The non-paraxial phase-space representation of diffraction of optical fields in any state of spatial coherence has been successfully modeled by assuming a discrete set of radiant point sources at the aperture plane instead of a continuous wave-front. More than a mere calculation strategy, this discreteness seems to be a physical feature of the field, independent from the sampling procedure of the modeling.
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The starting point of the conventional description of diffraction is the idea that the optical field at the aperture plane (AP) distributes over a continuous wave-front, used as input of the Fresnel-Kirchhoff diffraction integral in order to determine the wave-front at the observation plane (OP), placed at any distance from the AP . This fundamental idea is extended to the non-paraxial propagation of the spatial coherence state of the field. In this case, a continuous second-order wave-front represents the cross-spectral density of the field at the AP, and is used as input of the Wolf’s integral (a formula closely related to the Fresnel-Kirchhoff diffraction integral) in order to determine the cross-spectral density at the OP . Mathematically, the cross-spectral density is the solution of two coupled Hemholtz equations and therefore, it should be a continuous function of two variables (i.e., the positions of pair of points in space), and derivable at least to the second-order in each of them .
Interesting physical features of this mathematical continuum become apparent as the meaning of its values is analyzed. Recently, the input to the Wolf’s integral was modeled in terms of a discrete set of radiant point sources instead of a continuous wave-front . Successful results for diffraction were reported by using only 16 point sources. Moreover, such discreteness was presented as a requirement for assuring the correctness and accuracy of the calculation, mainly at very short propagation distances (i.e., distances comparable to the wavelength), a condition that the conventional methods cannot meet . Indeed, the field distribution it predicts after sub-wavelength propagation distances is in agreement with the assumed conditions at the aperture plane . It seems to be more than a simple algorithmic procedure of calculation because of its physical consequences, as discussed in this paper for the first time. Therefore, such method provides a strategy for looking for new physical features of the optical field, which seems to be its main value.
In the framework of the phase-space representation, the cross-spectral density of the field at the AP is modeled in terms of two separate layers of point sources, i.e., the radiant and the virtual layers respectively, with quite different physical properties [4,5]. The radiant power is emitted by the point sources of the radiant layer, independently of the spatial coherence state of the field, and such power is positive-definite and recordable by squared modulus detectors. In contrast, the point sources of the virtual layer emit the modulating power which is closely related to the spatial coherence state of the field (it nullifies for spatially incoherent fields, for instance), takes on positive and negative values and is not individually recordable by squared modulus detectors. The modulating power redistributes the radiant power at the OP in interference or diffraction patterns, without violating the conservation law of the total energy of the field .
In addition, the two layers are arranged together in such a way that pure virtual point sources be inserted at the midpoints between consecutive radiant point sources (these places are empty if the field is fully spatially incoherent, but become filled along the field propagation obeying the Van Cittert Zernike theorem) . Radiant and virtual point sources that coincide in position inside the aperture give dual point sources. Such distribution criteria deal to a continuum as the distance between consecutive radiant (or dual) and pure virtual point sources become arbitrary short, which assures that the cross-spectral density mathematically exists at any and all points on the AP and fulfill the continuity requirements of the coupled Helmholtz equations.
However, the radiant layer alone cannot be a continuum. Indeed, the distance between consecutive radiant point sources cannot nullify because of the requirement of pure virtual point sources inserted between any pair of consecutive radiant ones. This feature is independent from (and previous to) the sampling procedure used for the calculations. In other words, any sampling (fine or coarse) must account for the pure virtual point sources at the midpoints between the consecutive pair of radiant (or dual) point sources, no matter the density of the source set. Otherwise, the resulting description of the field propagation based on Wolf’s integral becomes erroneous, mainly at short propagation distances.
The statements above lead to the prediction of a novel physical structure of the second-order wave-front that represents the cross-spectral density at the AP, which is validated by appealing to well-known basic diffraction and interference experiments, because (i) intuition is accessible by them, (ii) their very few parameters assure the reliability of the proposed statements in a more clear way as by more sophisticated experiments, and (iii) they are the fundamentals for understanding such sophisticated experiments. Therefore, the aim of the work is to report a physical attribute whose importance is conferred by the consistency between the predictions and the results of basic experiments, but not to propose a non-paraxial phase-space calculation strategy, whose performance and accuracy should be compared with other methods.
