Application of the semi-analytical Fourier transform to electromagnetic modeling

Semi-analytic simulation of optical wave propagation through turbulence

Split-step wave-optical simulations are useful for studying optical propagation through random media like atmospheric turbulence. The standard method involves alternating steps of paraxial vacuum propagation and turbulent phase accumulation. We present a semianalytic approach to evaluating the Fresnel diffraction integral with one phase screen between the source and observation planes and another screen in the observation plane. Specifically, we express the first phase screen's transmittance as a Fourier series, which allows us to bring phase screen effects outside of the Fresnel diffraction integral, thereby reducing the numerical computations. This particular setup is useful for simulating astronomical imaging geometries and two-screen laboratory experiments that emulate real turbulence with phase wheels, spatial light modulators, etc. Further, this is a key building block in more general semianalytic split-step simulations that have an arbitrary number of screens. Compared with the standard angular-spectrum approach using the fast Fourier transform, the semianalytic method provides relaxed sampling constraints and an arbitrary computational grid. Also, when a limited number of observation-plane points is evaluated or when many time steps or random draws are used, the semianalytic method can compute faster than the angular-spectrum method.

Far field calculation of distorted Gaussian beams

Far field calculations of beams, such as laser beams, are often applied in optical engineering. Current beam propagation methods fail in certain range parameters due to high storage requirements of the algorithms. This paper presents a new beam propagation method for far field calculations of distorted Gaussian beams in a homogeneous medium including optical elements, such as lenses. The method works even in the case of a large distance from the observation plane to the beam waist and can be applied to larger divergence angles. This new simulation technique factors out the phase of the Gaussian TEM00 beam and solves the resulting partial differential equation by suitable finite difference or finite element discretization methods.

Shifted band-extended angular spectrum method for off-axis diffraction calculation

The shifted band-extended angular spectrum method (Shift-BEASM) is proposed to calculate free-space diffraction between two parallel planes with an off-axis offset. Off-axis numerical propagation is useful for simulating non-paraxial and large-scale fields. The proposed Shift-BEASM allow us to calculate the off-axis diffraction in a wide propagation range by extending the effective bandwidth using the nonuniform fast Fourier transform. The calculation accuracy is higher than that of existing techniques, such as the shifted-Fresnel method and shifted band-limited angular spectrum method, not only in the near field but also in the far field. Numerical examples and accuracy as well as theoretical formulation are presented to confirm validity of the proposed method.

Generalized Debye integral

The Debye integral is an essential technique in physical optics, commonly used to efficiently tackle the problem of focusing light in lens design. However, this approximate method is only valid for systems that are well designed and with high enough Fresnel numbers. Beyond this assumption, the integral formula fails to provide accurate results. In this work, we generalize the Debye integral to overcome some of its limitations. The theory explicitly includes aberrations and extends the integral to fields on tilted planes in the focal region. We show, using examples, that the new formulas almost reach the accuracy of a rigorous modeling technique while being significantly faster.

Light shaping by freeform surface from a physical-optics point of view

Modeling techniques for light-shaping systems with freeform surface are presented from a physical-optics point of view. We apply the modeling techniques to different light-shaping systems with freeform surfaces designed by "ray mapping method". The simulation results show that the design is not always valid. Diffraction effects occur, especially in paraxial situations. We discuss the accuracy of the design via physical-optics simulation, and find an explanation in the geometric-optics assumption of the design algorithm being sufficient only if the optical system results in homeomorphic behavior for the electric field between the input and target.

k-domain method for the fast calculation of electromagnetic fields propagating in graded-index media

A conceptually straightforward method for the fast calculation of electromagnetic fields propagating in graded-index media is presented. More specifically, in this method, we convert Maxwell's curl equations into the spatial-frequency domain to obtain an ordinary differential equation (ODE), and subsequently solve the ODE with the 4^th-order Runge-Kutta method. Compared to the traditional beam propagation methods, this method deals with vectorial fields accurately, without physical approximations, like the scalar field approximation or the paraxial approximation; numerically, this method takes advantage of the fast Fourier transform and the convolution theorem to achieve an efficient calculation.

Theory and algorithm of the homeomorphic Fourier transform for optical simulations

The introduction of the fast Fourier transform (FFT) constituted a crucial step towards a faster and more efficient physio-optics modeling and design, since it is a faster version of the Discrete Fourier transform. However, the numerical effort of the operation explodes in the case of field components presenting strong wavefront phases-very typical occurrences in optics- due to the requirement of the FFT that the wrapped phase be well sampled. In this paper, we propose an approximated algorithm to compute the Fourier transform in such a situation. We show that the Fourier transform of fields with strong wavefront phases exhibits a behavior that can be described as a bijective mapping of the amplitude distribution, which is why we name this operation "homeomorphic Fourier transform." We use precisely this characteristic behavior in the mathematical approximation that simplifies the Fourier integral. We present the full theoretical derivation and several numerical applications to demonstrate its advantages in the computing process.

Light shaping from a physical-optics point of view

In the design of optical element for light shaping, a geometric-optics assumption is usually used, where the validity of the assumption is rarely discussed in literature. In this work, the field tracing techniques for modeling light-shaping systems are presented, which reveals the optical element resulted from those geometric-base algorithm is not always accurate enough for the design task. An example is demonstrated with the functional embodiment of the element. The simulation result shows that diffraction effect may occur, especially in paraxial situation. However, the designed result start with the assumption is well-introduced initial guess for further optimization with the iterative Fourier transform algorithm (IFTA).