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  • mbenkerumass 9:00 am on January 7, 2020 Permalink | Reply
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    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).

    refraction

    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.

    tir

     

    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:

    prism2

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

    prism1

     

    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.

    beamsplit

    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.

     

     

     
  • mbenkerumass 6:00 am on January 5, 2020 Permalink | Reply
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    Microstrip Stepped Impedance Low-Pass Filter 

    Passes Freqeuncies below 10 GHz:

     

    steplpfilter1steplpfilter2steplpfilter3

     
  • mbenkerumass 6:00 am on January 4, 2020 Permalink | Reply
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    The P-N Junction 

    A P-N junction is created in a single semiconductor crystal by doping one side as a p-type and one as an n-type. The region where the two types converge is known as the p-n junction.

    The extra electrons that were added to the n-type semiconductor move towards the p-type junction side while the holes added through p-type doping are positioned closer to the n-type junction.

    pnj2

    As electrons leave the n-type region, it becomes positively charged. This process is called diffusion. The depletion region is the area between the p and n-type sides. The state of equilibrium in the p-n junction is the state of the depletion region without any external electrical potential applied. As mentioned before in a previous paper, the Fermi level is the average between the conduction band and the valence band. By altering the levels of electron holes and electrons in the p-type and n-type sections, holes drift toward the the n-type side and electrons move towards the p-type side, which causes both sections to be closer to the Fermi level in their regions of the material.

    pnj

    When voltage is applied to the pn junction, electrons and electron holes from either side tend towards equilibrium. If the positive potential is applied to the p-type and it is more positive than the n-type area, holes will travel towards the negative voltage. Through diffusion, electrons or electron holes may jump through the depletion layer. For the reason however that electron holes (positive charge) may only move in the direction of the n-type region and electrons (negative charge) may only move in the opposite direction. The direction of electron flow, due to their negative charge is opposite the conventional direction of current flow. Since electrons are only moving from the n-type region to the p-type region, it can be understood that current will only move in the direction going from the side of the p-type region towards the n-type region.pnj1

     

     
  • mbenkerumass 9:00 am on January 2, 2020 Permalink | Reply
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    P-Type and N-Type Semiconductors 

    N-Type Semiconductors are created when doping a semiconductor with impurities that adds extra valence electrons to the outermost shell to share free electrons with neighboring atoms. Phosphorous, arsenic and antimony are examples of atoms with five valence electrons, also known as pentavalent impurities, adding an extra electron for each doped atom. This does not mean however that an N-type semiconductor is negatively changed, because there will exist a balancing positive charge in the nucleus of the doped atom. An N-type semiconductor is a better conductor than intrinsic semiconductor materials.

    P-Type Semiconductors are formed by adding group 3 elements, known as trivalent impurity atoms such as boron, aluminum and indium to the semiconductor structure. These atoms have only three electrons in the outermost shell, producing an extra electron hole, which attracts neighboring electrons.

    p-type and n-type semiconductors

    To recap, N-type semiconductors:

    • possess pentavalent elements as impurity atoms to add a donor electron to the material.
    • do not have a negative charge since atom nucleus charge offsets added electrons, meaning they are electrically neutral.

    P-type semiconductors:

    • possess trivalent impurity elements as impurity atoms to add an electron hole.
    • are also electrically neutral.

     

     
  • mbenkerumass 9:00 am on December 31, 2019 Permalink | Reply
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    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.

    mirror1

    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.

    ellipticalmirror

    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.

    spmirror

    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.

    spmir2

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

    sp

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

     
  • mbenkerumass 10:36 pm on December 29, 2019 Permalink | Reply
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    025/100 Smith Chart Impedance Plotting 

    Example 3.4-1: Plot the impedance Z = 25 + j25 Ohm on the standard Smith Chart.

    025-1

    In order to plot a schematic simulation on a smith chart diagram, run a simulation.

