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  • mbenkerumass 6:00 am on March 2, 2020 Permalink | Reply
    Tags: Optics,   

    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>

    sumstates

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

    gensumstate

    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.

    entanglestate

    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.

    stateex

    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:

    bspli

    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.

    mz11

    mz2

    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.

    mz3

     

    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.

     
  • mbenkerumass 6:00 am on February 25, 2020 Permalink | Reply
    Tags: Optics, ,   

    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:

    waveguide2

    Optical Waveguides:

    waveguide1

    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:

    optical_transmitrecieve

    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.

    waveg1waveg2

    Rectangular Waveguides

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

    waveg3waveg4.png

    The following are useful waveguide geometries:

    waveg5

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

    waveg6

     

     

     

     
  • mbenkerumass 6:00 am on February 15, 2020 Permalink | Reply
    Tags: , Optics   

    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.

    hermitegau2hermitegau1

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

    hermitegaussian

     

    Laguerre-Gaussian Beams

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

    laguerregau1

    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.

    laguerregau2

     

    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).

    besselbeam

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

     
  • mbenkerumass 6:00 am on February 13, 2020 Permalink | Reply
    Tags: , Optics   

    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.

    beam8

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

    beam9

    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.

    beam10

    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.

    beam11

    Beams may also be relayed and expanded using lenses.

    relaylens

    beamexpansion

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

    beam12

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

    beam13

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

     
  • mbenkerumass 6:00 am on February 12, 2020 Permalink | Reply
    Tags: , Optics   

    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.

    gaussianbeam

    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.

    beam1

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

    beam2

    beam3

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

    beam4

    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
      beam5
    • ξ(z): Beam center point
      beam6
    • W0: Minimum Beam level, found at z = 0
      beam7

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

     
  • mbenkerumass 6:00 am on February 10, 2020 Permalink | Reply
    Tags: Optics   

    Wave Optics – Interference, Interferometers 

    Interference
    interference1

    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.

    interference2

    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.

    interference3

    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.

    interferometer1interferometer2

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

     
  • mbenkerumass 6:00 am on January 30, 2020 Permalink | Reply
    Tags: Optics   

    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.

    transplate

    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).

    tras

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

    prism3

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

    thinlens

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

    grin1

    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.

    grating1

    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:

    grating2

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

     
  • mbenkerumass 6:00 am on January 28, 2020 Permalink | Reply
    Tags: Optics   

    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.

    mirror2

     

    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.

    refract1

     

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

     
  • mbenkerumass 6:00 am on January 23, 2020 Permalink | Reply
    Tags: , Optics,   

    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.oscillator

    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.

    laser2

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

     

     
  • mbenkerumass 9:00 am on January 21, 2020 Permalink | Reply
    Tags: Optics   

    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:

    wavefunction

    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.

    intensity

    A monochromatic wave does not vary in intensity over time:monoc

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

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

    helmholtz

    wavenumber

    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.

    paraboloidal

    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. wavefronts

     

    The paraxial Helmholtz equation is as follows:

    paraxhelmholtz

    (1)

     
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