Abstract

We consider the field generated by a wavefront-folding interferometer which is illuminated by a stochastic electromagnetic beam. The specular property and anti-specular property are discussed in the vector case. Take electromagnetic Gaussian Schell-model beam as an example, we investigate the spectral density, the spectral degree of coherence, the spectral degree of polarization as well as the state of polarization of the polarized portion of the field on propagation. Results show that the polarization properties including the degree of polarization, the orientation angle and the degree of ellipse can be adjusted by the phase difference of the interferometer and the phase retardation introduced by the prism. The results may be applied in free-space optical communication.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Since the unified theory of coherence and polarization was formulated [1,2], many issues of statistical theory of stochastic electromagnetic beams have been solved. In particular, the changes in the spectral density, the spectral degree of coherence, and the spectral degree of polarization on propagation can be investigated with the help of the 2×2 cross-spectral density matrix (CSDM) of the stochastic electromagnetic beams [3–8]. Besides, the state of polarization, characterized by the shape and the orientation of the polarization ellipse of the polarized part of the field can also be determined by the CSDM [9,10]. Statistical properties of light beams and their manipulation have always received considerable interest for their applications in a broad area of optics [11–14]. A number of methods have been used to modulate the statistical properties of the random electromagnetic beams, such as random phase screen [15] and slit aperture [16].

The correlation function characterized by the peculiar specularity property, not encountered in the most commonly used source models, was found in a space-time modulated field [17]. This property permits an interesting application to the optical processing in which a hologram of a spatially noncoherent object is accomplished by using a wavefront-folding interferometer (WFI) [18]. The extension of specularity to the correlation function in the space-frequency domain, i.e., cross-spectral density function, was realized shortly [19]. Since then it has received little attention apart from the study on specular and anti-specular partially coherent vector solitons [20]. Recently, the specular and antispecular fields were experimentally implemented by launching a Gaussian Schell-model beam into a WFI [21]. The impacts of turbulent media on specular or anti-specular property of this class of beams were also investigated [22]. Besides, the potential applications of specularity and anti-specularity to free-space optical communication and micro-particle trapping have been discussed [23,24]. However, to the best of our knowledge, the researches on the specularity or anti-specularity are all within the framework of the scalar case, and the specular or anti-specular stochastic electromagnetic fields have not been studied in the literature.

In this paper, we consider the field generated by passing a stochastic electromagnetic beam through a WFI. The specularity and anti-specularity are discussed in the vector case. Take electromagnetic Gaussian Schell-model (EGSM) beam as an example, we investigate the spectral density, the spectral degree of coherence, the spectral degree of polarization and the state of polarization of the polarized part of the field during propagation. In particular, we demonstrate the influences on the polarization properties of the phase difference introduced by the interferometer and the phase retardation produced by the compensated right-angle prism.

2. Transformation of random electromagnetic beams by a wavefront-folding interferometer

Figure 1(a) exhibits the wavefront-folding interferometer [25], which is a Michelson-type interferometer mainly considered for the measurement of the spatial correlation properties of partially coherent fields. The key elements of this device are the two perpendicularly oriented right-angle prisms PRx and PRy in the arms, which respectively retroreflect the incident field in the x and y direction. When these prisms are slightly tilted with respect to the optical axis, spatial interference fringes can be seen in the observation plane. However, we assume the device perfectly aligned.

 figure: Fig. 1

Fig. 1 Wavefront-folding interferometer (a) and retroreflection by a compensated right-angle prism (b). S is the source, BS is a non-polarizing beam splitter, PRx and PRy are right-angle prisms, WP is a wave plate. The set of coordinate axes in (b) is used to describe the state of polarization of a light beam passing through a compensated right-angle prism.

Download Full Size | PPT Slide | PDF

Suppose that an electromagnetic wave with the electric field vector E0(x,y) is incident on the interferometer. To describe the electromagnetic vibrations conveniently, we use a moving set of Cartesian axes Ox, Oy, Oz with Ox parallel to the beam splitter and Oz along the direction of propagation of the beam [see Fig. 1(a)]. E0(x,y) can be expressed as the sum of two mutually orthogonal components, E0x(x,y) and E0y(x,y), which are respectively along the axes Ox and Oy. The vibration at the input of the interferometer gives rise to two vibrations at the output. Let E1(x,y) and E2(x,y) be the electric field vectors at the output of the interferometer, which are from the waves reflected by prisms PRx and PRy, respectively. The components of the electric field vector E1(x,y) are related to the components of the electric field vector E0(x,y), by the matrix form

[E1x(x,y)E1y(x,y)]=M1×[E0x(x,y)E0y(x,y)].
The components of the electric field vector E2(x,y) are related to the components of the electric field vector E0(x,y), by the matrix form
[E2x(x,y)E2y(x,y)]=M2×[E0x(x,y)E0y(x,y)].
Here M1=TPRand M2=ROPOT [26], with
R=[rx00ry],T=[tx00ty],P=[eiφ001],O=[0110],
where the coefficients rx and ry are the complex amplitude reflectances for the components along Ox and Oy, respectively; tx and ty are the analogous complex amplitude transmittances. φ represents the total phase retardation introduced between the two electric field components by the prism [see Fig. 1(b)]. Using Eq. (3), one readily finds that

M1=[rxtxeiφ00ryty],M2=[rxtx00rytyeiφ].

By denoting θ the phase difference between the two arms of the interferometer, the total electric field vector at the output can be expressed in the form

[Ex(x,y)Ey(x,y)]=[E1x(x,y)E1y(x,y)]eiθ+[E2x(x,y)E2y(x,y)].
On substituting the expressions for E1(x,y) and E2(x,y) respectively from Eqs. (1) and (2) into Eq. (5), and on using Eq. (4), one can express the components of the total electric field
Ex(x,y)=rxtxeiφeiθE0x(x,y)rxtxE0x(x,y),
Ey(x,y)=rytyeiθE0y(x,y)rytyeiφE0y(x,y).
From Eq. (6), one immediately finds that the x component of the output field is related to the x component and irrelevant to the y component of the input field. It is similar for the y component of the output field. This is caused by the orientations of the prisms PRx and PRy, whose roof ridges are along the y direction and x direction, respectively.

Consider E0(x,y) as a single realization of a stochastic electromagnetic field. We assume that the random fluctuations are statistically stationary, at least in the wide sense. The second-order correlation properties of such a field may be characterized by a 2×2 correlation matrix, i.e., the cross-spectral density matrix

W(r1,r2;ω)=[Wij(r1,r2;ω)]=[Ei(r1;ω)Ej(r2;ω)],(i=x,y;j=x,y).
Here the asterisk denotes the complex conjugate and the sharp brackets represent ensemble average. Substitute Eqs. (6) and (7) into Eq. (8) one then obtains
Wij(x1,y1,x2,y2)=Cij[DijW0ij(x1,y1,x2,y2)+EijW0ij(x1,y1,x2,y2)+FijW0ij(x1,y1,x2,y2)+GijW0ij(x1,y1,x2,y2)],
with Cij=ritirjtj. W0ij(x1,y1,x2,y2) and Wij(x1,y1,x2,y2) are the elements of the CSDM at the input and output of the interferometer, respectively. The coefficients Dij,Eij,Fij and Gij are given by Table 1.

Tables Icon

Table 1. Expressions for the coefficients Dij,Eij,Fij and Gij

It is noted from Eqs. (6), (7) and (9) that the total output field depends on the phase retardation φ produced by a right-angle prism. For incidence at an angle 45 and the prism of index n=1.5, the total phase retardation due to two total internal reflections inside a prism is φ74. To study the influence of the phase retardation φ on the statistical properties of the stochastic electromagnetic beams, we will adopt several values of φ. This can be achieved by cementing a phase plate to the entrance face of each right-angle prism [26], whose principal axes are parallel and perpendicular to the roof ridge. Therefore, the total phase retardation φ=2(φ1+φ1) with φ1 and φ1 being the phase retardation introduced by the total reflection once and by the wave plate once, respectively. In practice, one can use a zero-order wave plate, which consists of two plates cemented together with the fast axis of the one coinciding with the slow axis of the other, to construct a compensated prism.

One can find that Eq. (9) is analogous to Eq. (3) in [21], which is used to study the specularity and antispecularity of scalar fields. To study the specular and antispecular electromagnetic fields, we will consider two special cases φ=2mπ and φ=(2m+1)π (m is an integer).

Case 1. φ=2mπ. Under this condition Eq. (9) becomes

Wij(x1,y1,x2,y2)=Cij[W0ij(x1,y1,x2,y2)+W0ij(x1,y1,x2,y2)W0ij(x1,y1,x2,y2)eiθW0ij(x1,y1,x2,y2)eiθ].
From Eq. (10) one readily finds that
{Wij(x1,y1,x2,y2)=Wij(x1,y1,x2,y2)whenθ=2lπ,Wij(x1,y1,x2,y2)=Wij(x1,y1,x2,y2)whenθ=(2l+1)π.
Here l is an integer. As can be seen from Eq. (11), whatever the random electromagnetic field at the input is, the CSDM of the field at the output is anti-specular W(x1,y1,x2,y2)=W(x1,y1,x2,y2) when θ=2lπ and specular W(x1,y1,x2,y2)=W(x1,y1,x2,y2) when θ=(2l+1)π.

