Fundamental structure of Fresnel diffraction: natural sampling grid and the fractional Fourier transform

Özaktaş, Haldun M.; Arık, Sercan Ö.; Coşkun, Türker

doi:10.1364/ol.36.002524

Cited by 22 publications

(20 citation statements)

References 19 publications

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“…The derivation of these results are similar to those in [37] and are thus not repeated. On the output plane, the space-bandwidth product is σ ″ = Δ ″Δ ″ N x x .…”

Section: Transverse Sampling Spacingmentioning

confidence: 66%

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Optimal representation and processing of optical signals in quadratic-phase systems

Arık

Özaktaş

2016

Optics Communications

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a b s t r a c tOptical fields propagating through quadratic-phase systems (QPSs) can be modeled as magnified fractional Fourier transforms (FRTs) of the input field, provided we observe them on suitably defined spherical reference surfaces. Non-redundant representation of the fields with the minimum number of samples becomes possible by appropriate choice of sample points on these surfaces. Longitudinally, these surfaces should not be spaced equally with the distance of propagation, but with respect to the FRT order. The non-uniform sampling grid that emerges mirrors the fundamental structure of propagation through QPSs. By providing a means to effectively handle the sampling of chirp functions, it allows for accurate and efficient computation of optical fields propagating in QPSs.

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“…The derivation of these results are similar to those in [37] and are thus not repeated. On the output plane, the space-bandwidth product is σ ″ = Δ ″Δ ″ N x x .…”

Section: Transverse Sampling Spacingmentioning

confidence: 66%

“…The parameters Δx and σ Δ x defines s and δa, which mirrors the information content of the set of signals. The special value σ = Δ Δ s x/ x matches the structure of the grid to the natural propensity of the signal to diffract, as determined by its space and frequency extent [37]. Fig.…”

Section: Discussionmentioning

confidence: 99%

Optimal representation and processing of optical signals in quadratic-phase systems

Arık

Özaktaş

2016

Optics Communications

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“…Earlier, we showed that by appropriately choosing sample points on these reference surfaces, it is possible to represent the diffracted signals in a nonredundant manner [1]. Here we show that these reference surfaces should be spaced equally with respect to the fractional Fourier transform order, rather than with respect to the distance of propagation.…”

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confidence: 74%

“…In [1], we showed that if we choose s Δx∕Δσ x p , then the spatial and spatial frequency extents of the diffracted signal on the spherical reference surface are Δx 0 MΔx;…”

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confidence: 99%

“…This implies that the number of samples Δx 0 ∕ 1∕Δσ 0 x required to represent the diffracted signal is the same as that required to represent the original signal, but these samples are spaced further apart. In [1], we further discuss sampling and computation issues [7][8][9][10][11][12][13][14][15][16]. It is also interesting to note that this choice of s is the square root of the recently proposed "space-bandwidth ratio" [17].…”

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confidence: 99%

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Fundamental structure of Fresnel diffraction: longitudinal uniformity with respect to fractional Fourier order

2011

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Fresnel integrals corresponding to different distances can be interpreted as scaled fractional Fourier transformations observed on spherical reference surfaces. Transverse samples can be taken on these surfaces with separation that increases with propagation distance. Here, we are concerned with the separation of the spherical reference surfaces along the longitudinal direction. We show that these surfaces should be equally spaced with respect to the fractional Fourier transform order, rather than being equally spaced with respect to the distance of propagation along the optical axis. The spacing should be of the order of the reciprocal of the space-bandwidth product of the signals. The space-dependent longitudinal and transverse spacings define a grid that reflects the structure of Fresnel diffraction.

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Linear Canonical Domains and Degrees of Freedom of Signals and Systems

Oktem

Özaktaş

2016

Linear Canonical Transforms

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We discuss the relationships between linear canonical transform (LCT) domains, fractional Fourier transform (FRT) domains, and the space-frequency plane. In particular, we show that LCT domains correspond to scaled fractional Fourier domains and thus to scaled oblique axes in the space-frequency plane. This allows LCT domains to be labeled and monotonically ordered by the corresponding fractional order parameter and provides a more transparent view of the evolution of light through an optical system modeled by LCTs. We then study the number of degrees of freedom of optical systems and signals based on these concepts. We first discuss the bicanonical width product (BWP), which is the number of degrees of freedom of LCT-limited signals. The BWP generalizes the space-bandwidth product and often provides a tighter measure of the actual number of degrees of freedom of signals. We illustrate the usefulness of the notion of BWP in two applications: efficient signal representation and efficient system simulation. In the first application we provide a sub-Nyquist sampling approach to represent and reconstruct signals with arbitrary space-frequency support. In the second application we provide a fast discrete LCT (DLCT) computation method which can accurately compute a (continuous) LCT with the minimum number of samples given by the BWP. Finally, we focus on the degrees of freedom of first-order optical systems with multiple apertures. We show how to explicitly quantify the degrees of freedom of such systems, state conditions for lossless transfer through the system and analyze the effects of lossy transfer.

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Fundamental structure of Fresnel diffraction: natural sampling grid and the fractional Fourier transform

Cited by 22 publications

References 19 publications

Optimal representation and processing of optical signals in quadratic-phase systems

Optimal representation and processing of optical signals in quadratic-phase systems

Fundamental structure of Fresnel diffraction: longitudinal uniformity with respect to fractional Fourier order

Linear Canonical Domains and Degrees of Freedom of Signals and Systems

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