# Applications of the Paraxial Approximation

It was discussed in a previous article, Mirrors in Geometrical Optics, Paraxial Approximation that the paraxial approximation is used to consider an apparently imperfect or flawed system as a perfect system.

Paraxial Approximation

The paraxial approximation was proposed in response to a normal occurrence in optical systems where the focal point is inconsistent for incident rays of higher incidence angles.The focal point F for a spherical mirror is understood under the paraxial approximation to be half the radius of curvature. Without the paraxial approximation, the system becomes increasingly complicated, as the focal point is a varying trigonometric function of the angle of incidence. The paraxial approximation assumes that all incident angles will be small.

The paraxial approximation can be likened (and when analyzed fully, this is it exactly) to a case of a triangle of base B, hypotenuse H and angle θ. Consider a case where H/B is very close to 1. θ will also be very small. In this case, it is of little harm to consider such a triangle as a triangle with θ=0, virtually to lines on top of each other, H and B, and more explicitly, H=B. This is precisely what is done when using the paraxial approximation.

An interesting question to ask is, what angle should be the limit to which we allow a paraxial approximation? The answer would be, it depends on how accurate, or clear the image must be. When discussing optical systems, an aberration is a case in which rays are not precisely focused at the focal point of a mirror (or another type of optical system involving focusing). An aberration will actually cause the image clarity to be reduced at the output of the system. The following image would be an example of the result of an aberration to an image in an optical system:

Here is an example of a problem that makes clear an example of the issue of an aberration. Two rays appear to be correctly aligned to the focal point, however another ray with angle of incidence of 55 degrees is not focused at the focal point. A system that would allow a ray of incidence of 55 degrees may be acceptable under some circumstances, however one would expect to have an aberration or some level of blurriness to the image.

# Entanglement, Mach-Zehnder Interferometer

It may be that one wishes to describe a quantum state as the existence of two separate, non-interacting particles each in a particular state. Imagine particle 1 can be in either state |u1> or |u2> and particle 2 can be in either |v1> or |v2>. We wish to describe the state where particle 1 is in state |u1> and particle 2 is in state |u2>. The notation for this state is:

|u1> ⊗ |u2>

Given a probability for each state, a general state formula can be written to describe both particles:

To describe a superposition of particular states however will result in a dependency of the particles on each other. This is called an engtangled state.

The following outlines an entangled state example. This shows how the fate of two particles becomes intertwined in such an entangled state, where there will exist no other combination other than the complete state combinations made available by the definition of the state.

Einstein has objected to the entangled pair hypothesis. John Bell had proposed an experiment to test entanglement using a three directions, such that a correlation would be more presentable. The results of his experiments however confirmed the possibility of this sort of entanglement on the quantum level, which appears to deny classical mechanics.

Mach-Zehnder Interferometer Quantum Mechanical Calculation

Let us model the Mach-Zehnder Interferometer using photon probability state matricies. First, we will consider the operation of the beam splitter. When a photon enters the beamsplitter from one direction, there is a given probability that the photon will be present at either the transmitted or refracted position. Given that the beamsplitter is balanced, meaning that the photon has an equal change (1/2) of exiting either side, the beamsplitter is modeled below:

Next, using the beamsplitter matrix, the Mach-Zehnder Interferometer can be modeled. Interestingly, the photon appears to exit (100%) from the side opposite which it entered.

Let’s consider a case in which mirror 2 is blocked. Using the matrix for beamsplitter 1 and beamsplitter 2, the probabilities are calculated that the photon will either 1. be blocked by the concrete, 2. exit at detector 0 and 3. exit at detector 1.

Barton Zwiebach. 8.04 Quantum Physics I. Spring 2016. Massachusetts Institute of Technology: MIT OpenCourseWare, https://ocw.mit.edu. License: Creative Commons BY-NC-SA.