Case 2. φ=(2m+1)π. In this particular case Eq. (9) becomes

Wij(x1,y1,x2,y2)=LijCij[W0ij(x1,y1,x2,y2)+W0ij(x1,y1,x2,y2)+W0ij(x1,y1,x2,y2)eiθ+W0ij(x1,y1,x2,y2)eiθ],
with
Lij={1wheni=j,1whenij.
From Eq. (12) one can find that
{Wij(x1,y1,x2,y2)=Wij(x1,y1,x2,y2)whenθ=2lπ,Wij(x1,y1,x2,y2)=Wij(x1,y1,x2,y2)whenθ=(2l+1)π.
Equation (14) indicates that the CSDM of the field at the output of the interferometer is specular W(x1,y1,x2,y2)=W(x1,y1,x2,y2) when θ=2lπ and anti-specular W(x1,y1,x2,y2)=W(x1,y1,x2,y2) when θ=(2l+1)π.

Comparing these two cases, one immediately finds that the specular case and anti-specular case rely on both the phase retardation φ and the phase difference θ. The two special cases reveal that the output field of the interferometer is specular if one of φ and θ is odd times of πwhile the other is even times of π. For anti-specular field, the condition that φ and θ are both odd or even times of π is satisfied.

It is worth mentioning that Case 2 is in agreement with the specular or antispecular case of the scalar field [21].

3. The statistical properties of random electromagnetic beams propagating in free space

Let us consider the field at the output of the interferometer as a secondary source, which propagates close to the z axis from the output plane of the interferometer (z=0) into the half-space z>0 in free space. By employing the propagation law applicable to a paraxial ABCD system [2,27], one may obtain the expression for the elements of the CSDM at any two points (ρ1,z) and (ρ2,z) in any transverse plane z=const>0 from the elements of the CSDM at the output plane of the interferometer, viz.,

Wij(ρ1,ρ2,z;ω)=(k2πz)2d2ρ1d2ρ2Wij(0)(ρ1,ρ2;ω)×exp{ik2z[(ρ12ρ22)2(ρ1ρ1ρ2ρ2)+(ρ12ρ22)]}.

The spectral density S(ρ,z;ω) at a point (ρ,z), at frequency ω, is given by the expression (Ref [28], Sec. 9.2)

S(ρ,z;ω)=TrW(ρ,ρ,z;ω),
where Tr denotes the trace. The spectral degree of coherence is defined by the formula (Ref [28], Sec. 9.2)

η(ρ1,ρ2,z;ω)=TrW(ρ1,ρ2,z;ω)S(ρ1,z;ω)S(ρ2,z;ω).

Any statistically stationary light beam may be, at each point, expressed as the sum of contributions from a completely polarized and from a completely unpolarized beam. The ratio of the intensity of the polarized portion to the total intensity is called the degree of polarization, which is given by the expression [1]

P(ρ,z;ω)=14DetW(ρ,ρ,z;ω)[TrW(ρ,ρ,z;ω)]2,
where Det represents the determinant. In addition to the spectral degree of polarization, the polarization property of the field also includes the state of polarization of the polarized part, characterized by the shape and the orientation of the polarization ellipse, which can be obtained from the CSDM. The angle of orientation α(ρ,z;ω) (the angle between the major axis of the polarization ellipse and the x axis) is given by the formula [9]
α(ρ,z;ω)=12arctan(2Re[Wxy(ρ,z;ω)]Wxx(ρ,z;ω)Wyy(ρ,z;ω)),π/2απ/2.
The squares of the major semi-axis a(ρ,z;ω) and of the minor semi-axis b(ρ,z;ω) of the polarization ellipse are given by [9]
a2(ρ,z;ω)=12[(WxxWyy)2+4|Wxy|2+(WxxWyy)2+4[ReWxy]2],
b2(ρ,z;ω)=12[(WxxWyy)2+4|Wxy|2(WxxWyy)2+4[ReWxy]2].
The shape of the polarization ellipse is characterized by the degree of ellipticity, i.e.,
ε=b(ρ,z;ω)/a(ρ,z;ω),0ε1,
which is zero for linear polarization and unity for circular polarization.

4. An example: propagation of a modulated electromagnetic Gaussian Schell-model beam in free space

Suppose an EGSM beam incident on the interferometer, whose elements of the CSDM are given by (Ref [28], Sec. 9.4.2)

W0ij(ρ1,ρ2;ω)=AiAjBijexp(ρ124σi2ρ224σj2)exp(|ρ2ρ1|22δij2),(i=x,y;j=x,y),
where ρ1=(x1,y1) and ρ2=(x2,y2) denote the two-dimensional position vectors. Ai and Aj are the amplitudes of the electric field vector components, Bij is the correlation coefficient between the two components of the electric field vector, σi and σj denote the variance of the intensity distribution, δij denotes the variance of the correlation. The parameters Ai, Aj, Bij, σi, σj and δij are independent of position but may depend on the frequency. However, they may not all be chosen arbitrarily. In particular [29,30]
Bij=1wheni=j,
|Bij|1whenij,
Bij=Bji,
δji=δij.
Due to the non-negative definition of the correlation matrix, some additional restrictions on the variances δij and the coefficients Bij have to be satisfied [30]
max{δxx,δyy}δxymin{δxx|Bxy|,δyy|Bxy|}.
In addition, there are also conditions that the CSDM of the EGSM source must satisfy to generate a beam [31].

Substitute Eq. (23) into Eq. (9), one can express the elements of the CSDM at the output of the interferometer by the formula

Wij(ρ1,ρ2;ω)=2CijAiAjBijexp(x12+y124σi2x22+y224σj2)×{Hijexp[(x2x1)2+(y2y1)22δij2]+Kijexp[(x2+x1)2+(y2+y1)22δij2]},
where the coefficients Hij and Kij are given in Table 2. It should be noticed that Eq. (29) will reduce to Eq. (23) except for the coefficients 2Cij when φ=0 and θ=π/2.

Tables Icon

Table 2. Expressions for the coefficients Hij and Kij

Using Eqs. (15) and (29), the elements of the CSDM of the field in any transverse plane z>0 can be determined

Wij(ρ1,ρ2,z;ω)=2CijAiAjBijΔijexp{[ikz(βij+ikz)1Δij]ρρ}×{Hijexp(4αijρ2Δij)exp(γijρ2Δij)+Kijexp(αijρ2Δij)exp(4γijρ2Δij)},
where

ρ=ρ1+ρ22,ρ=ρ2ρ1,
Δij=(zk)2[16αijγij(βij+ikz)2],
αij=116(1σi2+1σj2),βij=14(1σi21σj2),γij=116(1σi2+1σj2)+12δij2.

Using Eqs. (16)-(22) and (30), one can investigate the statistical properties (i.e., the spectral density, the spectral degree of coherence, the spectral degree of polarization and the state of polarization of the polarized portion) of the modulated EGSM beams upon propagation in free space.

5. The spectral density, the spectral degree of coherence and the polarization properties of a modulated EGSM beam on propagation

In this section, we will demonstrate the behavior of the statistical properties of the modulated EGSM beams during propagation in free space. For convenience, we assume that the split ratio of the non-polarizing beam splitter for the light at a certain wavelength (for instance, 632.8nm) is 50:50. Consequently, it is reasonable to get rx=ry=tx=ty, and then Cij=C. In such a case the expression (30) can be used to describe the evolution of the statistical properties of the incident EGSM beam without passing through the WFI when φ=0 and θ=π/2, i.e., the modulated EGSM beam reduces to the normal EGSM beam. In particular we set σx=σy=σ0, implying that the polarization (including the degree of polarization, the orientation angle and the degree of ellipticity) of the incident EGSM beam is uniform across the source. Besides, we suppose that δxx=δyy=δ0.

Substituting Eq. (30) into Eq. (16) yields

S(ρ,z;ω)=2CΔxx{(Ax2+Ay2)exp(ρ22σI2(z))[Ax2cos(φ+θ)+Ay2cos(φθ)]exp(ρ22σA2(z))},
where σI(z)=Δxx/8αxxand σA(z)=Δxx/8γxx. σI(z)>σA(z) because αxx<γxx. From Eq. (34), one can find that a central peak or dip of width 2σA(z) is observed on a Gaussian background with the beam width 2σI(z) during propagation, as shown in Fig. 2(c). The width of the central peak or dip could be adjusted by varying the coherence length δ0, which will increase as δ0 increases [see Figs. 2(a) and 2(b)]. Equation (34) also reveals that the central distribution (peak or dip) is determined by the phase retardation φ and the phase difference θ, which is displayed in Figs. 3(d1), 3(d2) and 3(d3).

 figure: Fig. 2

Fig. 2 Intensity distributions of the modulated EGSM beams with φ=π at z=1000m. SGmax(z)=2C(Ax2+Ay2)/Δxx represents the maximum value of the normal Gaussian beam. (a) θ=0; (b) θ=π. We choose δ0=1mm in (c). The other parameters of the incident beam are chosen as follows: λ=632.8nm, Ax=2, Ay=1, Bxy=0.2exp(iπ/3), σ0=1cm.

