation of pulsed nonideal beams in a four-dimension domain,” Opt. Lett. 18(9), 669–671 (1993).
    [Crossref] [PubMed]
  14. P. A. Bélanger, “Beam propagation and the ABCD ray matrices,” Opt. Lett. 16(4), 196–198 (1991).
    [Crossref] [PubMed]
  15. P. Paakkonen, J. Turunen, P. Vahimaa, A. T. Friberg, and F. Wyrowski, “Partially coherent Gaussian pulses,” Opt. Commun. 204(1-6), 53–58 (2002).
    [Crossref]
  16. Q. Lin, L. Wang, and S. Zhu, “Partially coherent light pulse and its propagation,” Opt. Commun. 219(1-6), 65–70 (2003).
    [Crossref]
  17. L. G. Wang, Q. Lin, H. Chen, and S. Y. Zhu, “Propagation of partially coherent pulsed beams in the spatiotemporal domain,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 67(5), 056613 (2003).
    [Crossref] [PubMed]
  18. 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(11), 2117–2123 (2004).
    [Crossref]
  19. Y. Cai and Q. Lin, “The fractional Fourier transform for a partially coherent pulse,” J. Opt. A, Pure Appl. Opt. 6(4), 307–311 (2004).
    [Crossref]
  20. J. Lancis, V. Torres-Company, E. Silvestre, and P. Andrés, “Space-time analogy for partially coherent plane-wave-type pulses,” Opt. Lett. 30(22), 2973–2975 (2005).
    [Crossref] [PubMed]
  21. H. Lajunen, J. Turunen, P. Vahimaa, J. Tervo, and F. Wyrowski, “Spectrally partially coherent pulse trains in dispersive media,” Opt. Commun. 255(1-3), 12–22 (2005).
    [Crossref]
  22. H. Lajunen, P. Vahimaa, and J. Tervo, “Theory of spatially and spectrally partially coherent pulses,” J. Opt. Soc. Am. A 22(8), 1536–1545 (2005).
    [Crossref]
  23. V. Torres-Company, G. Mínguez-Vega, J. Lancis, and A. T. Friberg, “Controllable generation of partially coherent light pulses with direct space-to-time pulse shaper,” Opt. Lett. 32(12), 1608–1610 (2007).
    [Crossref] [PubMed]
  24. A. T. Friberg, H. Lajunen, and V. Torres-Company, “Spectral elementary-coherence-function representation for partially coherent light pulses,” Opt. Express 15(8), 5160–5165 (2007).
    [Crossref] [PubMed]
  25. V. Torres-Company, H. Lajunen, J. Lancis, and A. T. Friberg, “Ghost interference with classical partially coherent light pulses,” Phys. Rev. A 77(4), 043811 (2008).
    [Crossref]
  26. H. Lajunen, V. Torres-Company, J. Lancis, E. Silvestre, and P. Andrès, “Pulse-by-pulse method to characterize partially coherent pulse propagation in instantaneous nonlinear media,” Opt. Express 18(14), 14979–14991 (2010).
    [Crossref] [PubMed]
  27. C. Ding, L. Pan, and B. Lu, “Characterization of stochastic spatially and spectrally partially coherent electromagnetic pulsed beams,” N. J. Phys. 11(8), 083001 (2009).
    [Crossref]
  28. C. Ding and B. Lu, “Spectral shifts and spectral switches of diffracted spatially and spectrally partially coherent pulsed beams in the far field,” J. Opt. A, Pure Appl. Opt. 10(9), 095006 (2008).
    [Crossref]
  29. M. Kempe, U. Stamm, B. Wilhelmi, and W. Rudolph, “Spatial and temporal transformation of femtosecond laser pulses by lenses and lens systems,” J. Opt. Soc. Am. B 9(7), 1158–1165 (1992).
    [Crossref]
  30. Q. Lin, S. Wang, J. Alda, and E. Bernabeu, “Spatial-temporal coupling in grating-pair pulse compression system analysed by matrix optics,” Opt. Quantum Electron. 27(7), 679–692 (1995).
    [Crossref]
  31. G. Piquero, R. Borghi, and M. Santarsiero, “Gaussian Schell-model beams propagating through polarization gratings,” J. Opt. Soc. Am. A 18(6), 1399–1405 (2001).
    [Crossref]
  32. E. Wolf, “Unified theory of coherence and polarization of random electromagnetic fields,” Phys. Lett. A 312(5-6), 263–267 (2003).
    [Crossref]
  33. F. Gori, “Matrix treatment for partially polarized, partially coherent beams,” Opt. Lett. 23(4), 241–243 (1998).
    [Crossref]

2010 (1)

2009 (2)

