# Signal Processing And The Basics Of Fourier Transform Biology Essay

**Published:** **Last Edited:**

This essay has been submitted by a student. This is not an example of the work written by our professional essay writers.

The literature review has been divided into three parts. The first one deal with signal processing and the basics of Fourier transform. The next section is an account of knowledge acquired from published papers and the last part is on photodiodes.

Signal processing is concerned with the representation, transformation and manipulation of signals and the information they contain [1]. Before the 1960s, signal processing was utterly a continuous-time analog technology. But with the rapid growth of digital computers and other important theoretical developments such as the fast Fourier Transform, there has been a major shift to digital technology. This has given rise to the field of digital signal processing.

In digital signal processing, signals are represented by sequences of finite-precision numbers, and processing is implemented using digital computation. The more general term discrete-time signal processing includes digital signal processing as a special case, but also includes the possibility that sampled data are processed with other discrete-time technologies [1]. Even continuous-time signals can be processed using discrete-time technology. The continuous-time signal is first sampled to be converted in a discrete-time signal. The after processing, the output is converted back to a continuous-time signal.

Signal interpretation also forms part of signal processing. Here the aim of processing is not to obtain an output signal but to characterize the input signal. An example is the speech recognition system, where information is extracted from the input signal. For such a system, the signal will be digitally preprocessed (filtering, parameter estimation, etcâ€¦), followed by a pattern recognition system producing a symbolic representation, for example, a phonemic transcription of the speech. Symbolic manipulation of signal processing expressions is yet another category of signal processing. The foundation of sophisticated systems designed for signal expression processing are the fundamental signal processing concepts, theorems and properties. Signal modeling is fundamental in data compression and coding.

As we can see, signal processing is a very vast subject. It is a technology that is found in almost every field including entertainment, communications, space exploration, medicine, archaeology, science, military etcâ€¦ so only the term related to the project is considered here, that is, the Fourier Transform.

## 2.1.1 Fourier analysis

The origins of Fourier analysis can be traced to Ptolemy's decomposing celestial orbits into cycles an epicycles and Pythagoras' decomposing music to consonances. Its modern history began with the eighteenth century work of Bernoulli, Euler and Gauss on what later came to be known as the Fourier series [2]. In 1822, Joseph Fourier proposed his Théorie analytique de la chaleur in which he claimed that arbitrary periodic functions could be expanded in a trigonometric series. This claim was proved to be incorrect but it was not far from the truth.

The Fourier transform, also called Fourier analysis or harmonic analysis, is a fundamental mathematical tool in various fields. It can be viewed as the conversion of a signal from one domain, usually the time or space domain, to another domain, the frequency domain. It is used to decompose signals into fundamental components, providing shortcuts for the computation of complicated sums and integrals.

The Fourier series expresses a time-domain signal in terms of a d.c. component and a series of sinusoids of frequency being integer multiples of the fundamental frequency. It can be computed using the equation:

where, is the angular frequency, and is calculated using the fundamental frequency, as follows:

is the mean value of the signal and and are the Fourier coefficients. These three values can be calculated using

where T is the time period.

Below is a cosine function,.

Figure 2.1: Graph of cosine function [3]

Using the equation, its Fourier transform can be obtained. The results are as follows:

Figure2.2: Frequency spectrum of cosine function [3]

Fourier series is a mathematical method to transform a periodic signal from the time domain to its frequency domain by breaking it down to sinusoids of different frequencies. However, signals such as speech, music or video are not periodic. Thus Fourier series cannot be applied to these non-periodic signals but still their frequency spectrum can be obtained. This is done using Fourier transform.

## 2.1.2 Fourier transform

Non periodic signals do not repeat themselves so the Fourier transform is used to convert them to their transform domain. It uses exponentials and complex numbers and is expressed as follows:

where, is the Fourier transform, and

Ï‰ is the angular frequency.

