Based on the unified theory of coherence and polarization, the statistical properties of a partially coherent beam can be characterized by the cross-spectral density (CSD) [48]. For Schell-model fields, endowed with rectangular symmetry, the cross-spectral density (CSD) of the beam can be expressed as W 0   ( r 1 , r 2 ) = τ ∗ ( r 1 ) τ ( r 2 )   u ( | x 1 − x 2 | )   u ( | y 1 − y 2 | ) , (1) where r ₁ = (x ₁, y ₁), r ₂ = (x ₂, y ₂) are two arbitrary transverse position vectors, τ (r_i) denotes the transmission function of an arbitrary (complex) amplitude filter and u (|x ₁-x ₂|)u (|y ₁-y ₂|) is the spectral degree of coherence. When the beam carries a twist phase, the CSD is defined as W 0 u   ( r 1 , r 2 ) = τ ∗ ( r 1 ) τ ( r 2 )   u ( | x 1 − x 2 | )   u ( | y 1 − y 2 | ) exp ( − i k μ 0 r 1 × r 2 ) ,   (2) where the last exponential term represents the twist phase, with μ ₀ being the twist factor. According to the Refs. [44, 45], the CSD will be bona fide, if and only if the corresponding uniform source, defined as W u   ( r 1 , r 2 ) =   u ( | x 1 − x 2 | )   u ( | y 1 − y 2 | ) exp ( − i k μ 0 r 1 × r 2 ) , (3) is bona fide too. Then, we find that the uniform source satisfies the following formal integral relationship ∫ d 2 ρ W u ( r 1 , ρ ) T u ( ρ , r 2 ) =   ∫ d 2 ρ T u ( r 1 , ρ ) W u ( ρ , r 2 ) ,   (4) for any pair ( r ₁, r ₂), ρ = (ρ _x , ρ _y ) is an arbitrary transverse position vector, and T u = e x p ( − i k μ 0 r 1 × r 2 ) denotes the twist phase. To prove this, on substituting Eq. 3 into Equation 4, the l. h. s of Eq. 4 reads as follows ∫ d 2 ρ W u ( r 1 , ρ ) T u ( ρ , r 2 ) =   ∫ d 2 ρ   u ( | x 1 − ρ x | )   u ( | y 1 − ρ y | ) exp ( − i k μ 0 ( r 1 − r 2 ) × ρ ) ,   (5)

For the r. h. s of Eq. 4 we obtain: ∫ d 2 ρ T u ( r 1 , ρ ) W u ( ρ , r 2 ) = ∫ d 2 ρ   u ( | ρ x − x 2 | )   u ( | ρ y − y 2 | ) exp ( − i k μ 0 ( r 1 − r 2 ) × ρ ) , (6)

Then, taking into account that   exp [ − i k μ 0 ( r 1 − r 2 ) × ρ ] = exp [ − i k μ 0 ( r 1 − r 2 ) × ( ρ − ( r 2 − r 1 ) ) ] , Eq. 6 can be expressed as follows ∫ d 2 ρ T u ( r 1 , ρ ) W u ( ρ , r 2 ) = ∫ d 2 ρ   u ( | ρ x − ( x 2 − x 1 ) − x 1 | )   u ( | ρ y − ( y 2 − y 1 ) − y 1 | )           × exp [ − i k μ 0 ( r 1 − r 2 ) × ( ρ − ( r 2 − r 1 ) ) ] . (7)

On letting ρ = ρ − ( r ₂− r ₁), Eq. 7 can be recast as ∫ d 2 ρ T u ( r 1 , ρ ) W u ( ρ , r 2 ) = ∫ d 2 ρ   u ( | ρ x − x 1 | )   u ( | ρ y − y 1 | ) exp [ − i k μ 0 ( r 1 − r 2 ) × ρ ] . (8)

Thus, the Equation 4 has been proved. Moreover, it has been mentioned in [45], when the Equation 4 is satisfied, the uniform source W _u defined by Eq. 3 and the twist phase T _u share the same coherent modes, defined as Φ j , m ( r ) = u π [ ( j − | m | ) ! ( j + | m | ) ! ] 1 / 2 ( r u ) 2 | m | ⁡ exp ( i 2 m φ ) L j − | m | 2 | m | ( u r 2 ) exp ( − u r 2 2 ) , (9) with u = kμ ₀ , k = 2π/λ is the wavenumber with wavelength λ, and j = 0, 1/2, 1, … , m = -j, j+1, … , j. Here, L j − | m | 2 | m | is the Laguerre polynomials with the radial index j-|m| and the angular index 2|m|, while exp (i2mφ) represents the vortex phase.