C. Ding, L. Pan, and B. Lu, “Characterization of stochastic spatially and spectrally partially coherent electromagnetic pulsed beams,” N. J. Phys. 11(8), 083001 (2009).
[Crossref]

O. Korotkova, Y. Cai, and E. Watson, “Stochastic electromagnetic beams for LIDAR systems operating through turbulent atmosphere,” Appl. Phys. B 94(4), 681–690 (2009).
[Crossref]

2008 (3)

M. Yao, Y. Cai, H. T. Eyyuboğlu, Y. Baykal, and O. Korotkova, “Evolution of the degree of polarization of an electromagnetic Gaussian Schell-model beam in a Gaussian cavity,” Opt. Lett. 33(19), 2266–2268 (2008).
[Crossref] [PubMed]

C. Ding and B. Lu, “Spectral shifts and spectral switches of diffracted spatially and spectrally partially coherent pulsed beams in the far field,” J. Opt. A, Pure Appl. Opt. 10(9), 095006 (2008).
[Crossref]

V. Torres-Company, H. Lajunen, J. Lancis, and A. T. Friberg, “Ghost interference with classical partially coherent light pulses,” Phys. Rev. A 77(4), 043811 (2008).
[Crossref]

2007 (2)

2005 (5)

O. Korotkova and E. Wolf, “Effects of linear non-image forming devices on coherence and polarization properties of random electromagnetic beams. Part I. General theory,” J. Mod. Opt. 52, 2659–2671 (2005).
[Crossref]

O. Korotkova and E. Wolf, “Effects of linear non-image forming devices on spectra and on coherence and polarization properties ofstochastic electromagnetic beams. Part II. Examples,” J. Mod. Opt. 52, 2673–2685 (2005).
[Crossref]

J. Lancis, V. Torres-Company, E. Silvestre, and P. Andrés, “Space-time analogy for partially coherent plane-wave-type pulses,” Opt. Lett. 30(22), 2973–2975 (2005).
[Crossref] [PubMed]

H. Lajunen, J. Turunen, P. Vahimaa, J. Tervo, and F. Wyrowski, “Spectrally partially coherent pulse trains in dispersive media,” Opt. Commun. 255(1-3), 12–22 (2005).
[Crossref]

H. Lajunen, P. Vahimaa, and J. Tervo, “Theory of spatially and spectrally partially coherent pulses,” J. Opt. Soc. Am. A 22(8), 1536–1545 (2005).
[Crossref]

2004 (2)

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(11), 2117–2123 (2004).
[Crossref]

Y. Cai and Q. Lin, “The fractional Fourier transform for a partially coherent pulse,” J. Opt. A, Pure Appl. Opt. 6(4), 307–311 (2004).
[Crossref]

2003 (3)

Q. Lin, L. Wang, and S. Zhu, “Partially coherent light pulse and its propagation,” Opt. Commun. 219(1-6), 65–70 (2003).
[Crossref]

L. G. Wang, Q. Lin, H. Chen, and S. Y. Zhu, “Propagation of partially coherent pulsed beams in the spatiotemporal domain,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 67(5), 056613 (2003).
[Crossref] [PubMed]

E. Wolf, “Unified theory of coherence and polarization of random electromagnetic fields,” Phys. Lett. A 312(5-6), 263–267 (2003).
[Crossref]

2002 (2)

P. Paakkonen, J. Turunen, P. Vahimaa, A. T. Friberg, and F. Wyrowski, “Partially coherent Gaussian pulses,” Opt. Commun. 204(1-6), 53–58 (2002).
[Crossref]

Q. Lin and Y. Cai, “Tensor ABCD law for partially coherent twisted anisotropic Gaussian-Schell model beams,” Opt. Lett. 27(4), 216–218 (2002).
[Crossref]

2001 (1)

1998 (1)

1995 (1)

Q. Lin, S. Wang, J. Alda, and E. Bernabeu, “Spatial-temporal coupling in grating-pair pulse compression system analysed by matrix optics,” Opt. Quantum Electron. 27(7), 679–692 (1995).
[Crossref]

1993 (1)

1992 (1)

1991 (1)

1990 (2)

A. G. Kostenbauder, “Ray-pulse matrices: a rational treatment for dispersive optical systems,” IEEE J. Quantum Electron. 26(6), 1148–1157 (1990).
[Crossref]

S. P. Dijaili, A. Dienes, and J. S. Smith, “ABCD matrices for dispersive pulse propagation,” IEEE J. Quantum Electron. 26(6), 1158–1164 (1990).
[Crossref]

1989 (1)

O. E. Martinez, “Matrix formalism for dispersive laser cavities,” IEEE J. Quantum Electron. 25(3), 296–300 (1989).
[Crossref]

1988 (1)

O. E. Martinez, “Matrix formalism for pulse compressors,” IEEE J. Quantum Electron. 24(12), 2530–2536 (1988).
[Crossref]

Alda, J.

