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Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 9 — May. 6, 2013
  • pp: 11276–11293
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Phase–space non-paraxial propagation modes of optical fields in any state of spatial coherence

Román Castañeda and Hernán Muñoz  »View Author Affiliations


Optics Express, Vol. 21, Issue 9, pp. 11276-11293 (2013)
http://dx.doi.org/10.1364/OE.21.011276


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Abstract

The non-paraxial marginal power spectrum is decomposed in propagation modes, so that the zeroth-order mode is only emitted by the radiant point sources at the aperture plane, while the modes of higher orders than zero are only emitted by the virtual point sources. It allows representing the non-paraxial propagation of optical fields in arbitrary states of spatial coherence and along arbitrary distances from the aperture plane without approximations, by simply using the power distribution and the spatial coherence state at the aperture plane as entries. This modal expansion is potentially useful in micro-diffraction and spatial coherence modulation.

© 2013 OSA

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

1. K. Wang, C. Zhao, and X. Lu, “Nonparaxial propagation properties for a partially coherent anomalous hollow beam,” Optik (Stuttg.) 123(3), 202–207 (2012). [CrossRef]

, 2

2. X. Li and Y. Cai, “Nonparaxial propagation of a partially coherent dark hollow beam,” Appl. Phys. B 102(1), 205–213 (2011). [CrossRef]

], Lorentz and Lorentz-Gauss beams [3

3. H. Yu, L. Xiong, and B. Lü, “Nonparaxial Lorentz and Lorentz–Gauss beams,” Optik (Stuttg.) 121(16), 1455–1461 (2010). [CrossRef]

], Hermite-Gaussian and Airy beams [4

4. B. Tang and M. Jiang, “Propagation properties of vectorial Hermite–cosine–Gaussian beams beyond the paraxial approximation,” J. Mod. Opt. 56(8), 955–962 (2009). [CrossRef]

7

7. H. Laabs, “Propagation of Hermite-Gaussian-beams beyond the paraxial approximation,” Opt. Commun. 147(1-3), 1–4 (1998). [CrossRef]

]. 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

8. H. Bethe, “Theory of diffraction by small holes,” Phys. Rev. 66(7-8), 163–182 (1944). [CrossRef]

]. 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

9. J. H. Wu, “Modeling of near-field optical diffraction from a subwavelength aperture in a thin conducting film,” Opt. Lett. 36(17), 3440–3442 (2011). [CrossRef] [PubMed]

]. The diffraction of electromagnetic plane waves through small rectangular apertures was also analysed [10

10. K. Duan and B. Lü, “Nonparaxial diffraction of vectorial plane waves at a small aperture,” Opt. Laser Technol. 37(3), 193–197 (2005). [CrossRef]

], as well as the spectral shifts and spectral switches of diffracted non-paraxial beams [11

11. G. Zhao, E. Zhang, and B. Lü, “Spectral switches of partially coherent nonparaxial beams diffracted at an aperture,” Opt. Commun. 282(2), 167–171 (2009). [CrossRef]

], and the modulation instability of non-paraxial beams in self-focussing Kerr media [12

12. H. Wang and W. She, “Modulation instability and interaction of non-paraxial beams in self-focusing Kerr media,” Opt. Commun. 254(1-3), 145–151 (2005). [CrossRef]

]. Relative few contributions regarded the spatial coherence state of the non-paraxial field [1

1. K. Wang, C. Zhao, and X. Lu, “Nonparaxial propagation properties for a partially coherent anomalous hollow beam,” Optik (Stuttg.) 123(3), 202–207 (2012). [CrossRef]

, 2

2. X. Li and Y. Cai, “Nonparaxial propagation of a partially coherent dark hollow beam,” Appl. Phys. B 102(1), 205–213 (2011). [CrossRef]

, 11

11. G. Zhao, E. Zhang, and B. Lü, “Spectral switches of partially coherent nonparaxial beams diffracted at an aperture,” Opt. Commun. 282(2), 167–171 (2009). [CrossRef]

]. They attempted to solve the Wolf’s integral equation [13

13. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, 1995). Equation (4).4–25) is the Wolf’s integral equation.

] 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

14. 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]

]. 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

14. 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]

]). 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

15. M. Testorf, B. Hennelly, and J. Ojeda-Castaneda, Phase-space optics: fundamentals and applications (Mc Graw-Hill, New York, 2010).

