1. Introduction

Nowadays, the study of the non-paraxial features of the scalar and electromagnetic fields is of interest for beam characterization, micro-diffraction and light propagation through non-linear media, for instance. In this context, there are recent reports on the non-paraxial propagation of anomalous and dark hollow beams [1, 2], Lorentz and Lorentz-Gauss beams [3], Hermite-Gaussian and Airy beams [4–7]. The term micro-diffraction refers to the light behaviour in arrangements where the sizes of the apertures and the propagation distances are comparable with the wavelength. The pioneer work on this subject is due to Bethe, who analyzed the diffraction of electromagnetic radiation by a very small circular hole on a conducting screen and applied it to the description of coupled cavities [8]. His results were refined for propagation in the vicinity of the diffracting aperture in order to describe the modus operandi of the near-field scanning optical microscope [9]. The diffraction of electromagnetic plane waves through small rectangular apertures was also analysed [10], as well as the spectral shifts and spectral switches of diffracted non-paraxial beams [11], and the modulation instability of non-paraxial beams in self-focussing Kerr media [12]. Relative few contributions regarded the spatial coherence state of the non-paraxial field [1, 2, 11]. They attempted to solve the Wolf’s integral equation [13] by applying a linear approximation to the argument of the integral kernel (or propagator), which restricts the non-paraxial calculations to relative small numerical apertures.

Such limitations were recently overcoming by the exact calculation of the Wolf’s equation in the framework of the phase-space representation provided by the non-paraxial marginal power spectrum [14]. This strategy takes into account both the non-linear argument of the integral kernel and the inclination factor, and gives accurate results under conditions by which most the conventional procedures are not longer valid, for instance by high numerical apertures and aperture sizes and propagation distances comparable with the wavelength. It is stressed by comparing this phase-space representation with those based on Wigner distribution functions, WDF (it is worth noting that the non-paraxial marginal power spectrum is not a WDF although it becomes a WDF under paraxial approach in the far field propagation [14]). WDFs are widely used because they provide well-defined descriptions of the paraxial behaviour of scalar and electromagnetic fields on account of the linear argument of their integral kernel [15–18]. It allows calculating them by applying Fourier analysis tools. In spite of their approximation, WDFs are also proposed for non-paraxial fields [19]. However, their validity is strongly restricted when the non-linearity of the kernel argument and the effect of the inclination factor must be taken into account.

The phase-space strategy for the exact calculation of the Wolf’s equation significantly reduces the processing time and the calculation requirements too. However, such features can even be optimized by expanding the non-paraxial marginal power spectrum onto a finite and standard basis of non-paraxial modes, whose coefficients are determined by the power distribution and the spatial coherence state of the field at the aperture plane (AP). The definition of such modal expansion, the analysis of its mathematical features and its physical implications constitute the subject of the current work. Thus, it essentially is a theoretical development illustrated by a simple but representative one-dimensional diffraction arrangement with physical parameters, in order to encourage other researchers to apply it in more sophisticated experiments.

In order to specify the modes and to use them efficiently, the following features should be taken into account:

The non-paraxial and finite modal expansion is a more detailed, exhaustive and powerful tool than the direct estimation of the Wolf’s equation by sampling the continuum second-order wave-fronts. It is potentially useful in the design of very compact optical devices and arrangements (i.e. at the micro- and nano-scales) as well as in partially coherent imaging. An interesting challenge is to implement this modal expansion in multimode interferometric techniques for manufacturing partially coherent sources as superluminiscent LEDs for instance [23, 24].

2. Non-paraxial propagation modes in the phase-space

The non-paraxial marginal power spectrum for an optical field of frequency $\nu ,$ wave-number $k=2\pi /\lambda$ and wavelength $\lambda ,$ in any state of spatial coherence, has the following form, deduced in detail in [14]:

2.1. Definition of the modes

Let us regard the correlated pair of radiant point sources with ${\xi }_D=b$ that belongs to the structured spatial coherent support centred at ${\xi }_A=a$ on the AP, and introduce the function