The Fourier transform, being a linear transform, can be used to analyse the spectral components of a signal. The frequency, amplitude and phase of each sinusoid needed to make up any signal can be calculated using equation (2.6).

## 2.1.3 Discrete Fourier Transform

Fourier transform can also be used to analyse discrete signals. The equation below describes the discrete Fourier transform of a signal:

and the inverse of the discrete Fourier transform is given by:

Where, is the input sequence,

is the discrete Fourier transform,

2Ï€k is the angular frequency, and

N is the number of sequences.

## 2.1.4 Importance of Fourier transform

Fourier transform, being a mathematical tool, is fundamental in signal processing for the following reasons:

It provides an efficient approximate representation of the original signal.

Filtering is easier in the frequency domain as undesirable frequencies can be eliminated from the signal.

Some mathematical operations such as convolution are easier and faster to perform in the frequency domain rather than the time domain.

## 2.1.5 Applications of Fourier transform

Fourier transform is one of the most widely used signal analysis tool. It is found in a variety of fields and has a wide range of applications that are ever present in our modern life.

Some examples of Fourier transform applications are:

Scientific

Spectral analysis

Simulation and modeling

Data acquisition

Earthquake recording and analysis

Space

Space photograph enhancement

Data compression

Intelligent sensory analysis by remote space probes

Medical

Diagnostic imaging

Electrocardiogram analysis

Medical image storage/retrieval

Commercial

Image and sound compression for multimedia presentation

Movie special effects

Video conference calling

Telephone

Voice and data compression

Echo reduction

Signal multiplexing

Filtering

Military

Radar

Sonar

Ordnance guidance

Secure communication

Industrial

Oil and mineral prospecting

Process monitoring and control

Non destructive testing

CAD and design tools [4]

## 2.1.6 Spectral analysis and discrete Fourier transform

Information is encoded in the sinusoids that form a signal. This is true for all signals, either naturally occurring or created by man. For example, speech is the result of human vocal cords vibration; the brightness of stars and planets change as they rotate on their axes and revolve around each other; ship's propellers generate periodic displacement of water, and so on. In these signals, the shape of the time domain waveform is not important. The key information is in the frequency, phase and amplitude of the component sinusoids and discrete Fourier transform is used to extract this information [4].

## 2.2 Paper review

With the development and application of high-quality thin-film growth techniques, e.g., molecular-beam epitaxy (MBE) and organometallic vapour phase epitaxy (OMVPE), groups of new materials such as semiconductor heterostructures including superlattices (SLs), single quantum wells (SQWs), and multiple quantum wells (MQWs) have been produced. The reason behind it is their new physical properties and device applications [5]. To characterise these new materials, numerous methods have been applied to manipulate various physical data from them. These techniques include: photoluminescence (PL), photoluminescence excitation (PLE) spectroscopy, modulation spectroscopy, Raman and resonant Raman spectroscopy, absorption spectroscopy, transmission electron microscopy (TEM), Hall measurements among others. However, most of these schemes work under specific conditions such as low temperature, e.g., PL, PLE or need special sample preparation, e.g., TEM, etc. In experiments, the most popular processes must be simple and provide for relevant information. This criterion is satisfied by modulation spectroscopy.

Modulation spectroscopy is the branch of optical spectroscopy that deals with the measurements and interpretation of changes in optical spectra of a sample which are caused by modifying in some way the measurement conditions [6]. Because of its derivative like nature and sharp spectral features, it has been a powerful method for the study of the optical properties and microstructures of semiconductors [7]. Electro-modulation is the most widely used modulation technique. It can be carried out by either electroreflectance (ER), which is the application of an external periodic electric field or photoreflectance (PR) that is a photo-voltage induced by a chopped laser beam.

