The subtraction of mutually displaced Gaussian Schell-Model beams J. Carlos G. de Sande1, Massimo Santarsiero2, Gemma Piquero3, and Franco Gori2 1 Departamento de Teoŕıa de la Señal y Comunicaciones, Universidad Politécnica de Madrid, Campos Sur, 28031 Madrid, Spain 2 Dipartimento di Ingegneria, Università Roma Tre, via V. Volterra, 62; 00146 Rome, Italy 3 Departamento de Óptica, Universidad Complutense de Madrid, 28040 Madrid, Spain September 2, 2015 Abstract Using recently derived results about the difference of two cross-spectral densities, we consider a source whose correlation function is the difference of two mutually displaced Gaussian Schell- model cross-spectral densities. We examine the main features of this new cross-spectral density in terms of coherence and intensity distribution, both across the source plane and after free propagation. 1 Introduction In classical optics, coherence theory [1–3] is a subject that continuously receives innovative contributions from countless authors, both within the scalar [4–11] and vectorial realm [12–21], including nonstationary states [22–26]. A basic tool in this theory is given by the cross-spectral density (CSD) [1, 2]. Given the variety of physical situations one can encounter, different forms of the CSD are to be envisaged. From the mathematical viewpoint though, the CSD has to be a non-negative definite kernel [27], and this implies that its analytical structure cannot be chosen at will. Generally speaking, determining whether a kernel is endowed with the non-negativeness property is a difficult task, so that the risk of unphysical assumptions is to be tamed. Sufficient criteria to ensure correctness of alleged CSDs are quite useful. One such suf- ficient criterion was given not long ago [28, 29], and proved its usefulness in several in- stances [30–32]. In the course of time, efforts were produced to understand operations suitable to lead to new CSD forms. In some recent papers [33, 34] the possibility of constructing a new, genuine CSD by subtraction of two correlation functions was demon- strated. The subtraction of two coaxial Gaussian Schell-model (GSM) CSDs was studied in the case in which the shape-invariance condition is met [35]. As well known [2, 36] 1 this means that the effects of propagation are limited to a scale expansion of the CSD accompanied by the onset of a curvature phase term. Accordingly, studying the coherence properties across a certain plane, e.g. the waist plane, is sufficient to enlighten the beam features through the whole space. Here, we want to study a more general case of GSM CSDs subtraction. First, the two CSDs have different expansion coefficients, second, they are mutually displaced at any typical transverse plane except for planes in the far zone where a mutual displacement is replaced by different average inclination of the beam axes. In Section 2 the problem to be solved will be stated and, in Section 3, conditions for obtaining genuine CSD’s from subtraction of two mutually displaced GSM CSDs will be obtained and an experimental way to obtain the described source will be proposed. The CSD properties at the source plane will be described in Section 4 while the properties of the CSD after propagation will be studied in Section 5. Finally, concluding remarks will be presented. 