Q. Lin, S. Wang, J. Alda, and E. Bernabeu, “Spatial-temporal coupling in grating-pair pulse compression system analysed by matrix optics,” Opt. Quantum Electron. 27(7), 679–692 (1995).
[Crossref]

Q. Lin, S. Wang, J. Alda, and E. Bernabeu, “Transformation of pulsed nonideal beams in a four-dimension domain,” Opt. Lett. 18(9), 669–671 (1993).
[Crossref] [PubMed]

Andrés, P.

Andrès, P.

Baykal, Y.

Bélanger, P. A.

Bernabeu, E.

Q. Lin, S. Wang, J. Alda, and E. Bernabeu, “Spatial-temporal coupling in grating-pair pulse compression system analysed by matrix optics,” Opt. Quantum Electron. 27(7), 679–692 (1995).
[Crossref]

Q. Lin, S. Wang, J. Alda, and E. Bernabeu, “Transformation of pulsed nonideal beams in a four-dimension domain,” Opt. Lett. 18(9), 669–671 (1993).
[Crossref] [PubMed]

Borghi, R.

Cai, Y.

O. Korotkova, Y. Cai, and E. Watson, “Stochastic electromagnetic beams for LIDAR systems operating through turbulent atmosphere,” Appl. Phys. B 94(4), 681–690 (2009).
[Crossref]

M. Yao, Y. Cai, H. T. Eyyuboğlu, Y. Baykal, and O. Korotkova, “Evolution of the degree of polarization of an electromagnetic Gaussian Schell-model beam in a Gaussian cavity,” Opt. Lett. 33(19), 2266–2268 (2008).
[Crossref] [PubMed]

Y. Cai and Q. Lin, “The fractional Fourier transform for a partially coherent pulse,” J. Opt. A, Pure Appl. Opt. 6(4), 307–311 (2004).
[Crossref]

Q. Lin and Y. Cai, “Tensor ABCD law for partially coherent twisted anisotropic Gaussian-Schell model beams,” Opt. Lett. 27(4), 216–218 (2002).
[Crossref]

Chen, H.

L. G. Wang, Q. Lin, H. Chen, and S. Y. Zhu, “Propagation of partially coherent pulsed beams in the spatiotemporal domain,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 67(5), 056613 (2003).
[Crossref] [PubMed]

Dienes, A.

S. P. Dijaili, A. Dienes, and J. S. Smith, “ABCD matrices for dispersive pulse propagation,” IEEE J. Quantum Electron. 26(6), 1158–1164 (1990).
[Crossref]

Dijaili, S. P.

S. P. Dijaili, A. Dienes, and J. S. Smith, “ABCD matrices for dispersive pulse propagation,” IEEE J. Quantum Electron. 26(6), 1158–1164 (1990).
[Crossref]

Ding, C.

C. Ding, L. Pan, and B. Lu, “Characterization of stochastic spatially and spectrally partially coherent electromagnetic pulsed beams,” N. J. Phys. 11(8), 083001 (2009).
[Crossref]

C. Ding and B. Lu, “Spectral shifts and spectral switches of diffracted spatially and spectrally partially coherent pulsed beams in the far field,” J. Opt. A, Pure Appl. Opt. 10(9), 095006 (2008).
[Crossref]

Eyyuboglu, H. T.

Friberg, A. T.

V. Torres-Company, H. Lajunen, J. Lancis, and A. T. Friberg, “Ghost interference with classical partially coherent light pulses,” Phys. Rev. A 77(4), 043811 (2008).
[Crossref]

V. Torres-Company, G. Mínguez-Vega, J. Lancis, and A. T. Friberg, “Controllable generation of partially coherent light pulses with direct space-to-time pulse shaper,” Opt. Lett. 32(12), 1608–1610 (2007).
[Crossref] [PubMed]

A. T. Friberg, H. Lajunen, and V. Torres-Company, “Spectral elementary-coherence-function representation for partially coherent light pulses,” Opt. Express 15(8), 5160–5165 (2007).
[Crossref] [PubMed]

P. Paakkonen, J. Turunen, P. Vahimaa, A. T. Friberg, and F. Wyrowski, “Partially coherent Gaussian pulses,” Opt. Commun. 204(1-6), 53–58 (2002).
[Crossref]

Gori, F.

Kempe, M.

Korotkova, O.