In the presence of an electric field, the electro-modulation spectrum exhibits Franz-Keldysh oscillations (FKO). Even if it could be a simple and effective technique to study electric fields in various semiconductor structures, it has proven to be more complex in the past. This was due to non-uniform electric fields and impurity induced broadening which reduced the number of detectable Franz-Keldysh oscillations and introduced uncertainties into the measurements.

In 1989, a surface-undoped-doped (UN+/UP+) structure was developed [7]. In this new structure, a uniform electric field was established between the surface and the interface of the undoped/doped layer, allowing a large numbers of FKOs to be observed, thus, permitting accurate determination of electric fields. The photo-reflectance spectrum of the "Van Hoof" showed long extended FKOs together with interference beats, implying that contributions from heavy-holes and light-holes transitions were important in FKO line shape analysis.

Î´-doped semiconductors are formed by introducing a thin planar doping layer in the middle of an undoped layer. The photo-reflectance spectrum of Î´-doped semiconductor samples also exhibit Franz-Keldysh oscillations above the band gap energy. This is because in these samples, there is an undoped cap layer above the region of high planar doping [8]. Near the highly doped region, the Fermi level is close to the conduction band whereas the Fermi level at the surface is near the middle of the band gap, thus creating a uniform electric field in the undoped layer.

Figure 2.3: (a) Schematic diagram of layers in Van Hoof structure. (b) Schematic illustration of a semiconductor substrate and an epitaxial film with a Î´-doped layer.

The Fermi levels of surface-intrinsic-n+-type doped structures are as stated above. The surface states at the surface are filled with electrons from the n+ region [9]. Thus, the system acts as oppositely charged parallel plates, between which the electric field is known to be uniform. However, there is no evidence to prove this except by observing the behavior of the FKOs.

The FKO periods are related to the electric field, F and the effective mass, Âµ. As a result of different effective mass for heavy holes and light holes, the FKO exhibits two different periods. Applying line shape analysis or Fourier transformation to the spectrum, the electric field of the sample can be determined.

## 2.2.1 METHODS TO DETERMINE THE ELECTRIC FIELD, F

## 2.2.1.1 LINE SHAPE ANALYSIS

When the photon energy, E, exceeds the band gap energy, Eg, the asymptotic form of the electro-optic function [10] below can be used to describe the oscillations.

in which, d is the dimensionality of the critical point, and Ä§Î¸ is the electro-optic energy as given by

where, e is the electronic charge, F is the electric field in the depletion region, and Âµ is the reduced mass, given by

where i = heavy- or light-hole, mee is the effective mass of an electron and mei is the effective mass of heavy- or light-hole.

The normalized changes in reflectivity due to modulation parameter can also be described in terms of a change in the dielectric function Î”Îµ, as shown below [10]:

where, Î± and Î² are the Seraphin coefficients, and Î´Îµ1 and Î´Îµ2 are the modulation-induced changes in the real and imaginary parts of the complex dielectric function, respectively.

Near the band edge E0 of GaAs, Î² â‰ˆ 0 and Î”R/R â‰ˆ Î±Î´Îµ1.

For the flatband modulation at the electric field F, Î”Îµ is defined as

Near the band edge of GaAs,

where, = hh or lh, standing for the heavy- and light-hole contributions, respectively, and are parameters which contain the interband optical transition matrix elements.

The function G has a simple Airy functional form

where H is a unit step function and Ai, Bi, Ai' and Bi' are Airy function and their derivatives.

However, the photovoltaic effect from the pumping beam used cannot be neglected. The electron - hole pairs generated by the pumping beam will produce a photovoltage, Vs, which will oppose the original built-in voltage. In most cases, the modulation field, Î´F is assumed to be much smaller than F so that both the positions of the extrema of the FKOs and the deduced electric field are independent of Î´F [11]. But, Vs can be over 2/3 of the Fermi level, VF at low temperatures and thus, cannot be neglected even at room temperature. In this case, R.N. Bhattacharya, H. Shen, P. Parayanthal, F.H. Pollak, T. Coutts and H. Aharoni [12] [13] suggested that

Consequently, the interference beats in the FKO are not only due to HH and LH, but are also influenced by Î´F.