# Optical Waveguides

Just as a metallic strip connects the various components of an electrical integrated circuit, optical waveguides connects components and devices of an optical integrated circuit. However, optical waveguides differ from the flow of current in that the optical waves travel through the a waveguide in a spatial distribution of optical energy, or mode. In contrast to bulk optics, which guide optical waves through air, optical waveguides guide light through dielectric conduits.

Bulk Optical Circuit:

Optical Waveguides:

The use of waveguides allows for the creation of optical integrated circuits or photonic integrated circuits (PIC). Take for example, the following optical transmit and receive module:

Planar Waveguides

A planar waveguide is a structure that limits mobility in only one direction. If we consider the planar waveguide to be on the x axis, then the waveguide may limit the travel of light between two values on the x axis. In the y and z directions, light may travel infinitely. The planar waveguide does not serve many practical uses, however it’s concept is the basis for other tpyes of waveguides. Planar waveguides are also referred to as slab waveguides.Planar waveguides can be made out of mirrors or using a dielectric with a high refractive index slab. See also, Planar Boundaries, Total Internal Reflection, Beamsplitters.

Rectangular Waveguides

Rectangular waveguides can also be built either from mirrors or using a high refractive index rectangular waveguide.

The following are useful waveguide geometries:

Various combinations of waveguides may produce different and useful configurations of waveguides:

# Hermite-Gaussian, Laguerre-Gaussian and Bessel Beam

The Gaussian Beam [link] is not the only available solution to the Helmholtz equation [link]. The Hermite-Gaussian Beam also satisfies the Helmholtz equation and it shares the same wavefronts (shape) of the Gaussian Beam. Where it differs is in the distribution of intensity in the beam. The Hermite-Gaussian Beam distribution is a modulated Gaussian distribution in the x and y directions which can be seen as a number of functions in superposition. The below figures depict the cross-sections of ascending order intensity distributions for the Hermite-Gaussian Beam. Secondly, distribution orders zero through three are shown.

The Complex amplitude of the Hermite-Gaussian beam labeled by indexes l,m (orders):

Laguerre-Gaussian Beams

The Laguerre-Gaussian Beam is a solution to the Helmholtz equation in cylindrical coordinates.

The shape of the Laguerre-Gaussian Beam intensity distribution is of a toroid which increases in radius for orders where m = 0 and for orders m > 0, it takes the form of multiple rings.

The Bessel Beam

The Bessel Beam, by comparison to the Gaussian Beam differs in that it has a ripple effect by oscillation in addition to a similar gaussian curve. The complex amplitude of the Bessel Beam is an exact solution to the Helmholtz equation, while the complex amplitude of the Gaussian beam is an approximate solution (paraxial solution).

B. E. A. Saleh and M. C. Teich, Fundamentals of photonics. Hoboken: Wiley, 2019.

# Gaussian Beam Transmission Through Optical Components

The most important note about the transmission of a Gaussian Beam [link] through various optical components [link] is that the beam will remain Gaussian, given that the system is paraxial. The shape of the Gaussian beam will change according to the components, however.

The complex amplitude of the Gaussian beam (width) is adjusted to the width of an optical component, for example.

The Gaussian beam that emerges from the above lens takes the following formulas:

Lenses may be used to focus the a Gaussian beam. This is achieved by positioning the lense appropriately according to the location of the beam waist. For applications such as laser scanning and compact-disk burning, it is desired to focus the beam to the smalles size possible.

The focused waist W0′ and the distance of the focused waist z’ are a function of the waist of the original beam and the focal length f of the lens.

Beams may also be relayed and expanded using lenses.

A Gaussian beam, as do rays and waves behave differently for a plane mirror (i.e. spherical mirror with infinite radius) and spherical mirrors.

As is the case with geometrical ray optics, beam properties through a system can be modeled using the ABCD matrix method.

B. E. A. Saleh and M. C. Teich, Fundamentals of photonics. Hoboken: Wiley, 2019.

# The Gaussian Beam

Wave optics as previously discussed operated under an ideal assumption that light can be confined to a uniform, rectangular shape that moves through space. A more realistic understanding of a wave that propagates through space is the goal of beam optics, which instead describes a light wave as a distribution of light.