Download Full Size | PPT Slide | PDF

 figure: Fig. 3

Fig. 3 Changes with ρ/σP0 of the spectral degree of polarization, the orientation angle, the degree of ellipticity and the spectral density for several selected values of φ and θ in the output plane of the WFI. σP0=1/1/σ02+4/δm2 with δm=max{δij} represents the modulation width by θ and φ in the output plane of the interferometer. Rows 1-3 respectively correspond to φ=π/6,π/4,π/2. The values of θ are given beside the curves. The parameters are the same as Fig. 2(c), δxy=2mm. The polarization of the incident EGSM beam is uniform with P=0.621,α=3.797 and ε=0.113.

Download Full Size | PPT Slide | PDF

On substituting from Eq. (29) into Eqs. (18)-(22), the expressions can readily be derived for all the polarization properties of the beam at points located in the output plane of the WFI. The spectral degree of polarization is given by the expression

P(ρ,0;ω)={Ax2Ay2[Ax2cos(φ+θ)Ay2cos(φθ)]exp(2ρ2δ02)}2+4Ax2Ay2|Bxy|2[cosφcosθexp(2ρ2δxy2)]2{Ax2+Ay2[Ax2cos(φ+θ)+Ay2cos(φθ)]exp(2ρ2δ02)}2.
The orientation angle of the polarization ellipse is given by
α(ρ,0;ω)=12arctan(2AxAyRe[Bxy][cosφcosθexp(2ρ2δxy2)]Ax2Ay2[Ax2cos(φ+θ)Ay2cos(φθ)]exp(2ρ2δ02)),
and the degree of ellipticity is given by the formula
ε(ρ,0;ω)={{Ax2Ay2[Ax2cos(φ+θ)Ay2cos(φθ)]exp(2ρ2δ02)}2+4Ax2Ay2|Bxy|2[cosφcosθexp(2ρ2δxy2)]2{Ax2Ay2[Ax2cos(φ+θ)Ay2cos(φθ)]exp(2ρ2δ02)}2+4Ax2Ay2Re2[Bxy][cosφcosθexp(2ρ2δxy2)]2{Ax2Ay2[Ax2cos(φ+θ)Ay2cos(φθ)]exp(2ρ2δ02)}2+4Ax2Ay2|Bxy|2[cosφcosθexp(2ρ2δxy2)]2+{Ax2Ay2[Ax2cos(φ+θ)Ay2cos(φθ)]exp(2ρ2δ02)}2+4Ax2Ay2Re2[Bxy][cosφcosθexp(2ρ2δxy2)]2}1/2.
From Eqs. (35)-(37), it is evident that the input EGSM beam with uniform polarization (when φ=0 and θ=π/2) across the source will become non-uniform at the output of the WFI in general (see Fig. 3). It is worth mentioning that ε(ρ,0;ω)=0 and α(ρ,0;ω)=0 when cosφcosθexp(2ρ2/δxy2)=0 (i.e., Wxy(ρ,0;ω)=0), meaning that the polarized part of the field at certain point is linearly polarized along the x direction. Two special cases of this situation are considered: (i) if both φ and θ are odd times of π/2 then the polarized part of the field will be linearly polarized along the x direction at all points across the output plane of the WFI [for example, φ=π/2,θ=3π/2 in Figs. 3(b3) and 3(c3)]; (ii) if φ=θ then the field will be completely polarized with linear polarization along the x direction at point ρ=0 [for example, φ=θ=π/6 in Figs. 3(a1)-3(c1) and φ=θ=π/4 in Figs. 3(a2)-3(c2)].

Substituting from Eq. (30) into Eqs. (18)-(22), the expressions can be obtained for all the polarization properties of the beam at points in any transverse plane. It can readily be found that the above-mentioned cases (i) and (ii) still hold during propagation. In particular, we will discuss the degree of polarization, the orientation angle and the degree of ellipticity along the axis z(i.e., ρ=0), which are given by the formulae

P(0,z;ω)={Ax2Ay2[Ax2cos(φ+θ)Ay2cos(φθ)]}2+4Ax2Ay2|Bxy|2χ2(cosφcosθ)2{Ax2+Ay2[Ax2cos(φ+θ)+Ay2cos(φθ)]}2,
α(0,z;ω)=12arctan(2AxAyχRe[Bxy](cosφcosθ)Ax2Ay2[Ax2cos(φ+θ)Ay2cos(φθ)]),
and
ε(0,z;ω)={{Ax2Ay2[Ax2cos(φ+θ)Ay2cos(φθ)]}2+4Ax2Ay2|Bxy|2χ2(cosφcosθ)2{Ax2Ay2[Ax2cos(φ+θ)Ay2cos(φθ)]}2+4Ax2Ay2Re2[Bxy]χ2(cosφcosθ)2{Ax2Ay2[Ax2cos(φ+θ)Ay2cos(φθ)]}2+4Ax2Ay2|Bxy|2χ2(cosφcosθ)2+{Ax2Ay2[Ax2cos(φ+θ)Ay2cos(φθ)]}2+4Ax2Ay2Re2[Bxy]χ2(cosφcosθ)2}1/2,
with χ=Δxx/Δxy. It should be noticed that in the special case (iii) of φ=mπ, the on-axis polarization properties will be independent of the phase difference θ upon propagation. As the field propagates to the far-zone (z), χ=(1/4σ02+1/δ02)/(1/4σ02+1/δxy2), implying that the degree of polarization, the orientation angle and the degree of ellipticity along the beam axis will tend to constants, determined by the phase retardation φ and the phase difference θ.

Figure 4 shows the evolution of on-axis polarization properties for several selected θ when φ=π/4, in which curve θ=π/4 corresponds to case (ii). Figure 5 demonstrates the on-axis polarization properties for different φ and θ in the far-zone, in which curve φ=π corresponds to case (iii).

 figure: Fig. 4

Fig. 4 Changes with the propagation distance z of on-axis spectral degree of polarization, orientation angle and degree of ellipticity for several selected values of θ when φ=π/4. The values of θ are given beside the curves. The parameters are the same as Fig. 2(c), δxy=2mm.

Download Full Size | PPT Slide | PDF

 figure: Fig. 5

Fig. 5 Changes with θ/π of on-axis spectral degree of polarization, orientation angle and degree of ellipticity for some selected values of φ in the far-zone. Black solid curve, φ=π/6; red short dashed curve, φ=π/4; blue short dotted curve, φ=π/2; green short dashed-dotted curve, φ=π. The parameters are the same as Fig. 2(c), δxy=2mm.

Download Full Size | PPT Slide | PDF

In Fig. 6, we illustrate the spectral degree of coherence as a function of the phase difference θ at several propagation distances. It is evident that the coherence between two symmetrical points can be controlled ranging from complete incoherence to complete coherence. Curves φ=0 and φ=π shows that complete coherence is achieved when θ=lπ, i.e., the specular case or the anti-specular case. This result is identical to that of the scalar case (Ref [21], Fig. 5). As the propagation distance increases, the range of complete coherence with θ increases.

 figure: Fig. 6

Fig. 6 Changes with θ/π of the spectral degree of coherence between two symmetrical points ρ1=(ρ/2,0) and ρ2=(ρ/2,0) with ρ=1mm. (a) z=0; (b) z=200m; (c) z=1000m. Black solid curve, φ=0; red short dashed curve, φ=π/4; blue short dotted curve, φ=π/2; green short dashed-dotted curve, φ=π.

Download Full Size | PPT Slide | PDF

6. Conclusions

In conclusion, we have considered the field generated by a WFI which is illuminated by a stochastic electromagnetic beam. The concepts of specularity and anti-specularity are introduced to the vector case, which are discussed in two special cases. Take EGSM beam as an example, we have studied the spectral density, the spectral degree of coherence, the spectral degree of polarization as well as the state of polarization of the polarized portion of the field on propagation. We have found that the polarization properties including the degree of polarization, the orientation angle and the degree of ellipticity, in general, can be adjusted by varying the phase retardation produced by the compensated right-angle prism and the phase difference introduced by the interferometer. In particular, the on-axis polarization properties will tend to constants in the far-zone, determined by the phase retardation and the phase difference. A special situation has also been discussed under which the polarization properties are independent of the phase difference. Numerical results have shown that complete polarization and linear polarization can be realized from a partially polarized incident field with the polarized part being elliptical polarization. Therefore, the interferometer can be used as a kind of polarization modulator. The results obtained in this paper may find applications in free-space optical communication.

Funding

National Natural Science Foundation of China (NSFC) (11474253 and 11274273); Fundamental Research Funds for the Central Universities (2017FZA3005).