The conventional way to determine F from the FKOs is to make use of the asymptotic expressions of the Airy function in equation (2.15) above. The extrema of the FKOs are given by [10]

in which, is the index number of the th extremum, Ï† is an arbitrary phase factor, and En is the corresponding energy. A plot of (En - Eg)3/2 against n yields a straight line with a slope proportional to F [11].

Another way to determine the electric field in the sample is the fast Fourier transform which is discussed in the following section.

## 2.2.1.2 FAST FOURIER TRANSFORMATION

According to V. L. Alperovich, A. S. Jaroshevich, H. E. Scheiber, and A. S. Terekhov [14] and D. P. Wang and C. T. Chen [15], taking the fast Fourier transformation to the FKO line shapes would result in sharp HH and LH transition peaks.

In order to obtain the FKO period and the internal electric field, the fast Fourier transform has been applied to the spectrum in the energy region higher than the band gap [8]. Before applying the Fourier transformation, the spectrum must be renormalized by

substituting a new argument, z to deal with the periodic functions, and

multiplying the spectrum by

to compensate for inherent damping of FKO.

The resulting new function is

The Fourier transform is calculated using:

where and are the limits of the Fourier transform. The frequency Î© is directly related to the magnitude of the electric field by the following equation,

Ideally, each built-in electric field produces two peaks in the FFT corresponding to light and heavy hole-channels of the optical transitions [8]. But when the modulation field influences the FKO, it yields four frequencies. The HH peak breaks into and and the LH peaks breaks into and .However, only three peaks are observed. In their work, W. H. Chang, T. M. Hsu, W. C. Lee, and R. S. Chuang [7] suggested that the three peaks correspond to , and and that the mixes with and disappears.

## 2.2.2 ADVANTAGES OF FAST FOURIER TRANSFORMATION OVER THE CONVENTIONAL METHOD

An unambiguous reduced effective mass in the determination of F

Even if the signal is too poor to be analyzed, the FFT transformed spectra can still be resolved into two separated peaks related to heavy and and light hole transitions.

If there is more than one electric field existing in the signals, they can be resolved in the transformed spectra.

F can be determined for larger power densities of the pump beam even when the photovoltage effect cannot be neglected [8].

Determining the built-in electric field has allowed the determination of the potential barrier height VB between the surface and the Î´-doped region of the sample. The relation between F and VB is as follows,

where, L is the distance between the surface of the undoped layer and the plane of Î´-doping, assuming a non-degenerate case for the doping concentration.

For a uniform field in the s-i-n+ undoped layer, F and the bias voltage, Vbias are related by the following equation:

where d is the thickness of the undoped layer, Vbi is the built-in voltage, Vs is the photoinduced voltage, and kT/q is the potential induced from thermally excited carriers. Thus, from a plot of F against Vbias, d and (Vbi - Vs - kT/q) can be found.

For the uniformly doped sample, the relation between the surface electric field (Fs) and Vbias is given by [12,13]

where, k is the dielectric constant of the semiconductor under consideration, Îµ0 is the permittivity of free space, and q is the electronic charge. Therefore, a plot of against Vbias can be used to deduce the carrier concentration, n0 and (Vbi - Vs - kT/q).

Once the carrier concentration is known, it is used to determine the depletion zone width and other important parameters which are fundamental for the proper operation of photodiodes. The next section is an overview of photodiodes and its characteristics.

## 2.3 Photodiodes

Photodiodes are semiconductor devices which contain a p-n junction, and often an intrinsic (undoped) layer between n and p layers. They react to high energy particles and photons. Photodiodes operate by absorbing charged particles, usually known as photons to generate a flow of current in an external circuit. This generated current is proportional to the incident power. Photodiodes can be used to detect the presence or absence of infinitesimal quantities of light and can be calibrated for extremely accurate measurements from intensities below 1 pW/cm2 to intensities above 100 mW/cm2 [16].