2 Statement of the problem Let us start with a few words about the basic quantities we are going to use. The CSD at two points ρ1 and ρ2 of a given plane z = const will be denoted by Wz(ρ1,ρ2), omitting the explicit dependence from the temporal frequency. When the two points coincide, the CSD reduces to the optical intensity, i.e., Iz(ρ) = Wz(ρ,ρ). Furthermore, the spectral degree of coherence, is defined as [1] µz(ρ1,ρ2) = Wz(ρ1,ρ2)√ Iz(ρ1)Iz(ρ2) . (1) Symbols like W0, I0, and µ0 will denote (with a slight notation abuse) the quantities Wz, Iz, and µz when the choice z = 0 is done. We shall study partially coherent fields in whose correlation function two GSM CSDs are subtracted from one another and determine conditions under which such subtraction operation gives rise to a genuine new CSD. Let us then assume the CSD across the source plane to be W0(ρ1,ρ2) = W01(ρ1,ρ2)−W02(ρ1,ρ2), (2) where [36] W0j(ρ1,ρ2) = B0j exp [ −(ρ1 − vj)2 + (ρ2 − vj)2 4σ2 0j − (ρ1 − ρ2) 2 2δ20j ] , (j = 1, 2), (3) with positive constants B0j. The positive quantities σ2 0j and δ20j denote intensity and coherence variances respectively. It is seen that the two contributions to W0 are centered at possibly different positions specified by the vectors vj, (j = 1, 2). Nonetheless, since 2 we can always move the coordinate origin we shall let v1 = 0 and write W0(ρ1,ρ2) = B01 exp [ −ρ 2 1 + ρ22 4σ2 01 − (ρ1 − ρ2) 2 2δ201 ] −B02 exp [ −(ρ1 − v2)2 + (ρ2 − v2)2 4σ2 02 − (ρ1 − ρ2) 2 2δ202 ] . (4) The overall associated intensity is I0(ρ) = B01 exp ( − ρ2 2σ2 01 ) −B02 exp [ −(ρ− v2)2 2σ2 02 ] . (5) An alternative and useful way of writing Eq. (4) is W0(ρ1,ρ2) = B01 exp [−P01(ρ1 + ρ2) 2 −M01(ρ1 − ρ2) 2] −B02 exp [−P02(ρ1 + ρ2 − 2v2) 2 −M02(ρ1 − ρ2) 2] , (6) where P0j = 1 8σ2 0j ; M0j = 1 8σ2 0j + 1 2δ20j , (j = 1, 2). (7) Generally speaking, Eq. (4) does not represent a genuine CSD. A naive idea would be that of requiring that the intensity predicted by Eq. (5) be nowhere negative. Unfortu- nately, this condition alone does not guarantee that the CSD described by Eq. (4) is a genuine CSD. As a matter of fact, even if the optical intensity is non-negative through- out the whole space cases exist in which the modulus of the spectral degree of coherence can exceed one for certain pairs of points [34]. In the next Section, conditions on the parameters ensuring the CSD to be genuine will be derived. 3 Conditions for obtaining a genuine CSD In [28] it has been shown that the validity of the expression of a CSD is warranted whenever the latter can be expressed through a superposition integral of the form W0(ρ1,ρ2) = ∫ p(v)H∗0 (ρ1,v)H0(ρ2,v)d2v , (8) where p is a non-negative function that will be referred to as the weight function, and H0 is a suitable kernel. It has been also shown [29] that any valid CSD can be expressed through a representation of the form (8). We shall then use this representation as a guide for our analysis. We shall begin by choosing p of the form p(v) = p1(v)− p2(v), (9) 3 where p1 and p2 are two Gaussian functions, centered at the points 0 and v2, respectively, i.e., p(v) = A1 exp(−γ1v2)− A2 exp[−γ2(v − v2)2], (10) with positive constants A1, A2, and γ1, γ2. The kernel H0 that we will use is itself a Gaussian function H0(ρ,v) = exp[−α(ρ− v)2], (11) where α is a positive constant. We arrange the reference frame in such a way that v2 lies along the x-axis. Denoting by u and w the components of a typical v vector, the function p then becomes p(u,w) = A1 exp{−γ1(u2 + w2)} − A2 exp{−γ2[(u− u2)2 + w2]}. (12) We now derive the conditions to be satisfied by Aj and γj, (j = 1, 2) in order to have p(v) ≥ 0 for any v. First, in order to ensure a non-negative p when |v| tends to infinity the following condition has to be satisfied γ2 > γ1. (13) Let us now discuss the behavior of p(v) for finite values of v. In qualitative terms, the presence of the second Gaussian term digs a hole in the surface p(u,w). An example is given in Fig. 1, where a density plot of Eq. (12) is shown for A1 = 1, γ1 = 0.5, A2 = 0.95, γ2 = 0.75 and u2 = 0.1. All the parameters are in arbitrary units throughout the whole text and figures. The bottom of the hole digged in the surface p(u,w) is located at a minimum for p(u,w). At that point, two conditions