References and links

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

2. E. Wolf, “Correlation-induced changes in the degree of polarization, the degree of coherence, and the spectrum of random electromagnetic beams on propagation,” Opt. Lett. 28(13), 1078–1080 (2003). [CrossRef]   [PubMed]  

3. M. Salem, O. Korotkova, A. Dogariu, and E. Wolf, “Polarization changes in partially coherent electromagnetic beams propagating through turbulent atmosphere,” Waves Random Media 14(4), 513–523 (2004). [CrossRef]  

4. B. Lü and L. Pan, “Propagation of vector Gaussian–Schell-model beams through a paraxial optical ABCD system,” Opt. Commun. 205(1-3), 7–16 (2002). [CrossRef]  

5. O. Korotkova, B. G. Hoover, V. L. Gamiz, and E. Wolf, “Coherence and polarization properties of far fields generated by quasi-homogeneous planar electromagnetic sources,” J. Opt. Soc. Am. A 22(11), 2547–2556 (2005). [CrossRef]   [PubMed]  

6. G. P. Agrawal and E. Wolf, “Propagation-induced polarization changes in partially coherent optical beams,” J. Opt. Soc. Am. A 17(11), 2019–2023 (2000). [CrossRef]   [PubMed]  

7. M. Salem and E. Wolf, “Coherence-induced polarization changes in light beams,” Opt. Lett. 33(11), 1180–1182 (2008). [CrossRef]   [PubMed]  

8. Z. Tong and O. Korotkova, “Stochastic electromagnetic beams in positive- and negative-phase materials,” Opt. Lett. 35(2), 175–177 (2010). [CrossRef]   [PubMed]  

9. O. Korotkova and E. Wolf, “Changes in the state of polarization of a random electromagnetic beam on propagation,” Opt. Commun. 246(1-3), 35–43 (2005). [CrossRef]  

10. H. Wang, X. Wang, A. Zeng, and K. Yang, “Effects of coherence on anisotropic electromagnetic Gaussian-Schell model beams on propagation,” Opt. Lett. 32(15), 2215–2217 (2007). [CrossRef]   [PubMed]  

11. H. C. Kandpal, J. S. Vaishya, and K. C. Joshi, “Wolf shift and its application in spectroradiometry,” Opt. Commun. 73(3), 169–172 (1989). [CrossRef]  

12. D. F. V. James and E. Wolf, “Determination of field correlations from spectral measurements with application to synthetic aperture imaging,” Radio Sci. 26(5), 1239–1243 (1991). [CrossRef]  

13. B. Redding, M. A. Choma, and H. Cao, “Speckle-free laser imaging using random laser illumination,” Nat. Photonics 6(6), 355–359 (2012). [CrossRef]   [PubMed]  

14. E. Wolf and D. F. V. James, “Correlation-induced spectral changes,” Rep. Prog. Phys. 59(6), 771–818 (1996). [CrossRef]  

15. T. Shirai and E. Wolf, “Coherence and polarization of electromagnetic beams modulated by random phase screens and their changes on propagation in free space,” J. Opt. Soc. Am. A 21(10), 1907–1916 (2004). [CrossRef]   [PubMed]  

16. D. Zhao and X. Du, “Polarization modulation of stochastic electromagnetic beams on propagation through the turbulent atmosphere,” Opt. Express 17(6), 4257–4262 (2009). [CrossRef]   [PubMed]  

17. P. De Santis, F. Gori, G. Guattari, and C. Palma, “A space-time modulated field with specular coherence function,” Opt. Commun. 64(1), 9–14 (1987). [CrossRef]  

18. A. S. Marathay, “Noncoherent-object hologram: its reconstruction and optical processing,” J. Opt. Soc. Am. A 4(10), 1861–1868 (1987). [CrossRef]  

19. F. Gori, G. Guattari, C. Palma, and C. Padovani, “Specular cross-spectral density functions,” Opt. Commun. 68(4), 239–243 (1988). [CrossRef]  

20. S. A. Ponomarenko and G. P. Agrawal, “Asymmetric incoherent vector solitons,” Phys. Rev. E 69(3), 036604 (2004). [CrossRef]   [PubMed]  

21. H. Partanen, N. Sharmin, J. Tervo, and J. Turunen, “Specular and antispecular light beams,” Opt. Express 23(22), 28718–28727 (2015). [CrossRef]   [PubMed]  

22. Z. Zhou, M. Guo, and D. Zhao, “Influence of atmospheric turbulence on the properties of specular and antispecular beams,” Appl. Opt. 55(24), 6757–6762 (2016). [CrossRef]   [PubMed]  

23. M. Guo and D. Zhao, “Changes in radiation forces acting on a Rayleigh dielectric sphere by use of a wavefront-folding interferometer,” Opt. Express 24(6), 6115–6125 (2016). [CrossRef]   [PubMed]  

24. M. Guo and D. Zhao, “Interfering optical coherence lattices by use of a wavefront-folding interferometer,” Opt. Express 25(13), 14351–14358 (2017). [CrossRef]   [PubMed]  

25. Q. He, J. Turunen, and A. T. Friberg, “Propagation and imaging experiments with Gausian Schell-model beams,” Opt. Commun. 67(4), 245–250 (1988). [CrossRef]  

26. F. Roddier, C. Roddier, and J. Demarcq, “A rotation shearing interferometer with phase-compensated roof-prisms,” J. Opt. 9(3), 145–149 (1978). [CrossRef]  

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

28. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University, 2007).

29. O. Korotkova, M. Salem, and E. Wolf, “The far-zone behavior of the degree of polarization of electromagnetic beams propagating through atmospheric turbulence,” Opt. Commun. 233(4-6), 225–230 (2004). [CrossRef]  

30. H. Roychowdhury and O. Korotkova, “Realizability conditions for electromagnetic Gaussian Schell-model sources,” Opt. Commun. 249(4-6), 379–385 (2005). [CrossRef]  

31. O. Korotkova, M. Salem, and E. Wolf, “Beam conditions for radiation generated by an electromagnetic Gaussian Schell-model source,” Opt. Lett. 29(11), 1173–1175 (2004). [CrossRef]   [PubMed]  

References

  • View by:

  1. E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312(5-6), 263–267 (2003).
    [Crossref]
  2. E. Wolf, “Correlation-induced changes in the degree of polarization, the degree of coherence, and the spectrum of random electromagnetic beams on propagation,” Opt. Lett. 28(13), 1078–1080 (2003).
    [Crossref] [PubMed]
  3. M. Salem, O. Korotkova, A. Dogariu, and E. Wolf, “Polarization changes in partially coherent electromagnetic beams propagating through turbulent atmosphere,” Waves Random Media 14(4), 513–523 (2004).
    [Crossref]
  4. B. Lü and L. Pan, “Propagation of vector Gaussian–Schell-model beams through a paraxial optical ABCD system,” Opt. Commun. 205(1-3), 7–16 (2002).
    [Crossref]
  5. O. Korotkova, B. G. Hoover, V. L. Gamiz, and E. Wolf, “Coherence and polarization properties of far fields generated by quasi-homogeneous planar electromagnetic sources,” J. Opt. Soc. Am. A 22(11), 2547–2556 (2005).
    [Crossref] [PubMed]
  6. G. P. Agrawal and E. Wolf, “Propagation-induced polarization changes in partially coherent optical beams,” J. Opt. Soc. Am. A 17(11), 2019–2023 (2000).
    [Crossref] [PubMed]
  7. M. Salem and E. Wolf, “Coherence-induced polarization changes in light beams,” Opt. Lett. 33(11), 1180–1182 (2008).
    [Crossref] [PubMed]
  8. Z. Tong and O. Korotkova, “Stochastic electromagnetic beams in positive- and negative-phase materials,” Opt. Lett. 35(2), 175–177 (2010).
    [Crossref] [PubMed]
  9. O. Korotkova and E. Wolf, “Changes in the state of polarization of a random electromagnetic beam on propagation,” Opt. Commun. 246(1-3), 35–43 (2005).
    [Crossref]
  10. H. Wang, X. Wang, A. Zeng, and K. Yang, “Effects of coherence on anisotropic electromagnetic Gaussian-Schell model beams on propagation,” Opt. Lett. 32(15), 2215–2217 (2007).
    [Crossref] [PubMed]
  11. H. C. Kandpal, J. S. Vaishya, and K. C. Joshi, “Wolf shift and its application in spectroradiometry,” Opt. Commun. 73(3), 169–172 (1989).
    [Crossref]
  12. D. F. V. James and E. Wolf, “Determination of field correlations from spectral measurements with application to synthetic aperture imaging,” Radio Sci. 26(5), 1239–1243 (1991).
    [Crossref]
  13. B. Redding, M. A. Choma, and H. Cao, “Speckle-free laser imaging using random laser illumination,” Nat. Photonics 6(6), 355–359 (2012).
    [Crossref] [PubMed]
  14. E. Wolf and D. F. V. James, “Correlation-induced spectral changes,” Rep. Prog. Phys. 59(6), 771–818 (1996).
    [Crossref]
  15. T. Shirai and E. Wolf, “Coherence and polarization of electromagnetic beams modulated by random phase screens and their changes on propagation in free space,” J. Opt. Soc. Am. A 21(10), 1907–1916 (2004).
    [Crossref] [PubMed]
  16. D. Zhao and X. Du, “Polarization modulation of stochastic electromagnetic beams on propagation through the turbulent atmosphere,” Opt. Express 17(6), 4257–4262 (2009).
    [Crossref] [PubMed]
  17. P. De Santis, F. Gori, G. Guattari, and C. Palma, “A space-time modulated field with specular coherence function,” Opt. Commun. 64(1), 9–14 (1987).
    [Crossref]
  18. A. S. Marathay, “Noncoherent-object hologram: its reconstruction and optical processing,” J. Opt. Soc. Am. A 4(10), 1861–1868 (1987).
    [Crossref]
  19. F. Gori, G. Guattari, C. Palma, and C. Padovani, “Specular cross-spectral density functions,” Opt. Commun. 68(4), 239–243 (1988).
    [Crossref]
  20. S. A. Ponomarenko and G. P. Agrawal, “Asymmetric incoherent vector solitons,” Phys. Rev. E 69(3), 036604 (2004).
    [Crossref] [PubMed]
  21. H. Partanen, N. Sharmin, J. Tervo, and J. Turunen, “Specular and antispecular light beams,” Opt. Express 23(22), 28718–28727 (2015).
    [Crossref] [PubMed]
  22. Z. Zhou, M. Guo, and D. Zhao, “Influence of atmospheric turbulence on the properties of specular and antispecular beams,” Appl. Opt. 55(24), 6757–6762 (2016).
    [Crossref] [PubMed]
  23. M. Guo and D. Zhao, “Changes in radiation forces acting on a Rayleigh dielectric sphere by use of a wavefront-folding interferometer,” Opt. Express 24(6), 6115–6125 (2016).
    [Crossref] [PubMed]
  24. M. Guo and D. Zhao, “Interfering optical coherence lattices by use of a wavefront-folding interferometer,” Opt. Express 25(13), 14351–14358 (2017).
    [Crossref] [PubMed]
  25. Q. He, J. Turunen, and A. T. Friberg, “Propagation and imaging experiments with Gausian Schell-model beams,” Opt. Commun. 67(4), 245–250 (1988).
    [Crossref]
  26. F. Roddier, C. Roddier, and J. Demarcq, “A rotation shearing interferometer with phase-compensated roof-prisms,” J. Opt. 9(3), 145–149 (1978).
    [Crossref]
  27. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).
  28. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University, 2007).
  29. O. Korotkova, M. Salem, and E. Wolf, “The far-zone behavior of the degree of polarization of electromagnetic beams propagating through atmospheric turbulence,” Opt. Commun. 233(4-6), 225–230 (2004).
    [Crossref]
  30. H. Roychowdhury and O. Korotkova, “Realizability conditions for electromagnetic Gaussian Schell-model sources,” Opt. Commun. 249(4-6), 379–385 (2005).
    [Crossref]
  31. O. Korotkova, M. Salem, and E. Wolf, “Beam conditions for radiation generated by an electromagnetic Gaussian Schell-model source,” Opt. Lett. 29(11), 1173–1175 (2004).
    [Crossref] [PubMed]

2017 (1)

2016 (2)

2015 (1)

2012 (1)

B. Redding, M. A. Choma, and H. Cao, “Speckle-free laser imaging using random laser illumination,” Nat. Photonics 6(6), 355–359 (2012).
[Crossref] [PubMed]

2010 (1)

2009 (1)

2008 (1)

2007 (1)

2005 (3)

O. Korotkova and E. Wolf, “Changes in the state of polarization of a random electromagnetic beam on propagation,” Opt. Commun. 246(1-3), 35–43 (2005).
[Crossref]

O. Korotkova, B. G. Hoover, V. L. Gamiz, and E. Wolf, “Coherence and polarization properties of far fields generated by quasi-homogeneous planar electromagnetic sources,” J. Opt. Soc. Am. A 22(11), 2547–2556 (2005).
[Crossref] [PubMed]

H. Roychowdhury and O. Korotkova, “Realizability conditions for electromagnetic Gaussian Schell-model sources,” Opt. Commun. 249(4-6), 379–385 (2005).
[Crossref]

2004 (5)

O. Korotkova, M. Salem, and E. Wolf, “Beam conditions for radiation generated by an electromagnetic Gaussian Schell-model source,” Opt. Lett. 29(11), 1173–1175 (2004).
[Crossref] [PubMed]

O. Korotkova, M. Salem, and E. Wolf, “The far-zone behavior of the degree of polarization of electromagnetic beams propagating through atmospheric turbulence,” Opt. Commun. 233(4-6), 225–230 (2004).
[Crossref]

S. A. Ponomarenko and G. P. Agrawal, “Asymmetric incoherent vector solitons,” Phys. Rev. E 69(3), 036604 (2004).
[Crossref] [PubMed]

T. Shirai and E. Wolf, “Coherence and polarization of electromagnetic beams modulated by random phase screens and their changes on propagation in free space,” J. Opt. Soc. Am. A 21(10), 1907–1916 (2004).
[Crossref] [PubMed]

M. Salem, O. Korotkova, A. Dogariu, and E. Wolf, “Polarization changes in partially coherent electromagnetic beams propagating through turbulent atmosphere,” Waves Random Media 14(4), 513–523 (2004).
[Crossref]

2003 (2)

2002 (1)

B. Lü and L. Pan, “Propagation of vector Gaussian–Schell-model beams through a paraxial optical ABCD system,” Opt. Commun. 205(1-3), 7–16 (2002).
[Crossref]

2000 (1)

1996 (1)

E. Wolf and D. F. V. James, “Correlation-induced spectral changes,” Rep. Prog. Phys. 59(6), 771–818 (1996).
[Crossref]

1991 (1)

D. F. V. James and E. Wolf, “Determination of field correlations from spectral measurements with application to synthetic aperture imaging,” Radio Sci. 26(5), 1239–1243 (1991).
[Crossref]

1989 (1)

H. C. Kandpal, J. S. Vaishya, and K. C. Joshi, “Wolf shift and its application in spectroradiometry,” Opt. Commun. 73(3), 169–172 (1989).
[Crossref]

1988 (2)

F. Gori, G. Guattari, C. Palma, and C. Padovani, “Specular cross-spectral density functions,” Opt. Commun. 68(4), 239–243 (1988).
[Crossref]

Q. He, J. Turunen, and A. T. Friberg, “Propagation and imaging experiments with Gausian Schell-model beams,” Opt. Commun. 67(4), 245–250 (1988).
[Crossref]

1987 (2)

P. De Santis, F. Gori, G. Guattari, and C. Palma, “A space-time modulated field with specular coherence function,” Opt. Commun. 64(1), 9–14 (1987).
[Crossref]

A. S. Marathay, “Noncoherent-object hologram: its reconstruction and optical processing,” J. Opt. Soc. Am. A 4(10), 1861–1868 (1987).
[Crossref]

1978 (1)

F. Roddier, C. Roddier, and J. Demarcq, “A rotation shearing interferometer with phase-compensated roof-prisms,” J. Opt. 9(3), 145–149 (1978).
[Crossref]

Agrawal, G. P.

Cao, H.

B. Redding, M. A. Choma, and H. Cao, “Speckle-free laser imaging using random laser illumination,” Nat. Photonics 6(6), 355–359 (2012).
[Crossref] [PubMed]

Choma, M. A.

B. Redding, M. A. Choma, and H. Cao, “Speckle-free laser imaging using random laser illumination,” Nat. Photonics 6(6), 355–359 (2012).
[Crossref] [PubMed]

De Santis, P.

P. De Santis, F. Gori, G. Guattari, and C. Palma, “A space-time modulated field with specular coherence function,” Opt. Commun. 64(1), 9–14 (1987).
[Crossref]

Demarcq, J.

F. Roddier, C. Roddier, and J. Demarcq, “A rotation shearing interferometer with phase-compensated roof-prisms,” J. Opt. 9(3), 145–149 (1978).
[Crossref]

Dogariu, A.

M. Salem, O. Korotkova, A. Dogariu, and E. Wolf, “Polarization changes in partially coherent electromagnetic beams propagating through turbulent atmosphere,” Waves Random Media 14(4), 513–523 (2004).
[Crossref]

Du, X.

Friberg, A. T.

Q. He, J. Turunen, and A. T. Friberg, “Propagation and imaging experiments with Gausian Schell-model beams,” Opt. Commun. 67(4), 245–250 (1988).
[Crossref]

Gamiz, V. L.

Gori, F.

F. Gori, G. Guattari, C. Palma, and C. Padovani, “Specular cross-spectral density functions,” Opt. Commun. 68(4), 239–243 (1988).
[Crossref]

P. De Santis, F. Gori, G. Guattari, and C. Palma, “A space-time modulated field with specular coherence function,” Opt. Commun. 64(1), 9–14 (1987).
[Crossref]

Guattari, G.

