poon, banerjee - contemporary optical image processing with matlab

This book serves two purposes: first to introduce the readers to the concepts of geometrical optics, physical optics and techniques of optical imaging and image processing, and secondly,

Image Processing with MATLAB |

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Image Processing with MATLAB |

Ting-Chung Poon

Bradley Department of Electrical and Computer Engineering,

Virginia Polytechnic Institute and State University,

Blacksburg, VA, USA

Partha P Banerjee

Department of Electrical and Computer Engineering,

University of Dayton, Dayton, OH, USA

2001

ELSEVIER

A M S T E R D A M L O N D O N NEW Y O R K - O X F O R D PARIS S H A N N O N - TOKYO

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Contemporary optical image processing with MATLAB

1 MATLAB (Computer file) 2 Image processing - Computer

Q The paper used in this publication meets the requirements of ANSI/NISO Z39.48-1992 (Permanence of Paper)

Printed in The Netherlands

This book serves two purposes: first to introduce the readers to the concepts of geometrical optics, physical optics and techniques of optical imaging and image processing, and secondly, to provide them with experience in modeling the theory and applications using a commonly used software tool MATLAB It is a comprehensively revised and updated version of the authors' previous book Principles of Applied Optics

A sizeable portion of this book is based on the authors' own in-class presentations, as well as research in the area

Emphasis is placed on physical principles, on innovative ways of analyzing ray and wave propagation through optical systems using matrix and FFT methods that can be easily implemented using MATLAB The reason MATLAB is emphasized is because of the fact that it is now a widely accepted software tool, which is very routinely used in signal processing Furthermore, MATLAB is now commonly available in PC or workstation clusters in most universities, and student versions of it (Version 5) are available at the price of an average hardback textbook Although student versions of MATLAB do have limitations compared with the professional version, they are nevertheless very powerful for array and matrix manipulation, for performing FFTs and for easy graphing MATLAB code is very concise and complex mathematical operations can be performed using only a few lines of code In our book we provide several examples of analysis of optical systems using MATLAB and list MATLAB programs for the benefit of readers Since optical processing concepts are based on two dimensional linear system theories for the most part, we feel that this approach provides a natural bridge between traditional optics specialists and the signal and digital image processing community We stress however that we have chosen to use MATLAB as a supplement rather than a replacement of traditional analysis techniques Along with traditional problems, we have included a set of computer exercises at the end of each chapter Taking this approach enables instructors to teach the concepts without committing to the use of MATLAB alone

The book is primarily geared towards a senior/graduate level audience Since the purpose of the book is to bring out the systems aspect of optics, some of the traditional theories in physical optics such as the classical derivation of the Fresnel diffraction formula have been omitted Instead we emphasize the transfer function approach to optical propagation wherever possible, discuss the coherent and optical transfer function

of an image processing system etc In geometrical optics, we take the readers quickly to the matrix formalism, which can be easily used to simulate ray propagation in the absence of diffraction Emphasis is also placed on Gaussian beam optics, and the q- formulation is derived in a straightforward and simple way through the transfer function concept Holography and complex spatial filtering is introduced simultaneously since they are essentially similar Also novel in the book is the ray theory of hologram construction and reconstruction, which is elegant and simple to use in order to determine quickly the location and characteristics of the reconstructed image Of course the ray theory of holograms has its roots in the rigorous wave theory, this is pointed out clearly

in the text

vi Preface

Another novel feature in the book is the discussion of optical propagation through guided media like optical fibers and self-induced guiding using optical nonlinearities In each case, there are ample MATLAB simulations to show beam propagation in such media The reason for introducing nonlinearities in this book is because an increasingly large number of applications of optical nonlinearities exist in image processing, e.g., edge enhancemem and image correlation through phase conjugation Contemporary topics such as this, as well as scanning holography, bipolar incoherent image processing, image processing using acousto-optics, and dynamic holographic techniques for phase distortion correction of images are discussed in the book

A comment concerning units and notation: we mainly use the MKS system of units and the engineering convention for wave propagation, to be made more precise in the text Instructive problems and MATLAB assignments are included at the end of each Chapter Note that some of the examples given in the text may not work with the student version because of the size of the matrix We hope that the book will adequately prepare the interested readers to modem research in the area of image processing