The Gaussian Beam

The Gaussian beam is a common description of the distribution of a light beam which satisfies the Helmholtz equation. Light is concentrated towards the center of the beam in a Gaussian distribution.

The width of the beam is a minimum at what is termed the waist of the beam and the width increases at distances further from the waist. Eventually, the width of the beam would become very wide and the distribution of light would be wide enough, almost to approximate a spherical beam. In reference to the figure above, the leftmost distribution may for example be the distribution at the waist of the beam and the rightmost picture is the beam further from the waist. In a localized area, the beam exhibits similar characteristics to the ideal plane wave.

The width of the beam is determined by the following formula:

The complex amplitude of the Gaussian beam is described by the following formula:

Further parameters of the beam used in the above formula are the following:

• W(z): Beam width function (above)
• R(z): wavefront radius of curvature
• ξ(z): Beam center point
• W0: Minimum Beam level, found at z = 0

B. E. A. Saleh and M. C. Teich, Fundamentals of photonics. Hoboken: Wiley, 2019.

# Wave Optics – Interference, Interferometers

Interference

When two or more waves of the same frequency are present in the same location, the sum of their intensities may not equal the intensity of the total wavefunction. The interference is understood as the difference between the intensity of the total wavefunction and the sum of the individual wavefunction intensities.

The interference equation is used to talculate the intensity of the total wavefunction. The third term is the interference between the two waves, where φ is equal to the sum of the phases of the two waves.

When adding wavefunctions of different phases, these wavefunctions can be drawn as a superposition of vectors, where the intensity of the wavefunction in the magnitude and the phase is the angle of the wavefunction vector.

Consider the case in which two waves, represented by two vectors are equal in magnitude, but 180 degrees out of phase of each other. In this case, the intensity of the total wavefunction is zero. If there is no phase difference between the two wavefunction vectors, then the interference of the two waves is zero and the maximum intensity of the system is reached.

Interferometers

It has been mentioned that Wave Optics and Geometrical Optics are insufficient to take measurements of the intensity of rays and waves. However, by determining the level to which waves interfere with each other, a relative intensity can be measured. The interferometer is an instrument that detects the intensity of the a superposition of waves of a varied phase difference. A wave is split using a beamsplitter and each split wave is reflected after different (or possibly the same) distances and recombined. After recombination of the optical waves, the interference is measured by amount of loss in the system and subsequently the distances of the mirrors. Applications include metrology, measurements of refractive index and spectrometry.

Three prominent examples of interferometers are the Mich-Zehnder interferometer, the Michelson interferometer and the Sagnac interferometer.

B. E. A. Saleh and M. C. Teich, Fundamentals of photonics. Hoboken: Wiley, 2019.

# Wave Optics – Transmission, Grating and Lenses

Through a transparent plate with no angle of incidence, a plane wave continues to propagate through the plate, but with an altered wavenumber n*k0, where n is the refractive index and k0 is the wavenumber of the plane wave before transmission through the plate.

Where d is the thickness of the plate and U(x,y,z) is the complex amplitude of the wave, the transmittance of a plane wave in a homogenous, transparent plate is described as:
t(x,y) = U(x,y,d)/U(x,y,0) = exp(-j*n*k0*d).

Given the scenario of a plane wave with wavevector k and angle of incidence θ, the formula is altered and may be modeled using Snell’s Law:

sinθ = n*sinθ,

exp(-j*k1 • r) = exp[-jnk0(zcosθ1 + xsinθ1),

t(x,y) = exp(-j*n*k0*d*cosθ1).

A prism may be used in the following manner to direct the propagation of a plane wave:

Further, a thin lens may be used to focus a plane wave, converting it into a paraboloidal wave.

A graded-index (GRIN) lens may be used to produce the same effect:

Diffraction Grating is a method of modulating either the phase or amplitude of an incident wave. An incident plane wave is split into multiple plane waves. They may also be used as filters or spectrum analyzers.