## 2.3.1 Photodiode Construction

Photodiodes can be manufactured using a number of different technologies thus creating different types of photodiodes. One of them is the planar diffusion type. A planar diffused photodiode is simply a p-n junction. It can be formed by diffusing either a P-type impurity into an N-type wafer, or an N-type impurity into a P-type wafer. The photodiode active area is defined by the diffused area. Another impurity diffusion into the backside of the wafer is necessary to form an ohmic contact. The impurity is an N-type for P-type active region and P-type for an N-type active region. Contact pads are deposited on the front active region on specific areas, and on the backside, completely covering the device. An anti-reflection coating is applied to the active area to reduce the reflection of the light for a specific predefined wavelength. By controlling the thickness of bulk substrate, the speed and responsivity of the photodiode can be controlled.

Figure 2.4: planar diffused silicon photodiode [17]

Note: Photodiodes, when biased, must be operated in the reverse bias mode, i.e. a negative voltage applied to anode and positive voltage to cathode.

## 2.3.2 Principle of Operation

All semiconductors have band gap energy. It is the gap between the valence band and the conduction band, for example, the band gap energy of silicon at room temperature is 1.12eV.

Figure 2.5: Energy band diagram [17]

At absolute zero temperature, the valence band is completely filled with electrons while the conduction band is empty. As temperature increases, the thermal energy of the electrons increases. They become excited and are able to move from the valence band to the conduction band. The electrons can also move up to the conduction band by particles or photons with energy greater or equal to the band gap. The electrons in the conduction band are now free to carry an electric current.

When light is incident on a photodiode, electron-hole pairs are generated which are swept away by drift in the depletion region and are collected by diffusion from the neutral region. The current generated is proportional to the incident light. The light is absorbed exponentially with distance and is proportional to the absorption coefficient. The absorption coefficient is very high for shorter wavelengths in the UV region and is small for longer wavelengths. However, photons with energies smaller than the band gap are not absorbed at all [18].

Across a p-n junction, due to a concentration gradient, electrons from the n-type region diffuse to the p-type region and holes from the p-type region move to the n-type region. Due to this diffusion, the p-region acquires a negative potential compared to the n-region. The movement of holes in the opposite direction leads to a positive charge in the n-region. So there is a potential difference between the two regions, creating an electric field that opposes the diffusion of holes and electrons in either direction. This potential difference is called the built-in voltage, and is calculated using:

where, is the thermal energy,

with k: Boltzmann's constant,

T: temperature in Kelvin, and

q: electronic charge.

is the intrinsic concentration of the material,

is the concentration of free electrons in n-type material, and

is the concentration of holes in p-type material.

For n-type material with electrons only, the concentration of free electrons is equal to the concentration of donors, , and for p-type material with holes only, the concentration of holes is equal to the concentration of acceptors, .

So equation (2.26) can be written as

The movement of electrons and holes between the n and p regions results in a carrier free region, known as the depletion region.

Figure 2.6: depletion region in p-n junction [19]

## 2.3.3 Forward and reverse bias

Let be the bias voltage applied to a p-n junction. When a positive voltage is applied to the anode, relative to the cathode, the junction is said to be forward biased. This causes the potential difference across the semiconductor to decreases, thus decreasing the depletion layer width. When the applied voltage is negative, the junction is reverse biased with the anode being negative compared to the cathode. This negative voltage increases the potential difference across the junction causing the depletion layer width to increase. The total potential, across the semiconductor equals the difference in the built-in potential, and the applied voltage [20] or:

## 2.3.4 Biasing

A photodiode signal can be measured as either a voltage or a current. The advantages of current measurement over voltage measurement are better linearity, offset and bandwidth performance. Photodiodes can be operated with or without applying a reverse bias depending on the application requirements. These two modes are referred to as photoconductive (biased) or photovoltaic (unbiased).