18

18. R. Castañeda, R. Betancur, J. Herrera, and J. Carrasquilla, “Phase-space representation and polarization domains of random electromagnetic fields,” Appl. Opt. 47(22), E27–E38 (2008). [CrossRef] [PubMed]

]. It allows calculating them by applying Fourier analysis tools. In spite of their approximation, WDFs are also proposed for non-paraxial fields [19

19. C. J. Sheppard and K. G. Larkin, “Wigner function for nonparaxial wave fields,” J. Opt. Soc. Am. A 18(10), 2486–2490 (2001). [CrossRef] [PubMed]

]. 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.

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

  • The non-paraxial modal expansion describes the propagation of both the radiant and the modulating energies, provided by sets of point sources that distribute on separate but inserted layers at the AP, named the radiant and the virtual layer respectively [20

    20. 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]

    ]. The virtual point sources are turned on by pairs of correlated radiant point sources.
  • Each non-paraxial mode is associated to pairs of correlated radiant point sources with a given separation vector. The zeroth-order mode describes the radiant energy emitted by any individual radiant point source. The remaining modes are ordered and labelled with integer suffices according to the lengths of the separation vectors of the source pairs.
  • Virtual point sources are placed at the centres of corresponding structured spatial coherence supports [21

    21. R. Castañeda, H. Muñoz-Ossa, and G. Cañas-Cardona, “The structured spatial coherence support,” J. Mod. Opt. 58(11), 962–972 (2011). [CrossRef]

    ]. There are also structured supports that simultaneously contain radiant point sources at their centres. Therefore, any structured spatial coherence support can be individually characterized by the finite non-paraxial modal expansion, so that the energy emitted by the radiant point source, if any, is described by the zeroth-order mode, while the modulating energy emitted by the virtual point source is described by the set of modes with higher order than zero, each one describing the contribution of the corresponding pair of correlated radiant point sources that belongs to the structured support. This capability is reported by the first time in this work and allows accessing and manipulating specific structured spatial coherence supports individually, which is very useful in spatial coherence modulation for instance [22

    22. R. Castañeda, “The optics of spatial coherence wavelets,” in: Advances in imaging and electron physics164, P. W. Hawkes, ed. (Academic Press, 2010), pp. 29–255.

    ].
  • The set of all the correlated pairs of radiant point sources with the same separation vector across the AP constitutes the class of pairs with the order specified by its characteristic separation vector [22

    22. R. Castañeda, “The optics of spatial coherence wavelets,” in: Advances in imaging and electron physics164, P. W. Hawkes, ed. (Academic Press, 2010), pp. 29–255.

    ]. In this sense, the set of radiant point sources is the zeroth-order class. All the pairs of each class emit its energy contribution in the same characteristic mode, no matter they belong to different structured supports. However, the energy contribution of the whole class has the same shape as the characteristic mode of its pairs only in the far-field propagation (i.e. the Fraunhofer domain). For shorter propagation distances, the individual modes are shifted to each other, and therefore the shape of their superposition differs from the shape of any of them. The capability of identifying the classes of pairs allows applying novel procedures as the class filtering [22

    22. R. Castañeda, “The optics of spatial coherence wavelets,” in: Advances in imaging and electron physics164, P. W. Hawkes, ed. (Academic Press, 2010), pp. 29–255.

    ]. It gives new develop perspectives to the optical processing, useful in partially coherent imaging for instance.