T.-C Poon would like to thank his wife Eliza Lau and his children Christina and Justine for their encouragement, patience and love P.P Banerjee would like to thank his wife Noriko and his children Hans and Neil for their encouragement and support The authors would like to thank Taegeun Kim and Christina Poon for their help in writing some of the MATLAB codes, Christina Poon and Justine Poon for help in typing parts of the manuscript and drawing some of the figures, and Bill Davis for his assistance on the use of the word processing software We would like to acknowledge all our students who have contributed substantially to some of the work reported in this book, especially in Chapters 4 and 7 We would also like to express our gratitude to Professor Adrian Korpel of the University of Iowa for instilling in us the spirit of optics Last, but not least, we would like to thank our parems for their moral encouragement and sacrifice that made this effort possible

C O N T E N T S

Chapter 1: Introduction to Linear Systems

1.1 One and Two-dimensional Fourier Transforms

1.2 The Discrete Fourier Transform

1.3 Linear Systems, Convolution and Correlation

Chapter 2: Geometrical Optics

2.1 Fermat's Principle

2.2 Reflection and Refraction

2.3 Refraction in an Inhomogeneous Medium

2.4 Matrix Methods in Paraxial Optics

2.4.1 The ray transfer matrix

2.4.2 Illustrative examples

2.5 Ray Optics using MATLAB

Chapter 3: Propagation and Diffraction of Optical Waves

3.1 Maxwell's Equations: A Review

3.2 Linear Wave Propagation

3.2.1 Traveling-wave solutions

3.2.2 Intrinsic impedance, the Poynting vector,

and polarization

3.3 Spatial Frequency Transfer Function for Propagation

3.3.1 Examples of Fresnel diffraction

3.3.2 MATLAB example: the Cornu Spiral

3.3.3 MATLAB example: Fresnel diffraction

of a square aperture

3.3.4 Fraunhofer diffraction and examples

3.3.5 MATLAB example: Fraunhofer diffraction

of a square aperture

3.4 Fourier Transforming Property of Ideal Lenses

3.5 Gaussian Beam Optics and MATLAB Example

3.5.1 q-transformation of Gaussian beams

3.5.2 Focusing of a Gaussian beam

3.5.3 MATLAB example: propagation of a Gaussian beam

Chapter 4 : Optical Propagation in Inhomogeneous Media

4.1 Introduction: The Paraxial Wave Equation

4.2 The Split-step Beam Propagation Method

4.3 Wave Propagation in a Linear Inhomogeneous Medium

4.3.1 Optical propagation through graded index fiber

4.3.2 Optical propagation through step index fiber

viii Contents

4.4.2 Photorefractive Media

Chapter 5 Single and Double Lens Image Processing Systems

5.1 Impulse Response and Single Lens Imaging System

5.2 Two-Lens Image Processing System

5.3 Examples of Coherent Image Processing

5.4 Incoherent Image Processing and Optical Transfer Function

5.5 MATLAB Examples of Optical Image Processing

5.5.1 Coherent lowpass filtering

5.5.2 Coherent bandpass filtering

5.5.3 Incoherent spatial filtering

Chapter 6: Holography and Complex Spatial Filtering

6.1 Characteristics of Recording Devices

6.2 The Principle of Holography

6.3 Construction of Practical Holograms

6.4 Reconstruction of Practical Holograms and Complex Filtering

6.5 Holographic Magnification

6.6 Ray Theory of Holograms: Construction and Reconstruction

Chapter 7: Contemporary Topics in Optical Image Processing

7.1 Theory of Optical Heterodyne Scanning

7.1.1 Bipolar incoherent image processing

7.1.2 Optical scanning holography

7.2 Acousto-Optic Image Processing

7.2.1 Experimental and numerical simulations of 1-D

image processing using one acousto-optic cell

7.2.2 Improvement with two cascaded

Introduction to Linear Systems

1.1

1.2

1.3

One and Two-dimensional Fourier Transforms

The Discrete Fourier Transform

Linear Systems, Convolution and Correlation

In this Chapter, we introduce readers to mathematical basics that are often used throughout the rest of the book First, we review some of the properties of the Fourier transform and provide examples of two- dimensional Fourier transform pairs Next we introduce readers to discrete Fourier transforms since this serves as the basis for Fast Fourier transform algorithms that will be used for simulations using MATLAB Finally, we discuss properties of linear systems and the concept of

convolution and correlation

1.1 One and T w o - d i m e n s i o n a l Fourier

T r a n s f o r m s

The one-dimensional (I-D) spatial Fourier transform of a square- integrable function f (z) is given as [Banerjee and Poon (1991)]