The grating equation is as follows:
θq = θi + q*λ/Λ,
where θq is the angle of resultant wave(s),  θi is the angle of incidence, q is the diffraction order (0,1,2…) and Λ is the period of thickness variation in the diffraction grating. Since the device is dependent on the wavelength, it may be used to produce a polychromatic wave, separating it’s spectral components in the following manner:

(1) B. E. A. Saleh and M. C. Teich, Fundamentals of photonics. Hoboken: Wiley, 2019.

# Wave Optics – Reflection, Refraction

Reflection of Optical Waves on a mirror may be modeled using the Helmholtz equation, k1 • r = k2 • r, where r = (x,y,0) (following the example below on the z = 0 plane), k1 = (k0*sinθ1, k0*cosθ1) and k2 = (k0*sinθ2, k0*cosθ2). The wavenumbers k1 and k2 in this formula, is assumed to be equal as this is one property of reflected waves. This simplifies to the expression θ1 = θ2, which means that the angle of incidence is equal to the angle of reflection.

Refraction of Optical Waves at a planar boundary can be described using the Helmholtz equation and Snell’s Law. For the below scenario, where k1 is the incident wave, k2 is the refracted wave and k3 is the reflected wave, the Helmholtz equation is satisfied by the following vectors:

Helmholtz equation: k1 • r = k2 • r = k3 • r ,  for r = (x,y,0)

k1 = (n1*k0*sinθ1, 0, n1*k0*cosθ1)
k2 = (n2*k0*sinθ2, 0, n2*k0*cosθ2)
k3 = (n1*k0*sinθ3, 0, -n1*k0*cosθ3)

This relationship may be simplified to show that θ1 = θ3 (essentially proven in the previous example of reflected waves) and n1*sinθ1 = n2*sinθ2 (Snell’s Law). Note that it is not possible with Wave Optics to describe the magnitudes of the incident, reflrected and refracted waves and a more rigorous method, such as Electromagnetic Optics is required to explain such phenomenon.

(1)  B. E. A. Saleh and M. C. Teich, Fundamentals of photonics. Hoboken: Wiley, 2019.

# The Oscillator

The oscillator is an important concept used in a variety of applications. One basic use of an oscillator is that of signal generation.

An oscillator is a system with a gain and positive feedback. The gain must be greater than the loss in the feedback system, so that each time the signal goes through the aplifier in the system, a net gain is produced. The phase shift of a single round trip in the gain-feedback loop must also be a multiple of 2*pi so that a pure signal is repeatedly amplified.

When these conditions are satisfied, the system is unstable and oscillation begins. Eventually, the amplifier gain becomes saturated and rather than a further increase of amplification, the added gain only compensates for system losses.

Since the system is dependen upon a 2*pi phase shift (the period), an oscillator may be designed for a specific frequency. An oscillator generate a signal from noise by repeatedly amplifying the noise periodically.

Although there are many applications for oscillators, a laser is fundamentally an optical oscillator, an optical signal generator. The maser, which stands for microwave amplification by stiumulated emission of radiation was used before the laser. The saser is an acoustic version of the laser, in which instead of emitting a beam of photons or electromagnetic radiation, an acoustic beam or signal is generated.

The following outlines the operation of a laser; an optical amplifier placed inside of a resonator with a partially transmitting mirror as the output of the system.

B. E. A. Saleh and M. C. Teich, Fundamentals of photonics. Hoboken: Wiley, 2019.

# Wave Optics – Introduction

Wave optics describes light with a second order differential equation known as the wave equation. Using the wave equation, optical phenomena that fall outside of the scope of ray optics can be described, such as interference and diffraction. One limitation of wave optics is in it’s inability to describe polarization effects and other phenomena that require a vector formulation.

The wave equation used in wave optics is as follows:

Optical intensity I(r,t) is proportional to the squared wavefunction and is defined as watts/cm^2. This is also referred to as irradiance.

