Electromagnetic wave fields in the microdiffraction domain

Paraxial propagation theories are not suitable for describing the behavior of electromagnetic wave fields in any states of spatial coherence and polarization in the microdiffraction domain. The proposed nonparaxial theory overcomes such limitations by modeling (i) any planar source in terms of sets of point sources with tensor statistical behavior, and (ii) the transport of the wave field on scalar, deterministic, time-independent, and nonparaxial propagation modes, defined only by the geometry of the boundary conditions of the experimental setups. So the field emission by the source and the space structure due to the modes are independent from each other. The theory provides a unified framework for the power and the states of spatial coherence and polarization of the field, as well as for interference and diffraction, and describes the significant changes suffered by the wave fields in the microdiffraction domain.

Figure 1

Geometry of the nonparaxial setups for the microdiffraction domain.

Figure 2

Conceptual sketches of the planar source modeling in terms of pure radiant (r), pure virtual (v), and dual (d) point sources. (a) Circular-shaped and (b) square-shaped planar sources. The Cartesian axes in (b) determine the ${\xi }_A$ positions of the point sources.

Figure 3

Transport of the power emitted by three radiant point sources at the scale metric λ = 0.632 μm in the microdiffraction domain. Zero-order nonparaxial modes for (a) $0\le z\le 8\lambda$ and (b) $0\le z\le {10}^3\lambda$. (c, d) Cross-section profiles at specific propagation distances in (a) and (b), respectively. The contributions of the individual radiant point sources are resolved in (a) and (c), but not in (b) and (d). The modes are enhanced and the profiles are scaled for presentation purposes.

Figure 4

Modal expansion for the transport of the modulating power emitted by the three virtual point sources associated to the radiant point sources in Fig. 3, at the scale metric λ = 0.632 μm, for (a) $0\le z\le 8\lambda$ and (b) ${10}^2\lambda \le z\le {10}^3\lambda$. (c, d) Cross-section profiles at specific propagation distances in (a) and (b), respectively. The positions of the virtual point sources are represented by the small circles in (a), and the corresponding pairs of radiant point sources is indicated. The modal expansion is enhanced and the main maxima in (c) are truncated for presentation purposes.

Figure 5

Transport of the power spectrum provided by the strong correlated radiant point sources in Fig. 3 along (a) $0\le z\le 8\lambda$ and (b) ${10}^2\lambda \le z\le {10}^3\lambda$. (c, d) Power spectra profiles, corresponding to the cross sections of (a) and (b) at specific propagation distances. The positions of the radiant r and the virtual v point sources are represented by the small circles in (a). The graphs in (a) and (b) are enhanced and the main maxima in (c) are truncated for presentation purposes.

Figure 6

Transport of the power spectrum emitted by the partially correlated radiant point sources in Fig. 3 for (a) $0\le z\le 8\lambda$ and (b) ${10}^2\lambda \le z\le {10}^3\lambda$. (c, d) Power spectra profiles corresponding to the cross sections of (a) and (b) at specific propagation distances. The positions of the radiant r and the virtual v point sources are represented by the small circles in (a). The graphs in (a) and (b) are enhanced and the main maxima in (c) are truncated for presentation purposes.

Figure 7

Transport of the electromagnetic wave field provided by the planar source in the examples before. The three radiant point sources are linearly polarized with null phase differences in between. The state of the source at ${\mathit{\xi }}_A=-2.5\lambda$ is orthogonal to the states of the other two, which are mutually parallel. (a) Graph of the modulating power for $0\le z\le 8\lambda$ (λ = 0.632 μm). (b, c) Profiles of the modulating power at specific propagation distances. (d) Graph of the power spectrum for $0\le z\le 8\lambda$. (e, f) Profiles of the power spectrum at the same propagation distances as in (b) and (c), respectively.

Figure 8

Transport of the electromagnetic wave field provided by the planar source in the examples before. The three radiant point sources are mutually parallel linearly polarized, with null phase differences and spacing 0.8λ. (a) Graph of the modulating power for $0\le z\le 3\lambda$ (λ = 0.632 μm). (b, c) Profiles of the modulating power at specific propagation distances. (d) Graph of the power spectrum for $0\le z\le 3\lambda$. (e, f) Profiles of the power spectrum at the same propagation distances as in (b) and (c), respectively.

Figure 9

Transport of the electromagnetic wave field provided by the planar source in Fig. 8, but the polarization state of the mid-source was turned orthogonal to the polarization states of the other two. (a) Graph of the modulating power for $0\le z\le 3\lambda$ (λ = 0.632 μm). (b, c) Profiles of the modulating power at specific propagation distances. (d) Graph of the power spectrum for $0\le z\le 3\lambda$. (e, f) Profiles of the power spectrum at the same propagation distances as in (b) and (c), respectively.

Figure 10

Graphs of Eq. (24a) for the structured support centered at $x_A=0$. The spatially incoherent planar source is a line of 11 identical radiant point sources with pitch λ/2, at the scale metric λ = 0.632 μm. (a) Magnitude and (b) phase for $0\le z\le 8\lambda$; the same quantities are shown in (c) and (d), respectively, for ${10}^2\lambda \le z\le {10}^3\lambda$. The graphs in (a) and (c) are enhanced for presentation purposes.

Figure 11

Graphs of Eq. (24a) for the structured support centered at $x_A=500\lambda$ on the OP, provided by the same planar source in Fig. 10 for (a) $0\le z\le {10}^2\lambda$ and (b) ${10}^2\lambda \le z\le {10}^3\lambda$. The graphs are enhanced for presentation purposes.

Figure 12

Profiles of the moduli of the degree of spatial coherence in Figs. 10 and 11, and of the paraxial approached sinclike degree for ${10}^4\lambda \le z\le {10}^6\lambda$, i.e., such profiles remain invariant along this propagation segment.

Figure 13

Profiles of the interference patterns produced by three identical, spatially coherent, and linearly polarized radiant point sources in the far field $\left(z={10}^8\lambda ,\lambda =0.632\mu \mathrm{m}\right)$. (a) Mutually parallel polarization states and spacing 5λ. (b) Mutually parallel polarization states and spacing 500λ. The state of the source at ${\mathit{\xi }}_A=-2.5\lambda$ is turned orthogonal to the states of the other two in (c) and (d), and the spacing is the same as in (a) and (b), respectively.

Figure 14

Profiles of the diffraction patterns produced by identical, spatially coherent, and mutually parallel, linearly polarized radiant point sources in the far field ($z$ = 10${}^4$ μm). The exact nonparaxial pattern is provided by three sources with spacing 0.8λ in (a) and by 16 sources with spacing 0.3λ in (b) (λ = 0.632 μm). The paraxial approached pattern is provided by a slit of width 1.6λ in (a) and 4.5λ in (b).