## 2.3.4.1 Photoconductive Mode

When a reverse bias, that is, cathode positive and anode negative is applied to a photodiode, it is in the photoconductive mode. In this mode, the linearity of the device and its speed of response are improved due to increase in depletion region width resulting in a decrease in junction capacitance. However dark current and noise increases.

Figure 2.7: Photodiode in photoconductive mode [21].

Note: Bias voltages larger than the photodiode maximum reverse voltage should not be applied.

## 2.3.4.2 Photovoltaic Mode

When no voltage is applied to a photodiode, it is being operated in photovoltaic mode. When light is incident on a p-n junction, electron-hole pairs are formed in the depletion zone and the p- and n-regions. The electrons are accelerated to then-region due to the internal electric field and the holes move to the p-region. Connecting a load will result in a current flowing through it.

Figure 2.8: Photodiode in photovoltaic mode [21]

This mode of operation is preferred when a photodiode is being used in low frequency applications (up to 350 kHz) and ultra low light level applications. This mode offers a simple operational configuration and also the photocurrent generated has less variation in responsivity with temperature.

## 2.3.5 Density of states

The density of state is defined as the number of states that can be occupied by electrons. However not every state is occupied. The Fermi-Dirac distribution function below gives the probability that an electron occupies an energy level, E at a particular temperature.

Where, is the Fermi level,

k is Boltzmann constant, and

T is temperature in Kelvin.

Figure 2.9: Fermi function at different temperatures [22].

## 2.3.5.1 P-N junction in equilibrium

Equilibrium is reached when the tendency of carriers to diffuse is exactly balanced by their tendency to drift in the newly created electric field [23]. Once equilibrium is reached, the Fermi level is constant through the sample, as shown below.

Figure 2.10: Energy band diagram of a p-n junction in equilibrium [20].

## 2.3.6 Electric field and depletion region

The electrostatic analysis of a p-n junction is very useful as the electric field in the depletion region can be known. Before proceeding with this analysis, some assumptions are made. These are:

Depletion approximation: the electric field is confined to a particular region.

There are no free carriers in the depletion region.

The device is one-dimensional[24]

Starting with Poisson's equation in one dimensional form where the gradient of the electric field, F is proportional to the charge density, Ï yields:

where,

: permittivity of free space,

: permittivity of material, and

Where, and are the edges of the depletion region in the p- and n- type side respectively, measured from the physical junction between the two materials.

The electric field can then be computed using

Using the depletion approximation, which state that the electric field must go to zero at the boundary of the depletion regions, the integration constants and can be determined, resulting in

and

Figure 2.11: (a) Doping concentration in a p-n junction (The dotted lines are the actual net charge density and the solid line represents the assumed charge density in the depletion approximation.

(b) The electric field in a p-n junction [24].

The field is maximum at the junction between the p- and n-region. The electric field lines must be continuous across the interface, thus the electric field in the p-region and the n-region must be equal to each other at the interface, that is at x=0. Substituting x=0 in equation (2.36) and then equating both values of F gives:

This equation makes sense physically as it states that the total charge on one side of the junction must be equal to the total charge on the other. In other words, if the electric field is confined to the depletion region, the net charge in the depletion region must be zero.

Integrating the electric field in a particular region yields the voltage in that region.

To get the potential difference across the junction, one of its sides is set to zero. Here the voltage on the p-type side is set to zero such that at x=-xp, V(x)=0. Using this, C3 is obtained as follows:

Thus,

Using the fact that the potential on the n- and p-region are similar at the interface, C4 can be calculated, such that:

or simply,

Overall, V(x) can be written as:

Assuming that the voltage on the p-type side is zero, the voltage is maximum at x=xn. This voltage is also equal to the built-in voltage across the p-n junction. Thus,

Using equations (2.39) and (2.46), the following equation for xn and xp can be obtaind.

