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

     
  • jalves61 11:44 am on February 14, 2020 Permalink | Reply
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    ARRL Examination Study (Part II) 

    For part II of the ARRL examination study, we will study propagation of radio waves.

    Radio waves spread out when transmitted from an antenna in straight lines unless they are reflected or refracted by some object. Due to this spreading and scattering, the waves become weaker as they propagate farther into the air. This limits the “range” a radio transmission can communicate over. The curvature of the Earth creates a “radio horizon” that limits the range of radio propagation. “Line of sight” propagation is when radio waves are transmitted within direct sight of the receiver. This is commonly done in VHF frequencies and higher. Lower frequencies travel as “ground waves”.

    Radio waves are partially reflected when the medium through which the wave propagates changes due to a change in intrinsic impedance (a property defined by permittivity and permeability). Radio waves can even be reflected by change in weather patterns. The figure below shows the concept of diffraction (bending past an obstruction) of radio waves. Diffraction can also refer to spreading when a wave travels through a narrow medium into an open area.

    diffract.PNG

    Light waves also bend by “refraction” which is exactly how radio waves travel around the earth. The earth is curved and therefore the waves need to bend to propagate past “line of sight” distances. The shorter the wavelength (and hence higher frequency), the easier the wave can travel in and out of buildings by penetration of openings in solid objects.

    It interesting to note that different waves received by an antenna can interfere if they are out of phase (destructive interference). This is called “multipath” which is when antennas receive waves from different paths. Moving an antenna a few feet can counterract this. Multipath propagation results in irregular fading. VHF and UHF signals propagating with multipath propagation experience fluttering or “picket-fencing” which comes from rapid variation of the signal strength. Tropospheric propagation or “tropo” is propagation of VHF or higher frequencies assisted by atmospheric phenomena such as weather fronts or temperature inversions. It is not uncommon for Tropo signals to propagate over 300 miles. Reflections can also be caused by conductors such as airplanes. Satellites reflect waves with conductive plating.

    iono

    Thirty to 260 miles above the earth, the ionospheric layer resides. Atoms of nitrogen and oxygen are ionized by UV rays from the sun and become positively charged. The separation of the electrons and the creation of positive ions creates a weakly conductive region. The ionosphere is composed of many different regions. The E, F1 and F2 layers tend to reflect radio waves and the D and E regions tend to absorb waves.

    “Skip” or sky wave propagation is when HF waves are completely bent back towards the earth. The conductive surface of the earth reflects the wave back and the process repeats. These “hops” or reflections allow the waves to be received at farther distances. Lower frequencies are bent more than higher frequencies. For this reason, UHF signals are rarely heard beyond the radio horizon. The MUF (maximum usable frequency) and LUF (lowest usable frequency) are the highest and lowest frequencies that can be reflected by the ionosphere without absorption. When sunspot activity increases, the makes the ionosphere more conductive and increases the MUF.

    Sporadic or “E-Skip” propagation is when patches of the ionosphere become ionized enough to reflect frequencies as high as VHF and UHF. This is most common during early summer and mid winter months.

     

     
  • mbenkerumass 6:00 am on February 13, 2020 Permalink | Reply
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    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
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    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
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    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 7:13 pm on February 7, 2020 Permalink | Reply
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    The Bipolar Transistor, Modes of Operation 

    The transistor is a multifunction semiconductor device that, when used with other circuit elements has the ability to produce a current gain, voltage gain and signal-power gain. The transistor is referred to as a passive device, while the diode is passive. The three basic types of transistor technologies are the bipolar transistor, the metal-oxide-semiconductor field effect transistor (MOSFET) and the junction field effect transistor (JFET). The bipolar transistor most often functions as a voltage-controlled current source.

    The Bipolar Junction Transistor

    The BJT has three separately doped regions and two pn-junctions, which are close enough to interact between each other. The BJT can either be constructed as an NPN or PNP transistor, which stands for the arrangement of positive and negatively doped regions.

    pnpnpn

    The main connections of a BJT transistor are referred to as the collector, base and emitter. Generally, the emitter side is doped to a higher level than the collector. The result of this is that when a supplied a voltage, the electrons will flow in the direction from the emitter to the collector. The direction of current then will be from the collector to the emitter.

    npnpnp2pnp1

    BJT Modes of Operation

    There exist three modes of operation for the BJT transistor. In reference to the diagram below, when the Base-Emitter voltage is zero or reverse biased, the majority of carrier electrons from the emitter will not be injected into the base. This mode where all currents in the transistor are zero is referred to as cut-off. When the Base Emitter voltage is positive (forward biasing), an emitter current is generated. As the Base Emitter voltage increases, the collector current will continue to increase until a certain point at which both the Base Emitter and Base Collector junctions become forward biased. This mode is called saturation.

    npppnn

     
  • jalves61 6:07 pm on February 6, 2020 Permalink | Reply
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    Ferrimagnetic Materials – Circulators and Isolators and Ferrite Phase Shifters 

    When designing microwave and RF components, a non reciprocal device can be obtained by using ferrimagnetic components. Sometimes, it is a good thing for a device to be reciprocal (when the ports of the S parameter matrix are reversible), but in the case of RF devices such as circulators or isolators, it is important for power flow to only move in one direction or to have directional dependence. When directional dependence is present, permeability and permittivity become a tensor rather than a constant and the material is said to be anisotropic.