F. Gori, G. Guattari, C. Palma, and C. Padovani, “Specular cross-spectral density functions,” Opt. Commun. 68(4), 239–243 (1988).
[Crossref]

P. De Santis, F. Gori, G. Guattari, and C. Palma, “A space-time modulated field with specular coherence function,” Opt. Commun. 64(1), 9–14 (1987).
[Crossref]

Guo, M.

He, Q.

Q. He, J. Turunen, and A. T. Friberg, “Propagation and imaging experiments with Gausian Schell-model beams,” Opt. Commun. 67(4), 245–250 (1988).
[Crossref]

Hoover, B. G.

James, D. F. V.

E. Wolf and D. F. V. James, “Correlation-induced spectral changes,” Rep. Prog. Phys. 59(6), 771–818 (1996).
[Crossref]

D. F. V. James and E. Wolf, “Determination of field correlations from spectral measurements with application to synthetic aperture imaging,” Radio Sci. 26(5), 1239–1243 (1991).
[Crossref]

Joshi, K. C.

H. C. Kandpal, J. S. Vaishya, and K. C. Joshi, “Wolf shift and its application in spectroradiometry,” Opt. Commun. 73(3), 169–172 (1989).
[Crossref]

Kandpal, H. C.

H. C. Kandpal, J. S. Vaishya, and K. C. Joshi, “Wolf shift and its application in spectroradiometry,” Opt. Commun. 73(3), 169–172 (1989).
[Crossref]

Korotkova, O.

Z. Tong and O. Korotkova, “Stochastic electromagnetic beams in positive- and negative-phase materials,” Opt. Lett. 35(2), 175–177 (2010).
[Crossref] [PubMed]

O. Korotkova and E. Wolf, “Changes in the state of polarization of a random electromagnetic beam on propagation,” Opt. Commun. 246(1-3), 35–43 (2005).
[Crossref]

O. Korotkova, B. G. Hoover, V. L. Gamiz, and E. Wolf, “Coherence and polarization properties of far fields generated by quasi-homogeneous planar electromagnetic sources,” J. Opt. Soc. Am. A 22(11), 2547–2556 (2005).
[Crossref] [PubMed]

H. Roychowdhury and O. Korotkova, “Realizability conditions for electromagnetic Gaussian Schell-model sources,” Opt. Commun. 249(4-6), 379–385 (2005).
[Crossref]

O. Korotkova, M. Salem, and E. Wolf, “Beam conditions for radiation generated by an electromagnetic Gaussian Schell-model source,” Opt. Lett. 29(11), 1173–1175 (2004).
[Crossref] [PubMed]

M. Salem, O. Korotkova, A. Dogariu, and E. Wolf, “Polarization changes in partially coherent electromagnetic beams propagating through turbulent atmosphere,” Waves Random Media 14(4), 513–523 (2004).
[Crossref]

O. Korotkova, M. Salem, and E. Wolf, “The far-zone behavior of the degree of polarization of electromagnetic beams propagating through atmospheric turbulence,” Opt. Commun. 233(4-6), 225–230 (2004).
[Crossref]

Lü, B.

B. Lü and L. Pan, “Propagation of vector Gaussian–Schell-model beams through a paraxial optical ABCD system,” Opt. Commun. 205(1-3), 7–16 (2002).
[Crossref]

Marathay, A. S.

Padovani, C.

F. Gori, G. Guattari, C. Palma, and C. Padovani, “Specular cross-spectral density functions,” Opt. Commun. 68(4), 239–243 (1988).
[Crossref]

Palma, C.

F. Gori, G. Guattari, C. Palma, and C. Padovani, “Specular cross-spectral density functions,” Opt. Commun. 68(4), 239–243 (1988).
[Crossref]

P. De Santis, F. Gori, G. Guattari, and C. Palma, “A space-time modulated field with specular coherence function,” Opt. Commun. 64(1), 9–14 (1987).
[Crossref]

Pan, L.

B. Lü and L. Pan, “Propagation of vector Gaussian–Schell-model beams through a paraxial optical ABCD system,” Opt. Commun. 205(1-3), 7–16 (2002).
[Crossref]

Partanen, H.

Ponomarenko, S. A.

S. A. Ponomarenko and G. P. Agrawal, “Asymmetric incoherent vector solitons,” Phys. Rev. E 69(3), 036604 (2004).
[Crossref] [PubMed]

Redding, B.

B. Redding, M. A. Choma, and H. Cao, “Speckle-free laser imaging using random laser illumination,” Nat. Photonics 6(6), 355–359 (2012).
[Crossref] [PubMed]

Roddier, C.

F. Roddier, C. Roddier, and J. Demarcq, “A rotation shearing interferometer with phase-compensated roof-prisms,” J. Opt. 9(3), 145–149 (1978).
[Crossref]

Roddier, F.

F. Roddier, C. Roddier, and J. Demarcq, “A rotation shearing interferometer with phase-compensated roof-prisms,” J. Opt. 9(3), 145–149 (1978).
[Crossref]

Roychowdhury, H.

H. Roychowdhury and O. Korotkova, “Realizability conditions for electromagnetic Gaussian Schell-model sources,” Opt. Commun. 249(4-6), 379–385 (2005).
[Crossref]

Salem, M.

M. Salem and E. Wolf, “Coherence-induced polarization changes in light beams,” Opt. Lett. 33(11), 1180–1182 (2008).
[Crossref] [PubMed]

M. Salem, O. Korotkova, A. Dogariu, and E. Wolf, “Polarization changes in partially coherent electromagnetic beams propagating through turbulent atmosphere,” Waves Random Media 14(4), 513–523 (2004).
[Crossref]

O. Korotkova, M. Salem, and E. Wolf, “The far-zone behavior of the degree of polarization of electromagnetic beams propagating through atmospheric turbulence,” Opt. Commun. 233(4-6), 225–230 (2004).
[Crossref]

O. Korotkova, M. Salem, and E. Wolf, “Beam conditions for radiation generated by an electromagnetic Gaussian Schell-model source,” Opt. Lett. 29(11), 1173–1175 (2004).
[Crossref] [PubMed]

Sharmin, N.

Shirai, T.

Tervo, J.

Tong, Z.

Turunen, J.

H. Partanen, N. Sharmin, J. Tervo, and J. Turunen, “Specular and antispecular light beams,” Opt. Express 23(22), 28718–28727 (2015).
[Crossref] [PubMed]

Q. He, J. Turunen, and A. T. Friberg, “Propagation and imaging experiments with Gausian Schell-model beams,” Opt. Commun. 67(4), 245–250 (1988).
[Crossref]

Vaishya, J. S.

H. C. Kandpal, J. S. Vaishya, and K. C. Joshi, “Wolf shift and its application in spectroradiometry,” Opt. Commun. 73(3), 169–172 (1989).
[Crossref]

Wang, H.

Wang, X.

Wolf, E.

M. Salem and E. Wolf, “Coherence-induced polarization changes in light beams,” Opt. Lett. 33(11), 1180–1182 (2008).
[Crossref] [PubMed]

O. Korotkova, B. G. Hoover, V. L. Gamiz, and E. Wolf, “Coherence and polarization properties of far fields generated by quasi-homogeneous planar electromagnetic sources,” J. Opt. Soc. Am. A 22(11), 2547–2556 (2005).
[Crossref] [PubMed]

O. Korotkova and E. Wolf, “Changes in the state of polarization of a random electromagnetic beam on propagation,” Opt. Commun. 246(1-3), 35–43 (2005).
[Crossref]

T. Shirai and E. Wolf, “Coherence and polarization of electromagnetic beams modulated by random phase screens and their changes on propagation in free space,” J. Opt. Soc. Am. A 21(10), 1907–1916 (2004).
[Crossref] [PubMed]

M. Salem, O. Korotkova, A. Dogariu, and E. Wolf, “Polarization changes in partially coherent electromagnetic beams propagating through turbulent atmosphere,” Waves Random Media 14(4), 513–523 (2004).
[Crossref]

O. Korotkova, M. Salem, and E. Wolf, “The far-zone behavior of the degree of polarization of electromagnetic beams propagating through atmospheric turbulence,” Opt. Commun. 233(4-6), 225–230 (2004).
[Crossref]

O. Korotkova, M. Salem, and E. Wolf, “Beam conditions for radiation generated by an electromagnetic Gaussian Schell-model source,” Opt. Lett. 29(11), 1173–1175 (2004).
[Crossref] [PubMed]

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

E. Wolf, “Correlation-induced changes in the degree of polarization, the degree of coherence, and the spectrum of random electromagnetic beams on propagation,” Opt. Lett. 28(13), 1078–1080 (2003).
[Crossref] [PubMed]

G. P. Agrawal and E. Wolf, “Propagation-induced polarization changes in partially coherent optical beams,” J. Opt. Soc. Am. A 17(11), 2019–2023 (2000).
[Crossref] [PubMed]

E. Wolf and D. F. V. James, “Correlation-induced spectral changes,” Rep. Prog. Phys. 59(6), 771–818 (1996).
[Crossref]

D. F. V. James and E. Wolf, “Determination of field correlations from spectral measurements with application to synthetic aperture imaging,” Radio Sci. 26(5), 1239–1243 (1991).
[Crossref]

Yang, K.