F(k~) - f _ ~ f ( z ) e x p ( j k ~ z ) d z - f ~ { f ( z ) } (1.1-1) The inverse Fourier transform is

f ( x ) - ~ f ~ o F(k~)exp( - jk~x) dx - ~ ~ {F(/c~)} (1.1-2)

2 1 Introduction to Linear Systems

The definitions for the forward and backward transforms are consistent with the engineering convention for a traveling wave, as explained in Banerjee and Poon (1991) If f ( x ) denotes a phasor electromagnetic quantity, multiplication by exp(jwt) gives a collection or spectrum of forward traveling plane waves

The two-dimensional (2-D) extensions of Eqs (1.1-1), (1.1-2) are

1 OO

f (x,y;z) - -g~ f_o F(kx, ky, z) exp( - j k ~ x - jkyy) d x d y (1.1-5)

The usefulness of this transform lies in the fact that when substituted into the wave equation, one can reduce a three-dimensional (3-D) partial differential equation (PDE) to a one-dimensional ordinary differential equation (ODE) for the spectral amplitude F(k~, kv; z)

Typical properties and examples of two-dimensional Fourier transform appear in the Table below

sinc( kx k~ , ~ ) - sinc( ~ )sinc( ~),k~ kx where sinc(x) - sin(Trx)

where kxo=27r/xo, kyo=27r/yo

Table 1.1 Properties and examples of some two-dimensional Fourier Transforms

1.2 The Discrete Fourier T r a n s f o r m

Given a discrete function f ( n A ) , n - 0 , N - I , where A is the sampling interval in x, a corresponding periodic function fp(nA) with period N A can be formed as [Antoniou (1979)]"

O O

r = - o o The discrete function f ( n A ) may be formed by the discrete values of a continuous function f ( x ) evaluated at the points x - nL

The discrete Fourier transform (DFT) of fp(nA) is defined as

N - 1

F p ( m K ) - E f p ( n A ) e x p ( j m n K A ) K - , - N A " 2~ ( 1 2 - 2 )

n = 0 The inverse DFT is defined as

N - 1 fp(nA) - ~ F p ( m K ) e x p ( - j m n K A ) (1.2-3)

r n = 0 For properties of the DFT, e.g., linearity, symmetry, periodicity etc., as well as relationship to the z-transform, the Fourier transform and the Fourier series, the readers are referred to any standard book on digital signal processing [Antoniou (1979)]

4 1 Introduction to Linear Systems

For the purposes of this book, the DFT is a way of numerically approximating the continuous Fourier transform of a function The DFT

is of interest because it can be efficiently and rapidly evaluated by using standard fast Fourier transform (FFT) packages Note that the direct evaluation of the DFT requires N complex multiplications and N - 1 complex additions for each value of Fp(mK), and since there are N values to determine, N 2 multiplications and N ( N - 1) additions are necessary However, by using FFT algorithms, such as decimation in time or decimation in frequency, the number of multiplications can be reduced to (N/2)log2N For example, if N > 512, the number of multiplications is reduced to less than 1% of that required by direct evaluation Details of FFT algorithms can be found in any standard digital signal processing text, see for instance, Antoniou (1979) We will use FFT concepts in beam propagation problems in Chapter 4 and in image processing, in Chapters 5-7

The direct connection between the continuous Fourier transform and the DFT is given below For a function f(x) and its continuous Fourier transform F ( kx ) ,

In Eq (1.2-4), Fp(mK) is defined, as in Eq (1.2-2), to be the DFT of fp(nA) The equality holds for the fictitious case when the function is both approximately space and spatial frequency limited

1.3 Linear Systems, Convolution and

Correlation

A system is the mapping of an input or set of inputs into an output

or set of outputs A convenient representation of a system is a mathematical operator [Poularikas and Seely (1991)] For instance for a single-input single-output system,