O. Korotkova, Y. Cai, and E. Watson, “Stochastic electromagnetic beams for LIDAR systems operating through turbulent atmosphere,” Appl. Phys. B 94(4), 681–690 (2009).
[Crossref]

M. Yao, Y. Cai, H. T. Eyyuboğlu, Y. Baykal, and O. Korotkova, “Evolution of the degree of polarization of an electromagnetic Gaussian Schell-model beam in a Gaussian cavity,” Opt. Lett. 33(19), 2266–2268 (2008).
[Crossref] [PubMed]

O. Korotkova and E. Wolf, “Effects of linear non-image forming devices on coherence and polarization properties of random electromagnetic beams. Part I. General theory,” J. Mod. Opt. 52, 2659–2671 (2005).
[Crossref]

O. Korotkova and E. Wolf, “Effects of linear non-image forming devices on spectra and on coherence and polarization properties ofstochastic electromagnetic beams. Part II. Examples,” J. Mod. Opt. 52, 2673–2685 (2005).
[Crossref]

Kostenbauder, A. G.

A. G. Kostenbauder, “Ray-pulse matrices: a rational treatment for dispersive optical systems,” IEEE J. Quantum Electron. 26(6), 1148–1157 (1990).
[Crossref]

Lajunen, H.

Lancis, J.

Lin, Q.

Y. Cai and Q. Lin, “The fractional Fourier transform for a partially coherent pulse,” J. Opt. A, Pure Appl. Opt. 6(4), 307–311 (2004).
[Crossref]

L. G. Wang, Q. Lin, H. Chen, and S. Y. Zhu, “Propagation of partially coherent pulsed beams in the spatiotemporal domain,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 67(5), 056613 (2003).
[Crossref] [PubMed]

Q. Lin, L. Wang, and S. Zhu, “Partially coherent light pulse and its propagation,” Opt. Commun. 219(1-6), 65–70 (2003).
[Crossref]

Q. Lin and Y. Cai, “Tensor ABCD law for partially coherent twisted anisotropic Gaussian-Schell model beams,” Opt. Lett. 27(4), 216–218 (2002).
[Crossref]

Q. Lin, S. Wang, J. Alda, and E. Bernabeu, “Spatial-temporal coupling in grating-pair pulse compression system analysed by matrix optics,” Opt. Quantum Electron. 27(7), 679–692 (1995).
[Crossref]

Q. Lin, S. Wang, J. Alda, and E. Bernabeu, “Transformation of pulsed nonideal beams in a four-dimension domain,” Opt. Lett. 18(9), 669–671 (1993).
[Crossref] [PubMed]

Lu, B.

C. Ding, L. Pan, and B. Lu, “Characterization of stochastic spatially and spectrally partially coherent electromagnetic pulsed beams,” N. J. Phys. 11(8), 083001 (2009).
[Crossref]

C. Ding and B. Lu, “Spectral shifts and spectral switches of diffracted spatially and spectrally partially coherent pulsed beams in the far field,” J. Opt. A, Pure Appl. Opt. 10(9), 095006 (2008).
[Crossref]

Martinez, O. E.

O. E. Martinez, “Matrix formalism for dispersive laser cavities,” IEEE J. Quantum Electron. 25(3), 296–300 (1989).
[Crossref]

O. E. Martinez, “Matrix formalism for pulse compressors,” IEEE J. Quantum Electron. 24(12), 2530–2536 (1988).
[Crossref]

Mínguez-Vega, G.

Paakkonen, P.

P. Paakkonen, J. Turunen, P. Vahimaa, A. T. Friberg, and F. Wyrowski, “Partially coherent Gaussian pulses,” Opt. Commun. 204(1-6), 53–58 (2002).
[Crossref]

Pan, L.

C. Ding, L. Pan, and B. Lu, “Characterization of stochastic spatially and spectrally partially coherent electromagnetic pulsed beams,” N. J. Phys. 11(8), 083001 (2009).
[Crossref]

Piquero, G.

Rudolph, W.

Santarsiero, M.

Silvestre, E.

Smith, J. S.

S. P. Dijaili, A. Dienes, and J. S. Smith, “ABCD matrices for dispersive pulse propagation,” IEEE J. Quantum Electron. 26(6), 1158–1164 (1990).
[Crossref]

Stamm, U.

Tervo, J.

Torres-Company, V.

Turunen, J.

H. Lajunen, J. Turunen, P. Vahimaa, J. Tervo, and F. Wyrowski, “Spectrally partially coherent pulse trains in dispersive media,” Opt. Commun. 255(1-3), 12–22 (2005).
[Crossref]

P. Paakkonen, J. Turunen, P. Vahimaa, A. T. Friberg, and F. Wyrowski, “Partially coherent Gaussian pulses,” Opt. Commun. 204(1-6), 53–58 (2002).
[Crossref]

Vahimaa, P.