Zeng, A.

Zhao, D.

Zhou, Z.

Appl. Opt. (1)

J. Opt. (1)

F. Roddier, C. Roddier, and J. Demarcq, “A rotation shearing interferometer with phase-compensated roof-prisms,” J. Opt. 9(3), 145–149 (1978).
[Crossref]

J. Opt. Soc. Am. A (4)

Nat. Photonics (1)

B. Redding, M. A. Choma, and H. Cao, “Speckle-free laser imaging using random laser illumination,” Nat. Photonics 6(6), 355–359 (2012).
[Crossref] [PubMed]

Opt. Commun. (8)

O. Korotkova and E. Wolf, “Changes in the state of polarization of a random electromagnetic beam on propagation,” Opt. Commun. 246(1-3), 35–43 (2005).
[Crossref]

H. C. Kandpal, J. S. Vaishya, and K. C. Joshi, “Wolf shift and its application in spectroradiometry,” Opt. Commun. 73(3), 169–172 (1989).
[Crossref]

B. Lü and L. Pan, “Propagation of vector Gaussian–Schell-model beams through a paraxial optical ABCD system,” Opt. Commun. 205(1-3), 7–16 (2002).
[Crossref]

F. Gori, G. Guattari, C. Palma, and C. Padovani, “Specular cross-spectral density functions,” Opt. Commun. 68(4), 239–243 (1988).
[Crossref]

P. De Santis, F. Gori, G. Guattari, and C. Palma, “A space-time modulated field with specular coherence function,” Opt. Commun. 64(1), 9–14 (1987).
[Crossref]

O. Korotkova, M. Salem, and E. Wolf, “The far-zone behavior of the degree of polarization of electromagnetic beams propagating through atmospheric turbulence,” Opt. Commun. 233(4-6), 225–230 (2004).
[Crossref]

H. Roychowdhury and O. Korotkova, “Realizability conditions for electromagnetic Gaussian Schell-model sources,” Opt. Commun. 249(4-6), 379–385 (2005).
[Crossref]

Q. He, J. Turunen, and A. T. Friberg, “Propagation and imaging experiments with Gausian Schell-model beams,” Opt. Commun. 67(4), 245–250 (1988).
[Crossref]

Opt. Express (4)

Opt. Lett. (5)

Phys. Lett. A (1)

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

Phys. Rev. E (1)

S. A. Ponomarenko and G. P. Agrawal, “Asymmetric incoherent vector solitons,” Phys. Rev. E 69(3), 036604 (2004).
[Crossref] [PubMed]

Radio Sci. (1)

D. F. V. James and E. Wolf, “Determination of field correlations from spectral measurements with application to synthetic aperture imaging,” Radio Sci. 26(5), 1239–1243 (1991).
[Crossref]

Rep. Prog. Phys. (1)

E. Wolf and D. F. V. James, “Correlation-induced spectral changes,” Rep. Prog. Phys. 59(6), 771–818 (1996).
[Crossref]

Waves Random Media (1)

M. Salem, O. Korotkova, A. Dogariu, and E. Wolf, “Polarization changes in partially coherent electromagnetic beams propagating through turbulent atmosphere,” Waves Random Media 14(4), 513–523 (2004).
[Crossref]

Other (2)

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

E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University, 2007).

Cited By

Optica participates in Crossref's Cited-By Linking service. Citing articles from Optica Publishing Group journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)

Fig. 1
Fig. 1 Wavefront-folding interferometer (a) and retroreflection by a compensated right-angle prism (b). S is the source, BS is a non-polarizing beam splitter, PRx and PRy are right-angle prisms, WP is a wave plate. The set of coordinate axes in (b) is used to describe the state of polarization of a light beam passing through a compensated right-angle prism.
Fig. 2
Fig. 2 Intensity distributions of the modulated EGSM beams with φ = π at z = 1000 m . S G max ( z ) = 2 C ( A x 2 + A y 2 ) / Δ x x represents the maximum value of the normal Gaussian beam. (a) θ = 0 ; (b) θ = π . We choose δ 0 = 1 mm in (c). The other parameters of the incident beam are chosen as follows: λ = 632.8 n m , A x = 2 , A y = 1 , B x y = 0.2 exp ( i π / 3 ) , σ 0 = 1 c m .
Fig. 3
Fig. 3 Changes with ρ / σ P0 of the spectral degree of polarization, the orientation angle, the degree of ellipticity and the spectral density for several selected values of φ and θ in the output plane of the WFI. σ P0 = 1 / 1 / σ 0 2 + 4 / δ m 2 with δ m = max { δ i j } represents the modulation width by θ and φ in the output plane of the interferometer. Rows 1-3 respectively correspond to φ = π / 6 , π / 4 , π / 2 . The values of θ are given beside the curves. The parameters are the same as Fig. 2(c), δ x y = 2 mm . The polarization of the incident EGSM beam is uniform with P = 0.621 , α =3 .797 and ε = 0.113 .
Fig. 4
Fig. 4 Changes with the propagation distance z of on-axis spectral degree of polarization, orientation angle and degree of ellipticity for several selected values of θ when φ = π / 4 . The values of θ are given beside the curves. The parameters are the same as Fig. 2(c), δ x y = 2 mm .
Fig. 5
Fig. 5 Changes with θ / π of on-axis spectral degree of polarization, orientation angle and degree of ellipticity for some selected values of φ in the far-zone. Black solid curve, φ = π / 6 ; red short dashed curve, φ = π / 4 ; blue short dotted curve, φ = π / 2 ; green short dashed-dotted curve, φ = π . The parameters are the same as Fig. 2(c), δ x y = 2 mm .
Fig. 6
Fig. 6 Changes with θ / π of the spectral degree of coherence between two symmetrical points ρ 1 = ( ρ / 2 , 0 ) and ρ 2 = ( ρ / 2 , 0 ) with ρ = 1 m m . (a) z = 0 ; (b) z = 200 m ; (c) z = 1000 m . Black solid curve, φ = 0 ; red short dashed curve, φ = π / 4 ; blue short dotted curve, φ = π / 2 ; green short dashed-dotted curve, φ = π .

Tables (2)

Tables Icon

Table 1 Expressions for the coefficients D i j , E i j , F i j and G i j

Tables Icon

Table 2 Expressions for the coefficients H i j and K i j

Equations (40)

Equations on this page are rendered with MathJax. Learn more.