2. Layers of point sources and classes of radiator pairs
Let us regard the non-paraxial propagation of an optical field of frequency , wave-number and wavelength , from the AP to the OP placed at a distance to each other, and let us assume center and difference coordinates and at the AP and the OP respectively. These coordinates determine pairs of points that belong to the structured spatial coherence supports centered at the coordinate with suffix A and with separation vectors given by the coordinate with suffix D . The function gives the cross-spectral density of the field at the AP, with a short notation for the positions the illumination power spectrum at these positions, where denotes the complex transmission, and the complex degree of spatial coherence of the field at the AP . Therefore is the cross-spectral density at the OP, where symbolizes the transformation due to the non-paraxial Wolf’s integral .
The power spectrum at the OP is obtained by evaluating the cross-spectral density there for the marginal power spectrum that provides the non-paraxial phase-space representation of the field propagation . Although conventional phase-space representations in different fields of physics are based on Wigner distribution functions (WDF), it is worth noting that the non-paraxial marginal power spectrum is not a WDF, mainly because the non-linearity of the propagator argument (see Eq. (2b)) [6,7]. However, it becomes the WDF of optical fields in arbitrary states of spatial coherence in the paraxial approach for far-field propagation [4,6].
The non-paraxial marginal power spectrum can be expressed as , where denotes the radiant energy provided by the radiant point source placed at each specific position onto any position , while denotes the modulating energy provided by the virtual point source turned on at onto any position . This virtual point source is due to all the correlated pairs of radiant point sources that belong to the structured spatial coherence support centered at [4,5]. If there is only a pure radiant or a pure virtual point source at , is completely determined by the terms or respectively. If there is a dual point source, the addition of these two terms determines .
For the present purposes, it is useful to introduce the concept of class of radiator (radiant point source) pairs in order to describe the emission of the modulating energy at the AP . A class gathers all the correlated pairs with the same separation vector across the AP, each of them emitting in the same non-paraxial propagation mode (the modes are defined by Eqs. (2)). Thus, the modal expansion3]. While the class of order corresponding to emits its modulating energy in the non-paraxial mode
Figure 1 illustrates the non-paraxial propagation of the power spectrum provided by specific pairs of radiant point sources in different states of spatial coherence, along the distance . Each pair emits two zeroth-order modes (Eq. (2a)) for the radiant energy and only one mode for the modulating energy (Eq. (2b)), i.e., a low-order mode (source separation ) on the upper row, and a high-order mode (source separation ) on the bottom row. The radiant energy propagated by the zeroth-order modes (delimited by white dotted lines on the graph on column (a) bottom row) constitutes the power spectrum of the spatially incoherent case (column (a)). It is modulated by the modes emitted by the pure virtual point source at the midpoint between the radiant pair, in columns (b) for the partially coherent case , and (c) for the spatially coherent case . Both amplitude and spatial frequency modulations are apparent due to the Lorentzian envelope and the non-linear propagator argument respectively. Graphs on column (b) show low contrasted (low visibility ) fringes and a power minimum on the optical axis (the normal to the midpoint between the radiant pair) because , while fringes of graphs on column (c) are highly contrasted (maximum visibility) because of the complete spatial coherence, and a power maximum appears on the optical axis because .
It is interesting to regard the resolution or distinguishability of the contributions provided by the individual radiant point sources in each spatial coherence state . Vertical dotted lines at on the graphs of Fig. 1 determine the power spectrum profile in which such contributions can be resolved by applying the Rayleigh criterion by spatial incoherence (column (a)) . However, these profiles are strongly affected by the modulating energy by partial coherence (column (b)) and complete coherence (column (c)), i.e., the radiant energy redistribution impedes to distinguish the individual contributions, in such a way that the Rayleigh criterion is no longer applicable. A shorter distance was arbitrary chosen on the graphs of the bottom row, under the condition that the power lobes provided by the individual radiant sources even remain separate and their modulation is relatively soft (maxima smaller than the 10% of the lobe main maximum) and occurs mainly within each lobe, i.e., the redistributed energy is mainly provided by each specific radiant point source. Therefore, the individual contributions can be distinguished along the propagation distance . It is apparent that the higher the mode orders the longer this distinguishability range. For example, a sub-wavelength Rayleigh distance is determined for the source separation of (upper row), so that the distinguishability range is negligible at the wavelength scale.
