Geometric optics or ray optics deals with the light ray (no such thing in reality!). It's just a convenient model that works well in some cases but not all. Other models that extend further are wave optics (scalar waves, electromagnetic waves), quantum optics (photons)
Physical optics, or wave optics, involves study of phenomenon such as interference, diffraction, polarization for which the ray approximation of geometric optics is not valid. It doesn't include effects such as quantum noise in optical communication
Paraxial Optics: deal with cases when the distances of rays from the axis approach zero, as do the angles of the rays relative to the axis. The ray-tracing equations in the paraxial limit are linear in angle and in distance from the axis, hence the term first-order optics (height and angle of light rays are computed with terms to the first order only and higher-order terms are ignored), which is often considered equivalent to paraxial. Note that there are no zeorth-order terms since the expansion is taken about the axis. A ray with an initial height and angle of zero is just a ray along the axis. The linearity of the paraxial equations makes them expressible in matrix form
Gaussian Optics: applies to those aspects of paraxial optics discovered by Gauss (Gauss 1840), who recognized that all rotationally symmetric systems of lens elements can be described paraxially by certain system properties. In particular, lenses can be treated as black boxes described by two axial length parameters and the locations of special points, called cardinal points, also called Gauss points. Once a lens is characterized by these cardinal points, knowledge of its actual makeup is unnecessary for many purposes. Eg., given the object location, the image location and magnification are determined from the gaussian parameters. In the limit of small heights and angles, the equations of collineation are identical to those of paraxial optics
Collineation refers to a mathematical transformation that approximates the imaging action of a lens with homogeneous refractive indices in both object and image space. This transformation takes points to points, lines to lines, and planes to planes. With an actual lens, incoming rays become outgoing rays, so lines go exactly to lines. In general, however, rays that interest in object space do not intersect in image space, so points do not go to points, nor planes to planes. The collinear transformation is an approximate description of image geometry with the optical system as a black box. It is not a theory that describes the process of image formation.
The imaging described by collineation is, by definition, stigmatic everywhere, and planes are imaged without curvature. And for rotationally symmetric lenses, planes perpendicular to the axis are imaged without distortion. So the three conditions of Maxwellian perfection are satisfied for all conjugates. Consequently , collineation is often taken as describing ideal imaging of the entire object space. However, it is physically impossible for a lens to image as described by collineation, except for the special case of an afocal lens under the following condition:
$m = m_z = n/n'$ $\leftarrow$ important to ensure this if you are doing remote focusing
$m \rightarrow$ transverse magnification (lateral)
$m_z \rightarrow$ longitudinal magnification (axial)
$n \rightarrow$ refractive indices in object space
$n' \rightarrow$ refractive indices in image space
Since $m_\alpha m_z = m$, then $m_\alpha = 1$ (angular magnification)
In other words, angles in object space are preserved in image space i.e., same angles in both places.
Magnification: Transverse (lateral) magnification or simply referred to as magnification ($m$), longitudinal (axial) magnification ($m_z$) and angular magnification ($m_\alpha$) are related by:
$$m_\alpha m_z = m$$
This relationship hold for both focal and afocal lenses. Angular magnification is the ratio of outgoing to incoming ray angles and longitudinal (axial) magnification is the ratio of axial separations in image space to that in object space.
Afocal system: a system that produces no net convergence or divergence of a beam (i.e., a lens-system without focus). Also called telescopic lens. Examples include binoculars, telescopes, and beam expanders consisting of a system of lenses (focal lenses). Transverse and longitudinal magnifications are both constant (i.e., same for all rays) and are related as follows:
$$m_z = \frac{n'}{n} m^2$$
Note that the above condition is satisfied when $m = m_z = n/n'$ (condition for Collineation)
For afocal systems, the Sine condition and Herschel condition are identical.