H. Lajunen, J. Turunen, P. Vahimaa, J. Tervo, and F. Wyrowski, “Spectrally partially coherent pulse trains in dispersive media,” Opt. Commun. 255(1-3), 12–22 (2005).
[Crossref]

H. Lajunen, P. Vahimaa, and J. Tervo, “Theory of spatially and spectrally partially coherent pulses,” J. Opt. Soc. Am. A 22(8), 1536–1545 (2005).
[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. A 21(11), 2117–2123 (2004).
[Crossref]

P. Paakkonen, J. Turunen, P. Vahimaa, A. T. Friberg, and F. Wyrowski, “Partially coherent Gaussian pulses,” Opt. Commun. 204(1-6), 53–58 (2002).
[Crossref]

Wang, L.

Q. Lin, L. Wang, and S. Zhu, “Partially coherent light pulse and its propagation,” Opt. Commun. 219(1-6), 65–70 (2003).
[Crossref]

Wang, L. G.

L. G. Wang, Q. Lin, H. Chen, and S. Y. Zhu, “Propagation of partially coherent pulsed beams in the spatiotemporal domain,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 67(5), 056613 (2003).
[Crossref] [PubMed]

Wang, S.

Q. Lin, S. Wang, J. Alda, and E. Bernabeu, “Spatial-temporal coupling in grating-pair pulse compression system analysed by matrix optics,” Opt. Quantum Electron. 27(7), 679–692 (1995).
[Crossref]

Q. Lin, S. Wang, J. Alda, and E. Bernabeu, “Transformation of pulsed nonideal beams in a four-dimension domain,” Opt. Lett. 18(9), 669–671 (1993).
[Crossref] [PubMed]

Watson, E.

O. Korotkova, Y. Cai, and E. Watson, “Stochastic electromagnetic beams for LIDAR systems operating through turbulent atmosphere,” Appl. Phys. B 94(4), 681–690 (2009).
[Crossref]

Wilhelmi, B.

Wolf, E.

O. Korotkova and E. Wolf, “Effects of linear non-image forming devices on spectra and on coherence and polarization properties ofstochastic electromagnetic beams. Part II. Examples,” J. Mod. Opt. 52, 2673–2685 (2005).
[Crossref]

O. Korotkova and E. Wolf, “Effects of linear non-image forming devices on coherence and polarization properties of random electromagnetic beams. Part I. General theory,” J. Mod. Opt. 52, 2659–2671 (2005).
[Crossref]

E. Wolf, “Unified theory of coherence and polarization of random electromagnetic fields,” Phys. Lett. A 312(5-6), 263–267 (2003).
[Crossref]

Wyrowski, F.

H. Lajunen, J. Turunen, P. Vahimaa, J. Tervo, and F. Wyrowski, “Spectrally partially coherent pulse trains in dispersive media,” Opt. Commun. 255(1-3), 12–22 (2005).
[Crossref]

P. Paakkonen, J. Turunen, P. Vahimaa, A. T. Friberg, and F. Wyrowski, “Partially coherent Gaussian pulses,” Opt. Commun. 204(1-6), 53–58 (2002).
[Crossref]

Yao, M.

Zhu, S.

Q. Lin, L. Wang, and S. Zhu, “Partially coherent light pulse and its propagation,” Opt. Commun. 219(1-6), 65–70 (2003).
[Crossref]

Zhu, S. Y.

L. G. Wang, Q. Lin, H. Chen, and S. Y. Zhu, “Propagation of partially coherent pulsed beams in the spatiotemporal domain,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 67(5), 056613 (2003).
[Crossref] [PubMed]

Appl. Phys. B (1)

O. Korotkova, Y. Cai, and E. Watson, “Stochastic electromagnetic beams for LIDAR systems operating through turbulent atmosphere,” Appl. Phys. B 94(4), 681–690 (2009).
[Crossref]

IEEE J. Quantum Electron. (4)

O. E. Martinez, “Matrix formalism for pulse compressors,” IEEE J. Quantum Electron. 24(12), 2530–2536 (1988).
[Crossref]

O. E. Martinez, “Matrix formalism for dispersive laser cavities,” IEEE J. Quantum Electron. 25(3), 296–300 (1989).
[Crossref]

A. G. Kostenbauder, “Ray-pulse matrices: a rational treatment for dispersive optical systems,” IEEE J. Quantum Electron. 26(6), 1148–1157 (1990).
[Crossref]

S. P. Dijaili, A. Dienes, and J. S. Smith, “ABCD matrices for dispersive pulse propagation,” IEEE J. Quantum Electron. 26(6), 1158–1164 (1990).
[Crossref]

J. Mod. Opt. (2)

O. Korotkova and E. Wolf, “Effects of linear non-image forming devices on coherence and polarization properties of random electromagnetic beams. Part I. General theory,” J. Mod. Opt. 52, 2659–2671 (2005).
[Crossref]