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

2.1. Definition of the modes

Let us regard the correlated pair of radiant point sources with ξD=b that belongs to the structured spatial coherent support centred at ξA=a on the AP, and introduce the function
F(a+b/2,ab/2,rA;ν)=14λ2(z+z2+|rAa|2+|b|2/4+abrAb)(z+z2+|rAa|2+|b|2/4ab+rAb)×exp[ikz2+|rAa|2+|b|2/4+abrAb]z2+|rAa|2+|b|2/4+abrAbexp[ikz2+|rAa|2+|b|2/4ab+rAb]z2+|rAa|2+|b|2/4ab+rAb,
(2)
which has hermitic symmetry, i.e. F(a+b/2,ab/2,rA;ν)=F(ab/2,a+b/2,rA;ν), with the asterisk denoting complex conjugated. According to Eq. (1), the expression
S0(a+b/2)t(a+b/2)S0(ab/2)t*(ab/2)μ(a+b/2,ab/2)F(a+b/2,ab/2,rA;ν)+S0(ab/2)t(ab/2)S0(a+b/2)t*(a+b/2)μ(ab/2,a+b/2)F(ab/2,a+b/2,rA;ν)
(3)
represents the contribution of such pair of radiant point sources to the marginal power spectrum emitted at the centre of this support. It includes the two degrees of freedom in orientation of the separation vector ξD=b with the same weight, because the radiators of the pair are not distinguishable. On account of the hermitic symmetry of the complex degree of spatial coherence, μ(a+b/2,ab/2)=μ(ab/2,a+b/2), Eq. (3) becomes
2S0(a+b/2)|t(a+b/2)|S0(ab/2)|t(ab/2)||μ(a+b/2,ab/2)|×Re{F(a+b/2,ab/2,rA;ν)exp[iΔφ(a+b/2,ab/2)+iα(a+b/2,ab/2)]},
(4)
with Re denoting the real part, Δϕ(a+b/2,ab/2)=ϕ(a+b/2)ϕ(ab/2) and

Re{F(a+b/2,ab/2,rA;ν)exp[iΔϕ(a+b/2,ab/2)+iα(a+b/2,ab/2)]}=14λ2(z+z2+|rAa|2+|b|2/4+abrAbz2+|rAa|2+|b|2/4+abrAb)(z+z2+|rAa|2+|b|2/4ab+rAbz2+|rAa|2+|b|2/4ab+rAb).×cos[kz2+|rAa|2+|b|2/4+abrAbkz2+|rAa|2+|b|2/4ab+rAb+Δϕ+α]
(5)

The structured support of spatial coherence centred at ξA=a contains one and only one correlated radiator pair with separation vector ξD=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 2Re[F(a+b/2,ab/2,rA;ν)] 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 ξA=a. The coefficient S0(a+b/2)|t(a+b/2)|S0(ab/2)|t(ab/2)||μ(a+b/2,ab/2)| specifies the weight of this mode in such marginal power spectrum. The phases Δϕ(a+b/2,ab/2) and α(a+b/2,ab/2) 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 |ξD|=|b|. 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

20. 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]

]. It allows labelling the modes by entire order-suffixes n=0,1,2, 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 |ξD|=0, i.e. the individual point sources of the radiant layer. Taking into account that Δϕ(a,a)=0,|μ(a,a)|=1andα(a,a)=0 hold [13

13. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, 1995). Equation (4).4–25) is the Wolf’s integral equation.

], and stating b=0, Eq. (5) yields
2Re{F(a,rA;ν)}=12λ2(z+z2+|rAa|2z2+|rAa|2)2
(6)
for the zeroth-order propagation mode emitted by the radiant point source placed at ξA=a. This mode is called the free-space diffraction envelope [14

14. 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 is the unique mode in which any radiant point source emits. Its main features are:

  • It is real valued, positive definite and Lorentzian-shaped along any propagation distance.
  • Its modal expansion coefficient is given by S0(a)|t(a)|2, i.e. the radiant power emitted by the corresponding radiant point source and the absorbance of the AP at the source position.
  • Its maximum is placed at rA=a and takes the value 2Re{F(a,a;ν)}=2(1/λz)2, i.e. it decays according to the 1/z2-law [25

    25. M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon Press, 1993).

    ]. Consequently, the mode spreading over the OP increases along the field propagation in order to fulfil the conservation law of the radiant energy [25

    25. M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon Press, 1993).

    ].

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 |ξD|>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 b0, belonging to the structured spatial coherence support centred at ξA=a, is propagated by the non-paraxial mode 2Re[F(a+b/2,ab/2,rA;ν)] emitted by the virtual point source placed at ξA=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

14. 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]

]. 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/z2-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 1δ(ξD)+[1δ(ξD)] in the integral of Eq. (1), with δ(ξD) 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 ξA=a on the AP can be expressed in terms of the non-paraxial propagation modes as follows

S(a,rA;ν)=S0(a)|t(a)|2F(a,rA;ν)+2APξD0S0(a+ξD/2)|t(a+ξD/2)|S0(aξD/2)|t(aξD/2)||μ(a+ξD/2,aξD/2)|×Re{F(a+ξD/2,aξD/2,rA;ν)exp[iΔφ(a+ξD/2,aξD/2)+iα(a+ξD/2,aξD/2)]}d2ξD.
(7)

