WAVE OPTICS (FOURIER OPTICS)

Transcription

1 WAVE OPTICS (FOURIER OPTICS) ARNAUD DUBOIS October 01

2 INTRODUCTION... Chapter 1: INTRODUCTION TO WAVE OPTICS POSTULATES OF WAVE OPTICS MONOCHROMATIC WAVES Complex Wavefunction Complex Amplitude The Helmholtz Equation Optical Intensit Wavefronts ELEMENTARY WAVES The Plane Wave The Spherical Wave Fresnel Approximation of the Spherical Wave; The Paraboloidal Wave TRANSMISSION THROUGH OPTICAL COMPONENTS Transparent Plate of constant thickness Thin Transparent Plate of Varing Thickness Thin Lens Chapter : FOURIER OPTICS INTRODUCTION PROPAGATION OF LIGHT IN FREE SPACE Correspondence Spatial Harmonic Function / Plane Wave Transmission through optical components Transfer function of free space Impulse-response function of free space Summar OPTICAL FOURIER TRANSFORM... 8

3 3.1 Amplitude in the far-field Amplitude in the back focal plane of a lens DIFFRACTION Fresnel diffraction Fraunhofer diffraction SPATIAL FILTERING Image formation Impulse response function Image formation with coherent illumination Image formation with incoherent illumination Chapter 3: COHERENCE Statistical properties of random light Optical intensit Temporal coherence and spectrum Spatial coherence Gain of spatial coherence b propagation: Zernike and Van Cittert theorem Interference of partiall coherent light Interference and temporal coherence Interference and spatial coherence... 55

4 References 1. Fundamentals of Photonics (Wile Series in Pure and Applied Optics). Bahaa E. A. Saleh, Malvin Carl Teich. John Wile & Sons, 1st edition (August 15, 1991). Introduction to Fourier Optics, J.W. Goodman (Stanford Universit). Second edition Mc Graw-Hill (1996) 3. Principle of Optics, Born and Wolf (Pergamon Press 6 th edition) 1

5 INTRODUCTION Light is an electromagnetic wave phenomenon. Electromagnetic radiations are classified in different categories, depending on their frequenc. Light occupies a limited region of the electromagnetic spectrum. The range of optical wavelengths contains three bands: ultraviolet (10 to 390 nm), visible (390 to 760 nm), and infrared (760 nm to 1 mm). The corresponding range of optical frequencies stretches from Hz to Hz. The spectrum of electromagnetic waves

6 Since light is an electromagnetic wave, light can be described b the same theoretical principles that govern all forms of electromagnetic radiation. Electromagnetic radiation propagates in the form of two coupled vector waves, an electric-field wave and a magnetic-field wave. Although light is described b two wave vectors, it is possible to describe man optical phenomena using a scalar wave theor. In this theor, light is described b a single scalar wavefunction. This approximate wa of treating light is called scalar wave optics, or simpl wave optics. When light waves propagate through and around objects whose dimensions are much greater than the wavelength, the wave nature of light is not discernable. In that case, the behaviour of light can be described b ras. This model of light, called ra optics, obes a set of geometrical rules. Theoreticall, ra optics is the limit of wave optics when the wavelength is zero. However, the wavelength needs not actuall be equal to zero for the ra-optics theor to be useful. As long as the light waves propagate through and around objects whose dimensions are much greater than the wavelength, the ra theor suffices for describing most phenomena. Because the wavelength of visible light is much shorter than the dimensions of the visible objects encountered in our dail lives, manifestations of the wave nature of light are not apparent without careful observation. Thus, the electromagnetic theor of light encompasses wave optics, which in turn, encompasses ra optics. Ra optics and wave optics provide approximate models of light which derive their validit from their successes in producing results that approximate those based on rigorous electromagnetic theor. Although electromagnetic optics provides the most complete treatment of light within the confines of classical optics, there are certain optical phenomena that are characteristicall quantum mechanical in nature and cannot be explained classicall (spectrum of blackbod radiation, absorption and emission of light b matter at specific wavelengths, etc...). These phenomena are described b a quantum electromagnetic theor known as quantum electrodnamics. For optical phenomena, this theor is also referred to as quantum optics. Historicall, optical theor developed roughl in the following sequence: (1) ra optics; () wave optics; (3) electromagnetic optics; (4) quantum optics. Not surprisingl, these models are progressivel more difficult and sophisticated, having being developed to provide explanations for the outcomes of successivel more complex and precise optical experiments. 3

7 Theor of quantum optics provides an explanation of virtuall all optical phenomena. The electromagnetic theor of light provides the most complete treatment of light within the confines of classical optics. Wave optics is a scalar approximation of electromagnetic optics. Ra optics is the limit of wave optics when the wavelength is ver short 4