The structured support of spatial coherence centred at ${\xi }_A=a$ contains one and only one correlated radiator pair with separation vector ${\xi }_D=b,$ whose contribution given by Eq. (4) cannot be obtained as a combination of the contributions of the reminder pairs belonging to the same structured support. As a consequence, the function $2\mathrm{Re}\left[F\left(a+b/2,a-b/2,r_A;\nu \right)\right]$ defines the non-paraxial propagation mode of the contribution of the regarded radiator pair, determined by Eq. (4), to the marginal power spectrum emitted at ${\xi }_A=a.$ The coefficient $\sqrt{S_0\left(a+b/2\right)}\left|t\left(a+b/2\right)\right|\sqrt{S_0\left(a-b/2\right)}\left|t\left(a-b/2\right)\right|\left|\mu \left(a+b/2,a-b/2\right)\right|$ specifies the weight of this mode in such marginal power spectrum. The phases $\Delta \varphi \left(a+b/2,a-b/2\right)$ and $\alpha \left(a+b/2,a-b/2\right)$ only redefine the coordinate origin of the cosine factor in Eq. (5) at the OP.

Accordingly, each correlated radiator pair belonging to the regarded structured support emits in one and only one specific mode, characterized by the magnitude of the separation vector $\left|{\xi }_D\right|=\left|b\right|.$ The propagation modes associated to radiator pairs whose separation vectors have the same magnitude but different orientation, have the same shape but their fringe structures are orthogonal to the corresponding separation vector. Furthermore, the set of non-paraxial propagation modes should be discrete and finite because the set of radiant point sources must be discrete in order to insert pure virtual point sources at the midpoint between consecutive pairs of them [20]. It allows labelling the modes by entire order-suffixes $n=0,1,2,\cdots$ corresponding to the lengths of the separation vectors, i.e. the longer the separation vector the higher the mode order.

2.2. Non-paraxial modes for the radiant and the virtual point sources

The zeroth-order propagation mode is labelled by $n=0$ and is contributed by the pairs with $\left|{\xi }_D\right|=0$, i.e. the individual point sources of the radiant layer. Taking into account that $\Delta \varphi \left(a,a\right)=0,\left|\mu \left(a,a\right)\right|=1\text{and}\alpha \left(a,a\right)=0$ hold [13], and stating $b=0$, Eq. (5) yields

The non-paraxial modes of order $n>0$ describe the propagation of the modulating energies contributed by the correlated pairs of radiant point sources with $\left|{\xi }_D\right|>0,$ i.e. their superposition determine the propagation of the emissions provided by the point sources of the virtual layer. For instance, the modulating energy contributed by the correlated pair of radiant point sources with separation vector $b\ne 0,$ belonging to the structured spatial coherence support centred at ${\xi }_A=a,$ is propagated by the non-paraxial mode $2\mathrm{Re}\left[F\left(a+b/2,a-b/2,r_A;\nu \right)\right]$ emitted by the virtual point source placed at ${\xi }_A=a.$ According to Eq. (5) the non-paraxial propagation modes of order $n>0$ have an oscillatory structure with chirped frequency, conferred by the cosine factor with non-linear argument [14]. This structure changes along the propagation, so that the modes are not shape-invariant along the propagation, and in general they do not obey the $1/z^2$-law.

2.3. Modal expansion of the non-paraxial marginal power spectrum

It characterizes the propagation of the energy emitted by the radiant (if any) and the virtual point sources associated to any individual and specific structured spatial coherence support at the AP. After introducing the dimensionless function $1\equiv \delta \left({\xi }_D\right)+\left[1-\delta \left({\xi }_D\right)\right]$ in the integral of Eq. (1), with $\delta \left({\xi }_D\right)$ the dimensionless Dirac’s delta, and taking into account that the radiant point sources that constitute a correlated pair are undistinguishable to each other, the marginal power spectrum referred to the position ${\xi }_A=a$ on the AP can be expressed in terms of the non-paraxial propagation modes as follows