The power spectrum at the OP for the propagation distance z0 is given by S(rA;ν)=APS(ξA,rA;ν)d2ξA [14

14. 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 results from a linear combination of the non-paraxial propagation modes, obtained by integrating Eq. (7) over the positions ξA on the AP. Thus,

S(rA;ν)=Srad(rA;ν)+Svirt(rA;ν)holds,withSrad(rA;ν)=APSrad(ξA,rA;ν)d2ξA=APS0(ξA)|t(ξA)|2F(ξA,rA;ν)d2ξA 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
Svirt(rA;ν)=2APAPξD0S0(ξA+ξD/2)|t(ξA+ξD/2)|S0(ξAξD/2)|t(ξAξD/2)||μ(ξA+ξD/2,ξAξD/2)|×Re{F(ξA+ξD/2,ξAξD/2,rA;ν)exp[iΔϕ(ξA+ξD/2,ξAξD/2)+iα(ξA+ξD/2,ξAξD/2)]}d2ξDd2ξA
the modulating power provided by the point sources of the whole virtual layer at the AP [14

14. 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]

, 22

22. R. Castañeda, “The optics of spatial coherence wavelets,” in: Advances in imaging and electron physics164, P. W. Hawkes, ed. (Academic Press, 2010), pp. 29–255.

], i.e. the combination of the higher order modes (n>0) emitted by all the virtual point sources. The conservation law of the total energy of the field imposes the condition OPS(rA;ν)d2rA=APS0(ξA)|t(ξA)|2d2ξA, whose right member is the total energy of the field emerging from the AP, while the left member is the total energy of the field at the OP for any propagation distance, which takes the form OPS(rA;ν)d2rA=OPSrad(rA;ν)d2rA+OPSvirt(rA;ν)d2rA. On account of the independence of the non-paraxial propagation modes, the achievement of the conservation law of the total energy of the field implies that:

  • OPF(ξA,rA;ν)d2rA=1, regardless the position of the emitting radiant point source on the AP and the propagation distance z from the AP to the OP, i.e. all the zeroth-order modes are identically Lorentzian-shaped and normalized at a given propagation distance.
  • OPSvirt(rA;ν)d2rA=0,i.e.OPRe{F(ξA+ξD/2,ξAξD/2,rA;ν)}d2rA=0 must hold for any higher-order non-paraxial propagation mode, regardless the position of the emitting virtual point source on the AP (even regardless the specific correlated pair that contributes the mode) and the propagation distance to the OP. It means that the positive and negative modulating energies of each higher order mode must be symmetrically distributed onto the OP at any propagation distance, independently of the mode evolution due to the field propagation. It is assured by the cosine function in Eq. (5).

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

14. 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]

, 22

22. R. Castañeda, “The optics of spatial coherence wavelets,” in: Advances in imaging and electron physics164, P. W. Hawkes, ed. (Academic Press, 2010), pp. 29–255.

]) 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

22. R. Castañeda, “The optics of spatial coherence wavelets,” in: Advances in imaging and electron physics164, P. W. Hawkes, ed. (Academic Press, 2010), pp. 29–255.

]. Such access can be formalised with basis on the non-paraxial propagation modes too, by introducing the quantity (with power units) S(ξD,rA;ν), that fulfils the expression S(rA;ν)=APS(ξD,rA;ν)d2ξD 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 ξD across the AP, to the power spectrum at the OP. S(ξD,rA;ν) can be defined in terms of the following expressions:
S(0,rA;ν)=APS0(ξA)|t(ξA)|2F(ξA,rA;ν)d2ξA,
(8)
for ξD=0, and
S(b,rA;ν)=2APξD0S0(ξA+b/2)|t(ξA+b/2)|S0(ξAb/2)|t(ξAb/2)||μ(ξA+b/2,ξAb/2)|×Re{F(ξA+b/2,ξAb/2,rA;ν)exp[iΔϕ(ξA+b/2,ξAb/2)+iα(ξA+b/2,ξAb/2)]}d2ξA,
(9)
for ξD=b. The integrands of Eqs. (8) and (9) represent the classes of pairs of the order corresponding to its characteristic separation vector. Accordingly, Eq. (8) establishes the power contribution at the OP provided by the zeroth-order class, i.e. by the set of point sources of the whole radiant layer, so that S(0,rA;ν)=Srad(rA;ν)0 holds; while Eq. (9) establishes the (modulating) power contribution at the OP provided by the class of order corresponding to ξD=b0, i.e. by the set of point sources of the virtual layer associated to the structured supports that contain such radiator pairs.