According to the analysis above, Eqs. (2a) and (2b) formalize the physical meanings of the non-paraxial propagation modes. The zeroth-order mode is exclusive for the radiant energy propagation, while the remaining modes propagate the modulating energy emitted by each virtual point source (both the pure ones and the virtual component of dual point sources). Therefore, modulating modes are responsible for diffraction and interference. Summarizing, the modal expansion characterizes the energy emitted by any point source, both radiant and virtual. These energies can be individually manipulated, by spatial coherence modulation or by class filtering for instance, in order to produce specific effects on the power spectrum distribution [4,5]. Such energy characterization and the possibility of its manipulation is mainly what this modal expansion physically implies. It is also worth noting that the modal expansion can be referred to each individual structured support of spatial coherence, whose virtual point source emits the modulating energy only in the modes corresponding to the correlated radiant pairs that belong to the structured support. Thus, low order modes are contributed by pairs with relatively short separations, usually belonging to the coherence patch (i.e., the small area around the support center that encloses highly coherent pairs) while the high order modes are contributed by correlated pairs with longer separations than the patch size. In this sense, the modal expansion provides progressively better estimate of partially coherent propagation.
Equation (3) is the key for characterizing the physical discreteness of the set of point sources on the radiant layer by keeping the mathematical continuity of the second-order wave-front. Indeed, by using it, the analysis is supported by the power spectrum distribution at the OP, which is an observable quantity recorded by conventional squared modulus detector at such plane.
3. Continuous wave-front and discrete set of radiant point sources
Let us analyze the behavior of Eq. (3) by assuming the discreteness of the set of radiant point sources as initial condition and taking the limit when their separations tends to null. To this aim, let us consider the one-dimensional diffraction of a spatially coherent and uniform plane wave through a slit. This simple experiment is chosen because (i) intuition is accessible by it, (ii) its very few parameters assure the reliability of the proposed statements in a more clear way as by more sophisticated diffraction experiments, and (iii) it is fundamental for understanding such experiments. It is worth noting that all the non-paraxial propagation modes associated to the set of radiant point sources are used if the set is spatially coherent. By partially coherent sets, the mode weights are modulated (and eventually the modes can be filtered) by the degree of coherence . But once the phase differences between the correlated pairs are fixed, the shape and behavior of the modes along the field propagation are independent of the spatial coherence state of the field (Fig. 1), i.e., these features in the partially coherent cases are the same as in the spatially coherent case. For this reason, the spatially coherent case provides the most general description. Accordingly, let us assume that , with a the slit width. The radiant layer within the slit can be modeled as a line of identical and equidistant point sources of pitch b, and the virtual layer as a second line of equidistant point sources of pitch (Fig. 2).
From them, are pure virtual point sources and are the virtual components of the dual point sources. It is worth noting that, although the radiant point sources are identical, the virtual point ones are not because classes of different orders contribute to each one, i.e., the modulating power emitted by each virtual point source results from the superposition of a specific set of modes (Figs. 3(a) to 3(c) for odd-order modes and 3(d) to 3(e) for even order modes). Furthermore, the number of virtual point sources is determined by the spatial coherence state of the field. Actually, it fulfills the condition for spatially incoherent and for partially coherent optical fields respectively.
The limit condition of (with and a fixed) is realized under two different requirements that meet the mathematical continuity of the second-order wave-front at the AP:
- • By keeping the set of pure virtual point sources, so that the second-order wave-front is characterized by the continuous sequence of point sources, distributed as r-v-d-v-d-…-d-v-r, with r: pure radiant, v: pure virtual and d: dual point sources (Fig. 2). Thus, the set of radiant point sources remains discrete through the limit procedure on account of the pure virtual point sources inserted between consecutive pairs of radiant (or dual) point sources.