Equation (7) has the form $S\left(a,r_A;\nu \right)=S_{rad}\left(a,r_A;\nu \right)+S_{virt}\left(a,r_A;\nu \right)$ [14, 22], whose radiant component $S_{rad}\left(a,r_A;\nu \right)=S_0\left(a\right){\left|t\left(a\right)\right|}^{2}F\left(a,r_A;\nu \right)$ is emitted by the radiant point source placed at ${\xi }_A=a$, while its virtual component $S_{virt}\left(a,r_A;\nu \right)$ is emitted by the virtual point source placed at the same position. This last component has the form of a modal expansion on modes of order $n>0$. It is worth noting that (i) $S\left(a,r_A;\nu \right)=S_{rad}\left(a,r_A;\nu \right)$ if a pure radiant point source is placed at ${\xi }_A=a$, (ii) $S\left(a,r_A;\nu \right)=S_{virt}\left(a,r_A;\nu \right)$ if a pure virtual point source is placed at this position, and (iii) $S\left(a,r_A;\nu \right)=S_{rad}\left(a,r_A;\nu \right)+S_{virt}\left(a,r_A;\nu \right)$ if a dual point source is placed there. Furthermore, the individual access to the structured spatial coherence support centred at ${\xi }_A=a,$ assured by Eq. (7), also allows manipulating the radiant and/or the modulating energies emitted only by its associated radiant and/or virtual point sources, by simply changing the weight coefficients in such modal expansion. It can be performed by systematically modifying the power distribution of the field at the AP, its complex degree of spatial coherence or the aperture transmission there. Such capability is not offered by conventional mode expansions and is potentially useful in spatial coherence modulation, applied to partially coherent imaging for instance.

The power spectrum at the OP for the propagation distance $z\ge 0$ is given by $S\left(r_A;\nu \right)={\displaystyle \underset{AP}{\int }S\left({\xi }_A,r_A;\nu \right)d^2{\xi }_A}$ [14]. It results from a linear combination of the non-paraxial propagation modes, obtained by integrating Eq. (7) over the positions ${\xi }_A$ on the AP. Thus,

$S\left(r_A;\nu \right)=S_{rad}\left(r_A;\nu \right)+S_{virt}\left(r_A;\nu \right)\text{holds,}\text{with}{S}_{rad}\left(r_A;\nu \right)={\displaystyle \underset{AP}{\int }{S}_{rad}\left({\xi }_A,r_A;\nu \right)d^2{\xi }_A}={\displaystyle \underset{AP}{\int }{S}_0\left({\xi }_A\right){\left|t\left({\xi }_A\right)\right|}^{2}F\left({\xi }_A,r_A;\nu \right)d^2{\xi }_A}$ the radiant power of the field provided by the whole radiant layer at the AP, i.e. the combination of the zeroth-order modes emitted by all the point sources of the radiant layer, and

The analysis above means that only the zeroth-order modes are involved in the achievement of the conservation law of the total energy of the field on propagation, and therefore that the particular behaviour of the higher order modes (i.e. the redistribution of the radiant energy on the OP at any propagation distance [14, 22]) cannot affect this law.

2.4. Non-paraxial propagation modes and classes of correlated radiant point sources

In applications of optical information processing, it could be more important to access a specific class of correlated pairs of radiant point sources across the AP than to access an individual structured spatial coherence support on this plane. It allows implementing class filtering procedures in order to systematically affect the optical field at the OP [22]. Such access can be formalised with basis on the non-paraxial propagation modes too, by introducing the quantity (with power units) $S\left({\xi }_D,r_A;\nu \right),$ that fulfils the expression $S\left(r_A;\nu \right)={\displaystyle \underset{AP}{\int }S\left({\xi }_D,r_A;\nu \right)d^2{\xi }_D}$ for the power spectrum at the OP, i.e. it gathers the contributions of the correlated pairs of radiant point sources with the same separation vector ${\xi }_D$ across the AP, to the power spectrum at the OP. $S\left({\xi }_D,r_A;\nu \right)$ can be defined in terms of the following expressions:

Nevertheless, Eqs. (8) and (9) cannot be considered as modal expansions in the sense of Eq. (7), because all the members of a given class emit in the same non-paraxial propagation mode. In other words, the integrands of such Eqs. contain only the mode corresponding to the zeroth-order class and to the class of the order specified by ${\xi }_D=b\ne 0,$ respectively. Moreover, $S\left(0,r_A;\nu \right)\text{and}S\left(b,r_A;\nu \right)$ reach the same shapes of the characteristic modes of their classes when they propagate in Fraunhofer domain.

By modifying the power distribution of the field at the AP, its complex degree of spatial coherence or the aperture transmission there, the weight coefficients of the non-paraxial propagations modes in Eqs. (8) and (9) can be changed. This capability can be applied in order to enhance, to reduce or eventually to remove the contribution of the class to the field on propagation, procedures called class filtering. For instance, class filtering based on spatial coherence modulation has been successfully used in beam shaping [26].