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

26. R. Betancur and R. Castañeda, “Spatial coherence modulation,” J. Opt. Soc. Am. A 26(1), 147–155 (2009). [CrossRef] [PubMed]

].

3. Micro-diffraction and spatial coherence modulation

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
Fig. 1 Distribution of radiant, virtual and dual point sources for modelling micro-diffraction of spatially coherent light (λ=0.632μm) by a slit.
). The attributes of the virtual segment can be changed by adjusting the spatial coherence state of the field at the slit. Thus, Nrad=(L/b)+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 0Nvirt2Nrad3 is the number of virtual point sources, whose value depends on the spatial coherence state of the field, i.e. Nvirt=0 for spatially incoherent fields and Nvirt=2Nrad3 for spatially coherent fields, for instance. Diffraction is assured under the conditions L>λandbλ [14

14. 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]

]. For λ<b<L the arrangement of radiant point sources behaves as an interference grating, and for b<L<λ it behaves closely similar to an isolated radiant point source [14

14. 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]

]. 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.

Thus, the following parameters were assumed: λ=0.632μm,b=0.3μmandNrad=10,sothatL=2.7μmandz25.5μm, i.e. there are 10 identical radiant point sources within a slit about 4λ wide, distributed with a pitch of about λ/2, as sketched in Fig. 1, and a maximum propagation distance of about 40λ is considered. These parameters configure a micro-diffraction situation. The array of radiant point sources turns on an array of Nvirt17 point sources on the virtual layer. If Nvirt=17 stands, the array of virtual point sources is uniformly distributed on a segment of length 2.4μm, with pitch 0.15μm (Fig. 1).

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

13. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, 1995). Equation (4).4–25) is the Wolf’s integral equation.

], i.e. |t(ξA+ξD/2)|=|t(ξAξD/2)|=1,ϕ(ξA+ξD/2)=ϕ(ξAξD/2)=0,|μ(ξA+ξD/2,ξAξD/2)|=exp((ξDnb)22σ2) and α(ξA+ξD/2,ξAξD/2)=0withσ2 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 σ=0andσ= denote the ideal spatially incoherent and spatially coherent illuminations).

Let us also label (with 1n9):

  • Mn(ξA,xA)=2Re[F(ξA+nb/2,ξAnb/2,xA;ν)] the non-paraxial propagation mode of n-order, emitted by the virtual or dual point source at the position ξA on the AP.
  • Gn(ξA,xA)S0(ν)exp((ξDnb)22σ2)m=n2Nn21Mn(ξAmb,xA) the n-order class of radiator pairs. It is the set of n-order modes emitted by all the virtual point sources belonging to this class.
  • S(nb,xA;ν)=APξD0Gn(ξA,xA)d2ξA the contribution of the n-order class at the AP to the power spectrum at the OP.

Only one odd- and one even-order class were arbitrary chosen in three spatial coherence states, labelled by σ=3,6,inf (Figs. 4
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 σ. 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 G3(ξA,xA) is the third-order mode M3(ξA,xA) emitted by the pure virtual point source placed at the ξA-coordinate of the corresponding mode.
and 5
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 G8(ξA,xA) is the eighth-order mode M8(ξA,xA) emitted by the virtual component of the dual point source placed at the ξA-coordinate of the corresponding mode.
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 ξA–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 0.7μmz25μm 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

26. R. Betancur and R. Castañeda, “Spatial coherence modulation,” J. Opt. Soc. Am. A 26(1), 147–155 (2009). [CrossRef] [PubMed]

], 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(3b,xA;ν), for the different values of σ. This behaviour is potentially useful in beam shaping for lithography and optical tweezers for instance.

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 nGn(ξA,xA), and the modulating energy profile provided by the whole virtual layer is obtained from the expression APnGn(ξA,xA)dξA.