If the uniform source W _u is bona fide, the sufficient condition is that the eigenvalue sequence {λ_j,m} should be nonnegative. These eigenvalues are defined as λ j , m = ∬ d 2 r 1 r 2 W u ( r 1 , r 2 ) Φ j , m ( r 1 ) Φ j , m ∗ ( r 2 ) . (10)

On substituting equations 3 and (9) into Equation 10, and on letting r 1 − r 2   =   r , we have λ j , m = ∫ d 2 r u ( x ) u ( y ) ∫ d 2 r 2 ⁡ exp ( − i k μ 0 r × r 2 ) Φ j , m ( r + r 2 ) Φ j , m ∗ ( r 2 ) . (11)

Use the Following Expression [45] ∫ d 2 r 2 ⁡ exp ( − i k μ 0 r × r 2 ) Φ j , m ( r + r 2 ) Φ j , m ∗ ( r 2 ) = ℒ j + m ( u r 2 ) , (12) with ℒ n ( x ) = L n ( x ) exp ( − x / 2 ) , and λ j , m = λ j + m = Λ b . After some operation, we have Λ b = ∑ k 0 = 0 b ∫ u ( x ) L k 0 − 1 / 2 ( u x 2 ) exp ( − u x 2 2 ) d x ∫ u ( y ) L b − k 0 − 1 / 2 ( u y 2 ) exp ( − u y 2 2 ) d y . (13)

Eq. 13 is one of the main results of this paper, which can be used to assess the conditions when the Schell-model beams with rectangular symmetry can carry the twist phase.

In this section, we introduce a new kind of twisted partially coherent beam with nonconventional correlation function, named the twisted Hermite-Gaussian correlated Schell-model (THGCSM) beam. As a natural extension of the Hermite-Gaussian correlated Schell-model (HGCSM) beam [22], the CSD in the source plane (z = 0) is defined as W ( r 1 , r 2 ) = exp ( r 1 2 + r 2 2 4 σ 0 2 ) H 2 m [ ( x 2 − x 1 ) / 2 δ 0 ] H 2 m [ 0 ] exp [ − ( x 2 − x 1 ) 2 2 δ 0 2 ] × H 2 n [ ( y 2 − y 1 ) 2 δ 0 ] H 2 n [ 0 ] exp [ − ( y 2 − y 1 ) 2 2 δ 0 2 ] exp ( − i k μ 0 r 1 × r 2 ) , (14) where r ₁ = (x ₁, y ₁), r ₂ = (x ₂, y ₂) are two arbitrary transverse position vectors in the source plane, σ₀ and δ₀ represent the beam width and the spatial coherence width, respectively. H _2m and H _2n are the Hermite polynomial of order 2m and 2n, respectively. After some algebra, Equation 14 can be expressed in the following alternative form W ( r 1 , r 2 ) = m ! π Γ ( m + 1 / 2 ) n ! π Γ ( n + 1 / 2 ) exp ( r 1 2 + r 2 2 4 σ 0 2 ) L m − 1 / 2 [ − ( x 2 − x 1 ) 2 2 δ 0 2 ] exp [ − ( x 2 − x 1 ) 2 2 δ 0 2 ] × L n − 1 / 2 [ − ( y 2 − y 1 ) 2 2 δ 0 2 ] exp [ − ( y 2 − y 1 ) 2 2 δ 0 2 ] exp ( − i k μ 0 r 1 × r 2 ) . (15)

