Relationship between the beam propagation method and linear canonical and fractional Fourier transforms.

The beam propagation method (BPM) can be viewed as a chain of alternating convolutions and multiplications, as filtering operations alternately in the space and frequency domains or as multiplication operations sandwiched between linear canonical or fractional Fourier transforms. These structures provide alternative models of inhomogeneous media and potentially allow mathematical tools and algorithms associated with these transforms to be applied to the BPM. As an example, in the case where quadratic approximation is possible, it is shown that the BPM can be represented as a single LCT system, leading to significantly faster computation of the output field.

Linear algebraic theory of partial coherence: continuous fields and measures of partial coherence.

This work presents a linear algebraic theory of partial coherence for optical fields of continuous variables. This approach facilitates use of linear algebraic techniques and makes it possible to precisely define the concepts of incoherence and coherence in a mathematical way. We have proposed five scalar measures for the degree of partial coherence.

Phase-space window and degrees of freedom of optical systems with multiple apertures.

We show how to explicitly determine the space-frequency window (phase-space window) for optical systems consisting of an arbitrary sequence of lenses and apertures separated by arbitrary lengths of free space. If the space-frequency support of a signal lies completely within this window, the signal passes without information loss. When it does not, the parts that lie within the window pass and the parts that lie outside of the window are blocked, a result that is valid to a good degree of approximation for many systems of practical interest.

Beyond Nyquist sampling: a cost-based approach.

A sampling-based framework for finding the optimal representation of a finite energy optical field using a finite number of bits is presented. For a given bit budget, we determine the optimum number and spacing of the samples in order to represent the field with as low error as possible. We present the associated performance bounds as trade-off curves between the error and the cost budget.

Representation of optical fields using finite numbers of bits.

We consider the problem of representation of a finite-energy optical field, with a finite number of bits. The optical field is represented with a finite number of uniformly spaced finite-accuracy samples (there is a finite number of amplitude levels that can be reliably distinguished for each sample). The total number of bits required to encode all samples constitutes the cost of the representation.

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

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.

Teaching science, technology, and society to engineering students: a sixteen year journey.

The course Science, Technology, and Society is taken by about 500 engineering students each year at Bilkent University, Ankara. Aiming to complement the highly technical engineering programs, it deals with the ethical, social, cultural, political, economic, legal, environment and sustainability, health and safety, reliability dimensions of science, technology, and engineering in a multidisciplinary fashion. The teaching philosophy and experiences of the instructor are reviewed.

Full-complex amplitude modulation with binary spatial light modulators.

Imperfections and nonrobust behavior of practical multilevel spatial light modulators (SLMs) degrade the performance of many proposed full-complex amplitude modulation schemes. We consider the use of more robust binary SLMs for this purpose. We propose a generic method, by which, out of K binary (or 1 bit) SLMs of size M×N, we effectively create a new 2(K)-level (or K bit) SLM of size M×N.

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

Fresnel integrals corresponding to different distances can be interpreted as scaled fractional Fourier transformations observed on spherical reference surfaces. We show that by judiciously choosing sample points on these curved reference surfaces, it is possible to represent the diffracted signals in a nonredundant manner. The change in sample spacing with distance reflects the structure of Fresnel diffraction.

Synthesis of three-dimensional light fields with binary spatial light modulators.

Computation of a binary spatial light modulator (SLM) pattern that generates a desired light field is a challenging quantization problem for which several algorithms have been proposed, mainly for far-field or Fourier plane reconstructions. We study this problem assuming that the desired light field is synthesized within a volumetric region in the non-far-field range after free space propagation from the SLM plane. We use Fresnel and Rayleigh-Sommerfeld scalar diffraction theories for propagation of light.

Fast and accurate algorithm for the computation of complex linear canonical transforms.

A fast and accurate algorithm is developed for the numerical computation of the family of complex linear canonical transforms (CLCTs), which represent the input-output relationship of complex quadratic-phase systems. Allowing the linear canonical transform parameters to be complex numbers makes it possible to represent paraxial optical systems that involve complex parameters. These include lossy systems such as Gaussian apertures, Gaussian ducts, or complex graded-index media, as well as lossless thin lenses and sections of free space and any arbitrary combinations of them.

Equivalence of linear canonical transform domains to fractional Fourier domains and the bicanonical width product: a generalization of the space-bandwidth product.

Linear canonical transforms (LCTs) form a three-parameter family of integral transforms with wide application in optics. 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 ordered by the corresponding fractional order parameter and provides insight into the evolution of light through an optical system modeled by LCTs.

Fast and accurate computation of two-dimensional non-separable quadratic-phase integrals.

We report a fast and accurate algorithm for numerical computation of two-dimensional non-separable linear canonical transforms (2D-NS-LCTs). Also known as quadratic-phase integrals, this class of integral transforms represents a broad class of optical systems including Fresnel propagation in free space, propagation in graded-index media, passage through thin lenses, and arbitrary concatenations of any number of these, including anamorphic/astigmatic/non-orthogonal cases. The general two-dimensional non-separable case poses several challenges which do not exist in the one-dimensional case and the separable two-dimensional case.

Efficient computation of quadratic-phase integrals in optics.

We present a fast NlogN time algorithm for computing quadratic-phase integrals. This three-parameter class of integrals models propagation in free space in the Fresnel approximation, passage through thin lenses, and propagation in quadratic graded-index media as well as any combination of any number of these and is therefore of importance in optics. By carefully managing the sampling rate, one need not choose N much larger than the space-bandwidth product of the signals, despite the highly oscillatory integral kernel.

Complex signal recovery from multiple fractional Fourier-transform intensities.

The problem of recovering a complex signal from the magnitudes of any number of its fractional Fourier transforms at any set of fractional orders is addressed. This problem corresponds to the problem of phase retrieval from the transverse intensity profiles of an optical field at arbitrary locations in an optical system involving arbitrary concatenations of lenses and sections of free space. The dependence of the results on the number of orders, their spread, and the noise is investigated.

Fractional free space, fractional lenses, and fractional imaging systems.

Continuum extensions of common dual pairs of operators are presented and consolidated, based on the fractional Fourier transform. In particular, the fractional chirp multiplication, fractional chirp convolution, and fractional scaling operators are defined and expressed in terms of their common nonfractional special cases, revealing precisely how they are interpolations of their conventional counterparts. Optical realizations of these operators are possible with use of common physical components.

Linear algebraic theory of partial coherence: discrete fields and measures of partial coherence.

A linear algebraic theory of partial coherence is presented that allows precise mathematical definitions of concepts such as coherence and incoherence. This not only provides new perspectives and insights but also allows us to employ the conceptual and algebraic tools of linear algebra in applications. We define several scalar measures of the degree of partial coherence of an optical field that are zero for full incoherence and unity for full coherence.

Optimization of orders in multichannel fractional Fourier-domain filtering circuits and its application to the synthesis of mutual-intensity distributions.

Owing to the nonlinear nature of the problem, the transform orders in fractional Fourier-domain filtering configurations have usually not been optimized but chosen uniformly. We discuss the optimization of these orders for multi-channel-filtering configurations by first finding the optimal filter coefficients for a larger number of uniformly chosen orders, and then maintaining the most important ones. The method is illustrated with the problem of synthesizing desired mutual-intensity distributions.