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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 11 — Jun. 3, 2013
  • pp: 12964–12975
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Discreteness of the set of radiant point sources: a physical feature of the second-order wave-fronts

Román Castañeda, David Vargas, and Esteban Franco  »View Author Affiliations


Optics Express, Vol. 21, Issue 11, pp. 12964-12975 (2013)
http://dx.doi.org/10.1364/OE.21.012964


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Abstract

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.

© 2013 OSA

1. Introduction

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 [3

3. R. Castañeda and J. Garcia-Sucerquia, “Non-approximated numerical modeling of propagation of light in any state of spatial coherence,” Opt. Express 19(25), 25022–25034 (2011). [CrossRef] [PubMed]

]. 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 [1

1. M. Born, and E. Wolf, Principles of Optics (Pergamon Press, 2005); Eq. (17) in Sec. 8.3.2 is the Fresnel-Kirchhoff diffraction formula.

]. Indeed, the field distribution it predicts after sub-wavelength propagation distances is in agreement with the assumed conditions at the aperture plane [3

3. R. Castañeda and J. Garcia-Sucerquia, “Non-approximated numerical modeling of propagation of light in any state of spatial coherence,” Opt. Express 19(25), 25022–25034 (2011). [CrossRef] [PubMed]

]. 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 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) [2

2. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, 1995); Eq. (4.4-25) is the Wolf’s integral equation.

]. 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.

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 ν, wave-number k=2π/λ and wavelength λ, from the AP to the OP placed at a distance z0 to each other, and let us assume center and difference coordinates (ξA,ξD) and (rA,rD) 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 [4

4. R. Castañeda, The Optics of Spatial Coherence Wavelets, Vol 164 of Advances in Imaging and Electron Physics, P. W. Hawkes, ed. (Academic, 2010), pp. 29–255.

]. The function W(+,;ν)=S0(+;ν)t(+)S0(;ν)t*()μ(+,;ν) gives the cross-spectral density of the field at the AP, with ± a short notation for the positions ξA±ξD/2,S0(±;ν) the illumination power spectrum at these positions, where t(±)=|t(±)|exp[iφ(±)] denotes the complex transmission, and μ(+,)=|μ(+,)|exp[iα(+,)] the complex degree of spatial coherence of the field at the AP [2

2. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, 1995); Eq. (4.4-25) is the Wolf’s integral equation.

]. Therefore W(rA+rD/2,rArD/2;ν)=P[W(+,;ν)] is the cross-spectral density at the OP, where P[] symbolizes the transformation due to the non-paraxial Wolf’s integral [2

2. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, 1995); Eq. (4.4-25) is the Wolf’s integral equation.

].

The power spectrum at the OP is obtained by evaluating the cross-spectral density there for rD=0,i.e.S(rA;ν)=W(rA,rA;ν)=P[W(+,;ν)]rD=0=APS(ξA,rA;ν)d2ξA,withS(ξA,rA;ν) the marginal power spectrum that provides the non-paraxial phase-space representation of the field propagation [3

3. R. Castañeda and J. Garcia-Sucerquia, “Non-approximated numerical modeling of propagation of light in any state of spatial coherence,” Opt. Express 19(25), 25022–25034 (2011). [CrossRef] [PubMed]

]. 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

6. M. Testorf, B. Hennelly, and J. Ojeda-Castaneda, Phase-Space Optics: Fundamentals and Applications (Mc Graw-Hill, 2010).

,7

7. K. Wolf, M. Alonso, and G. Forbes, “Wigner functions for Helmholtz wave fields,” J. Opt. Soc. Am. A 16(10), 2476–2487 (1999). [CrossRef]

]. However, it becomes the WDF of optical fields in arbitrary states of spatial coherence in the paraxial approach for far-field propagation [4

4. R. Castañeda, The Optics of Spatial Coherence Wavelets, Vol 164 of Advances in Imaging and Electron Physics, P. W. Hawkes, ed. (Academic, 2010), pp. 29–255.