The diagrams of nGn(ξA,xA) are shown on the top rows and the profiles of APnGn(ξA,xA)dξA 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 σ. 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 σ) include all the classes, some of them (beginning with the high-order classes) are dropped out as σ 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

21. R. Castañeda, H. Muñoz-Ossa, and G. Cañas-Cardona, “The structured spatial coherence support,” J. Mod. Opt. 58(11), 962–972 (2011). [CrossRef]

, 22

22. R. Castañeda, “The optics of spatial coherence wavelets,” in: Advances in imaging and electron physics164, P. W. Hawkes, ed. (Academic Press, 2010), pp. 29–255.

], 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

26. R. Betancur and R. Castañeda, “Spatial coherence modulation,” J. Opt. Soc. Am. A 26(1), 147–155 (2009). [CrossRef] [PubMed]

].

Figures 9
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 λ=0.632μm, diffracted by a slit of width L=2.7μm, along the propagation distance 0.1μmz0.5μm<λ<L. Units of axes xA and z are μm, and of the vertical axis are arbitrary.
to 11 show the profiles of the radiant and modulating powers and the power spectrum at the OP for λ=0.632μm,b=0.3μmandL=2.7μm, with the propagation distances: 0.1μmz0.5μm<λ<L(Fig.9),b<0.5μmz3.5μm(Fig.10)andb<λ<L<<8μmz14μm(Fig.11). They reveal interesting features of the optical field evolution in the micro-diffraction domain.

Figure 9 points out that

  • i) This methodology is able to determine power distributions at sub-wavelength propagation distances, i.e. it overcomes the limitations of conventional procedures at such distances [25

    25. M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon Press, 1993).

    ]. In fact, the maxima of the Lorentzian profiles of the zeroth-order modes provided by the radiant point sources determine the positions of these sources, as shown in Fig. 9(a) for z=0.1,0.2,0.3μm. This capability is a consequence of the inclusion of pure virtual point sources at the midpoints between the pairs of consecutive radiant point sources. It is a necessary condition for the accurate description of the field at such propagation distances, even by low spatial coherence [14

    14. 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]

    ].
  • ii) The existence of modulating power at these propagation distances points out that the interactions between pairs of radiant point sources, that give raise the interference and diffraction patterns, begin before the field propagates in the wave superposition region. These linkages between such pairs characterize the spatial coherence state of the light at the AP, and are described by the high-order modes. Their superposition oscillates between positive and negative values and does not clearly obey the 1/z2-law (Fig. 9(b)). In contrast, the profiles in Fig. 9(a) are positive-definite and obey such law of propagation.
  • iii) Because of the modulating power, the power spectrum differs from the radiant power distribution except if the optical field is fully spatially incoherent. However, the profile for z=0.1μm accounts for the discreteness of the set of radiant point sources at the AP. Furthermore, all the power spectrum profiles are positive definite and fulfil the 1/z2-law (Fig. 9(c)), which is in accordance with the fact that the modulating power only redistribute the radiant power without changing its physical attributes.

The individual zeroth-order modes are not resolvable in the profiles of the radiant power shown in Fig. 10
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 0.5μmz3.5μm. Units of axes xA and z are μm, and of the vertical axis are arbitrary.
, 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/z2-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/z2-law of propagation.

The radiant power profiles in Fig. 11(a)
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 λ<L<<8μmz14μm, i.e. in the Fraunhofer domain. Units of axes xA and z are μm, and of the vertical axis are arbitrary.
are propagation invariant, positive definite and follow the 1/z2-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/z2-law.

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=14μm. 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.

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.

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.

K. Wang, C. Zhao, and X. Lu, “Nonparaxial propagation properties for a partially coherent anomalous hollow beam,” Optik (Stuttg.) 123(3), 202–207 (2012). [CrossRef]

2.

X. Li and Y. Cai, “Nonparaxial propagation of a partially coherent dark hollow beam,” Appl. Phys. B 102(1), 205–213 (2011). [CrossRef]

3.

H. Yu, L. Xiong, and B. Lü, “Nonparaxial Lorentz and Lorentz–Gauss beams,” Optik (Stuttg.) 121(16), 1455–1461 (2010). [CrossRef]

4.

B. Tang and M. Jiang, “Propagation properties of vectorial Hermite–cosine–Gaussian beams beyond the paraxial approximation,” J. Mod. Opt. 56(8), 955–962 (2009). [CrossRef]

5.

A. V. Novitsky and D. V. Novitsky, “Nonparaxial Airy beams: role of evanescent waves,” Opt. Lett. 34(21), 3430–3432 (2009). [CrossRef] [PubMed]

6.