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9 Chapter 1: INTRODUCTION TO WAVE OPTICS Wave optics is a scalar approximation of electromagnetic optics. The vectorial nature of light is therefore ignored in the wave optics theor. Wave optics can be used for describing man optical phenomena that cannot be described b ra optics, including interference and diffraction. However, wave optics cannot describe optical phenomena that require a vector formulation, such as polarization effects. The wave optics theor is also not capable of providing a complete picture of reflection and refraction of light at the boundaries between dielectric materials. 1. POSTULATES OF WAVE OPTICS The theor of wave optics can be built from a set of postulates. These postulates can be justified b the electromagnetic theor of light : - Light propagates in the form of waves. - In free space, light waves travel with speed c 0. - A homogeneous transparent medium (such as glass) is characterized b a single constant, its refractive index n ( 1). In a medium of refractive index n, light waves travel with a reduced speed c c0 / n. - An optical wave is described mathematicall b a real function of position r ( x,, z) and time t, denoted u( r, t) and known as the wavefunction. It satisfies the wave equation: 1 u u c t where is the Laplacian operator. It is a scalar differential operator. In Cartesian coordinates, u u u it is defined as u. x z 0 An function satisfing the wave equation represents a possible optical wave. Because the wave equation is linear, the principle of superposition applies; i.e., if u1( r, t) and u( r, t) represent optical waves, then u( r, t) u1( r, t) u( r, t) also represents a possible optical wave. 6

10 The wave equation is approximatel applicable to media with position-dependent refractive indices, provided that the variation is slow within distances of a wavelength. The medium is then said to be locall homogeneous. For such media, n and c are simpl replaced b position-dependent functions n() r and c() r, respectivel.. MONOCHROMATIC WAVES In the wave optics theor, a monochromatic wave is represented b a wavefunction with harmonic time dependence, where u( r, t) a( r)cos t ( r ), a() r = amplitude () r = phase = frequenc (ccles/s or Hz) = = angular frequenc (radians/s). Both the amplitude and the phase are generall position dependent, but the wavefunction is a harmonic function of time with frequenc at all positions. As seen in the introduction, the frequenc of optical waves lies in the range to Hz..1 Complex Wavefunction It is usual and convenient to represent the real wavefunction u( r, t) in terms of a complex function so that U( r, t) a( r)exp j ( r ) exp( j t), * u( r, t) Re U( r, t) 1/ U ( r, t) U ( r, t). The function U( r, t), known as the complex wavefunction, describes the wave completel; the wavefunction u( r, t) is simpl its real part. Like the wavefunction u( r, t), the complex wavefunction U( r, t) must also satisf the wave equation 1 U U c t 0.. Complex Amplitude The complex wavefunction can be written in the form 7

11 U( r, t) U( r )exp( j t), where the time-independent factor U( ) a( )exp j ( ) r r r is referred to as the complex amplitude. The wavefunction u( r, t) is therefore related to the complex amplitude b u( r, t) Re U( r )exp( j t) At a given position r, the complex amplitude U() r is a complex variable whose magnitude r r is the amplitude of the wave and whose argument U r U( ) a( ) arg ( ) ( r ) is the phase..3 The Helmholtz Equation The monochromatic wave must satisf the wave equation. Substituting U( r, t) U( r )exp( j t) into the wave equation, we obtain the differential equation k U ( r ) 0 called the Helmholtz equation, where k / c / c is referred to as the wavenumber..4 Optical Intensit The optical intensit of a monochromatic ware is defined as the absolute square of its complex amplitude I( r) U( r ) The intensit of a monochromatic wave does not var with time..5 Wavefronts The wavefronts are the surfaces of equal phase, () r = constant. The constants are often taken to be multiples of, ( r ) q, where q is an integer. 8

12 Summar A monochromatic wave of frequenc is described b a complex wavefunction U( r, t) U( r )exp( j t), which satisfies the wave equation. The complex amplitude U() r satisfies the Helmholtz equation; its magnitude U() r and argument arg U ( r ) are the amplitude and phase of the wave, respectivel. The optical intensit is I( r) U( r ). The wavefronts are the surfaces of constant phase, ( r) arg U( r ) q ( q = integer). The wavefunction u(, t) r is the real part of the complex wavefunction, u(, t) Re U (, t) The wavefunction also satisfies the wave equation. r r. 3. ELEMENTARY WAVES We consider here two special monochromatic waves: the plane wave and the spherical wave. These waves are the simplest solutions to the Helmholtz equation in a homogeneous medium. 3.1 The Plane Wave The plane wave has a complex amplitude U( r ) Aexp j k. r Aexp j kxx k kzz, where A is a constant called the complex envelope, and kx, k, kz wavevector. k is called the For the plane wave to satisf the Helmholtz equation, k k k k, so that the magnitude of the wavevector k is the wavenumber k. Since the phase arg U( ) arga x z r k r, the wavefronts obe kr kxx k kzz q arga ( q = integer). This is the equation describing parallel planes perpendicular to the wavevector k (hence the name "plane wave"). These planes are separated b a distance /k, so that c/ where is called the wavelength. 9

