## Direct-Bandgap & Indirect-Bandgap Semiconductors

Direct Semiconductors

When light reaches a semiconductor, the light is absorbed if the photon energy is greater than or equal to the band gap, creating electron-hole pairs. In a direct semiconductor, the minimum of the conduction band is aligned with the maximum of the valence band.

One example of a direct semiconductor is GaAs. The band diagram for GaAs is shown to

the right. As the gap between the valence band and conduction band is 1.42eV, if a

photon of same or greater energy is applied to the semiconductor, a hole-electron pair is created for each photon. This is termed the photo-excitation of semiconductors. The photon is thereby absorbed into the semiconductor.

Indirect Semiconductors and Phonons

For an indirect semiconductor to absorb a photon, the process must be mediated by phonons, which are quanta of sound and in this case refer to the acoustic vibration of crystal lattice. A phonon is also used to provide energy for radiative recombination. When understanding the essence of a phonon, one should recall that sound is not necessarily within hearing range (20 – 20kHz). In fact, the sound vibrations in a semiconductor may well be in the Terrahertz range. The diagram to the right shows how an indirect semiconductor band would appear and also the use of phonon energy to mediate the process of allowing the indirect semiconductor to behave as a semiconductor.

Excitons

Excitons are bound electron-hole pairs that are created in pure semiconductors when a photon with bandgap energy or larger is absorbed. In bulk semiconductors, these excitons will dissipate rapidly. In quantum wells however, the excitons may remain, even at room temperature. The effect of the quantum well is to force an electron and hole to be very close to each other. This allows for a strong bonding effect to take place and allows the quantum well the ability to generate light as a semiconductor laser.

# Quiz

The band structure of a semiconductor is given by:

Where mc = 0.2 * m0 and mv = 0.8 * m0 and Eg = 1.6 eV. Sketch the E-k Diagram.

## Gas Laser and Semiconductor Lasers

The Gas Laser

In laboratory settings, gas lasers (shown right) are often used to eveluate waveguides and other interated optical devices. Essentially, an electric charge is pumped through a gas in a tube as shown to produce a laser output. Gasses used will determine the wavelength and efficiency of the laser. Common choices include Helium, Neon, Argon ion, carbon dioxide, carbon monoxide, Excimer, Nitrogen and Hydrogen. The gas laser was first invented in 1960. Although gas lasers are still frequently used in lab setting sfor testing, they are not practical choices to encorperate into optical integrated circuits. The only practical light sources for optical integrated circuits are semiconductor lasers and light-emitting diodes.

The Laser Diode

The p-n junction laser diode is a strong choice for optical integrated circuits and in fiber-optic communications due to it’s small size, high reliability nd ease of construction. The laser diode is made of a p-type epitaxial growth layer on an n-type substrate. Parallel end faces may functions as mirrors to provide the system with optical feedback.

The Tunnel-Injection Laser

The tunnel-injection laser enjoys many of the best features of the p-n junction laser in it’s size, simplicity and low voltage supply. The tunnel-injection laser however does not make use of a junction and is instead made in a single crystal of uniformly-doped semiconductor material. The hole-electron pairs instead are injected into the semiconductor by tunneling and diffusion. If a p-type semiconductor is used, electrons are injected through the insulator by tunneling and if the semiconductor is n-type, then holes are tunneled through the insulator.

## 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.

## 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.

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