As mentioned in section 2.1, the THGCSM beam will be bona fide, if and only if the corresponding uniform source, defined as W u 0 ( r 1 , r 2 ) = L m − 1 / 2 [ − ( x 2 − x 1 ) 2 2 δ 0 2 ] exp [ − ( x 2 − x 1 ) 2 2 δ 0 2 ] × L n − 1 / 2 [ − ( y 2 − y 1 ) 2 2 δ 0 2 ] exp [ − ( y 2 − y 1 ) 2 2 δ 0 2 ] exp ( − i k μ 0 r 1 × r 2 ) . (16) is bona fide too. On substituting Equation 15 into Equation 10, and after some tedious integrations and algebraic manipulations, we have Λ b = ∑ k 0 = 0 b 2 δ 0 2 1 + u δ 0 2 Γ ( m + k 0 + 1 / 2 ) Γ ( n + b − k 0 + 1 / 2 ) m ! k ! n ! ( b − k 0 ) ! ( u δ 0 2 1 + u δ 0 2 ) m + n ( 1 − u δ 0 2 1 + u δ 0 2 ) b × 2 F 1 [ − m , − k 0 ; − m − k 0 + 1 2 ; − ( 1 + u δ 0 2 ) ( 1 − u δ 0 2 ) ] × 2 F 1 [ − n , k 0 − b ; k 0 − b − n + 1 2 , − ( 1 + u δ 0 2 ) 1 − u δ 0 2 ] , (17) where ₂ F ₁ is a hypergeometric function. So, in order for W _u to be bona fide (i.e., λ _j,m should be nonnegative), the parameter u = kμ ₀ in Eq. 17 must be bounded by the following inequality: u δ 0 2 ≤ 1 . (18)

Thus, the THGCSM beam will be bona fide, if and only if the Equation 18 is satisfied. Then, the propagation of the THGCSM beam through an ABCD optical system can be investigated with the help of the generalized Collins formula [38, 40]. W ( ρ 1 , ρ 2 ; z ) = 1 ( λ B ) 2 exp [ − i k D 2 B ( ρ 1 2 − ρ 2 2 ) ] ∫ − ∞ ∞ ∫ − ∞ ∞ W ( r 1 , r 2 ) × exp [ − i k A 2 B ( r 1 2 − r 2 2 ) ] exp [ i k B ( r 1 ⋅ ρ 1 − r 2 ⋅ ρ 2 ) ] d 2 r 1 d 2 r 2 , (19) where ρ ₁ = ( ρ _x1, ρ _y1) and ρ ₂ = ( ρ _x2, ρ _y2) are two arbitrary transverse position vectors in the observation plane, and A, B, C, D are the transfer matrix elements of an optical system. On substituting Equation 14 into Equation 19, we obtain the analytical formulae for the CSD of a THGCSM beam in the output plane as follows: W ( ρ 1 , ρ 2 ; z ) = π 2 λ 2 B 2 α 1 α 2 exp [ − β x 2 + β y 2 4 α 2 ] exp [ − i k D 2 B ( ρ 1 2 − ρ 2 2 ) ] exp [ − k 2 4 α 1 B 2 ( ρ 1 − ρ 2 ) 2 ] × ∑ k 1 = 0 m ∑ k 2 = 0 n ∑ p 1 = 0 k 1 ∑ p 2 = 0 k 2 m ! π Γ ( m + 1 / 2 ) n ! π Γ ( n + 1 / 2 ) ( 2 k 1 ) ( 2 k 1 − 2 p 1 ) ! p 1 ! ( 2 k 2 ) ( 2 k 2 − 2 p 2 ) ! p 2 ! ( − 1 ) k 1 + k 2 k 1 ! k 2 ! × ( n − 1 2 n − k 2 ) ( m − 1 2 m − k 1 ) ( 1 2 δ 0 2 ) k 1 + k 2 ( β x 2 α 2 ) 2 k 1 ( β y 2 α 2 ) 2 k 2 ( α 2 β x 2 ) p 1 ( α 2 β y 2 ) p 2 , (20) with α 1 = 1 2 σ 0 2 ; α 2 = 1 8 σ 0 2 + 1 2 δ 0 2 + k 2 μ 0 2 4 α 1 + k 2 A 2 4 α 1 B 2 , (21) β x = i k ( ρ x 1 + ρ x 2 ) 2 B + k 2 μ 0 ( ρ y 1 − ρ y 2 ) 2 α 1 B + k 2 A ( ρ x 1 − ρ x 2 ) 2 α 1 B 2 (22) β y = i k ( ρ y 1 + ρ y 2 ) 2 B − k 2 μ 0 ( ρ x 1 − ρ x 2 ) 2 α 1 B + k 2 A ( ρ y 1 − ρ y 2 ) 2 α 1 B 2 (23)