have to be met. First, both partial derivatives of p(u,w) have to vanish. Second, the Hessian must be positive. We can dispose of the second condition, being evident by direct inspection that the point (at a finite distance from the origin) where both first partial derivatives vanish corresponds to a minimum. The vanishing of the partial derivatives gives the two equations ∂p(u,w) ∂u = −2A1γ1ue −γ1(u2+w2) + 2A2γ2(u− u2)e−γ2[(u−u2) 2+w2] = 0, (14) ∂p(u,w) ∂w = −2A1γ1we −γ1[u2+w2] + 2A2γ2we −γ2[(u−u2)2+w2] = 0. (15) In addition, since p(v) is to be non-negative everywhere, its minimum is required to be non-negative as well. Equation (15) is satisfied along the whole u-axis (line w = 0). Letting w = 0 we derive from Eq. (14) A1γ1ue −γ1u2 = A2γ2(u− u2)e−γ2[(u−u2) 2]. (16) Let us refer to the limiting case in which the minimum of p is a zero. Then, Eq. (12) along the line w = 0 gives A1 exp ( −γ1u2 ) = A2 exp [ −γ2(u− u2)2 ] , (17) 4 Figure 1: Weight function associated with Eq. (12), with A1 = 1, γ1 = 0.5, A2 = 0.95, γ2 = 0.75 and u2 = 0.1. which, when inserted into Eq. (14) (at w = 0), leads to γ1u = γ2(u− u2), (18) or u = γ2u2 γ2 − γ1 . (19) When this value of v is inserted into Eq. (17) we obtain A1 = A2 exp ( γ1γ2 γ2 − γ1 u22 ) . (20) Obviously, if A1 is greater than the right-hand side of this equation, the minimum of p will be positive. In conclusion, the following conditions guarantee that Eqs. (8), (10), and (11) furnish a genuine CSD A1 ≥ A2 exp ( γ1γ2 γ2 − γ1 u22 ) ; γ2 > γ1. (21) We note that, when v2 = 0, such conditions reduce to those already obtained in [34], where the case of the subtraction of two centered GSM CSDs was studied. Behaviors of the weight function p(u,w = 0) can be observed in Fig. 2 for several values of the displacement u2 and the maximum allowed value of A2 (a), and for u2 = 0.1 and several values of A2 (b). 5 Figure 2: Weight function associated with Eq. (12), with A1 = 1, γ1 = 0.5, γ2 = 0.75. (a) as a function of u with w = 0, A2 = A (max) 2 = A1 exp ( − γ1γ2 γ2−γ1u 2 2 ) and several values of u2; (b) as a function of u with w = 0, u2 = 0.1 and several values of A2. Before concluding this section, we want to suggest an experimental procedure aimed at producing partially coherent sources of the type discussed up to now. Production of several partially coherent fields with prescribed CSD have been obtained in recent times [10] using sophisticated methods of spatial light modulation. In our case, as we shall see, a simple optical system can be employed. The basic idea comes directly from the integral representation given in Eq. (8). Imagine a partially coherent field described by a certain CSD Win(v,v′) is sent to the input plane of a linear system characterized by a coherent impulse response H0(ρ,v). The CSD across the output plane of the system will be [2] W0(ρ1,ρ2) = ∫∫ Win(v,v′)H∗0 (ρ1,v)H0(ρ2,v ′)d2v d2v′, (22) both integrals being extended to the source plane. Now, we introduce the hypothesis that the input field is spatially incoherent, so that its CSD is specified by the delta-like structure Win(v,v′) = p(v)δ(v−v′) where p(v) is the intensity distribution. On inserting this condition into Eq. (22) we obtain the same relation as Eq. (8). Therefore, we can say that the CSD described by Eq. (8) is displayed at the output plane of an optical system fed by an intensity distribution of the form given in Eq. (10) and endowed with the coherent impulse response in Eq. (11). The optical device can simply be the 4f telescopic system of Fig. 3 [37]. The incoherent intensity distribution (10) (produced through uniform illumination of a suitable filter) is put across the input plane (left). In the pupil plane (common focal plane of the two 6 v r ρ f f f f Gaussian amplitude filter Incoherent source Image Figure 3: Schematic of a 4f telescopic system. lenses) a Gaussian filter with transmission function exp[−π2r2/(λ2f 2α)] is put to produce the impulse response (11). The synthesized source appears at the exit plane (right) as the image of the system. 