Y. Zhang, “Nonparaxial propagation analysis of elliptical Gaussian beams diffracted by a circular aperture,” Opt. Commun. 248(4-6), 317–326 (2005). [CrossRef]

7.

H. Laabs, “Propagation of Hermite-Gaussian-beams beyond the paraxial approximation,” Opt. Commun. 147(1-3), 1–4 (1998). [CrossRef]

8.

H. Bethe, “Theory of diffraction by small holes,” Phys. Rev. 66(7-8), 163–182 (1944). [CrossRef]

9.

J. H. Wu, “Modeling of near-field optical diffraction from a subwavelength aperture in a thin conducting film,” Opt. Lett. 36(17), 3440–3442 (2011). [CrossRef] [PubMed]

10.

K. Duan and B. Lü, “Nonparaxial diffraction of vectorial plane waves at a small aperture,” Opt. Laser Technol. 37(3), 193–197 (2005). [CrossRef]

11.

G. Zhao, E. Zhang, and B. Lü, “Spectral switches of partially coherent nonparaxial beams diffracted at an aperture,” Opt. Commun. 282(2), 167–171 (2009). [CrossRef]

12.

H. Wang and W. She, “Modulation instability and interaction of non-paraxial beams in self-focusing Kerr media,” Opt. Commun. 254(1-3), 145–151 (2005). [CrossRef]

13.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, 1995). Equation (4).4–25) is the Wolf’s integral equation.

14.

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]

15.

M. Testorf, B. Hennelly, and J. Ojeda-Castaneda, Phase-space optics: fundamentals and applications (Mc Graw-Hill, New York, 2010).

16.

A. Torre, Linear ray and wave optics in the phase-space (Elsevier, 2005).

17.

R. Castañeda, “Phase-space representation of electromagnetic radiometry,” Phys. Scr. 79, 035302 (10pp) (2009).

18.

R. Castañeda, R. Betancur, J. Herrera, and J. Carrasquilla, “Phase-space representation and polarization domains of random electromagnetic fields,” Appl. Opt. 47(22), E27–E38 (2008). [CrossRef] [PubMed]

19.

C. J. Sheppard and K. G. Larkin, “Wigner function for nonparaxial wave fields,” J. Opt. Soc. Am. A 18(10), 2486–2490 (2001). [CrossRef] [PubMed]

20.

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]

21.

R. Castañeda, H. Muñoz-Ossa, and G. Cañas-Cardona, “The structured spatial coherence support,” J. Mod. Opt. 58(11), 962–972 (2011). [CrossRef]

22.

R. Castañeda, “The optics of spatial coherence wavelets,” in: Advances in imaging and electron physics164, P. W. Hawkes, ed. (Academic Press, 2010), pp. 29–255.

23.

Z. Zang, T. Minato, P. Navaretti, Y. Hinokuma, M. Duelk, C. Velez, and K. Hamamoto, “High-power (>110 mW) superluminescent diodes by using active multimode interferometer,” IEEE Photon. Technol. Lett. 22(10), 721–723 (2010). [CrossRef]

24.

Z. Zang, K. Mukai, P. Navaretti, M. Duelk, C. Velez, and K. Hamamoto, “High Power and Stable High Coupling Efficiency (66%) Superluminescent Light Emitting Diodes by Using Active Multi-Mode Interferometer,” IEICE Trans. Elec. E94-C, 862–864 (2011).

25.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon Press, 1993).

26.

R. Betancur and R. Castañeda, “Spatial coherence modulation,” J. Opt. Soc. Am. A 26(1), 147–155 (2009). [CrossRef] [PubMed]

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: December 3, 2012
Revised Manuscript: January 23, 2013
Manuscript Accepted: January 30, 2013
Published: May 1, 2013

Citation
Román Castañeda and Hernán Muñoz, "Phase–space non-paraxial propagation modes of optical fields in any state of spatial coherence," Opt. Express 21, 11276-11293 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-9-11276