A monochromatic wave does not vary in intensity over time:

The time-independent equation is known as the complex amplitude of the wave:

And the above formula is the solution to the Helmholtz equation, another important formula used in wave optics:

Where k is the wavenumber.

Plane Waves, Spherical Waves, Paraboloidal Waves are the main wave formations.  A plane wave is seen as a wave that continues infinitely along a continuous plane with a constant intensity. The vector along which the plane exists is known as the wavevector. The spherical wave consists of spheres centered about a single point. It may also originate from the centered point or travel inwardly towards the center. A paraboloidal wave is an approximation to the spherical wave using the Fresnel approximation. A paraboloidal approximation is useful for simplifying the interaction of a spherical wave in calculations for diffraction and other situations. Paraxial Waves are a ideal approximation of the interaction of these waves that simplifies many calculations. The approximation leading to the paraboloidal wave is done by considering the spherical wave at a sufficiently great distance, at which only curvatures in the wave are detected. For points very far from the center of the sphere, waves may be treated as plane waves.

Paraxial optics is an idealization of the directionality of optical rays that allows for approximated results of a number of optical systems. A wave is a paraxial wave under the condition that the wavefront normals are paraxial rays. Under the paraxial approximation of waves, optical rays are perpendicular to optical waves.

The paraxial Helmholtz equation is as follows:

(1)

# Ray Optics – Graded-Index Fibers, Matrix Optics

Guiding light rays with multiple lenses or mirrors is possible, however this may result in great loss of optical power due to refraction in a system if there are many lenses or mirrors. Using total internal reflection however, rays may be transmitted over long distances without these losses. Glass fibers are the primary choice for guiding light in this manner using total internal reflection. Glass fibers consist of a glass wire with a cladding. The refractive index of the outer cladding will be smaller than the glass core. This allows for a consistent total internal reflection throughout the wire.

A graded-index material (GRIN) has a refractive index that varies throughout the material. When a ray moves through a graded-index material, the variance in refractive index causes the ray to bend and curve according to how the graded index is laid out.

The path of an optical ray in graded-index material is determined by Fermat’s principle, which states that the path of a ray is the integral of the refractive index (a function of position on the material) between two points, equated to zero. The ray equation can solve this problem, however for simplification, a paraxial approach is taken to give the paraxial ray equation.

Ray Equation:

Paraxial Ray Equation:

A graded index glass fiber is modeled below:

Matrix Optics

A paraxial ray is described by a coordinate and angle. Using this approximation, the output paraxial ray going through a system can be written in matrix form:

,

An optical system can be modeled using the 2×2 ABCD matrix. Matrices of systems may also be cascaded to describe the effect of many systems on a ray.

# Planar Boundaries, Total Internal Reflection, Beamsplitters

Refraction is an important effect in ray optics. The refractive index of a material influences how rays react when entering or leaving a boundary. For instance, if the ray is exiting a medium of smaller refractive index and entering a medium with a higher refractive index, the angle will tend towards being perpendicular to the boundary line. The angle of refraction is also greater than the angle of incidence. This case is called external refraction (n1 < n2) and (θ1 > θ2). If the ray is exiting a medium of higher refractive index into a medium with a lower refractive index, the rays will tend towards being closer to parallel with the medium boundary. This case is referred to as internal refraction (n1 > n2) and (θ2 > θ1). Both of these situations are governed by Snell’s Law:

n1*sin(θ1) = n2*sin(θ2)

When the rays are paraxial, the relation between θ1 and θ2 is linear (n1*θ1 = n2*θ2).

The critical angle occurs when n1*sin(θ1) = n2*sin(pi/2) = n2. θ1 in this case is then equal to the critical angle. If θ1 is greater than the critical angle θC, refraction cannot occur and the situation is characterized by a phenomenon known as total internal reflection (TIR). Total internal reflection is the basis for many optical systems and devices. Systems with total internal reflection are understood to be highly efficient even under more rigorous approaches to optics such as electromagnetic optics.