The spectral density of the THGCSM beam at point ρ in the receiver plane is defined as S ( ρ , z ) = W ( ρ , ρ , z ) . (24)

The spectral degree of coherence (SDOC) of the THGCSM beam at a pair of transverse points with position vectors ρ ₁ and ρ ₂ in the output plane can be expressed by the formula η ( ρ 1 , ρ 2 ; z ) = W ( ρ 1 , ρ 2 ; z ) W ( ρ 1 , ρ 1 ; z ) W ( ρ 2 , ρ 2 ; z ) (25)

Based on the obtained formulae above, we can study the propagation properties of a THGCSM beam in a convenient way.

In this section, we study the paraxial propagation of the THGCSM beam through an ABCD optical system by applying the formulae derived in section 2. In the following examples, we consider the beam propagating in free space after passing through a lens with focal length f = 400mm, which is located at z   =   0 . The parameters of the beam and the transfer matrix are defined as λ = 632.8 nm, σ ₀ = 1 mm, δ ₀ = 0.5 mm, A   =   1 − z / f ,   B   =   z , C   =   − 1 / f , and D   =   1 . According to Equation 24, we calculate the normalized spectral density of a focused THGCSM beam at several propagation distances with different values of the twist factor μ ₀, as shown in Figures 1, 2. One can find that when μ ₀ = 0 m⁻¹ (see the first row), the THGCSM beam reduces to a HGCSM beam, and with the increase of the propagation distance, the intensity distributions of the HGCSM beam gradually change from one beam spot into two spots or four beam spots as expected [22]. From the second and third rows of Figures 1, 2 , it is interesting to find that a twist phase does not seem to cause the beam to rotate during the transmission, no matter what the value of the parameters m n is. This result is quite different from that obtained in former works [37, 38], where it was shown that the twist phase would induce the beam to rotate on propagation. This phenomenon will be explained in Figure 8, by investigating the self-reconstruction characteristics of the beam spectral density. In addition, the twist phase has the effect of hindering the beam spot to split on propagation, and the larger the value of the twist factor is, the beam spot will split more difficult. This means that the twist phase plays a role preventing deterioration of the intensity distribution. Thus, the twist phase can be used to control the intensity distribution of a THGCSM beam on propagation in free space.

Then, the evolution properties the SDOC of a focused THGCSM beam on propagation also have been investigated. Figures 3, 4 show the modulus of the SDOC between two points ρ and -ρ (i.e., |η(ρ,-ρ)|), at several propagation distances with different values of the twist factor μ ₀. The first row of the Figures 3, 4 show the variation of the SDOC of a focused HGCSM beam versus the propagation distances z. It is found that the distribution of the SDOC of the HGCSM beam exhibits an array distribution in the source plane (i.e., z = 0 mm), and the number of the beamlets increase as the values of the beam order m or n increase.

When the propagation distance z increases, the profile of the SDOC firstly remains invariant, and then becomes one beam spot in the focal plane. This means that the information regarding the SDOC is increasingly lacking with increasing propagation distance. The second and the third rows of the Figures 3, 4 show the influence of the twist phase on the evolution properties of the SDOC. We find that the twist phase induces a rotation of the SDOC on propagation, such that when μ ₀ > 0, the distribution of the SDOC rotates anti-clockwise, and when μ ₀ < 0, the distribution of the SDOC rotates clockwise. The SDOC rotates faster with increasing twist factor μ ₀, and the rotation angle varies between −π/2 or π/2 in the focal plane. These phenomena can be explained by the fact that the twist phase imposes angular momentum on the beam. Further, one can still determine the structure of the SDOC even in the focal plane. Thus, the twist phase can be used to maintain the beam’s information of the correlation function.