,6

6. M. Testorf, B. Hennelly, and J. Ojeda-Castaneda, Phase-Space Optics: Fundamentals and Applications (Mc Graw-Hill, 2010).

].

The non-paraxial marginal power spectrum can be expressed as S(ξA,rA;ν)=Srad(ξA,rA;ν)+Svirt(ξA,rA;ν), where Srad(ξA,rA;ν) denotes the radiant energy provided by the radiant point source placed at each specific position ξA onto any position rA, while Svirt(ξA,rA;ν) denotes the modulating energy provided by the virtual point source turned on at ξA onto any position rA. 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 ξA [4

4. R. Castañeda, The Optics of Spatial Coherence Wavelets, Vol 164 of Advances in Imaging and Electron Physics, P. W. Hawkes, ed. (Academic, 2010), pp. 29–255.

,5

5. R. Castañeda, G. Cañas-Cardona, and J. Garcia-Sucerquia, “Radiant, virtual, and dual sources of optical fields in any state of spatial coherence,” J. Opt. Soc. Am. A 27(6), 1322–1330 (2010). [CrossRef] [PubMed]

]. If there is only a pure radiant or a pure virtual point source at ξA, S(ξA,rA;ν) is completely determined by the terms Srad(ξA,rA;ν) or Svir(ξA,rA;ν) respectively. If there is a dual point source, the addition of these two terms determines S(ξA,rA;ν) [5

5. R. Castañeda, G. Cañas-Cardona, and J. Garcia-Sucerquia, “Radiant, virtual, and dual sources of optical fields in any state of spatial coherence,” J. Opt. Soc. Am. A 27(6), 1322–1330 (2010). [CrossRef] [PubMed]

].

Figure 1
Fig. 1 Power spectrum propagation along 0z10μm of the field of λ=0.632μm emitted by two radiant point sources at the AP (z = 0), separated by 1 μm (upper row) and 5 μm (bottom row), for (a) |μ(+,)|=0 (spatial incoherence), (b) μ(+,)=0.3,α(+,)=π (partial coherence) and (c) μ(+,)=1,α(+,)=0 (spatial coherence).
illustrates the non-paraxial propagation of the power spectrum (λ=0.632μm) provided by specific pairs of radiant point sources in different states of spatial coherence, along the distance 0z10μm. 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 1μm) on the upper row, and a high-order mode (source separation 5μm) 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 (μ(+,)=0.3,α(+,)=π), and (c) for the spatially coherent case (μ(+,)=1,α(+,)=0). 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 [1

1. M. Born, and E. Wolf, Principles of Optics (Pergamon Press, 2005); Eq. (17) in Sec. 8.3.2 is the Fresnel-Kirchhoff diffraction formula.

]) fringes and a power minimum on the optical axis (the normal to the midpoint between the radiant pair) because α(+,)=π, 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 α(+,)=0.

It is interesting to regard the resolution or distinguishability of the contributions provided by the individual radiant point sources in each spatial coherence state [1

1. M. Born, and E. Wolf, Principles of Optics (Pergamon Press, 2005); Eq. (17) in Sec. 8.3.2 is the Fresnel-Kirchhoff diffraction formula.

]. Vertical dotted lines at z=R 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)) [1

1. M. Born, and E. Wolf, Principles of Optics (Pergamon Press, 2005); Eq. (17) in Sec. 8.3.2 is the Fresnel-Kirchhoff diffraction formula.

]. 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 z=D 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 0zD. 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 1μm (upper row), so that the distinguishability range is negligible at the wavelength scale.

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 [4

4. R. Castañeda, The Optics of Spatial Coherence Wavelets, Vol 164 of Advances in Imaging and Electron Physics, P. W. Hawkes, ed. (Academic, 2010), pp. 29–255.