O. Korotkova and E. Wolf, “Effects of linear non-image forming devices on spectra and on coherence and polarization properties ofstochastic electromagnetic beams. Part II. Examples,” J. Mod. Opt. 52, 2673–2685 (2005).
[Crossref]

J. Opt. A, Pure Appl. Opt. (2)

Y. Cai and Q. Lin, “The fractional Fourier transform for a partially coherent pulse,” J. Opt. A, Pure Appl. Opt. 6(4), 307–311 (2004).
[Crossref]

C. Ding and B. Lu, “Spectral shifts and spectral switches of diffracted spatially and spectrally partially coherent pulsed beams in the far field,” J. Opt. A, Pure Appl. Opt. 10(9), 095006 (2008).
[Crossref]

J. Opt. Soc. Am. A (3)

J. Opt. Soc. Am. B (1)

N. J. Phys. (1)

C. Ding, L. Pan, and B. Lu, “Characterization of stochastic spatially and spectrally partially coherent electromagnetic pulsed beams,” N. J. Phys. 11(8), 083001 (2009).
[Crossref]

Opt. Commun. (3)

H. Lajunen, J. Turunen, P. Vahimaa, J. Tervo, and F. Wyrowski, “Spectrally partially coherent pulse trains in dispersive media,” Opt. Commun. 255(1-3), 12–22 (2005).
[Crossref]

P. Paakkonen, J. Turunen, P. Vahimaa, A. T. Friberg, and F. Wyrowski, “Partially coherent Gaussian pulses,” Opt. Commun. 204(1-6), 53–58 (2002).
[Crossref]

Q. Lin, L. Wang, and S. Zhu, “Partially coherent light pulse and its propagation,” Opt. Commun. 219(1-6), 65–70 (2003).
[Crossref]

Opt. Express (2)

Opt. Lett. (7)

Opt. Quantum Electron. (1)

Q. Lin, S. Wang, J. Alda, and E. Bernabeu, “Spatial-temporal coupling in grating-pair pulse compression system analysed by matrix optics,” Opt. Quantum Electron. 27(7), 679–692 (1995).
[Crossref]

Phys. Lett. A (1)

E. Wolf, “Unified theory of coherence and polarization of random electromagnetic fields,” Phys. Lett. A 312(5-6), 263–267 (2003).
[Crossref]

Phys. Rev. A (1)

V. Torres-Company, H. Lajunen, J. Lancis, and A. T. Friberg, “Ghost interference with classical partially coherent light pulses,” Phys. Rev. A 77(4), 043811 (2008).
[Crossref]

Phys. Rev. E Stat. Nonlin. Soft Matter Phys. (1)

L. G. Wang, Q. Lin, H. Chen, and S. Y. Zhu, “Propagation of partially coherent pulsed beams in the spatiotemporal domain,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 67(5), 056613 (2003).
[Crossref] [PubMed]

Other (3)

S. M. Wang and D. M. Zhao, Matrix Optics, (Springer, 2000).

For the collection of original papers by R. C. Jones see W. Swindel, Polarized light, (Dowden, Hutchinson & Ross, (Stroudsburg, Pennsylvania, 1975).

H. Mueller, “The foundations of optics,” J. Opt. Soc. Am. 38, 661–661 (1948); for account of the Mueller’s theory see also E. Collett, Polarized Light: Fundamentals and Applications (Marcel Dekker, Inc., New York, 1993). Chap. 5.

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Figures (11)

Fig. 1
Fig. 1 Propagation scheme for a pulse beam through dispersive optical system and free space.
Fig. 2
Fig. 2 Illustrating notation for reflection of a pulse from a reflecting grating.
Fig. 3
Fig. 3 Dependence of the degree of polarization of a REMPB on the propagation distance z and the pulse duration σ t 0 for different diffraction orders m with σ I 0 = 1 m m , δ I 0 x x = 1 m m , δ t 0 = 2 p s .
Fig. 6
Fig. 6 Dependence of the degree of polarization P on the propagation distance z and the r.m.s. width of spatial degree of coherence δ I 0 x x for different diffraction orders m with σ I 0 = 1 m m , σ t 0 = 2 p s , δ t 0 = 2 p s .
Fig. 4
Fig. 4 Dependence of the degree of polarization P of a REMPB on the propagation distance z and the r.m.s. width σ I 0 of the intensity for different diffraction orders m with δ I 0 x x = 1 m m , σ t 0 = 2 p s , δ t 0 = 2 p s .
Fig. 5
Fig. 5 Dependence of the degree of polarization P on the propagation distance z and the r.m.s. width of temporal degree of coherence δ t 0 for different diffraction orders m with σ I 0 = 1 m m , δ I 0 x x = 1 m m , σ t 0 = 2 p s .
Fig. 7
Fig. 7 Dependence of the degree of polarization P on the propagation distance z for different diffraction orders m with σ I 0 = 1 m m , δ I 0 x x = 1 m m , σ t 0 = 2 p s , δ t 0 = 2 p s .
Fig. 11
Fig. 11 Dependence of the degree of polarization P on the r.m.s. width of spatial degree of coherence δ I 0 x x for different diffraction orders m at z = 200 m with σ I 0 = 1 m m , σ t 0 = 2 p s , δ t 0 = 2 p s .
Fig. 9
Fig. 9 Dependence of the degree of polarization P on the r.m.s. width σ I 0 of the intensity for different diffraction orders m at z = 200m with δ I 0 x x = 1 m m , σ t 0 = 2 p s , δ t 0 = 2 p s .
Fig. 8
Fig. 8 Dependence of the degree of polarization P on the pulse duration σ t 0 for different diffraction orders m at z = 200m with σ I 0 = 1 m m , δ I 0 x x = 1 m m , δ t 0 = 2 p s .
Fig. 10
Fig. 10 Dependence of the degree of polarization P on the r.m.s. width of temporal degree of coherence δ t 0 for different diffraction orders m at z = 200m with σ I 0 = 1 m m , δ I 0 x x = 1 m m , δ t 0 = 2 p s .