In order to investigate the influence of the spatial coherence width and the twist phase on the beam propagation properties, the density plots of the normalized spectral density and the modulus of the SDOC have been studied, as shown in Figures 5, 6. In Figure 5, we calculated the density plots of the normalized spectral density and the modulus of the SDOC (|η(ρ,−ρ)|) of a focused HGCSM beam (i.e., THGCSM beam with μ ₀ = 0) for different values of the spatial coherence width δ₀ in the focal plane. One can find that with the increase of the spatial coherence width, the beam profile of the HGCSM beam will evolve from the original array beam shape to a Gaussian beam profile (i.e., the self-splitting properties of the HGCSM beam on propagation gradually disappear) as expected in [22].

Moreover, regardless of the value of the spatial coherence width, the distribution of the SDOC is always maintained as a beam spot. Therefore, the ability to obtain information about the correlation function of the beam in the far field (or focal plane) cannot be improved by changing the coherence length or the order m n of the beam.

Further, we calculated the modulus of the SDOC (|μ(ρ,-ρ)|) of the THGCSM beam in the source and focal plane, respectively. It is interesting to find that with the increase of the twist factor, the strength of the sidelobes of the SDOC are enhanced. This exciting new finding may help us to find a new way to improve the reliability of the correlation function.

In this section, we focus on the Self-reconstruction behavior of the THGCSM beam when the THGCSM beam is partially blocked by a sector-shaped opaque obstacle (SSOO) in the source plane. Figure 7 shows the illustration of a THGCSM beam self-healing process. A beam in the input plane (z = 0) is disturbed by a partially opaque obstacle in the input plane and is propagating through an ABCD optical system consisting of a lens and the free space behind the lens.

Figure 8 shows the changes of the density plot of the normalized spectral density of a focused THGCSM beam obstructed by a SSOO for different values of the twist factor at several propagation distances. From the first row of the Figure 8, one can see that with the increase of the propagation distance z, due to the effect of the correlation function, the beam splits into four beamlets as expected in [22]. The second and third row of the Figure 8 show the effect of the twist phase on the normalized spectral density. By comparing the condition μ ₀ = 0.2 m⁻¹ and the condition μ ₀ = −0.2 m⁻¹, we can find that the twist phase would induce the rotation of the beam on propagation: the distribution of the spectral density rotates clockwise when μ ₀ > 0, the distribution of the spectral density rotates anti-clockwise when μ ₀ < 0. This phenomenon is consistent with former results [37, 38]. Therefore, the twist phase actually still causes the beam to rotate during propagation, but it is not noticeable when the beam is intact (Figures 1, 2).

Moreover, Figure 9 shows the Self-reconstruction characteristics of the normalized spectral density and the modulus of the SDOC (|η ρ,−ρ)|) when the beam is obstructed by a SSOO for different values of the center angle α in the focal plane. To assess the influences of the twist parameter on the self-reconstruction capability quantitatively, a parameter named the degree of self-reconstruction D_P (i.e., D p = [ ∬ 〈 I w t ( ρ ) 〉 〈 I o b ( ρ ) 〉 d 2 ρ ] 2 ∬ 〈 I w t ( ρ ) 2 〉 d 2 ρ ∬ 〈 I o b ( ρ ) 2 〉 d 2 ρ , with I _wt and I _ob stand for the beam intensities without and with obstruction, respectively) is used to characterize it [41]. It is interesting to find that even if the beam has been obstructed by a SSOO, one still can determine the information of the correlation function of an obstructed THGCSM from its SDOC distribution in the focal plane. In addition, with the increase of the center angle, the self-reconstruction capability also decreased (see the evolution of the quantity D_P in the left upper corner of the figure). Our results can find application in information transmission and recovery.