4 Properties of the CSD across the source plane According to Eqs. (8) and (10), the CSD at the source plane (z = 0) is given by W0(ρ1,ρ2) = A1 ∫ exp(−γ1v2) exp[−α(ρ1 − v)2] exp[−α(ρ2 − v)2]d2v −A2 ∫ exp[−γ2(v − v2)2] exp[−α(ρ1 − v)2] exp[−α(ρ2 − v)2]d2v (23) where the constants satisfy conditions (21). By straightforward evaluation of the integrals we obtain W0(ρ1,ρ2) = πA1 γ1 + 2α exp [ −αγ1 + α2 γ1 + 2α (ρ21 + ρ22) + 2α2 γ1 + 2α ρ1 · ρ2 ] − πA2 γ2 + 2α exp [ −αγ2 + α2 γ2 + 2α [ (ρ1 − v2)2 + (ρ2 − v2)2 ] + 2α2 γ2 + 2α (ρ1 − v2) · (ρ2 − v2) ] . (24) Therefore, each component of the overall CSD is of the GSM form. On using the following relations: B0j = A0j π γj + 2α ; 1 4σ2 0j = αγj γj + 2α ; 1 2δ20j = α2 γj + 2α , (j = 1, 2), (25) 7 Eq. (24) is easily transformed into Eq. (4). Furthermore, using the the definitions in Eq. (7), it turns out that P0j = αγj 2γj + 4α , (j = 1, 2); M01 = M02 = M = α/2 . (26) Note that the choice of a kernel H0 in the form given by Eq. (11) gives rise to the class of sources described in Theorem 2 of Ref. [34], (i.e., the difference of two GSM sources with the same values of the parameter M) with the only difference that now we are considering the possibility of a lateral displacement of the two sources. This implies that the inequalities in Eq. (21) represent necessary and sufficient conditions in order for Eq. (4) to express a genuine CSD. Using the relations in Eq. (25) and in Eq. (26), inequalities (21) can be transformed into B01 ≥ B02 M − P01 M − P02 exp ( 4P01P02 P02 − P01 u22 ) ; P02 > P01. (27) or, in terms of intensity and coherence variances, as B01 ≥ B02 ( δ02 δ01 )2 exp [ u22 2 (σ2 01 − σ2 02) ] ; δ02 > δ01, (28) where, taking Eq. (7) into account, σ01 > σ02. It is interesting to enquire about the spectral degree of coherence. In the case of shape-invariant subtraction of coaxial GSM CSDs [35] it was found that, for any choice of ρ1, there are points ρ2 such that the degree of coherence between ρ1 and ρ2 vanishes. In particular, there is a circle over which we can move ρ2 always obtaining a zero degree of coherence with respect to point ρ1. Such a circle is not centered at ρ1 except in the case ρ1 = 0. We wonder whether, for a typical value of ρ1, positions ρ2 can be found such that µ(ρ1,ρ2) = 0. In other words, using Eq. (6), we wonder whether we can solve the equation B01 exp [−P01(ρ1 + ρ2) 2 −M01(ρ1 − ρ2) 2] = B02 exp [−P02(ρ1 + ρ2 − 2v2) 2 −M02(ρ1 − ρ2) 2] , (29) or, dividing side by side and passing to the logarithms, ln β = P01(ρ1 + ρ2) 2 − P02(ρ1 + ρ2 − 2v2) 2, (30) where we let B01/B02 = β, (31) and the equality M01 = M02 has been taken into account. We take, as before, w2 = 0 and u2 6= 0. By simple manipulation Eq. (30) can be changed into( ρ1 + ρ2 − 2P02 P02 − P01 v2 )2 = 4P01P02 (P02 − P01)2 u22 − ln β P02 − P01 . (32) 8 This equation requires its left-hand side to be non-negative, and this happens whenever u22 > (P02 − P01) ln β 4P01P02 . (33) The above inequality can be expressed as B01 < B02 exp ( 4P01P02 P02 − P01 u22 ) , (34) which is clearly incompatible with the inequalities in Eq. (27). The conclusion is that no point pairs exist for which the degree of correlation is zero. 5 Propagation Starting from the expression (6) for the CSD across the plane z = 0 we can write the CSD at a plane z = const 6= 0 upon paraxial propagation [36] as Wz(ρ1,ρ2) = Bz1 exp [ ik 2Rz1 (ρ22 − ρ21)− Pz1(ρ1 + ρ2) 2 −Mz1(ρ1 − ρ2) 2 ] −Bz2 exp { ik 2Rz2 [ ρ22 − ρ21 − 2 (ρ2 − ρ1) · v2 ] −Pz2(ρ1 + ρ2 − 2v2) 2 −Mz2(ρ1 − ρ2) 2 } , (35) where the parameters in the