]. 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 S0(±;ν)=S0(ν),|t(±)|=1,φ(±)=0,|μ(+,)|=1andα(+,)=0standfor0|ξD|a, with a the slit width. The radiant layer within the slit can be modeled as a line of N2 identical and equidistant point sources of pitch b, and the virtual layer as a second line of 2N3 equidistant point sources of pitch b/2,i.e.a=(N1)b (Fig. 2
Fig. 2 Distribution of radiant, virtual and dual point sources for modeling diffraction of spatially coherent light by a slit.
).

From them, N1 are pure virtual point sources and N2 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)
Fig. 3 Conceptual diagram of the non-paraxial propagation modes emitted by pure virtual point sources (a) to (c) and dual point sources (d) to (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)).
to 3(c) for odd-order modes and 3(d) to 3(e) for even order modes). Furthermore, the number of virtual point sources Nvirt is determined by the spatial coherence state of the field. Actually, it fulfills the condition 0Nvirt2N3,withNvirt=0 for spatially incoherent and 0<Nvirt<2N3 for partially coherent optical fields respectively.

The limit condition of b0 (with N 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 2N1 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 N 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 |t(+)||t()| in Eqs. (1) determines the positions of the point sources at the AP. It takes the forms: δ(ξD)n=0N1δ(ξAnb) for the radiant point sources, δ(ξD2nb)m=nNn1δ(ξAmb) for the virtual components of the dual point sources, and δ(ξA(2n+1)b)m=nNn2δ(ξA(m+1/2)b) for the pure virtual point sources, with δ() the Dirac’s delta.

S(xA;ν)=S(0,xA;ν)+n=1PS(2nb,xA;ν)=S0(ν){n=0N1M(nb,nb,xA)+n=1Pm=nNn1M((m+n)b,(mn)b,xA)}.
(5b)

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 b<λ<<a. Therefore, let us compare the predicted power spectra by Eqs. (5), S(xA;ν)andS(xA;ν), to each other, with the following physical parameters in both of them:

  • Slit width a = 10 μm.
  • N=20 identical radiant point sources, uniformly distributed with pitch b=0.5263μm.
  • Spatially coherent optical field of λ=0.632μm, so that b=0.83λ=0.0526a and λ=0.063a stand, which assures the accomplishment of the condition b<λ<<a.
  • Propagation distances 0.055μmz103μm.

Figure 5
Fig. 5 (Media 1) Comparing the diffraction of the spatially coherent optical field of λ=0.632μm, emitted by 20 identical r-sources uniformly distributed in a slit of width 10 μm, regarding a discrete radiant layer on the upper row and a continuous radiant layer on the bottom row. Propagation distances 0.055μmz103μm. The blue and the green profiles in the graphs on the right column denote the radiant power and the contribution of the virtual components of the d-sources to the modulating power. The red profile in the upper graph is the contribution of the pure v-sources to the modulating power.
(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 z=103μm 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
Fig. 6 Rms-error of the power spectrum profile at the bottom row on the mid-column in Fig. 4 and the figure of merit (see the text), for (a) z102μm and (b) z103μm.
for (a) z102μm and (b) z103μm. 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 102z103μm, 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 z<102μm, 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
Fig. 7 (Media 2) Comparing the interference produced by 10 identical and spatially coherent r-sources uniformly distributed on an array of length 10 μm, that emit at λ=0.632μm. Propagation distances 0.055μmz103μm. The upper row model includes pure virtual point sources, while the bottom row model excludes them.
(Media 2) for 10 identical and spatially coherent r-sources uniformly distributed on an array of length 10 μ m, that emit at λ=0.632μm for the same propagation distances as before, i.e., 0.055μmz103μm. Indeed, the rms-error is about 13% for 102z103μm and fluctuates between 8% and 32% for z<102μm, as depicted in Figs. 8 (a)
Fig. 8 Rms- error of the power spectrum profile at the bottom row on the mid-column in Fig. 6 and the profile at the upper row on the same column. The last one is taken as merit figure, (a) for z102μm and (b) for z103μm.
and 8(b).

4. Discussion and concluding remarks

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.

Acknowledgments

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

1.

M. Born, and E. Wolf, Principles of Optics (Pergamon Press, 2005); Eq. (17) in Sec. 8.3.2 is the Fresnel-Kirchhoff diffraction formula.