Sine condition: Also known as Abbe-Sine condition. Applies to object and image planes perpendicular to the axis in regions about the axis. It establishes the linear proportionality of the image space angle ($\theta'$) and the object space angle ($\theta$) as:
$ m = n \sin \theta/n' \sin \theta' $
This condition needs to be met if sharp images of points along a given transverse plane are to be captured. Note that it looks very similar to Snell's law except for the transverse magnfication $m$ showing up in the relationship.
This relationship is true not only for the extreme rays passing through the lens, but for rays at all angles. This condition doesn't apply when there are discontinuities in ray behavior, for example, in devices such as Fresnel lenses or diffraction-based devices like zone plates.
Herschel condition: This relationship holds if the imaging is stigmatic for nearby points along the axis:
$$m = \frac {n \sin \frac{\theta}{2}} {n'\sin \frac{\theta'}{2}}$$
or, equivalently
$$m_z = \frac {n \sin^2 \frac{\theta}{2}} {n' \sin^2 \frac{\theta'}{2}}$$
The Herschel condition is inconsistent with the Abbe-Sine condition unless $ m = \pm n/n'$. So, in general, stigmatic imaging in one plane precludes that in other planes. That is, perfect imaging of a volume is not possible.
For afocal systems, Sine and Herschel conditions are identical.
Applied external fields are quantified in terms of $\mathcal{\vec E}$ and $\mathcal{\vec H}$, since both are measurable quantities
$\mathcal{\vec E}$ and $\mathcal{\vec B}$ are the fundamental fields, since their sources are ALL charges (free + bound) and ALL current densities (free + bound) respectively. Maxwell's equations written in terms of $\mathcal{\vec E}$ and $\mathcal{\vec B}$ are expressions involving ALL charges and current densities
$\mathcal{\vec D}$ and $\mathcal{\vec H}$ are auxiliary fields that allow Maxwell's equations to be expressed entirely in terms of free charges and current
In free space (no charge or current), $\mathcal{\vec E}$ and $\mathcal{\vec D}$ are linearly related as $\mathcal{\vec D} = \epsilon_o \mathcal{\vec E}$. Similarly, $\mathcal{\vec B}$ and $\mathcal{\vec H}$ are linearly related as ${\vec H} = \mu_o ^{-1}\mathcal{\vec B}$
Unit: V/m
From the unit of $\mathcal{\vec E,}$ you can envision a voltage (measurable) applied across two parallel plates that are some distance apart (measurable). From this perspective, it's easy to see why this is directly measurable, and thus the reason we quantify the strength of the applied field using $\mathcal{\vec E}$
Unit: A/m
From the unit of $\mathcal{\vec H,}$ you can envision a current (measurable) running through the length of a wire (measurable). From this perspective, it's easy to see why this is directly measurable, and thus the reason we quantiy the strength of the applied field using $\mathcal{\vec H}$
Represents electric field density (notice the m$^2$ in the Unit)
$\mathcal{\vec D}$ (unlike $\mathcal{\vec E}$) is not a quantity you measure directly in a lab
Represents magnetic field density (notice the m$^2$ in the Unit)
$\mathcal{\vec B}$ (unlike $\mathcal{\vec H}$) is not a quantity you measure directly in a lab, although we frequently use Tesla to quantify magnetic fields
Note: Section 6.3.1 of D. J. Griffiths Textbook "Introduction to Electrodynamics" discusses the distinction between $\mathcal{\vec B}$ and $\mathcal{\vec H}$