From these equations, the maximum electric field and the total depletion width,xd can be determined.

## 2.3.7 Electrical Characteristics

Figure 2.12: Small-signal photodiode model [25]

A photodiode can be modeled by a current source representing current generated by incident radiation in parallel with an ideal diode representing the p-n junction.

## 2.3.7.1 Shunt resistance, Rsh

Rsh is the shunt resistance that is the resistance of the photodiode depletion region. It is the slope of the current-voltage curve of the photodiode at the origin. It is used to determine the noise current in photovoltaic mode. The shunt resistance of an ideal photodiode is infinite. However, typical values range from 10s to 1000s of mega ohms.

and are the resistances in the depletion zone of then- and p-regions respectively.

where

and

where

A is the area of the depletion zone, and are the resistivity of the device in the depletion zone in the n- and p-region respectively and and is the mobility of electrons and holes respectively.

## 2.3.7.2 Series resistance, Rs

It arises from the resistance of the electrical contacts and the resistance of the undepleted p and n layers and is used to determine the linearity of the photodiode in the photovoltaic mode. Ideal photodiodes should have no series resistance but actual values range from 10 to 1000 ohms.

## 2.3.7.3 Junction capacitance, Cj

The boundaries of the depletion region act as the plates of a parallel plate capacitor. This capacitance can be calculated from the series connection of the capacitances of each region, by adding both depletion layer widths and the width of the undoped region as stated in the equation below.

where, Îµs is the product of the relative dielectric constant and permittivity of free space,

xd is the depletion zone width, and

A is the active area.

## 2.3.7.4 Rise/Fall time

The rise time and fall time of a photodiode is defined as the time for the signal to rise or fall from 10% to 90% or 90% to 10% of the final value respectively [25].

There are three factors defining the response time of a photodiode:

tDRIFT, the charge collection time of the carriers in the depleted region of the photodiode.

tDIFFUSED, the charge collection time of the carriers in the undepleted region of the photodiode.

tRC, the RC time constant of the diode-circuit combination.

tRC is determined by:

where R, is the sum of the diode series resistance and the load resistance (RS + RL), and

C, is the sum of the photodiode junction and the stray capacitances (Cj+CS).

Since the junction capacitance (Cj) is dependent on the diffused area of the photodiode and the applied reverse bias, faster rise times are obtained with smaller diffused area photodiodes, and larger applied reverse biases. In addition, stray capacitance can be minimized by using short leads, and careful lay-out of the electronic components. The total rise time is determined by:

Generally, in photovoltaic mode of operation, rise time is dominated by the diffusion time for diffused areas less than 5 mm2 and by RC time constant for larger diffused areas for all wavelengths. When operated in photoconductive mode, if the photodiode is fully depleted, such as high speed series, the dominant factor is the drift time. In non-fully depleted photodiodes, however, all three factors contribute to the response time.

## 2.3.7.5 Bandwidth

The bandwidth of a photodiode is the maximum frequency that a photodiode can detect without making essential errors. The transit time of the carriers and the inherent capacitance of the p-n junction are the two basic mechanisms that restrict the bandwidth of a photodiode.

The bandwidth of a photodiode depends on:

depletion zone width

material used to manufacture the photodiode

mobility of carriers in the device

active area of the photodiode

load connected to the device

Therefore, for xd greater than 1, it can be assumed that

Hence bandwidth of photodiodes is mostly related to the transit time of carriers in the junction.

## 2.3.8 Optical characteristics

## 2.3.8.1 Responsitivity

Responsitivity is the ratio of an optical detector's electrical output to its optical input [26]. In this case it is the ratio of the photocurrent, Ip to its optical power, Popt.

The photocurrent is given by:

where R is the reflectivity of the surface,

Î± is the coefficient of absorption,

xd is the depletion region width,

IPmax is the maximum photocurrent.