3. Micro-diffraction and spatial coherence modulation

The capabilities of modal expansions on non-paraxial propagation modes can be estimated by analysing (i) the field propagation when both the aperture size and the propagation distance are comparable with the wavelength (i.e. micro-diffraction, development of optical devices at the micro- and nano-scales), and (ii) the effects on the field features produced when individual structured supports at the AP are modified or specific classes of pairs are filtered (spatial coherence modulation, applied for instance to beam shaping and quality improvement of the partially coherent imaging). To this aim, let us regard the simplest but representative case of non-paraxial micro-diffraction through a slit. Although it is a one-dimensional particular case, many other micro-diffraction arrangements can be modelled by properly changing the parameters of this modal expansion and by considering it along the orthogonal axes in two dimensional separable situations, and multiplying the expansions thereafter. Physical values are given to the arrangement parameters, in order to encourage other researchers to apply it in more sophisticated experiments, too.

In order to build the modal expansion of the field emitted by the slit, both the radiant and the virtual layers are modelled in terms of linear arrangements of equidistant point sources, of lengths $L\text{and}L-b$ respectively, with $L$ the slit width and $b$ the pitch of the arrangement of radiant point sources.

Thus, b/2 is the pitch of the arrangement of virtual point sources if all the possible virtual sources are turned on (for instance when the field is spatially coherent, as conceptually sketched in Fig. 1). The attributes of the virtual segment can be changed by adjusting the spatial coherence state of the field at the slit. Thus, $N_{rad}=\left(L/b\right)+1$ is the number of radiant point sources, where $L/b$ should be integer in order to assure the location of a radiant point source at each slit edge, and $0\le N_{virt}\le 2N_{rad}-3$ is the number of virtual point sources, whose value depends on the spatial coherence state of the field, i.e. $N_{virt}=0$ for spatially incoherent fields and $N_{virt}=2N_{rad}-3$ for spatially coherent fields, for instance. Diffraction is assured under the conditions $L>\lambda \text{and}b\le \lambda$ [14]. For $\lambda <b<L$ the arrangement of radiant point sources behaves as an interference grating, and for $b<L<\lambda$ it behaves closely similar to an isolated radiant point source [14]. Arbitrary but physically realizable values were chosen for achieving the diffraction conditions for the micro-diffraction of a uniform field. It is clear that such values can be changed in order to adjust them to specific situations of interest, as sub-wavelength slits for instance.

 

Fig. 1 Distribution of radiant, virtual and dual point sources for modelling micro-diffraction of spatially coherent light $\left(\lambda =0.632\mu m\right)$ by a slit.

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Thus, the following parameters were assumed: $\lambda =0.632\mu m,b=0.3\mu m\text{and}{N}_{rad}=10,\text{so}\text{that}L=2.7\mu m\text{and}z\le 25.5\mu m,$ i.e. there are 10 identical radiant point sources within a slit about $4\lambda$ wide, distributed with a pitch of about $\lambda /2,$ as sketched in Fig. 1, and a maximum propagation distance of about $40\lambda$ is considered. These parameters configure a micro-diffraction situation. The array of radiant point sources turns on an array of $N_{virt}\le 17$ point sources on the virtual layer. If $N_{virt}=17$ stands, the array of virtual point sources is uniformly distributed on a segment of length $2.4\mu m,$ with pitch $0.15\mu m$ (Fig. 1).

Therefore, the complete array of 27 point sources includes: 2 pure radiant point sources at the slit edges, 9 pure virtual point sources placed at the midpoints between consecutive radiant point sources, and 8 dual point sources. The classes of pairs of radiant point sources, whose modes characterize the emission of modulating energy by the virtual point sources, are sketched in Fig. 2 and their specifications appear in Table 1. The column on the left of Fig. 2 shows the even order classes and the column on the right shows the odd order classes. It is worth noting that (i) only virtual components of dual point sources are associated to the even order classes for $n>0,$ while only pure virtual point sources are associated to the odd order classes, and (ii) the modal expansions emitted by the dual point sources contain only even order non-paraxial propagation modes and include the zeroth-order mode, while those emitted by the pure virtual point sources contain only odd order modes, as conceptually sketched in Fig. 3.