$\mathcal{\vec D} = \epsilon_o \mathcal{\vec E} + \mathcal{\vec P}$
$ \mathcal{\vec H} = {\mu_o}^{-1} {\mathcal{\vec B}} - \mathcal{\vec M}$
$\mathcal{\vec P}$ is the induced polarization density (dipole moment per unit volume) due to a slight separation of bound charges in the material in response to the applied electric field $\mathcal{\vec E}$. $\mathcal{\vec P}$ points in the opposite direction of $\mathcal{\vec E}$
$\mathcal{\vec M}$ is magnetization density of the material (permanent or induced dipole moment per unit volume) resulting from the bound current densities
$\mathcal{\vec M}$ plays a similar role in magnetostatics (field created by steady current) to $\mathcal{\vec P}$ in electrostatics (field created by static charges)
The interaction of applied fields $\mathcal{\vec H}$ and $\mathcal{\vec E}$ with the medium (dielectric material) result in $\mathcal{\vec M}$ and $\mathcal{\vec P}$ respectively. As a consequence, $\mathcal{\vec B}$ and $\mathcal{\vec D}$ are the result
In optical materials (in the UV, visible and IR wavelengths), magnetization $\mathcal{\vec M}$ is negligible, and as such $\mathcal{\vec H}$ and $\mathcal{\vec B}$ are related directly by a multiplicative factor (no additive term). Optics textbooks therefore use $\mathcal{\vec H}$ and $\mathcal{\vec B}$ liberally and often call $\mathcal{\vec H}$ the magnetic field
Also discusses higher-order Hermite-Gauss and Laguerre-Gauss modes besides the familiar TEM_00 Gaussian beam
Key approximations and validity of this theory:
Notation:
Real quantities are curled:
Complex quantities:
Light is an electromagnetic wave with coupled electric ($\mathcal{\vec E}$) and magnetic fields ($\mathcal{\vec B}$) traveling through space. In a homogeneous and isotropic medium, such as free space or a lens with constant refractive index, the electric and magnetic field vectors form a right-handed orthogonal triad with the direction of propagation. Disregarding polarization, each componenent of the fields, namely $\mathcal{E_x}$, $\mathcal{E_y}$, $\mathcal{E_z}$, $\mathcal{H_x}$, $\mathcal{H_y}$, $\mathcal{H_z}$, can be described by a real, harmonic, scalar, time-dependent function $\mathcal{U}(\mathbf{r},t)$ called a wave-function:
$\mathcal{U}(\mathbf{r},t) = a(\mathbf{r})cos(\phi \mathbf{r} - \omega t)$
It's called a wave-function as it is a solution to the homogeneous wave-equation (no current source or charge):
$ \left[\nabla ^{2} - \frac{1}{c^2} \frac{\partial^2}{\partial t^2} \right] \mathcal{U}(\mathbf{r},t) = 0$
To simply calculations, a complex wave-function $U(\mathbf{r},t)$ is defined from the real wave-function $\mathcal{U}(\mathbf{r},t)$. It is represented as:
$U(\mathbf{r},t) = a(\mathbf{r}) e^{j[\phi(\mathbf{r})-\omega t]} = U(\mathbf{r}) e^{-j \omega t}$
The real wave-function and the complex-wave functions are related as follows:
$\mathcal{U}(\mathbf{r},t) = \operatorname{Re} \left[ U(\mathbf{r},t) \right]$
Note that we've assumed that the spatial and temporal components of a complex-wave function $U(\mathbf{r}, t)$ are separable into a purely spatial term and a purely temporal term. This separation of variables is what gives us the time-independent Helmholtz equation from the full wave-equation. Here, $ \omega$ denotes the angular frequency of light, and the complex amplitude $U(\mathbf{r})$ is purely spatial.