Pxv{afil (x, y) + bfi2(x, y)}

= a P x y { f i l (x, y ) ) + bP~y{f~2(x, y ) } , (1.3-2) that is, the overall output is the weighted sum of the outputs due to inputs fil and fi2 This feature is particularly useful in constructing the output for a given input, knowing the output for an elementary input like the delta function

For a delta function input of the form ~5(x- x ' , y - y~), the

A output Pzy {6(x - x', y - y')} - h(x, y, x', y') is called the impulse

functions, we know that an arbitrary function fi(x, y) can be represented

a s

(1.3-3)

i.e., fi(x, y) can be regarded as a linear combination of weighted and shifted delta functions We can then write the output fo(X, y) of the linear system as

- L ~ L ~ f ~ ( ~ ' , y')h(~, y, ~', y') d~'ay' (1.3-4)

Now, a linear system is called space-invariant if the impulse response

h(x, y, x', y') depends only on x - x', y - y', that is,

6 1 Introduction to Linear Systems

Now the convolution 9(x, y) of two functions 91 (x, y) and g2(x, y) is defined as

9(x, y) f_~oof_~o~gl (x', y')g2(x - x', y - y') dx'dy'

Using this definition, Eq (1.3-6) can be reexpressed as

fo(~, v) - f~(x, v),h(~, v) - h(~, v ) , f~(~, v) (1.3-8)

It can be readily shown that the Fourier transform G(kx, ky)of

(1.3-9)

Hence, using this property, it follows that

where Fo(k~, kv), F~(k~, ky) and H(kx, lcy) are the Fourier transforms of

S(kx, =s; (kx, k )S:(kx, (1.3-13)

where S(kx, kv), Sl(]cx, ~y) and S2(k~, ky) are the Fourier transforms of

s(x, y), s~ (x, y) and s 2 (x, y), respectively

We will use properties of convolution in our discussion on transfer functions for propagation in Chapter 3, and properties of correlation in connection with matched filtering and image processing in Chapter 6

Problems

1.1 Verify the Fourier transform pairs 9, 10 and 11 in Table 1.1

1.2 From first principles find the Fourier transforms of:

(a) the signumfunction sgn(x, y) - sgn(x)sgn(y), where

(c) sech(x/Xo)Sech(y/yo )

1.3 Verify that fi(x, y).h(x, y) - h(x, y) fi(x, y)

1.4 Find the Fourier transforms of (i) [91 (x, y)*92(x, y)]g3(x, y), (ii)

[91(x, y)g2(x, g)]*g3(x, y) Express your results in terms of the Fourier transforms of 91,2,3

1.5 Prove Parseval's theorem:

f _ ~ f _ ~ l f ( x , y)[2dxdy - ~ ~ f_~f_~[F(kx, kyl2dkxdk,

8 1 Introduction to Linear Systems

1.6 Find the DFTs o f the following periodic functions, defined over a

Reflection and Refraction

Refraction in an lnhomogeneous Medium

Matrix Methods in Paraxial Optics

2.4.1 The ray transfer matrix

2.4.2 Illustrative examples

Ray Optics using MATLAB

In geometrical optics, we view light as particles of energy

traveling through space The trajectory of these particles follows along paths that we call rays We can describe an optical system comprising

elements such as mirrors and lenses by tracing the rays through the system In vacuum or free space, the speed of light particles is a constant approximately given by c - 3 x 10 8 m/s The speed of light in a transparent homogeneous material, which we term v, is again a constant but less than c This constant is a physical characteristic or signature of the material The ratio c/v is called the refractive index n of the material

We can derive the laws of geometrical optics, namely reflection and refraction, using a simple axiom known as Fermat's principle This

is an extremum principle from which we can trace the rays in a general optical medium Based on the laws of reflection and refraction, we will introduce a matrix approach to analyze ray propagation through an optical system

Geometrical optics is a special case of wave or physical optics,

which will be mainly our focus through the rest of the Chapters in the

10 2 Geometrical Optics

book Specifically, it can be shown that we can recover geometrical optics by taking the limit in which the wavelength of light approaches zero In this limit, diffraction and the wave nature of light is absent

2.1 Fermat's Principle

In classical mechanics, Hamilton's principle of least action provides a recipe to find the optimum displacement of a conservative system from one coordinate to another [Goldstein (1950)] Similarly, in optics, we have Fermat's principle which states that the path a ray o f light follows is an extremum in comparison with the nearby paths In Section 2.2, we will use Fermat's principle to derive the laws of geometrical optics