 

Fig. 2 Conceptual sketch of the classes of radiator pairs of even orders (including the 0-order class) in the column on the left and of odd orders in the column on the right.

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Table 1. Classes of pairs of radiant point sources for the diffraction model in Fig. 1. Even and odd orders are shown in separate groups. The term class population refers to the number of pairs of the each class across the slit.

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Fig. 3 Conceptual sketch of the non-paraxial propagation modes emitted by pure virtual point sources (a-c) and dual point sources (d-e). Each emitting source has a twin source that does not appear in the sketch, placed at the symmetric position with respect to the array midpoint (except the point source at the midpoint of the array, in c).

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For the sake of simplicity and without loss of generality, let us assume that the model above represents the diffraction of a uniform Gaussian Schell-model field [13], i.e. $\left|t\left({\xi }_A+{\xi }_D/2\right)\right|=\left|t\left({\xi }_A-{\xi }_D/2\right)\right|=1,\varphi \left({\xi }_A+{\xi }_D/2\right)=\varphi \left({\xi }_A-{\xi }_D/2\right)=0,\left|\mu \left({\xi }_A+{\xi }_D/2,{\xi }_A-{\xi }_D/2\right)\right|=\exp \left(-\frac{{\left({\xi }_D-nb\right)}^2}{2{\sigma }^2}\right)$ and $\alpha \left({\xi }_A+{\xi }_D/2,{\xi }_A-{\xi }_D/2\right)=0\text{with}{\sigma }^2$ the variance that adjusts the width of the Gaussian function (i.e. the size of the structured spatial coherence supports within the slit, in such a way that $\sigma =0\text{and}\sigma =\infty$ denote the ideal spatially incoherent and spatially coherent illuminations).

Let us also label (with $1\le n\le 9$):

Only one odd- and one even-order class were arbitrary chosen in three spatial coherence states, labelled by $\sigma =3,6,\inf$ (Figs. 4 and 5 respectively), for illustration purposes. The corresponding analysis is applicable to the remainder classes of the same order in other spatial coherence states, too. Because of graph construction, the origin of the ${\xi }_A$–coordinate was shifted to the midpoint of this axis. Figure 4 (Media 1) shows the third order class of radiator pairs on the top row and the profile of its contribution to the power spectrum at the OP on the bottom row, for the regarded degrees of spatial coherence, as example of the odd-order classes. Media 1 is corresponding to the field propagation along $0.7\mu m\le z\le 25\mu m$ with non-uniform steps, because the mode evolution becomes slower with the field propagation. The strong influence of the spatial coherence degree on the mode spreading across the OP along the field propagation is apparent too. It realizes the concept of spatial coherence modulation [26], i.e. the change in the power spectrum at the OP due to variations in the spatial coherence state of the field at the OP, which is manifested in the changes of the modulating power contribution of the class $S\left(3b,x_A;\nu \right),$ for the different values of $\sigma .$ This behaviour is potentially useful in beam shaping for lithography and optical tweezers for instance.

 

Fig. 4 (Media 1) Example of the odd-order classes: the third-order class for three states of spatial coherence denoted by the values of $\sigma$. The diagrams of its modes are on the top row and the profiles of its contribution to the power spectrum at the OP are on the bottom row. Each vertical structure of $G_3\left({\xi }_A,x_A\right)$ is the third-order mode $M_3\left({\xi }_A,x_A\right)$ emitted by the pure virtual point source placed at the ${\xi }_A$-coordinate of the corresponding mode.

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Fig. 5 (Media 2) Example of the even-order classes: the eighth-order class for the same states of spatial coherence and propagation distance as in Fig. 4. The diagrams of its modes are on the top row and the profiles of its contribution to the power spectrum at the OP are on the bottom row. Each vertical structure of $G_8\left({\xi }_A,x_A\right)$ is the eighth-order mode $M_8\left({\xi }_A,x_A\right)$ emitted by the virtual component of the dual point source placed at the ${\xi }_A$-coordinate of the corresponding mode.