- • By dropping out the set of pure virtual point sources, so that the second-order wave-front is characterized by the continuous sequence of point sources, distributed as r- d- d-…-d-r after applying the limit procedure. Thus, the continuous second-order wave-front is only constituted by radiant power values.
It is clear that the result of applying the limit is independent from the sampling procedure applied for the calculation. Consequently, any (fine or coarse) sampling procedure must sample all the pure virtual point sources in the first case, while the finest sampling procedure only samples dual point sources in the last case. In order to estimate the ability of each case in predicting the physical behavior of light, their respective power spectrum predictions at the OP are compared to each other and also to the well-known experimental diffraction patterns, obtained under the same physical conditions, as merit figures.
The factor in Eqs. (1) determines the positions of the point sources at the AP. It takes the forms: for the radiant point sources, for the virtual components of the dual point sources, and for the pure virtual point sources, with the Dirac’s delta.Fig. 4). Therefore, the modal expansion for the analysis under the first requirement is given by Eq. (4), i.e.,Eq. (5a), i.e.,
It is worth noting that Eq. (5b) removes the interference provided by the radiator pairs that belong to the odd-order classes, particularly the immediate neighbors, although a fully spatially coherent field has been assumed.
The limit is mathematically realized by taking the pitch b arbitrary short, which imposes the increase of the source density (i.e., N grows by keeping a fixed) in order to maintain the uniformity of the source distribution. A numerical good (or at least enough) approach to any limit situation for diffraction is reached by making . Therefore, let us compare the predicted power spectra by Eqs. (5), to each other, with the following physical parameters in both of them:
- • Slit width a = 10 μm.
- • identical radiant point sources, uniformly distributed with pitch .
- • Spatially coherent optical field of , so that and stand, which assures the accomplishment of the condition .
- • Propagation distances .
Figure 5 (Media 1) allows performing this comparison. Indeed, 39 point sources, uniformly distributed with pitch b/2 = 0.2632μm, configure the second-order wave-front that emerges from the slit in the case on the upper row, while only 20 uniformly distributed point sources with pitch b = 0.5263μm configure that wave-front in the case on the bottom row. The point sources in the graphs on the upper row include 2 pure r- sources (placed at the slit edges), 19 v- sources and 18 d-sources, in the sequence r-v-d-v-d-…-d-v-r, so that the pitch of the point sources of the radiant layer cannot be nullified because of the inserted pure virtual point sources. So, the set of radiant point sources remains discrete through the limit procedure in this case. On the bottom row, the graphs include 2 pure r- sources and 18 d-sources, in the sequence r-d-d-…-d-d-r. Thus, the pitch of the dual point source distribution can be arbitrary reduced, which means that the set of radiant point sources becomes a continuum as it tends to null.
The marginal power spectrum and the power spectrum at the OP are shown on the left and the mid-column respectively, for the same propagation distance. The radiant and modulating power components are separately sketched on the right column, even the modulating component is split into the contribution of the virtual components of the dual point sources and the contribution from the pure virtual point sources. The last component does not appear in the profiles of the bottom graph, because the pure virtual point sources were dropped out in this case. The profiles are scaled for presentation purposes.
Under the assumed conditions, the far-field paraxial diffraction pattern experimentally produced by a slit at is squared-sinc shaped. It is closely similar to that on the upper row mid-column of Fig. 5 at such propagation distance, but is quite different from that on the bottom row mid-column. Taking into account that the profiles in Fig. 5 result from the exact calculation of the non-paraxial Wolf’s integral, it clearly means that the power spectrum pattern on the upper row mid-column is not only the best prediction but also the correct prediction, and therefore it is used as figure of merit in order to calculate the root-mean-squared (rms) error of the pattern on the bottom row mid-column for all the propagation distances. The result is sketched in Fig. 6 for (a) and (b) . In addition to the shape differences between such patterns, which can be appreciated in Media 1, the rms-error between them stabilizes about 10% for , which is a significant per cent taking into account that the rms-errors in diffraction predictions are usually smaller than 1%. Furthermore, the rms-error fluctuates between 15% and 35% for , which means that the power spectrum predicted by Eq. (5b) cannot describe the field distribution on the OP at short distances from the AP, and it is not reliable for reproducing the assumed conditions on this plane. In contrast, the power spectrum predicted in Eq. (5a) fits this field distribution, which implies that the set of point sources on the radiant layer must be discrete, even regarding the second-order waver-front as a continuum with inserted pure virtual point sources.