We now give a mathematical enunciation of Fermat's principle Let n(x, y, z) represent a position-dependent refractive index Then

C /1"~ ~ C

represents the time taken to traverse the geometric path ds in a medium

of refractive index n as c is the speed of light in free space Thus, the time taken by the ray to traverse a path C between points A and B (see Figure 2.1) is ! f c C n(x, y z) ds ~ "

The integral above is called the optical p a t h length (OPL) According to Fermat's principle, the ray follows the path for which the OPL is an extremum"

The variation of the integration means that we find the partial differentials of the integral with respect to the free parameters in the integral This will become clear in the next Section where we derive the laws of reflection and refraction when we have a common boundary between two media of different refractive indices In a h o m o g e n e o u s

straight lines

We can also restate Fermat's principle as a principle o f least time

To see this, we divide Eq (2.1-1 a) by c to get

We remark that Eq (2.1-1b) is incorrectly called the least time principle To quote Feynmann [Feynmann (1963)], Eq (2.1-1 b) really means that "if we make a small change in the ray in any manner whatever, say in the location at which it comes to the mirror, or the shape of the curve, or anything, there will be no first order change in the time; there will be only a second order change in the time."

2.2 Reflection and Refraction

When a ray of light is incident on the boundary MM' separating two different media, as in Figure 2.2, observation shows that part of the light is reflected back into the first medium, while the rest of the light is refracted as it enters the second medium The directions taken by these rays are described by the laws of reflection and refraction We will now use Fermat's principle to derive the two laws

Figure 2.2 Reflected and refracted rays for light incident at the interface of two media

Consider a reflecting surface as shown in Figure 2.3 Light from point A is reflected from the reflecting surface to point B, forming the angle of incidence 0i and the angle of reflection Or, measured from the normal to the surface The time required for the ray of light to travel the path AO + OB is given by t - (AO + OB)/v, where v is the velocity of light in the medium containing the points AOB The medium is considered isotropic for convenience From the geometry, we find

According to the least time principle, light will find a path that extremizes t(z) with respect to variations in z We thus set d t ( z ) / d z = 0

plane o f incidence

Let us now use Fermat's principle to analyze refraction as illustrated in Figure 2.4 Ot and 0i are the angles of transmission and incidence, respectively, measured once again from the normal to the interface The time taken by the light to travel the distance AOB is

B

Figure 2.4 Incident (AO) and transmitted or refracted (OB) rays

Using the geometry of the problem, we conclude that

"V 1 V 2

Now v~,~ - c/n~,~ where n~,~ are the refractive indices of media 1 and 2,

respectively Equation (2.2-6a) may be restated as

s i n O t ~ n 1 '

where n ~ / n I is the relative refractive index of medium 2 with respect to

medium 1 Equation (2.2-6) is called Snell's law o f refraction Again,

as in reflection, the incident ray, the refracted ray, and the normal all lie

in the same plane of incidence Snell's law shows that when a light ray passes obliquely from a medium of smaller refractive index into one that has a larger refractive index, it is bent toward the normal Conversely, if the ray of light travels into a medium with a lower refractive index, it is bent away from the normal For the latter case, it is possible to visualize

a situation where the refracted ray is bent away from the normal by

exactly 90 ~ Under this situation, the angle of incidence is called the

critical angle Oc, given by

When the incident angle is greater than the critical angle, the ray originating in medium 1 is totally reflected back into medium 1 This phenomenon is called total internal reflection (TIR) The optical fiber uses this principle of total reflection to guide light, and the mirage on a hot summer day is a phenomenon due to the same principle

2.3 Refraction in an I n h o m o g e n e o u s M e d i u m

In the last Section, we have discussed refraction between two media with different refractive indices, i.e., possessing a discrete inhomogeniety in the simplest case Consider, now, a medium comprising a continuous set of thin slices of media of different refractive indices as shown in Figure 2.5 At every interface, the light ray satisfies Snell's law according to

i n h o m o g e n e o u s m e d i u m , the piecewise linear trajectory of the ray becomes a continuous curve, as shown in Figure 2.6 If d s represents the infinitesimal arc length along the curve, then