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Figure 5 (Media 2) illustrates the even-order classes by showing the eighth order class of radiator pairs on the top row and the profile of its contribution to the power spectrum at the OP on the bottom row, for the same degrees of spatial coherence and propagation distance as in the former case. The modes behave in a similar fashion as those in the former case, but with the following particularities: i) both the modes and the power contribution of this class are more oscillating than the corresponding quantities of the third-order class, because the oscillation frequencies are inversely proportional to the separation pair of the class, and ii) they reach their propagation invariant forms at a longer propagation distance z as those of the third order class.

The effect of the spatial coherence modulation is apparent in this case too. In general, the higher the order of the modes (or of the class) the more oscillating they are and the longer the propagation distance they require to become shape-invariant. These features must be considered in the design of optical devices at the micro- and nano-scales.

The non-paraxial propagation of the complete modulating energy, emitted by each point source of the virtual layer at the AP, is described by the summation ${\displaystyle \sum _n{G}_n\left({\xi }_A,x_A\right)},$ and the modulating energy profile provided by the whole virtual layer is obtained from the expression ${\displaystyle \underset{AP}{\int }{\displaystyle \sum _n{G}_n\left({\xi }_A,x_A\right)}d{\xi }_A}.$

The diagrams for both the odd- and even-orders of the considered array are separately shown on Figs. 6, (Media 3) and Fig. 7, (Media 4) respectively, for the same states of spatial coherence and propagation distances as in Figs. 4 and 5.

 

Fig. 6 (Media 3) Modal expansion of the modulating energy emitted by the pure virtual point sources, in the same states of spatial coherence and for the same propagation distances in Figs. 4 and 5. Each vertical structure of their diagrams on the top row results from the superposition of the high odd-order modes $\left(n=1,3,\cdots ,9\right)$ emitted by the pure virtual point source, placed at the ${\xi }_A$-coordinate. The profiles of the modulating power contributions of all the pure virtual point sources at the AP are shown on the bottom row.

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Fig. 7 (Media 4) Modal expansion of the modulating energy emitted by the virtual components of the dual point sources, in the same states of spatial coherence and for the same propagation distances in Fig. 6. Each vertical structure of their diagrams on the top row results from the superposition of the high even-order modes $\left(n=2,4,\cdots ,8\right)$ emitted by the virtual component of the dual point source, placed at the ${\xi }_A$-coordinate. The profiles of the modulating power contributions of the virtual components of all the dual point sources at the AP are shown on the bottom row.

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The diagrams of ${\displaystyle \sum _n{G}_n\left({\xi }_A,x_A\right)}$ are shown on the top rows and the profiles of ${\displaystyle \underset{AP}{\int }{\displaystyle \sum _n{G}_n\left({\xi }_A,x_A\right)}d{\xi }_A}$ are shown on the bottom rows of such Figs. It is worth remarking the evolution of both the modes and the contributions of the classes along the field propagation under micro-diffraction conditions, as well as the effects of spatial coherence modulation due to variations of $\sigma .$ For Fraunhofer diffraction, all the modes are shape invariant and the class contribution has the same shape as any of their modes. Although in this example, the sizes of the structured spatial coherence supports (determined by the values of $\sigma$) include all the classes, some of them (beginning with the high-order classes) are dropped out as $\sigma$ becomes small enough, because their radiator pairs cannot be included within any structured spatial coherence support. Thus, the structured spatial coherence support behaves as a filter of classes of radiator pairs in two modalities [21, 22], i.e. (i) the modes emitted by specific pairs of any class can be individually filtered by using the individual access to given structured supports; (ii) all the modes emitted by the whole class of pairs across the AP can be also filtered, for instance by manipulating the complex degree of spatial coherence of Schell-model fields. Because of this capability, the class filtering is a very important tool for optical processing based on spatial coherence modulation [26].