Equations (22)

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Γ 0 α β ( r 01 , τ 01 ; r 02 , τ 02 ) = A α A β B α β exp [ r 01 2 + r 02 2 4 σ I 0 α β 2 ( r 01 r 02 ) 2 2 δ I 0 α β 2 ]                                     × exp [ τ 01 2 + τ 02 2 2 σ τ 0 α β 2 ( τ 01 τ 02 ) 2 2 δ τ 0 α β 2 ] exp [ i ω 0 ( τ 01 τ 02 ) ] ,
Γ 0 α β ( r ¯ 0 ) = A α A β B α β exp [ i k 0 2 r ¯ 0 T Q ¯ 0 α β 1 r ¯ 0 ] exp [ i ω 0 ( τ 01 τ 02 ) ] ,    ( α = x , y ; β = x , y )
Q ¯ 0 α β 1 = ( i k 0 ( 1 2 σ I 0 α β 2 + δ I 0 α β 2 ) I 0 21 i k 0 δ I 0 α β 2 I 0 21 0 12 i k 0 ( σ τ 0 α β 2 + δ τ 0 α β 2 ) 0 12 i k 0 δ τ 0 α β 2 i k 0 δ I 0 α β 2 I 0 21 i k 0 ( 1 2 σ I 0 α β 2 + δ I 0 α β 2 ) I 0 21 0 12 i k 0 δ τ 0 α β 2 0 12 i k 0 ( σ τ 0 α β 2 + δ τ 0 α β 2 ) )
Γ 1 α β ( r ¯ 1 ) = A α A β B α β [ det ( A ¯ 1 + B ¯ 1 Q ¯ 0 α β 1 ) ] 1 / 2 exp [ i k 0 2 r ¯ 1 T Q ¯ 1 α β 1 r ¯ 1 ] ,
Q ¯ 1 α β 1 = ( C ¯ 1 + D ¯ 1 Q ¯ 0 α β 1 ) ( A ¯ 1 + B ¯ 1 Q ¯ 0 α β 1 ) 1 .
Q ¯ 1 α β 1 = ( q 11 q 12 q 13 q 14 q 15 q 16 q 21 q 22 q 23 q 24 q 25 q 26 q 31 q 32 q 33 q 34 q 35 q 36 q 41 q 42 q 43 q 44 q 45 q 46 q 51 q 52 q 53 q 54 q 55 q 56 q 61 q 62 q 63 q 64 q 65 q 66 ) .
A ¯ 1 = ( A ˜ 11 0 33 0 33 A ˜ 12 ) ,   B ¯ 1 = ( B ˜ 11 0 33 0 33 B ˜ 12 * ) ,   C ¯ 1 = ( C ˜ 11 0 33 0 33 C ˜ 12 * ) ,   D 1 = ( D ˜ 11 0 33 0 33 D ˜ 12 )
Γ 2 α β ( r ¯ 2 ) = A α A β B α β [ det ( A ¯ 1 + B ¯ 1 Q ¯ 0 α β 1 ) ] 1 / 2 [ det ( A ¯ 2 + B ¯ 2 Q ¯ 1 α β 1 ) ] 1 / 2 exp [ i k 0 2 r ¯ 2 T Q ¯ 2 α β 1 r ¯ 2 ] ,
A ˜ 21 = D ˜ 21 = ( 1 0 0 0 1 0 0 0 1 ) ,   B ˜ 21 = ( z 0 0 0 z 0 0 0 1 ) ,   C ˜ 21 = ( 0 0 0 0 0 0 0 0 0 ) .
W j α β ( r j 1 , ω 1 ; r j 2 , ω 2 ) = 1 2 π + + Γ j α β ( r j 1 , τ j 1 ; r j 2 , τ j 2 ) exp [ ( ω 1 τ j 1 ω 2 τ j 2 ) ] d τ j 1 d τ j 2 ,
P j ω ( r j , ω ) = 1 4 D e t [ W j ( r j , ω , r j , ω ) ] { T r [ W j ( r j , ω , r j , ω ) ] } 2 ,