Figure 2.5 Rays in a layered medium in which the refractive index is piecewise

Combining (2.3-2), (2.3-3) and (2.3-4), we obtain

Example 2.1 Homogeneous Medium

18 2 Geometrical Optics

approximation (i.e., for small launching angles), all the ray paths have approximately the same period

The case discussed above approximately explains the mechanism

in more detail in Chapter 4

2.4 Matrix Methods in Paraxial Optics

In this Section, we consider how matrices may be used to describe ray propagation through optical systems comprising, for instance, a succession of spherical refracting and/or reflecting surfaces all centered on the same axis, which is called the optical axis Unless otherwise stated, we will take the optical axis to be along the z-axis As

we will see shortly, the "coordinates" of a ray at a certain plane perpendicular to the optical axis can be specified by a vector, which contains the information of the position of the ray and its direction It would therefore be convenient if, given this information, we can find the

"coordinates" of the ray at any other plane, again normal to the optical axis, by means of successive operators acting on the initial ray coordinate vector, with each operator characteristic of the optical element through which the ray travels We can represent these operators by matrices The advantage of this matrix formalism is that any ray can be tracked during its propagation through the optical system by successive matrix multiplications, which can be easily programmed on a digital computer This representation of geometrical optics is elegant and powerful, and is widely used in optical element designs

We will only consider paraxial rays, implying rays that are close to the optical axis, and whose angular deviation from it is small enough such that the sine and tangent of the angles may be approximated by the angles themselves The reason for this paraxial approximation is that all paraxial rays starting from a given object point intersect at another point after passage through the optical system We call this point the image point Nonparaxial rays may not give rise to a single image point This

optics

In what follows, we will first develop the matrix formalism for paraxial ray propagation or ray transfer, and examine some of the properties of ray transfer matrices We then consider several illustrative

examples For instance, we examine the imaging properties of lenses, and derive the rules for ray tracing through an optical system

2.4.1 The ray transfer matrix

Consider the propagation of a paraxial ray through an optical system

as shown in Figure 2.7 Restricting ourselves to one transverse direction (z), a ray at a given cross-section or plane may be specified by its height

z from the optical axis and by its angle or slope which it makes with the axis Thus the quantities (z,O) represent the coordinates of the ray for a given z-constant plane However, instead of specifying the angle the ray makes with the z-axis, it is customary to replace the corresponding by

v = n0, where r~ is the refractive index at the z-constant plane

t /

X1

j s S " ~ , ~ ~ ~ , ~ , , ~ ' ,,,B /

/4

X2

z (Optical axis)

Figure 2.7 Reference planes in an optical system

In Figure 2.7, the ray passes through the input plane with coordinates (Zl, Vl - n101), then through the optical system, and finally

paraxial approximation, the corresponding output quantities are linearly dependent on the input quantities We can, therefore, represent the transformation from the input to the output in matrix form as

20 2 Geometrical Optics

The A B C D matrix above is called the ray transfer matrix, and as we shall see later, it can be made up of many matrices to account for the effects of a ray passing through various optical elements We can consider these matrices as operators successively acting on the input ray coordinate vector We state here that the determinant of the ray transfer matrix equals unity, i.e., A D - BC = 1 This will become clear after

we derive the translation, refraction and reflection matrices

Let us now investigate the general properties of an optical system from the A B C D matrix

Property 1: If D = 0, we have from Eq (2.4-1) that v2 = Czi This means that all rays crossing the input plane at the same point viz z i, emerge at the output plane making the same angle with the axis, no matter at what angle they enter the system The input plane is called the

frontfocalplane of the optical system [see Figure 2.8(a)]

P r o p e r t y 2 : I f / 3 = 0, z2 = Azl (from Eq (2.4-1)) This means that all rayspassing through the input plane at the same point (z l) will pass through the same point (z2) in the output plane {see Figure 2.8(b)] The input and output planes are called the object and image planes,

respectively In addition, A - z2/zi gives the magnification produced

by the system

Furthermore, by inverting the A B C D matrix and the fact that

A D - BC = 1, we note from Eq (2.4-1) that zx = Dz2 since B = 0 The implication of this is that the point z2 is imaged at Zl with magnification 1/A Hence, the two planes containing z l a n d z2 are called conjugate planes Moreover, if A - 1, i.e., the magnification between the two conjugate planes is unity, these planes are called the

unit or principal planes The points of intersection of the unit planes with the optical axis are the unit orprincipalpoints The principal points constitute one set of cardinal points