Both the complex wave-function $U(\mathbf{r},t)$ and the scalar wave-function $\mathcal{U}(\mathbf{r},t)$ satisfy the wave-equation:
$ \left[\nabla ^{2} - \frac{1}{c^2} \frac{\partial^2}{\partial t^2} \right] U(\mathbf{r}, t) = 0$
and, the complex amplitude function $U(\mathbf{r})$ satisfies the Helmholtz equation:
$ \left[\nabla ^{2} + \left( \frac{\omega}{c}\right)^2\right] {U}(\mathbf{r}) = 0$
$ \left[\nabla ^{2} + k^2\right] {U}(\mathbf{r}) = 0$
The Helmholtz equation follows directly from Maxwell equations under the condition of a homogeneous medium and an absence of sources. $\nabla^2$ is the Laplace operator i.e., partial second derivative wrt spatial co-ordinates, $\frac{\partial}{\partial x^2} + \frac{\partial}{\partial y^2} + \frac{\partial}{\partial z^2}$
Further reading:
A treatment of "Scalar theory of diffraction" can be found on this Appendix of Applied Digital Optics (2009)
Rayleigh-Sommerfeld Diffraction vs Fresnel-Kirchhoff, Fourier Propagation, and Poisson’s Spot link
The Fresnel-Kirchoff formulation is a scalar theory of diffraction and enables us to predict diffraction effects under these conditions:
Outside this regime, the predictions of this formula is no longer accurate. Additionally, Fresnel-Kirchhoff’s formula does not correctly explain Poisson spot, a bright spot at the center observed in the near-field when light is blocked by a circular screen
The Rayleigh-Sommerfeld formulation is another scalar theory of diffraction and applies when $a >>\lambda$, typically by a factor of 10x or more. It makes a slightly different assumption about the field and its derivative at the boundaries of the aperture. As such, it overcomes the mathematical inconsistencies encountered by Fresnel-Kirchoff formulation and is able to explain the Poisson's spot as well as reproduce the diffracted field right behind the aperture ($z \gtrsim \lambda$), which Fresnel-Kirchoff fails to do. For this reason, the Rayleigh-Sommerfeld formulation is considered to be a "full" scalar solution.
The Rayleigh–Sommerfeld diffraction integral is believed to be more accurate than the Fresnel-Kirchhoff formulation because of its mathematical consistency, and also because of its ability to reproduce closely the diffracted field right behind the aperture. However, it has been shown experimentally that the Fresnel–Kirchhoff diffraction formulation gives more accurate results than Rayleigh–Sommerfeld theory (assuming that we are many wavelengths away form the diffracting aperture). Moreover, the Rayleigh–Sommerfeld theory is limited to a plane surface, which is a severe limitation, since we usually deal with curved surfaces in optics." From Appendix B: "Scalar Theory of Diffraction," Applied Digital Optics (2009)
Neither Fresnel-Kirchoff nor Rayleigh-Sommerfield take into account the coupled electric and magnetic vector fields, the vector nature of which cannot be ignored when $a \sim \lambda$ or $z < \lambda$
Full Solution = [Rayleigh-Sommerfield diffraction]. However, there is no general analytical solution for the calculation of an exact RSD. Therefore, numerical methods have to be used
If distance between planes > a “few wavelengths” then Scalar Kirchoff Diffraction. This is Huygen’s Secondary Spherical wavelets plus obliquity factor due to 2-D whole in plane
If distance between planes large and planes are small, then make small angle approximation to get Fresnel Diffraction, which replaces Spherical waves with Parabolic and ignores obliquity factor. Fresnel diffraction= multiplication in Fourier plane by a parabolic phase term. See this link for more
Fresnel number, $F =a^2/\lambda z$ coarsely determines the near/far diffraction regimes:
For example: With a 1 inch aperture and a 500nm beam, $a^2/\lambda$= 0.2 miles. So, the observation distance $z$ must be larger than 2 miles (an order of magnitude larger) for the Fraunhofer approximation to be valid!