2.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, 1995); Eq. (4.4-25) is the Wolf’s integral equation.

3.

R. Castañeda and J. Garcia-Sucerquia, “Non-approximated numerical modeling of propagation of light in any state of spatial coherence,” Opt. Express 19(25), 25022–25034 (2011). [CrossRef] [PubMed]

4.

R. Castañeda, The Optics of Spatial Coherence Wavelets, Vol 164 of Advances in Imaging and Electron Physics, P. W. Hawkes, ed. (Academic, 2010), pp. 29–255.

5.

R. Castañeda, G. Cañas-Cardona, and J. Garcia-Sucerquia, “Radiant, virtual, and dual sources of optical fields in any state of spatial coherence,” J. Opt. Soc. Am. A 27(6), 1322–1330 (2010). [CrossRef] [PubMed]

6.

M. Testorf, B. Hennelly, and J. Ojeda-Castaneda, Phase-Space Optics: Fundamentals and Applications (Mc Graw-Hill, 2010).

7.

K. Wolf, M. Alonso, and G. Forbes, “Wigner functions for Helmholtz wave fields,” J. Opt. Soc. Am. A 16(10), 2476–2487 (1999). [CrossRef]

8.

E. Marchand and E. Wolf, “Walther’s definition of generalized radiance,” J. Opt. Soc. Am. 64(9), 1273–1274 (1974). [CrossRef]

9.

R. Castañeda, “Generalised radiant emittance in the phase-space representation of planar sources in any state of spatial coherence,” Opt. Commun. 284(19), 4259–4262 (2011). [CrossRef]

OCIS Codes
(030.1640) Coherence and statistical optics : Coherence
(030.4070) Coherence and statistical optics : Modes

ToC Category:
Coherence and Statistical Optics

History
Original Manuscript: November 2, 2012
Revised Manuscript: December 18, 2012
Manuscript Accepted: December 19, 2012
Published: May 20, 2013

Citation
Román Castañeda, David Vargas, and Esteban Franco, "Discreteness of the set of radiant point sources: a physical feature of the second-order wave-fronts," Opt. Express 21, 12964-12975 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-11-12964


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References

  1. M. Born, and E. Wolf, Principles of Optics (Pergamon Press, 2005); Eq. (17) in Sec. 8.3.2 is the Fresnel-Kirchhoff diffraction formula.
  2. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, 1995); Eq. (4.4-25) is the Wolf’s integral equation.
  3. R. Castañeda and J. Garcia-Sucerquia, “Non-approximated numerical modeling of propagation of light in any state of spatial coherence,” Opt. Express19(25), 25022–25034 (2011). [CrossRef] [PubMed]
  4. R. Castañeda, The Optics of Spatial Coherence Wavelets, Vol 164 of Advances in Imaging and Electron Physics, P. W. Hawkes, ed. (Academic, 2010), pp. 29–255.
  5. R. Castañeda, G. Cañas-Cardona, and J. Garcia-Sucerquia, “Radiant, virtual, and dual sources of optical fields in any state of spatial coherence,” J. Opt. Soc. Am. A27(6), 1322–1330 (2010). [CrossRef] [PubMed]
  6. M. Testorf, B. Hennelly, and J. Ojeda-Castaneda, Phase-Space Optics: Fundamentals and Applications (Mc Graw-Hill, 2010).
  7. K. Wolf, M. Alonso, and G. Forbes, “Wigner functions for Helmholtz wave fields,” J. Opt. Soc. Am. A16(10), 2476–2487 (1999). [CrossRef]
  8. E. Marchand and E. Wolf, “Walther’s definition of generalized radiance,” J. Opt. Soc. Am.64(9), 1273–1274 (1974). [CrossRef]
  9. R. Castañeda, “Generalised radiant emittance in the phase-space representation of planar sources in any state of spatial coherence,” Opt. Commun.284(19), 4259–4262 (2011). [CrossRef]

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