K. Thyagarajan

Lasers in Science

In this chapter, we discuss some experiments in physics and chemistry (and related areas) which have become possible only because of the availability of highly coherent and intense laser beams.

Digital Filter Structures

What we have learnt so far is how to design either an IIR or FIR digital filter to satisfy a given set of specifications in the frequency domain. We have also seen examples based on MATLAB wherein filtering operations are carried out by specific functions. We really don’t know how these functions really work. If you are a S/W or H/W engineer and want to implement a digital filter in software or hardware, you should be able to describe the flow of signal from the input to the output. Thus, a digital filter structure describes the flow of signal as it propagates from the input to the output sample by sample. This filtering operation is described by a signal flow graph, which is a block diagram with blocks corresponding to the arithmetic operations of addition, multiplication, and unit delays. The blocks are connected by lines with arrows pointing in the direction of signal flow. In digital filter terminology, an adder has two inputs and one output, as shown in Fig. 8.1a. Similarly, a multiplier accepts an input signal and multiplies it by a coefficient a to produce an output, as shown in Fig. 8.1b. A unit delay block is a register, which can hold a sample from its input. The sample can be read from its output after one sample interval. Figure 8.1c illustrates a unit delay element. Note that the unit delay operation in the Z-domain is denoted by z−1. Finally, Fig. 8.1d shows how a signal is tapped into. So, these are the basic building blocks of a digital filter structure. Let us look at a simple example.

IIR Digital Filters

In this chapter we will describe the design of infinite impulse response (IIR) digital filters. The impulse response of an IIR digital filter has an infinite extent or length or duration, hence the name IIR filters. Design of an IIR filter amounts to the determination of its impulse response sequence {h[n]} in the discrete-time domain or to the determination of its transfer function H(ejΩ) in the frequency domain. The design can also be accomplished in the Z-domain. In fact, this is the most commonly used domain. The theory of analog filters preceded that of digital filters. Elegant design techniques for analog filters in the frequency domain were developed much earlier than the development of digital filters. As a result, we will adopt some of the techniques used to design analog filters in designing an IIR digital filter. In order to facilitate the design of an IIR digital filter, one must specify certain parameters of the desired filter. These parameters can be in the discrete-time domain or in the frequency domain. Once the parameters or specifications are known, the task is to come up with either the impulse response sequence or the transfer function that approximates the specifications of the desired filter as closely as possible. In the discrete-time domain, one of the design techniques is known as the impulse invariance method. In the frequency domain, the design will yield a Butterworth or Chebyshev or elliptic filter. These three design procedures will result in a closed-form solution. Similarly, the impulse invariance technique will also result in a closed-form solution to the design of IIR digital filters. In addition to these analytical methods, an IIR digital filter can also be designed using iterative techniques. These are called the computer-aided design. Let us first describe the impulse invariance method of designing an IIR digital filter. We will then deal with the design in the frequency domain and the computer-aided design.

Discrete Fourier Transform

Discrete Fourier transform (DFT) is a frequency domain representation of finite-length discrete-time signals. It is also used to represent FIR discrete-time systems in the frequency domain. As the name implies, DFT is a discrete set of frequency samples uniformly distributed around the unit circle in the complex frequency plane that characterizes a discrete-time sequence of finite duration. DFT is also intrinsically related to the DTFT, as we will see in this chapter. Because DFT is a finite set of frequency samples, it is a computational tool to perform filtering and related operations. There is an efficient algorithm known as the fast Fourier transform (FFT) to perform filtering of long sequences, power spectrum estimation, and related tasks. We will learn about the FFT in this chapter as well.

Discrete-Time Signals and Systems

Continuous-time or analog signals are processed using analog devices such as amplifiers, filters, etc. It is impossible to process signals multiplexed from various sources using a single hardware system in the analog domain. On the other hand, digital signals can be processed using both special-purpose hardware and software systems. Worldwide use of Internet, mobile communications, etc. demands all kinds of data such as video, audio, graphics, etc. In order to receive this information on a single device, computer, for instance, it is impossible to use analog signals and techniques. In order to be able to design and implement digitally based systems, it is absolutely necessary to have an understanding of digital signals and systems. Digital signals are discrete in time and amplitude. However, we will assume discrete-time signals to have a continuum of amplitude in order to be able to analyze such signals and systems mathematically. In this chapter we will describe typical discrete-time signals mathematically and then use them to describe and analyze linear time-invariant discrete-time systems. To help the readers understand the mathematical details, we will work out examples followed by MATLAB-based examples. Since digital signals are obtained from analog sources, we will also discuss the conversion of continuous-time signals into digital signals using analog-to-digital converters (ADC).

FIR Digital Filters

A finite impulse response (FIR) digital filter, as the name implies, has an impulse response sequence that is of finite duration as opposed to an IIR digital filter, which has an impulse response that is of infinite duration. Therefore, the Z-transform of the impulse response of an FIR digital filter in general can be written as

Laser-Induced Fusion

It is well known that the enormous energy released from the sun and the stars is due to thermonuclear fusion reactions, and scientists have been working for over 40 years to devise methods to generate fusion energy in a controlled manner. Once this is achieved, one will have an almost inexhaustible supply of relatively pollution-free energy. A thermonuclear reactor based on laser-induced fusion offers great promise for the future. With the tremendous effort being expended on fabrication of extremely high-power lasers, the goal appears to be not too far away, and once it is practically achieved, it would lead to the most important application of the laser.

Einstein Coefficients and Light Amplification

In this chapter we discuss interaction of radiation and atoms and obtain the relationship between absorption and emission processes. We show that for light amplification a state of population inversion should be created in the atomic system. We also obtain an expression for the gain coefficient of the system. This is followed by a discussion of two-level, three-level, and four-level systems using the rate equation approach. Finally a discussion of various mechanisms leading to broadening of spectral lines is discussed.

Laser Rate Equations

In Chapter 4 we studied the interaction of radiation with matter and found that under the action of radiation of proper frequencies, the atomic populations of various energy levels change. In this chapter, we will be studying the rate equations which govern the rate at which populations of various energy levels change under the action of the pump and in the presence of laser radiation. The rate equations approach provides a convenient means of studying the time dependence of the atomic populations of various levels in the presence of radiation at frequencies corresponding to the different transitions of the atom.

Lasers in Industry

In Chapter 10 we discussed the special properties possessed by laser light, namely its extreme directionality, its extreme monochromaticity, and the large intensity associated with some laser systems. In the present chapter, we briefly discuss the various industrial applications of the laser.