The modal expansion of the non-paraxial marginal power spectrum (Eq. (7)), for the fully spatially coherent state and propagation distances $0.7\mu m\le z\le 25\mu m$, is shown in Fig. 8 (Media 5). Its radiant component $S_{rad}\left({\xi }_A,x_A;\nu \right)$ is corresponding to the zeroth-order class, whose diagram $G_0\left({\xi }_A,x_A\right)\equiv S_0\left(\nu \right){\displaystyle \sum _{m=0}^{N-1}{M}_0\left({\xi }_A-mb,x_A\right)}$ is on the top of the left column. Each vertical structure of this diagram is determined by the zeroth-order non-paraxial mode $M_0\left({\xi }_A,x_A\right)=2\mathrm{Re}\left[F\left({\xi }_A,x_A;\nu \right)\right]$ (i.e. the Lorentzian free-space diffraction envelope in Eq. (6)) emitted by the radiant point source placed at the corresponding position on the ${\xi }_A$-axis. The profile of the radiant power provided by the whole class (or equivalently by the radiant component of the non-paraxial power spectrum), $S_{rad}\left(x_A;\nu \right)=S\left(0,x_A;\nu \right)={\displaystyle \underset{AP}{\int }{G}_0\left({\xi }_A,x_A\right)d^2{\xi }_A},$ is shown on the bottom of the column.

 

Fig. 8 (Media 5) Modal expansion of the non-paraxial marginal power spectrum under fully spatially coherent illumination: the radiant component is sketched on the left column, the virtual component is on the mid-column and whole expansion is on the right column. The diagrams of the respective expansions are shown on the top row. The profiles of the radiant and the modulating powers as well as the power spectrum at the OP are shown on the bottom row.

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The modal expansion of the virtual component $S_{virt}\left({\xi }_A,x_A;\nu \right)$ is determined by the superposition of the high-order modes $\left(n=1,2,\cdots ,9\right)$ emitted by both the pure virtual point sources and the virtual components of the dual point sources. So, it is corresponding to ${\displaystyle \sum _{n=1}^9{G}_n\left({\xi }_A,x_A\right)},$ whose diagram is on the top of the mid-column. Each vertical structure of the diagram is determined by the superposition of the high-order modes emitted by the virtual point source placed at the corresponding position ${\xi }_A.$ The expression ${\displaystyle \underset{AP}{\int }{\displaystyle \sum _{n=1}^9{G}_n\left({\xi }_A,x_A\right)}{d}^2{\xi }_A}$ determines the profile of the modulating power provided by the whole layer of virtual point sources, shown on the bottom of the column, i.e. $S_{virt}\left(x_A;\nu \right)={\displaystyle \underset{AP}{\int }{S}_{virt}\left({\xi }_A,x_A;\nu \right)d^2{\xi }_A}.$

The diagram of the whole non-paraxial marginal power spectrum $S\left({\xi }_A,x_A;\nu \right)=S_{rad}\left({\xi }_A,x_A;\nu \right)+S_{virt}\left({\xi }_A,x_A;\nu \right)$ is shown on the top of the right column. Each vertical structure of the diagram results from the modal expansion of the structured spatial coherence support centred at the corresponding position ${\xi }_A.$ It gathers the modes emitted by both the radiant (if any) and the virtual point sources placed at such position, i.e. ${\displaystyle \sum _{n=0}^9{G}_n\left({\xi }_A,x_A\right)}.$ It is worth noting that the individual access to a specific structured spatial coherence support is realized by the capability of manipulating any vertical structure of $S\left({\xi }_A,x_A;\nu \right)$ individually, which is crucial for applications based on spatial coherence modulation as beam shaping and partially coherent imaging for instance. The profile of the power spectrum at the OP, $S\left(x_A;\nu \right)=S_{rad}\left(x_A;\nu \right)+S_{virt}\left(x_A;\nu \right)={\displaystyle \underset{AP}{\int }S\left({\xi }_A,x_A;\nu \right)d^2{\xi }_A},$ is shown on the bottom of the column.

Figures 9 to 11 show the profiles of the radiant and modulating powers and the power spectrum at the OP for $\lambda =0.632\mu m,b=0.3\mu m\text{and}L=2.7\mu m,$ with the propagation distances: $0.1\mu m\le z\le 0.5\mu m<\lambda <L\text{(Fig}\text{.}\text{9)}\text{,}b<0.5\mu m\le z\le 3.5\mu m(\text{Fig}\text{.10})\text{and}b<\lambda <L<<8\mu m\le z\le 14\mu m(\text{Fig}\text{.11}).$ They reveal interesting features of the optical field evolution in the micro-diffraction domain.