Prisms are common applications of refraction. A prism of apex angle α and refractive index n deflects a ray incident at an angle of θ:

This is taken by using Snell’s law twice along two planar boundaries.

A beamsplitter is an optical component that divides a ray into a reflected and refracted (or transmitted) ray. The proportions of reflected to refractive light is a problem dealt with in electromagnetic optics. Beamsplitters are also used to combine two rays.

Beam directors apply Snell’s law and the rules governing refraction to direct rays in different directions. Three methods of directing waves are the biprism, the Fresnel biprism and the axicon.

# Mirrors in Geometrical Optics, Paraxial Approximation

The main types of mirrors used as simple optical components are planar mirrors, paraboloidal mirrors, spherical mirrors and elliptical mirrors.

Planar Mirrors reflect rays in a manner that the apparent object location reflects rays from a position that forms a reflected angle (Snell’s law) with the angle between the point of reference and the mirror.

Paraboloidal Mirrors focus all incident rays to a single point, the focus or focal point. The distance from the end of the paraboloidal mirror to the focal point is the focal length. Paraboloidal mirrors are used in telescopes to collect light. Paraboloidal mirrors are also used in flashlight bulbs and light-emitting diodes to direct rays in one direction from a source of light.

Elliptical Mirrors reflect all rays from one source point to another point. Hero’s principle concludes that any path traveled from either point to another will be equal in distance, no matter the direction.

Spherical Mirrors will direct all rays in varying directions. Spherical mirrors may be concave and convex. A spherical mirror acts like a paraboloidal mirror of focal length f = radius/2.

Rays that make small angles with the mirrors axis are called paraxial rays. For paraxial rays, a spherical mirror exhibits a focusing property similar to an elliptical mirror and an imaging property as present in elliptical mirrors. The paraxial approximation considers only paraxial rays and therefore allows spherical mirrors to be considered for the above tendencies. Paraxial Optics is an approach to optics which operates under a set of rules derived from the paraxial approximation. Paraxial Optics is also referred to as first-order optics or Gaussian optics.

In spherical mirrors, considering the paraxial approximation, a focal point is assigned for each source point. All rays that are emitted from a a very far distance (approaching infinite distance) are focused to a point at distance f = (-R)/2.

The following is an example of a use of a paraxial approximation for an image formation using a spherical mirror:

Images are credit of Fundamentals of Photonics, Wiley Series in Pure and Applied Optics

# Postulates of Ray Optics

The following principles of ray optics may be used to describe many optical systems. The numbering system is of no significance.

1. Light travels in the form of a ray. This means that light will travel from a source and is observed when reaching a detector.

2. Optical rays are vector which point in the direction of energy flow.

3. An optical medium is characterized by a refractive index, n = c0 / c, where c0 is the speed of light in free space and c is the speed of light in the medium. The time taken by light to travel a distance d is d/c = nd/c0. The optical pathlength is n*d.

4. In an inhomogeneous medium, the refractive index n(r) is a function of the position r(x,y,z). The optical pathlength along a path between A and B is the integral of A to B of n(r)*ds.

5. Fermat’s Principle states that optical rays travel from A to B following the path that requires the least amount of travel time.

6. Hero’s Principle states that light travels in straight lines in a homogeneous medium. A homogeneous medium means that the refractive index is consistent throughout.

7. Light reflects from mirrors in accordance with the law of reflection: The angle of reflection equals the angle of incidence and the reflected ray lies in the plane of incidence. This may be proven using Hero’s principle.

8. At a boundary between two mediums of different refracting indexes, a ray is split in two. One resulted ray is a reflected ray and the other is a refracted or transmitted ray. The reflected ray is shown in figure (b) above as vector C, while the refracted ray is C’.

9. The refracted ray lies in the place of incidence. The angle of refraction is related to the angle of incidence by Snell’s Law:

10. The proportion of reflected light to refracted light is not dealt with in ray optics.