    025-2025-3

     
  • mbenkerumass 9:00 am on December 28, 2019 Permalink | Reply
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    Fermi Level in Semiconductor Materials 

    The Fermi level in a semiconductor is the probability that energy levels in a valence band and conduction band in the atoms are occupied. At absolute zero temperature, a semiconductor acts as a perfect insulator. As the temperature increases, free electrons are made available.  An intrinsic semiconductor is a pure crystal with no impurities or defect atoms. In an intrinsic semiconductor, the probability of occupation of energy levels in either the conduction band or the valence band are equal. The Fermi level of an intrinsic semiconductor lies between the valence band and the conduction band. This area between both bands is known as the forbidden band.

    fermi

    Where KB is the Boltzmann constant (1.3806503 × 10-23 m2 kg s-2 K-1), T is the absolute temperature of the intrinsic semiconductor, Nv is the density of states in the valence band, the hole concentration in the valence band is:

    hole

    Where Nc is the density of states in the conduction band, the electron concentration in the conduction band is calculated:

    electron

    The Fermi level for an intrinsic semiconductor is given as the average of the conduction band level and the valence band level.

    fermilev

     

    Electron Doping

    A intrinsic semiconductor may be altered by adding controlled amounts of specific atoms, called dopants to the crystal. This alters the number of electrons in the conduction bands or electron holes in the valence bands.

     
  • mbenkerumass 9:00 am on December 26, 2019 Permalink | Reply
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    Crystal Structures 

    Crystal Structures

    Crystalline structures are noted by their regular, predictable and periodic arrangement of atoms or molecules. The  arrangement of atoms and molecules for crystal structures is called a lattice. Crystalline materials include many metals, chemical salts and semiconductors.

    copper

    Solid crystals are classified by the cohesive forces that hold the lattice together and the shape or arrangement of the atoms in the material. Different arrangements include a simple cubic crystal, a face-centered cubic structure and a body-centered cubic structure.

    cubic

    In metals, each atom contributes at least one loosely bound electron to build an electron gas of nearly free electrons that move throughout the lattice structure. When an electric field E is applied to a metal, a current flows in the direction of the field. The flow of charges is described in terms of a current density J, or current per unit cross-sectional area. The current density is proportional to the applied electric field by a factor of the electrical conductivity σ of the material.

    J = σ*E

    The electrons in the lattice material experience a force F = -e*E due to the field and become accelerated. The velocity of electrons in the lattice is known as the drift velocity.

     

    Bonding and the formation of Semiconductors

    In atomic structures, different types of molecules have a varying number of electrons in the outer atomic rings or shells (valence electrons). Ionic bonding is performed by electrons present in the outermost shell, easily forming a positive ion by releasing the outer electron (net positive charge) or enter the outermost shell of another atom to make it a negative ion (net negative charge). Metallic bonding uses a loosely bound electron in an outermost shell to contribute to the crystal as a whole, creating a metallic crystal.

    periodic table

    The method of bonding for Ge, C and Si can be quite different however, since they have four valence electrons in the outermost shell. These four electrons can be shared with four neighboring molecules. The bonding force that results from this phenomenon is covalent bonding. In this formation however, electrons belonging to the same bond do not have a definite position in any one atom, meaning they may move between atoms that are bonded. Compound semiconductors such as GaAs (Gallium Arsenide), AlAs (Aluminum Arsenide) and InP (Indium Phosphide) have mixed bonding including both covalent and ionic bonding. These bonding characteristics and the ability for electrons to both move throughout atoms in the structure and to form ionic bonds are the basis for the use of semiconductor materials.

     
  • mbenkerumass 9:00 am on December 24, 2019 Permalink | Reply
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    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.

    planeofincidence

    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:

    snell

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

    ray1

     
  • mbenkerumass 8:49 am on December 22, 2019 Permalink | Reply
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    Review of Fourier Series 

    The French mathematician Fourier discovered that any periodic waveform can be expressed as a series of harmonically related sinusoids.

    Any periodic waveform can be expressed as the following:

    series

    The first term a0/2 is the constant DC or average component of f(t). The terms with coefficients a1 and b1 represent the fundamental frequency components of f(t). Coefficients a2 and b2 are the second harmonic components at frequency 2w. The frequency doubling on the second order harmonic is computed as a result of the multiplication of sinusoids.

    harmonics

    In order to determine the coefficients of a harmonic series ai and bi, multiply both sides of the above formula by 2sin(2wt). In this case for simplicity, let w = 1.

    calcaibi

    Next, integrate from zero to 2*pi.

    sereis2

    The following relations are then found:

    seriesrelations

    The Fourier series are often expressed in exponential form:

    series3

    The MATLAB function int(f,t,a,b) is often a useful tool, where f is the function, t is the symbolic variable, and a and b are the bounds of integration.

    matlabint

     

     

     
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