Generally, it is expressed in amperes per watt or volts per watt of incident radiant power. Also it changes with energy bandgap of the photodiode material and wavelength of the incident radiation, as shown below.

Figure 2.13: Graph of responsitivity against wavelength [27]

## 2.3.8.2 Quantum Efficiency

The aim of a photodiode is to absorb photons and generate charge carriers that contribute to produce a photocurrent. Quantum efficiency is the ratio of the number of emitted electrons, Ne to the number of absorbed photons, Np and is represented using the symbol Î·. Ideal devices would have quantum efficiencies of 1 but the actual value is less than 1.

Quantum efficiency too is a function of the wavelength, Î» and is given by:

Quantum efficiency can be calculated using:

or

To increase the quantum efficiency, reflections must be suppressed by an anti-reflection coating.

## 2.3.8.3 Dark current

Dark current is the current that will flow when the photodiode is shielded from radiation, that is, when the photodiode is in the dark. It is also known as the leakage current.

The magnitude of the dark current varies with temperature, terminal voltage and junction area. Increasing the reverse bias across the photodiode causes an increase in the dark current. It also determines the minimum level of radiation that can be detected. The desired photon-generated current must be larger than the dark current before it can be detected reliably.

## 2.3.8.4 I-V characteristics

When no light is incident on a photodiode, its current-voltage characteristic is similar to a rectifying diode. Current increases exponentially when the photodiode is forward biased and a small reverse saturation current appears when it is reversed biased. For ideal photodiodes,

where, is the photodiode dark current,

is the reverse saturation current,

q is the electronic charge,

is the applied bias voltage,

k is Boltzmann's constant and

T is the absolute temperature.

However, photodiodes are not ideal and the above equation changes to

where, n is the ideality factor [28].

From equation (2.67), three states can be defined:

V=0, in this state, the current becomes the reverse saturation current.

V=+V, the current increases exponentially. This state is also known as the forward bias mode.

V= -V, when a reverse bias is applied to a photodiode, the current behaves as shown in the figure.

Figure 2.14: Characteristic I-V curves of a photodiode for photoconductive and photovoltaic modes of operation (P0 and P1 represent different light levels) [18].

Illuminating the photodiode with optical radiation causes the I-V curve to shift by the amount of photocurrent, Ip.

As the applied reverse bias increases, there is a sharp increase in the photodiode current. The applied reverse bias at this point is referred to as the breakdown voltage [18]. This is the maximum reverse bias voltage that can be applied to the photodiode.

## 2.3.8.5 Noise

In the operation photodiode, three main types of noises can be identified, namely:

Quantum noise

Shot noise

Johnson noise/Thermal noise

## Quantum noise

Quantum noise is almost insignificant in optical devices due to its negligibly small value. For it to become a significant factor, the flux density must fall to very small levels.

where, is the power of incident light at which quantum noise is significant,

is the reduced Planck's constant,

c is the speed of light,

is the noise bandwidth,

is the wavelength, and

is the quantum efficiency.

## Shot noise

Shot noise arises due to statistical fluctuations in photocurrent and dark current. These fluctuations are due to random generation and recombination processes in the device. Shot noise current can be estimated using the following equation:

where, is the noise bandwidth,

q is the electronic charge,

is the dark current, and

is the photocurrent.

Shot noise is dominant when operating the photodiode in the photoconductive mode.

## Thermal noise

This is the noise generated by thermally agitated charge carriers. It is dominant when operating the photodiode in the photovoltaic mode. Its magnitude depends on temperature of operation, T, shunt resistance of the photodiode, Rsh, noise bandwidth, , and load resistance,Rl.

where, k is Boltzmann's constant.

## Net noise

The net noise, In can be defined as follows:

## Noise Equivalent Power

It is the incident light power on a phodetector, which generates a photocurrent equal to the noise current. It is defined as:

where, R is the responsivity, and

is the total noise current.