These v-sources contribute with the red profile of modulating power depicted in the graph on the upper row right column, that is absent in the graph on the bottom row right column. So, the rms- error of the power spectrum profile on the bottom row with respect to the figure of merit estimates the inaccuracy introduced by dropping the pure virtual point sources out of the second-order wave-front configuration.
The same analysis remains valid for the interference patterns produced when the distance between consecutive radiant point sources is longer than the wavelength. It is sketched in Fig. 7 (Media 2) for 10 identical and spatially coherent r-sources uniformly distributed on an array of length 10 μ m, that emit at for the same propagation distances as before, i.e., . Indeed, the rms-error is about 13% for and fluctuates between 8% and 32% for , as depicted in Figs. 8 (a) and 8(b).
4. Discussion and concluding remarks
The mathematical continuity of both the cross-spectral density at the AP and the non-paraxial propagation kernel of the Wolf’s integral equation assure the mathematical continuity of the marginal power spectrum. In this context, continuity means that these functions must have mathematical values in any and all points of its domain denoted by the coordinate, and that they fulfill basic limit conditions. The question arises on the physical nature of such values. On our knowledge, this question was not formulated before in the conventional context of the second-order theory of optical coherence. On the contrary, it is assumed that all the values of these functions have the same physical nature, related to the field correlation at two points and its propagation in space. The analysis above contributes a different scope by regarding that the physical nature of the marginal power spectrum values (and of the second-order wave-front values too) is determined by the nature of the energy emitted at each point . It is not the same for all points, even taking into account the continuity of the function by low spatial coherence.
Let us consider any three consecutive values of these functions at the AP that fulfill the mathematical continuity requirements, in such a way that the first and the third ones are radiant energy values. Then, the second one in between must be a modulating energy value. If it nullifies because of the spatial incoherence of the adjacent radiant sources, an empty place should be regarded in order to fulfill the requirements of the Van Cittert – Zernike theorem along the field propagation. This feature assures the interference contribution due to consecutive pairs or radiant point sources (actually the contributions of all the odd-order classes of radiator pairs at the AP), which inevitably is destroyed if the inserted pure virtual point sources are dropped out. In this sense, the presence of such inserted v-sources becomes a physical property and not a mere algorithmic requirement. Thus, the discreteness of the set of radiant point sources becomes apparent as the radiant layer is regarded alone. Because this topological condition, the continuous structure of the involved functions is altered if the set of radiant point sources is required as continuum, because the modulating energy values emitted by the v-sources are removed from the topology of the functions as such sources are dropped out.
The statements above were validated by appealing to well-known basic experimental results of interference and diffraction without regarding any sampling procedure. It allows concluding that any sampling must account for the topological structure of the analysed functions, mainly for the inserted pure virtual point sources. It means that the discreteness of the set of radiant point sources is previous to and must be respected by any sampling procedure. This physical feature seems to be connected with some reported properties of the generalized radiance and the generalized radiant emittance of planar sources, specifically their local negative values [8,9]. The rules of thumb for the density of sources needed to accurately represent the field and the computational cost of this approach in comparison with other propagation techniques are beyond the aim of this work and will be treated in next papers.
This work was partially supported by the Patrimonio Autónomo Fondo Nacional de Financiamiento para la Ciencia, la Tecnología y la Innovación, Francisco José de Caldas, Colciencias Grant number 111852128322, and by the Universidad Nacional de Colombia, Vicerrectoría de Investigación grants numbers 12932 and 12934. The authors also acknowledge the support of DIME (Dirección de Investigación Medellín UNAL) and DINAIN (Dirección Nacional de Investigación, UNAL).
References and links
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