Rayleigh-Sommerfeld: $a >> \lambda$ and $z \gtrsim \lambda$
Fresnel-Kirchoff: $a >> \lambda$ and $z >> \lambda$
Fresnel: $z >> a >> \lambda$
Fraunhofer: $z >> a^2/\lambda$
For Fresnel and Fraunhofer: observation plane close to the optic axis such that for any observation point on the plane, $r \approx z$
Typically a factor or 10x or more when $>>$
Fresnel diffraction: Field in the image plane is a 2D Fourier transform of the field at the aperture multiplied by a quadratic phase factor
Fraunhofer diffraction: Field in the image plane is a 2D Fourier transform of the field at the aperture
LightPipes is a set of software tools for simulation of propagation, diffraction and interference of coherent light. Algorithms are based on the scalar theory of diffraction, Fresnel-Kirkhoff, to be exact. The toolbox includes spectral, FFT-based and finite-difference based propagation models. Special tools have been developed for the propagation through lenses with coordinate transforms, simulation of any combination of Zernike aberrations, mode analysis in laser resonators, interferometers, inverse problems, waveguides and propagation in media with non-uniform distribution of refraction index
Physical Optics Propagation in Python (POPPY) is a Python package that simulates physical optical propagation including diffraction. It implements a flexible framework for modeling Fraunhofer and Fresnel diffraction and point spread function formation, particularly in the context of astronomical telescopes. It was developed as part of a simulation package for the James Webb Space Telescope, but is more broadly applicable to many kinds of imaging simulations. It is not, however, a substitute for high fidelity optical design software such as Zemax or Code V, but rather is intended as a lightweight alternative for cases for which diffractive rather than geometric optics is the topic of interest.
It does NOT do the following:
Able to import lenses from Edmund Optics catalogue
pyOpTools is being developed almost exclusively using “Debian derivative” Linux distributions (eg., Ubuntu)
Other inactive projects
Primary modules:
For digital hologram and light scattering applications
By Manoharan Lab: Soft matter, Biophysics, Optics
3D EM wave FDTD simulator; Has an optional PyTorch backend, enabling FDTD simulations on a GPU
FDTD algorithm solves both electric and magnetic fields in temporal and spatial domain using the full-vector differential form of Maxwell’s coupled curl equations. This allows for arbitrary model geometries and places no restriction on the material properties of the devices. It can thus model light propagation, scattering, diffraction, reflection, polarization, and non-linear effects.
It can also model material anisotropy and dispersion without any pre-assumption of field behavior such as the slowly varying amplitude approximation. The method allows for the effective and powerful simulation and analysis of sub-micron devices with very fine structural details. A sub-micron scale implies a high degree of light confinement and correspondingly, the large refractive index difference of the materials (mostly semiconductors) to be used in a typical device design.
FDTD is applicable in wide-range of applications including surface plasmon resonance, photonic band gaps, nanoparticles, cells and tissue, diffractive micro-optics devices and lens, non-linear materials etc
xrt (XRayTracer) is a python software library for ray tracing and wave propagation in x-ray regime. It is primarily meant for modeling synchrotron sources, beamlines and beamline elements. Includes a GUI for creating a beamline and interactively viewing it in 3D.
Classical ray tracing and wave propagation via Kirchhoff integrals, also freely intermixed. No further approximations, such as thin lens or paraxial. The optical surfaces may have figure errors, analytical or measured. In wave propagation, partially coherent radiation is treated by incoherent addition of coherently diffracted fields generated per electron.
Discusses most standard topics of traditional physical and geometrical optics through Python and PyQt5
Provides visualizations and in-depth descriptions of Python’s programming language and simulations
Includes simulated laboratories where students are provided a "hands-on" exploration of Python software
Python and PyQt5 (chapters 1-2)
Code-V (Synopsis) is >$10k for an annual subscription
OpticStudio Premium (Zemax) is $7800 for an annual subscription
OSLO (Lambda Research) is $3500 for a permanent USB single-license
OpTaliX full version (educational purpose): 800 euros
Five Seidel (third order) aberrations: Spherical aberration, Coma, Astigmatism, Field Curvature, Distortion. Higher order terms, particularly Chromatic aberation (longitudinal and lateral) bring the total parameters to consider for optimization to seven
Numerical evaluation methods trace many light rays from object to image space. These methods assume basic laws of reflection and refraction (Snell's law). For each ray, new ray parameters are calcuated at each surface. Sequential ray-tracing evaluates as light travels from surface to surface in a defined order, whereas non-Sequential ray-tracing doesn't assume a pre-defined path and as such, a ray hitting a surface along its path is allowed to reflect, refract, diffract and scatter off
Zemax uses a right-handed co-ordinate system, where Z-axis is the optical axis and light initially moves in the +Z direction
Resources for Optical Design