[ E 1 x ( x , y ) E 1 y ( x , y ) ] = M 1 × [ E 0 x ( x , y ) E 0 y ( x , y ) ] .
[ E 2 x ( x , y ) E 2 y ( x , y ) ] = M 2 × [ E 0 x ( x , y ) E 0 y ( x , y ) ] .
R = [ r x 0 0 r y ] , T = [ t x 0 0 t y ] , P = [ e i φ 0 0 1 ] , O = [ 0 1 1 0 ] ,
M 1 = [ r x t x e i φ 0 0 r y t y ] , M 2 = [ r x t x 0 0 r y t y e i φ ] .
[ E x ( x , y ) E y ( x , y ) ] = [ E 1 x ( x , y ) E 1 y ( x , y ) ] e i θ + [ E 2 x ( x , y ) E 2 y ( x , y ) ] .
E x ( x , y ) = r x t x e i φ e i θ E 0 x ( x , y ) r x t x E 0 x ( x , y ) ,
E y ( x , y ) = r y t y e i θ E 0 y ( x , y ) r y t y e i φ E 0 y ( x , y ) .
W ( r 1 , r 2 ; ω ) = [ W i j ( r 1 , r 2 ; ω ) ] = [ E i ( r 1 ; ω ) E j ( r 2 ; ω ) ] , ( i = x , y ; j = x , y ) .
W i j ( x 1 , y 1 , x 2 , y 2 ) = C i j [ D i j W 0 i j ( x 1 , y 1 , x 2 , y 2 ) + E i j W 0 i j ( x 1 , y 1 , x 2 , y 2 ) + F i j W 0 i j ( x 1 , y 1 , x 2 , y 2 ) + G i j W 0 i j ( x 1 , y 1 , x 2 , y 2 ) ] ,
W i j ( x 1 , y 1 , x 2 , y 2 ) = C i j [ W 0 i j ( x 1 , y 1 , x 2 , y 2 ) + W 0 i j ( x 1 , y 1 , x 2 , y 2 ) W 0 i j ( x 1 , y 1 , x 2 , y 2 ) e i θ W 0 i j ( x 1 , y 1 , x 2 , y 2 ) e i θ ] .
{ W i j ( x 1 , y 1 , x 2 , y 2 ) = W i j ( x 1 , y 1 , x 2 , y 2 ) when θ = 2 l π , W i j ( x 1 , y 1 , x 2 , y 2 ) = W i j ( x 1 , y 1 , x 2 , y 2 ) when θ = ( 2 l +1 ) π .
W i j ( x 1 , y 1 , x 2 , y 2 ) = L i j C i j [ W 0 i j ( x 1 , y 1 , x 2 , y 2 ) + W 0 i j ( x 1 , y 1 , x 2 , y 2 ) + W 0 i j ( x 1 , y 1 , x 2 , y 2 ) e i θ + W 0 i j ( x 1 , y 1 , x 2 , y 2 ) e i θ ] ,
L i j = { 1 when i = j , 1 when i j .
{ W i j ( x 1 , y 1 , x 2 , y 2 ) = W i j ( x 1 , y 1 , x 2 , y 2 ) when θ = 2 l π , W i j ( x 1 , y 1 , x 2 , y 2 ) = W i j ( x 1 , y 1 , x 2 , y 2 ) when θ = ( 2 l +1 ) π .
W i j ( ρ 1 , ρ 2 , z ; ω ) = ( k 2 π z ) 2 d 2 ρ 1 d 2 ρ 2 W i j ( 0 ) ( ρ 1 , ρ 2 ; ω ) × exp { i k 2 z [ ( ρ 1 2 ρ 2 2 ) 2 ( ρ 1 ρ 1 ρ 2 ρ 2 ) + ( ρ 1 2 ρ 2 2 ) ] } .
S ( ρ , z ; ω ) = T r W ( ρ , ρ , z ; ω ) ,
η ( ρ 1 , ρ 2 , z ; ω ) = T r W ( ρ 1 , ρ 2 , z ; ω ) S ( ρ 1 , z ; ω ) S ( ρ 2 , z ; ω ) .
P ( ρ , z ; ω ) = 1 4 Det W ( ρ , ρ , z ; ω ) [ T r W ( ρ , ρ , z ; ω ) ] 2 ,
α ( ρ , z ; ω ) = 1 2 arc tan ( 2 Re [ W x y ( ρ , z ; ω ) ] W x x ( ρ , z ; ω ) W y y ( ρ , z ; ω ) ) , π / 2 α π / 2 .
a 2 ( ρ , z ; ω ) = 1 2 [ ( W x x W y y ) 2 + 4 | W x y | 2 + ( W x x W y y ) 2 + 4 [ Re W x y ] 2 ] ,
b 2 ( ρ , z ; ω ) = 1 2 [ ( W x x W y y ) 2 + 4 | W x y | 2 ( W x x W y y ) 2 + 4 [ Re W x y ] 2 ] .
ε = b ( ρ , z ; ω ) / a ( ρ , z ; ω ) , 0 ε 1,
W 0 i j ( ρ 1 , ρ 2 ; ω ) = A i A j B i j exp ( ρ 1 2 4 σ i 2 ρ 2 2 4 σ j 2 ) exp ( | ρ 2 ρ 1 | 2 2 δ i j 2 ) , ( i = x , y ; j = x , y ) ,
B i j = 1 when i = j ,
| B i j | 1 when i j ,
B i j = B j i ,
δ j i = δ i j .
max { δ x x , δ y y } δ x y min { δ x x | B x y | , δ y y | B x y | } .
W i j ( ρ 1 , ρ 2 ; ω ) = 2 C i j A i A j B i j exp ( x 1 2 + y 1 2 4 σ i 2 x 2 2 + y 2 2 4 σ j 2 ) × { H i j exp [ ( x 2 x 1 ) 2 + ( y 2 y 1 ) 2 2 δ i j 2 ] + K i j exp [ ( x 2 + x 1 ) 2 + ( y 2 + y 1 ) 2 2 δ i j 2 ] } ,
W i j ( ρ 1 , ρ 2 , z ; ω ) = 2 C i j A i A j B i j Δ i j exp { [ i k z ( β i j + i k z ) 1 Δ i j ] ρ ρ } × { H i j exp ( 4 α i j ρ 2 Δ i j ) exp ( γ i j ρ 2 Δ i j ) + K i j exp ( α i j ρ 2 Δ i j ) exp ( 4 γ i j ρ 2 Δ i j ) } ,
ρ = ρ 1 + ρ 2 2 , ρ = ρ 2 ρ 1 ,
Δ i j = ( z k ) 2 [ 16 α i j γ i j ( β i j + i k z ) 2 ] ,
α i j = 1 16 ( 1 σ i 2 + 1 σ j 2 ) , β i j = 1 4 ( 1 σ i 2 1 σ j 2 ) , γ i j = 1 16 ( 1 σ i 2 + 1 σ j 2 ) + 1 2 δ i j 2 .
S ( ρ , z ; ω ) = 2 C Δ x x { ( A x 2 + A y 2 ) exp ( ρ 2 2 σ I 2 ( z ) ) [ A x 2 cos ( φ + θ ) + A y 2 cos ( φ θ ) ] exp ( ρ 2 2 σ A 2 ( z ) ) } ,
P ( ρ , 0 ; ω ) = { A x 2 A y 2 [ A x 2 cos ( φ + θ ) A y 2 cos ( φ θ ) ] exp ( 2 ρ 2 δ 0 2 ) } 2 + 4 A x 2 A y 2 | B x y | 2 [ cos φ cos θ exp ( 2 ρ 2 δ x y 2 ) ] 2 { A x 2 + A y 2 [ A x 2 cos ( φ + θ ) + A y 2 cos ( φ θ ) ] exp ( 2 ρ 2 δ 0 2 ) } 2 .
α ( ρ , 0 ; ω ) = 1 2 arc tan ( 2 A x A y Re [ B x y ] [ cos φ cos θ exp ( 2 ρ 2 δ x y 2 ) ] A x 2 A y 2 [ A x 2 cos ( φ + θ ) A y 2 cos ( φ θ ) ] exp ( 2 ρ 2 δ 0 2 ) ) ,
ε ( ρ , 0 ; ω ) = { { A x 2 A y 2 [ A x 2 cos ( φ + θ ) A y 2 cos ( φ θ ) ] exp ( 2 ρ 2 δ 0 2 ) } 2 + 4 A x 2 A y 2 | B x y | 2 [ cos φ cos θ exp ( 2 ρ 2 δ x y 2 ) ] 2 { A x 2 A y 2 [ A x 2 cos ( φ + θ ) A y 2 cos ( φ θ ) ] exp ( 2 ρ 2 δ 0 2 ) } 2 + 4 A x 2 A y 2 Re 2 [ B x y ] [ cos φ cos θ exp ( 2 ρ 2 δ x y 2 ) ] 2 { A x 2 A y 2 [ A x 2 cos ( φ + θ ) A y 2 cos ( φ θ ) ] exp ( 2 ρ 2 δ 0 2 ) } 2 + 4 A x 2 A y 2 | B x y | 2 [ cos φ cos θ exp ( 2 ρ 2 δ x y 2 ) ] 2 + { A x 2 A y 2 [ A x 2 cos ( φ + θ ) A y 2 cos ( φ θ ) ] exp ( 2 ρ 2 δ 0 2 ) } 2 + 4 A x 2 A y 2 Re 2 [ B x y ] [ cos φ cos θ exp ( 2 ρ 2 δ x y 2 ) ] 2 } 1 / 2 .
P ( 0 , z ; ω ) = { A x 2 A y 2 [ A x 2 cos ( φ + θ ) A y 2 cos ( φ θ ) ] } 2 + 4 A x 2 A y 2 | B x y | 2 χ 2 ( cos φ cos θ ) 2 { A x 2 + A y 2 [ A x 2 cos ( φ + θ ) + A y 2 cos ( φ θ ) ] } 2 ,
α ( 0 , z ; ω ) = 1 2 arc tan ( 2 A x A y χ Re [ B x y ] ( cos φ cos θ ) A x 2 A y 2 [ A x 2 cos ( φ + θ ) A y 2 cos ( φ θ ) ] ) ,
ε ( 0 , z ; ω ) = { { A x 2 A y 2 [ A x 2 cos ( φ + θ ) A y 2 cos ( φ θ ) ] } 2 + 4 A x 2 A y 2 | B x y | 2 χ 2 ( cos φ cos θ ) 2 { A x 2 A y 2 [ A x 2 cos ( φ + θ ) A y 2 cos ( φ θ ) ] } 2 + 4 A x 2 A y 2 Re 2 [ B x y ] χ 2 ( cos φ cos θ ) 2 { A x 2 A y 2 [ A x 2 cos ( φ + θ ) A y 2 cos ( φ θ ) ] } 2 + 4 A x 2 A y 2 | B x y | 2 χ 2 ( cos φ cos θ ) 2 + { A x 2 A y 2 [ A x 2 cos ( φ + θ ) A y 2 cos ( φ θ ) ] } 2 + 4 A x 2 A y 2 Re 2 [ B x y ] χ 2 ( cos φ cos θ ) 2 } 1 / 2 ,

Metrics