P r o p e r t y 3: If C = 0, v2 = DVl This means that all the rays entering the system parallel to one another will also emerge parallel, albeit in a new direction [see Figure 2.8(c)] In addition, D(r~l/n2) - 02/01 gives the angular magnification produced by the system

If D = n2/nl, we have unity angular magnification, i.e., 02/01 = 1 In this case, the input and output planes are referred to as the

nodal planes The intersections of the nodal planes with the optical axis are called the nodal points[see Figure 2.8(d)] The nodal points constitute the other set of cardinal points

P r o p e r t y 4: If A = 0, x2 = B y 1 This means that all rays entering the system at the same angle will pass through the same point at the output plane The output plane is the back focal plane of the system [see Figure 2.8(e)]

(c)

_ _ _ _ - - : = = - Z T _ _ _ 0 2

" N o d a l p l a n e s "

(d)

(e)

/>

F i g u r e 2.8 Rays at input and output planes for (a) D - 0, (b) B - 0, (c) C - 0, (d)

the case when the planes are nodal planes, and (e) A - 0

When a ray passes through an optical system, there are usually two types of processes, translation and refraction (and, sometimes, reflection; this is treated later), that we need to consider in order to determine the ray's progress As the rays propagate through a

22 2 Geometrical Optics

homogeneous medium, they undergo a translation process In order to specify the translation, we need to know the thickness of the medium and its refractive index However, when a ray strikes in interface between two regions of different refractive indices, it undergoes refraction To determine how much bending the ray undergoes, we need to know the radius of curvature of the boundary and the values of the refractive indices of the two regions We shall investigate the effect each of these two processes have on the coordinates of a ray between the input and the output planes In fact, we will derive the ray transfer matrices for the two processes

Figure 2.9 shows a ray traveling a distance d in a homogeneous medium of refractive index n Since the medium is homogeneous, the ray travels in a straight line The set of equations of translation by a distance d is

F i g u r e 2.9 A ray in a h o m o g e n e o u s m e d i u m o f refractive index n

These equations relate the output coordinates of the ray with its input coordinates We can express this transformation in a matrix form as

ix2/ ( i v 2 0 J )l/Xl/ l

The 2 x 2 ray transfer matrix, called the translation matrix T , is defined

as

T - 0 1 "

Note that its determinant is unity

We n o w adopt the following convention: when light rays travel across a distance d from the plane z - Zl to the plane z = z2(see Figure 2.9), z 2 - Zl will be taken to be positive for a ray traveling in the +z direction and negative for a ray traveling in the -z direction Therefore,

in the latter case, we take the refractive index of the medium to be negative so that the value of (z2 - Z l ) / n in the translation matrix will remain positive

We next study the effect of a spherical surface separating two regions of refractive indices nl and n2 as shown in Figure 2.10 The center of the curved surface is at C and its radius of curvature is R The ray strikes the surface at the point A and gets refracted Note that the radius of curvature of the surface will be taken as positive (negative) if the center C o f curvature lies to the right (left) of the surface Let x be the distance from A to the axis Then the angle r subtended at the center

24 2 Geometrical Optics

We see that in this case, the height of the ray at A, before and after the refraction, is the same, i.e., x2 = Xl We therefore need to obtain the relationship for v2 in terms of Xl and Vl Applying Snell's law [see Eq (2.2-6)] and using the paraxial approximation, we have

measured in meters, the unit of p is called diopters If an incident ray is made to converge (diverge) by a surface, the power will be assumed to

be positive (negative) in sign The (2 x 2) transfer matrix is called the refraction matrix 7-r and it describes refraction at A for the spherical surface:

Note that the determinant of 7~ is also unity

2.4.2 Illustrative examples

Example 2.3 Plane Parallel Layers

Consider a medium of thickness d, divided into two regions of thicknesses di and d2, and having the same refractive index n, as shown

in Figure 2.11 The system of equations relating the output coordinates

to the input coordinates are given by

Figure 2.11 Plane parallel layers of thicknesses dl and d2

The overall system transfer matrix T can be written as

as expected Note that the overall system matrix T is expressed in terms

of the product of the two individual matrices Ti and T2 written in order from right to left The order of the matrix multiplication is important, as matrix multiplication is not commutative in general