right-hand side vary according to the formulas Bzj = B0j F 2 j (z) ; Pzj = P0j F 2 j (z) ; Mzj = M0j F 2 j (z) ; Rzj = z [ 1 + ( πs20j λz )2 ] ; (j = 1, 2), (36) s0j being the spot-size of the underlying modes [38], whose square is given by s20j = 1√ 4P0jM0j , (j = 1, 2) , (37) and is related to the Rayleigh range of each of the two GSM beams, namely [39] zRj = πs20j λ , (j = 1, 2). (38) The functions Fj(z) are given by Fj(z) = √ 1 + ( λz πs20j )2 = √ 1 + ( z zRj )2 , (j = 1, 2) , (39) 9 Figure 4: Intensity across transverse planes at z = 0, z = 0.5zR1, z = zR1, and z = 2zR1 from left to right and from top to bottom. Furthermore, σ01 = 1, σ02 = 0.85, δ02 = 0.5, u2 = 0.1, B01 = 1, and B02 = B (max) 02 . and play the role of expansion factors. Taking into account the relations between the P , M parameters and the σ, δ quantities, it is seen in fact that both the intensity and the coherence spot-sizes increase upon propagation, being multiplied by Fj(z). The mutual displacement of the two component CSDs, represented by the vector v2 though remains unchanged, and becomes less and less significant when propagation proceeds. A few examples are given in Figs. 4, 5, and 6, where plots of the intensity profile and of the absolute value of the degree of coherence are shown. The source parameters are σ01 = 1, σ02 = 0.85, δ02 = 0.5, u2 = 0.1, B01 = 1, (note that δ01 ' 0.4941 is determined by Eq. (26) and the definition of M0j), while B02 takes the maximum value allowed by Eq. (28), i.e., B (max) 02 = ( δ01 δ02 )2 exp [ −u22 2 (σ2 01 − σ2 02) ] . (40) It is worth considering explicitly the behavior in the far zone. There, the CSD is known to be proportional (up to scale and amplitude factors) to the Fourier transform of W0 evaluated at (ν1,−ν2), where ν1 and ν2 are the conjugate variables of ρ1 and ρ2, respectively. Up to this convention we write W (∞)(ν1,ν2) = W̃0(ν1,−ν2) = ∫∫ W0(ρ1,ρ2)e 2πi(ρ1·ν1−ρ2·ν2)d2ρ1d 2ρ2, (41) 10 Figure 5: Absolute value of the degree of coherence µ(ρ1,ρ2 = 0), in logarithmic scale, across transverse planes at z = 0, z = 0.5zR1, z = zR1, and z = 2zR1 from left to right and from top to bottom. Furthermore, σ01 = 1, σ02 = 0.85, δ02 = 0.5, u2 = 0.1, B01 = 1, and B02 = B (max) 02 . 11 Figure 6: Intensity (a) and modulus of the degree of coherence (b) along the x and y direction at different propagation distances. Other parameters are σ01 = 1, σ02 = 0.85, δ02 = 0.5, u2 = 0.1, B01 = 1, and B02 = B (max) 02 . On replacing W0 in the integral by the expression given by Eq. (6), it turns out that W (∞)(ν1,ν2) = πB01 MP01 exp [ − π2 2M (ν21 + ν22) ] exp [ −π2 (M − P01) 4MP01 (ν1 − ν2) 2 ] − πB02 MP02 exp [ − π2 2M (ν21 + ν22) ] exp [ −π 2 (M − P02) 4MP02 (ν1 − ν2) 2 ] exp[i(ν1 − ν2) · v2], (42) which can be written as W (∞)(ν1,ν2) = √ I(∞)(ν1) √ I(∞)(ν2) µ (∞)(ν1 − ν2) , (43) with I(∞)(ν) = π M ( B01 P01 − B02 P02 ) exp ( − πν 2 M ) , (44) and µ(∞)(ν) = B01P02 exp ( −π 2ν2 4P01 ) −B02P01 exp ( −π 2ν2 4P02 + iν · v2 ) B01P02 −B02P01 × exp ( π2ν2 4M ) . (45) 12 Figure 7: Absolute value of the degree of coherence (a) along x1 direction for several x2 values and (b) along y1 direction for several y2 values at different propagation distances: z = 0, z = zR1, and z = 4zR1 from left to right. Other parameters are σ01 = 1, σ02 = 0.85, δ02 = 0.5, u2 = 0.1, B01 = 1, and B02 = B (max) 02 . Equation (43) shows that the CSD of the beam in the far field is of the Schell-model type, even though this does not happen at the source plane and for finite values of z. The intensity profile is Gaussianly shaped [Eq. (44)], while the degree of coherence, although not Gaussian [Eq. (45)], depends only on the difference of the coordinates of the two points. Such feature of the degree of coherence is made evident in Fig. 7, where the modulus of µ is plotted