A ˜ 11 = ( sin ϕ sin ψ 0 0 0 1 0 cos ψ cos ϕ sin ψ 0 1 ) ,    B ˜ 11 = C ˜ 11 = 0 ,    D ˜ 11 = ( sin ψ sin ϕ 0 cos ψ cos ϕ sin ϕ 0 1 0 0 0 1 ) ,
d G ( cos ϕ cos ψ ) = m λ 0 ,
Γ 1 α β ( r ¯ 1 ) = A α A β B α β sin ψ sin ϕ exp [ i k 0 2 r ¯ 1 T Q ¯ 1 α β 1 r ¯ 1 ] ,
Q ¯ 1 α β 1 = ( q 11 0 q 13 q 14 0 q 16 0 q 22 0 0 q 25 0 q 31 0 q 33 q 34 0 q 36 q 41 0 q 43 q 44 0 q 46 0 q 52 0 0 q 55 0 q 61 0 q 63 q 64 0 q 66 ) .
q 11 = q 44 = i g 1 2 2 k 0 ( σ I 0 α β 2 + 2 δ I 0 α β 2 ) i g 2 2 k 0 ( σ τ 0 α β 2 + δ τ 0 α β 2 ) , q 13 = q 31 = q 46 = q 64 = i g 2 k 0 ( σ τ 0 α β 2 + δ τ 0 α β 2 ) , q 14 = q 41 = i k 0 ( g 1 2 δ I 0 α β 2 + g 2 2 δ τ 0 α β 2 ) , q 16 = q 34 = q 43 = q 61 = i g 2 k 0 δ τ 0 α β 2 , q 22 = q 55 = i 2 k 0 ( σ I 0 α β 2 + 2 δ I 0 α β 2 ) , q 25 = q 52 = i k 0 δ I 0 α β 2 , q 33 = q 66 = i k 0 ( σ τ 0 α β 2 + δ τ 0 α β 2 ) , q 36 = q 63 = i k 0 δ τ 0 α β 2 .
Γ 2 α β = A α A β B α β g 1 [ 1 + z 2 4 k 0 2 σ I 0 α β 2 ( 4 δ I 0 α β 2 + σ I 0 α β 2 ) ] 1 / 2            × { 1 + z 2 4 k 0 2 ( g 1 2 σ I 0 α β 2 + 2 g 2 2 σ τ 0 α β 2 ) [ g 1 2 ( 4 δ I 0 α β 2 + σ I 0 α β 2 ) + 2 g 2 2 ( 2 δ τ 0 α β 2 + σ τ 0 α β 2 ) ] } 1 / 2 ,
P τ ( z ) = 1 4 D e t ( Γ 2 ) T r ( Γ 2 ) = | Γ 2 x x Γ 2 y y Γ 2 x x + Γ 2 y y | = | A x 2 Δ x x A y 2 Δ y y A x 2 Δ x x + A y 2 Δ y y |
Δ α β = [ 1 + z 2 4 k 0 2 σ I 0 α β 2 ( 4 δ I 0 α β 2 + σ I 0 α β 2 ) ] 1 / 2        × { 1 + z 2 4 k 0 2 ( g 1 2 σ I 0 α β 2 + 2 g 2 2 σ τ 0 α β 2 ) [ g 1 2 ( 4 δ I 0 α β 2 + σ I 0 α β 2 ) + 2 g 2 2 ( 2 δ τ 0 α β 2 + σ τ 0 α β 2 ) ] } 1 / 2 .
Δ α β = [ 1 + z 2 4 k 0 2 σ I 0 α β 2 ( 4 δ I 0 α β 2 + σ I 0 α β 2 ) ] 1 .
lim z P τ ( z ) = | A x 2 Ω x x A y 2 Ω y y A x 2 Ω x x + A y 2 Ω y y | ,
Ω α β = { σ I 0 α β 2 ( 4 δ I 0 α β 2 + σ I 0 α β 2 ) ( g 1 2 σ I 0 α β 2 + 2 g 2 2 σ τ 0 α β 2 ) } 1 / 2           × { [ g 1 2 ( 4 δ I 0 α β 2 + σ I 0 α β 2 ) + 2 g 2 2 ( 2 δ τ 0 α β 2 + σ τ 0 α β 2 ) ] } 1 / 2

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