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References

  1. K. Wang, C. Zhao, and X. Lu, “Nonparaxial propagation properties for a partially coherent anomalous hollow beam,” Optik (Stuttg.)123(3), 202–207 (2012). [CrossRef]
  2. X. Li and Y. Cai, “Nonparaxial propagation of a partially coherent dark hollow beam,” Appl. Phys. B102(1), 205–213 (2011). [CrossRef]
  3. H. Yu, L. Xiong, and B. Lü, “Nonparaxial Lorentz and Lorentz–Gauss beams,” Optik (Stuttg.)121(16), 1455–1461 (2010). [CrossRef]
  4. B. Tang and M. Jiang, “Propagation properties of vectorial Hermite–cosine–Gaussian beams beyond the paraxial approximation,” J. Mod. Opt.56(8), 955–962 (2009). [CrossRef]
  5. A. V. Novitsky and D. V. Novitsky, “Nonparaxial Airy beams: role of evanescent waves,” Opt. Lett.34(21), 3430–3432 (2009). [CrossRef] [PubMed]
  6. Y. Zhang, “Nonparaxial propagation analysis of elliptical Gaussian beams diffracted by a circular aperture,” Opt. Commun.248(4-6), 317–326 (2005). [CrossRef]
  7. H. Laabs, “Propagation of Hermite-Gaussian-beams beyond the paraxial approximation,” Opt. Commun.147(1-3), 1–4 (1998). [CrossRef]
  8. H. Bethe, “Theory of diffraction by small holes,” Phys. Rev.66(7-8), 163–182 (1944). [CrossRef]
  9. J. H. Wu, “Modeling of near-field optical diffraction from a subwavelength aperture in a thin conducting film,” Opt. Lett.36(17), 3440–3442 (2011). [CrossRef] [PubMed]
  10. K. Duan and B. Lü, “Nonparaxial diffraction of vectorial plane waves at a small aperture,” Opt. Laser Technol.37(3), 193–197 (2005). [CrossRef]
  11. G. Zhao, E. Zhang, and B. Lü, “Spectral switches of partially coherent nonparaxial beams diffracted at an aperture,” Opt. Commun.282(2), 167–171 (2009). [CrossRef]
  12. H. Wang and W. She, “Modulation instability and interaction of non-paraxial beams in self-focusing Kerr media,” Opt. Commun.254(1-3), 145–151 (2005). [CrossRef]
  13. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, 1995). Equation (4).4–25) is the Wolf’s integral equation.
  14. 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]
  15. M. Testorf, B. Hennelly, and J. Ojeda-Castaneda, Phase-space optics: fundamentals and applications (Mc Graw-Hill, New York, 2010).
  16. A. Torre, Linear ray and wave optics in the phase-space (Elsevier, 2005).
  17. R. Castañeda, “Phase-space representation of electromagnetic radiometry,” Phys. Scr.79, 035302 (10pp) (2009).
  18. R. Castañeda, R. Betancur, J. Herrera, and J. Carrasquilla, “Phase-space representation and polarization domains of random electromagnetic fields,” Appl. Opt.47(22), E27–E38 (2008). [CrossRef] [PubMed]
  19. C. J. Sheppard and K. G. Larkin, “Wigner function for nonparaxial wave fields,” J. Opt. Soc. Am. A18(10), 2486–2490 (2001). [CrossRef] [PubMed]
  20. 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]
  21. R. Castañeda, H. Muñoz-Ossa, and G. Cañas-Cardona, “The structured spatial coherence support,” J. Mod. Opt.58(11), 962–972 (2011). [CrossRef]
  22. R. Castañeda, “The optics of spatial coherence wavelets,” in: Advances in imaging and electron physics164, P. W. Hawkes, ed. (Academic Press, 2010), pp. 29–255.
  23. Z. Zang, T. Minato, P. Navaretti, Y. Hinokuma, M. Duelk, C. Velez, and K. Hamamoto, “High-power (>110 mW) superluminescent diodes by using active multimode interferometer,” IEEE Photon. Technol. Lett.22(10), 721–723 (2010). [CrossRef]
  24. Z. Zang, K. Mukai, P. Navaretti, M. Duelk, C. Velez, and K. Hamamoto, “High Power and Stable High Coupling Efficiency (66%) Superluminescent Light Emitting Diodes by Using Active Multi-Mode Interferometer,” IEICE Trans. Elec.E94-C, 862–864 (2011).
  25. M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon Press, 1993).
  26. R. Betancur and R. Castañeda, “Spatial coherence modulation,” J. Opt. Soc. Am. A26(1), 147–155 (2009). [CrossRef] [PubMed]

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