for points along the x and y axes as a function of the variable (x1 − x2) and (y1− y2), respectively, for several values of the propagation distance and for different values of x2 or y2. On increasing the propagation distance, less and less differences can be found among the shifted versions of the degree of coherence and, when z is much greater than the Rayleigh distance, µ turns out to be practically shift invariant. Figure 8 shows the absolute value of the far-field degree of coherence given in Eq. (45). It can be noted that it is nearly circularly symmetric, with higher values at the center followed by a ring of relatively low coherence sorrounded by another ring of higher degree of coherence. Except for a scale factor and differences along the low degree of coherence ring, this figure is quite similar to that in the lower right part of Fig. 5, as it could be expected. Let us analyze the possibility of finding zero values of the far field degree of coherence for finite values of ν. The first exponential in the right-hand side of Eq. (45) is always real valued, while in the second term a complex factor appears. Then, the degree of coherence only can be zero if this complex factor becomes real, that is, if the argument 13 Figure 8: Absolute value of the degree of coherence in the far field, in logarithmic scale. η and ξ are the Cartesian coordinates of ν. Source parameters are σ01 = 1, σ02 = 0.85, δ02 = 0.5, u2 = 0.1, B01 = 1, and B02 = B (max) 02 . 14 of the complex exponential is of the form ν · v2 = 2nπ, with integer n. This surely happens if v2 = 0, i.e., if the two subtracted sources are not mutually displaced. In such a case, it is easily seen that µ(∞)(ν) = 0 if B01P02 exp ( −π 2ν2 4P01 ) = B02P01 exp ( −π 2ν2 4P02 ) , (46) which gives rise to ν2 = 4 π2 P01P02 P02 − P01 ln ( B01 B02 P02 P01 ) . (47) This represents a circumference centered at the origin and with a radius defined by the source parameters. Note that conditions of Eq. (27) guarantee the positiveness of the right-hand side of Eq. (47). Conversely, if v2 6= 0 the function µ(∞)(ν) vanishes only in a discrete set of points of the ν-plane. In fact, in addition to Eq. (47), the condition ν · v2 = 2nπ (with integer n) has to be met. The latter condition identifies a set of straight lines orthogonal to the vector v2, evenly spaced with period 2π/v2, whose intersections with the circumference correspond to the points where µ(∞)(ν) = 0. Two of such points are always located along the line passing through the circumference center (n = 0) as can be observed in the bottom right part of Fig. 7. In Figure 8, it can be appreciated that the two intersection points between the η axis and the ring of relatively low degree of coherence are darker than its surroundings. Additional intersection points are present if the radius of the circumference or the shift parameter v2 are large enough. 6 Conclusions In this work a set of conditions under which the function obtained by subtracting two GSM CSD (arbitrarily displaced across the source plane) is itself a valid CSD has been found and discussed. The properties of the resulting CSD have then been studied both at the source plane and in propagation. It has been shown that the obtained CSD is not of the SM type, because the degree of coherence is not shift invariant, and the intensity profile is not Gaussian. This happens at any finite propagation distance. However, on approaching the far-field conditions, the degree of coherence tends to become shift invariant (although not Gaussian) and the intensity profile tends to become Gaussian. Finally, an experimental procedure to obtain this kind of CSDs has been proposed, making use of a 4f telescopic optical system and amplitude filters. 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