 

Fig. 9 Illustrating the propagation of a) the radiant power, b) the modulating power and c) the power spectrum of a uniform and spatially coherent field of $\lambda =0.632\mu m,$ diffracted by a slit of width $L=2.7\mu m,$ along the propagation distance $0.1\mu m\le z\le 0.5\mu m<\lambda <L.$ Units of axes x_A and z are $\mu m,$ and of the vertical axis are arbitrary.

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Figure 9 points out that

The individual zeroth-order modes are not resolvable in the profiles of the radiant power shown in Fig. 10, because of their spreading across the OP. Their superposition evolves to a Lorentzian profile along the field propagation. Nevertheless, they remain positive definite and accomplish the $1/z^2$-law of propagation (Fig. 10(a)). The modulating power profiles maintain the attributes of this type of power but their oscillations diminish with the propagation in such a way that their significant values tend to concentrate within the central region of the pattern (Fig. 10(b)). The shapes of the modulating power profiles strongly influence the shape of the power spectrum as appreciated by comparing Figs. 10(b) and 10(c). It is also apparent that the power spectrum is positive definite and fulfil the $1/z^2$-law of propagation.

 

Fig. 10 Illustrating the propagation of a) the radiant power, b) the modulating power and c) the power spectrum of a uniform and spatially coherent field of the same attributes as in Fig. 9, along the propagation distance $0.5\mu m\le z\le 3.5\mu m$. Units of axes x_A and z are $\mu m,$ and of the vertical axis are arbitrary.

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The radiant power profiles in Fig. 11(a) are propagation invariant, positive definite and follow the $1/z^2$-law. Their shapes are Lorentzian-like because the individual zeroth-order modes are not resolvable and practically coincide to each other. Although the modulating power spreads over the whole OP, its main values concentrate around the central maximum, that also decreases with the propagation (Fig. 11(b)). This decay is due to the condition that the modulating power redistributes the radiant power by achieving the conservation law of the total energy of the field and by assuring that the power spectrum be a positive definite quantity, taking into account that that radiant power fulfils the $1/z^2$-law.

 

Fig. 11 Illustrating the propagation of a) the radiant power, b) the modulating power and c) the power spectrum of a uniform and spatially coherent field of the same attributes as in Fig. 9, along the propagation distances $\lambda <L<<8\mu m\le z\le 14\mu m$, i.e. in the Fraunhofer domain. Units of axes x_A and z are $\mu m,$ and of the vertical axis are arbitrary.

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The modulating power patterns mainly determine the shape of the power spectrum at the OP (Fig. 11(c)), which acquires the well-known form of the squared circular sinus at $z=14\mu m$. It is characteristic of the Fraunhofer diffraction of a spatially coherent plane wave by a slit, and therefore it suggests novel physical implications taking into account that such power spectrum distribution is provided by a discrete set of only 10 radiant and 17 virtual point sources instead of a continuous wave-front.

It is worth taking into account that the algorithm for the calculations above is assembly on the conventional Mathlab® platform. It performs a matrix implementation of the modal expansions by determining 1024x1024 matrices, once the distribution of point sources and its degree of spatial coherence are given as entries to the algorithm. A vectored calculation of the propagation integrals allows processing times of order 25 ms in average. The modes emitted by the classes of pairs and also by the individual point sources are displayed on separate matrices in order to perform the modal analysis of the field on propagation.

4. Conclusion

The modal expansion of the non-paraxial marginal power spectrum in terms of non-paraxial propagation modes has been reported and analysed for the first time (to our knowledge) in this work. Taking into account that they are a standard set of modes applicable to any diffraction or interference set up with optical fields in arbitrary states of spatial coherence, the modal expansion constitutes an accurate and exhaustive tool for analysis, numerically calculations and simulations. Indeed, instead of using a sampled continuous second-order wave-front as entry to the non-paraxial propagation integral, a finite series of discrete non-paraxial propagation modes is applied, whose coefficients are determined by the power distribution and the spatial coherence state of the field at the aperture plane. Such modal expansion also allows (i) accessing specific structured spatial coherence supports individually, (ii) specifying the contributions of the classes of radiator pairs, and (iii) determining the marginal power spectrum and the power distribution of the field at any propagation distance, for instance at sub-wavelength distances. Such capabilities offer an important support for micro-diffraction applications based on spatial coherence modulation and class filtering, such as beam shaping, design of optical devices at the micro- and nano-scales and partially coherent imaging for instance.