    Ferrimagnets are different than ferromagnets such as iron or steel in the sense that ferrimagnets have high resistivity and directional dependence at mictrowave frequencies. Both are very strongly magnetic.

    A circulator is a three port device which can be matched at all ports and lossless at the same time. It can couple power in direction or the other, but not both directions. If the reverse direction is desired, the Scattering matrix can be transposed. For a ferrimagnetic circulator this is achieved by changing the polarity of the magnetic bias field. Most of the time a permanent magnet is used, but an electromagnetic can be used for the circulator to function as an SPDT switch.

    An isolator is a two port device which only functions in a single direction. The scattering matrix shown below, implies that the device is nonreciprocal (asymmetric matrix) and lossy due to disobedience to the unitary matrix properties.

    isolator.jpg

    An isolator can prevent damage to a high power source by forcing the power to flow only from the source to load. Any reflected power due to an impedance mismatch will be absorbed by the isolator.  The two main types of ferrite isolators are resonance and field displacement isolators.

    Another two port nonreciprocal RF device is the ferrite phase shifters. Phase shifters are generally used in test and measurement systems or in phased array antennas where the antenna beam can be steered using the device. It is also possible to design a reciprocal phase shifter. In fact most phase shifters are reciprocal in the sense that they provide an equal phase shift in both directions.

     
  • jalves61 5:36 pm on February 5, 2020 Permalink | Reply
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    Impedance Matching – Single Stub Tuning 

    One way to avoid impedance matching a transmission line to a load with lumped circuit elements is to implement Stub matching. Stubs are sections of transmission line that are terminated by either an open circuit or a short circuit. They can be connected in series or in parallel to the transmission line a certain distance from the load. For microstrip and stripline circuits, parallel configurations are preferred whereas series configurations are preferred for slotline or coplanar waveguides.

    Two variables when designing a stub are: the reactance of the stub and the distance between the stub and the load. The idea is that at a specific distance, the susceptance or (reciprocally) the reactance of the load should be cancelled out by the stub. This leads to the cancellation of reflection from the load. This reactance/susceptance value is determine by the length of the stub. The difference in length between a short and open stub is a quarter of the wavelength (which can be confirmed by the Smith Chart).

    The Smith Chart can be used to identify the length required for the stub admittance to equal 1+jb by traveling from the right hand side of the chart clockwise until the reactive parts are equal and opposite. You can also draw an SWR circle using the load impedance/admittance and find where the circle intersects the 1+jb circle. Then the same process for finding the stub length can be used. On a Smith Chart, lengths are always a function of the wavelength.

    stub.png

     

     
  • jalves61 4:55 pm on February 4, 2020 Permalink | Reply
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    Limitations of Impedance matching Networks and the Bode-Fano Criterion 

    When designing an impedance matching network for an RF application, it is important to know the limitations of the design. For example, the maximum reflection coefficient is ideally quite small, but the bandwidth should be large. The Bode-Fano Criterion can specify the limitations of various load configurations to specify how exactly this tradeoff can occur. In addition, complexity of the circuit must be taken into account. The equations differ depending on the configuration of the load impedance.

    bodefano

    The lossless matching networks are passive and lossless. These equations lead to the conclusion that increasing the bandwidth of the matching network can be achieved by increasing the maximum reflection efficient in the passband. In addition, a reflection of zero cannot possibly be achieved unless the bandwidth is zero. This means that a perfect match can only be achieved at a finite number of discrete frequencies (you can’t have a straight line of zero reflection within the passband, only at specific points in the passband). It also shows that as R and C increase, match quality decreases. Circuits with higher Quality factors (store energy longer) are harder to match than low Q circuits.

     
  • mbenkerumass 6:00 am on February 1, 2020 Permalink | Reply
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    The Schottky Diode 

    The Schottky diode, unlike the PN-junction diode is not made of a n-type and n-type semiconductor junction. Instead, is consists of a highly conductive silicide material or metal compound with an n-type semiconductor silicon material. Different metal compounds will allow for varying forward voltage drops, generally between 0.3 and 0.5 volts.

    schottky1

    IV curves are often referred to when understanding the function of a diode. One difference that can be inferred from the IV curve comparison is that the Schottky forward current can be much larger, making it useful in applications with higher levels of current.

    schottkyiv

     
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