Example 2.4 A Single Lens

Consider a single lens as shown in Figure 2.12 It is evident that the system matrix for the lens consists of two refraction matrices and a translation matrix"

For a thin lens in air, d ~ 0 and nl - 1

notational convenience, Equation (2.4-14) becomes

Writing n2 n for

degenerates into a unit matrix Equation (2.4-15) can be rewritten as

0, and the ray transfer matrix for the thin lens, as in Eq (2.4-16) It follows that for f>(<)0, the ray bends towards (away from) the axis upon

converging (convex) lens, while in the second case, we have a diverging (concave) lens

Example 2.5 Ray Tracing through a Single Thin Lens

In the previous example, we mentioned how rays parallel to the axis of a thin lens in air was bent towards or away from the axis after passing through a thin lens, depending upon whether the lens was converging or diverging To carry this idea on a little further, consider the following cases:

28 2 Geometrical Optics

(a) Ray traveling parallel to the axis:

The input ray vector is (z l, 0) r, hence the output ray vector is given, using Eq (2.4-1) and Eq (2.4-16), as (Zl, - z ~ / f ) r This ray now travels in a straight line at an angle - 1 / f with the axis This means that if Z l is positive (or negative), the ray after refraction through the lens intersects the optical axis at a point a distance f behind the lens

if the lens is converging (f>0) This justifies why f is called the focal length of the lens All rays parallel to the optical axis in front of the lens converge behind the lens to a point called the back focus [see Figure 2.13(a)] In the case of a diverging lens (f<0), the ray after refraction diverges away from the axis as if it were coming from a point on the axis

a distance f in front of the lens This point is called the front focus This

is also shown in Figure 2.13(a)

(b) Ray traveling through the center of the lens:

The input ray vector is ray (0, Vl) T, hence the output ray vector

is given using Eq (2.4-6), Eq (2.4-16) as (0, vl)T This means that a ray traveling through the center of the lens will pass undeviated as shown in

F i gure 2.13 (b)

(c) Ray passing through the front focus of a converging lens"

The input ray vector is given by (x~, x~/f) T, so that the output ray vector is (Zl, 0) T This means that the output ray will be parallel to the axis, as shown in Figure 2.13(c)

In a similar way, we can also show that for an input ray on a diverging lens appearing to travel toward its back focus, the output ray will be parallel to the axis

Figure 2.13 Ray tracing through thin converging and diverging lenses

Example 2.6 Imaging by a Single Thin Lens

Consider an object OO' located a distance do in front of a thin lens of focal length f , as shown in Figure 2.14 Assume that (Zo, vo)T

represents the coordinates originally from point O', and traveling towards the lens Then the output ray coordinates (z, v) at a distance z behind the lens can be written in terms of the input ray coordinates, two translation matrices and the transfer matrix for the thin lens as:

30 2 Geometrical Optics

(;) - 0 1 - 1 / f 1 0 1 Vo

- 1 - d o / f Vo 9 (2.4-18)

Assume for a moment that the object is a point source on the axis, i.e.,

the axis, the other at an angle % Upon refraction through the lens, the on-axis ray will emerge undeviated, while the other ray will emerge at an angle v We define the image of the point object to be the point where these two rays meet Clearly, in this case, the image will be a point on the axis We can calculate the distance z - di along the axis behind the lens where the image will form by setting Xo and x equal to zero in Eq (2.4-

1 1 _ 1 ( 2 4 - 1 9 )

do + ~ - 7"

The sign convention for do and di is as follows, do is positive (negative)

if the object is to the left (right) of the lens If di is positive (negative), the image is to the right (left) of the lens and it is real (virtual)

Now, returning to Eq (2.4-18), we have, corresponding to the image plane, the relation

image is erect (inverted)

If M >0 (<0), the

Example 2.7 Two-lens Combination

Consider two thin lenses of focal lengths fl = f2 = f, separated

distance of 2 f units to the left of lens L1 The matrix chain from the object to the image becomes

from which we can find the position of the image by setting

B = - 2 f + di = 0 Hence, di = 2 f and the magnification (A) is then equal to 1, implying a real erect image of the same size as the object