13 The plane wave has a constant intensit I() r A everwhere in space so that it carries infinite power. This wave is clearl an idealization since it exists everwhere and at all times. If the z axis is taken in the direction of the wavevector k, then U( r) Aexp jkz the corresponding wavefunction is u( r, t) Acos t kz arg A A cos t z / c arg A and The wavefunction is therefore periodic in time with period 1/, and periodic in space with period /k, which is equal to the wavelength. Since the phase of the complex wavefunction, U r t t z c A z / c arg (, ) / arg, varies with time and position as a function of the variable t (see Fig), c is called the phase velocit of the wave. u( z, t ) A plane wave travelling in the z direction is a periodic function of z with spatial period and a periodic function of t with temporal period 1/. In a medium of refractive index n : - the phase velocit is c c0 / n - the wavelength is c/ c0 / n, so that 0 / n where 0 c0/ is the wavelength in free space. For a given frequenc, the wavelength in the medium is reduced relative to that in free space b the factor n. - as a consequence, the wavenumber k / increases relative to that in free space k / ) b the factor n. ( 0 0 In summar: as a monochromatic wave propagates through media of different refractive indices its frequenc remains the same, but its velocit, wavelength, and wavenumber are modified. 3. The Spherical Wave Another simple solution of the Helmholtz equation is the spherical wave. The complex amplitude of a spherical wave is r ( / )exp U A r jkr 10

14 where r is the distance from the origin and k / c / c is the wavenumber. The intensit of a spherical wave, I( r ) A / r, is inversel proportional to the square of the distance. Taking arg 0 A for simplicit, the wavefronts are the surfaces kr q or r q, where q is an integer. These are concentric spheres separated b a radial distance /k that advance radiall at the phase velocit c (see Figure). Cross section of the wavefronts of a spherical wave A spherical wave originating at the position r 0 has a complex amplitude U(r) = (A/ r r 0 )exp(- jk r r 0 ). Its wavefronts are spheres centered about r 0. Note that a wave with complex amplitude U ( A/ r)expjkr traveling toward the origin instead of awa from the origin. r is a spherical wave 3.4 Fresnel Approximation of the Spherical Wave; The Paraboloidal Wave Let us consider a spherical wave at points r ( x,, z) sufficientl close to the z axis but far x z. from the origin, so that 1/ a a = x + 0 m z x / z. B using an approximation based on the Talor series expansion: We note 1 1/ 1/ r x z z x z1... z 1 z 8 z 4 Substituting x r z into the phase, and r z into the magnitude of Ur (), we obtain z 11

15 A x U( r) exp jkzexp jk z z A more accurate value of r was used in the phase since the sensitivit to errors of the phase is greater. This is called the Fresnel approximation. This approximation plas an important role in simplifing the theor of transmission of optical waves through apertures (theor of diffraction). With the Fresnel approximation, the complex amplitude of a spherical wave ma be viewed as representing a plane wave Aexp jkz modulated b the factor 1 x exp jk z z, x which involves a phase k z. This phase factor serves to bend the planar wavefronts of the plane wave into paraboloidal surfaces (see Fig.), since the equation of a paraboloid of x revolution is = constant. Thus the spherical wave is approximated b a paraboloidal z wave. When z becomes ver large, the phase approaches kz and the magnitude varies slowl with z, so that the spherical wave eventuall resembles the plane wave exp jkz illustrated below., as A spherical wave can be approximated at points near the z axis and sufficientl far from the origin b a paraboloidal wave. For ver far points, the spherical wave approaches the plane wave. The condition of validit of the Fresnel approximation is not simpl that / 1. Although the third term of the series expansion, 4 /8 ma be ver small in comparison with the second and first terms, when multiplied b kz it ma become comparable to. The approximation is therefore valid when 4 kz /8, or 3 x 4z. For points x, ling within a circle of radius a centered about the z axis, the validit condition is a 4 4z 3 or N F 4 m 1 where a/ z is the maximum angle and m 1

16 N F a z is known as the Fresnel number. EXERCISE 1: Validit of the Fresnel Approximation. Determine the radius of a circle within which a spherical wave of wavelength = 633 nm, originating at a distance 1 m awa, ma be approximated b a paraboloidal wave. Determine the maximum angle m and the Fresnel number N F. 4. TRANSMISSION THROUGH OPTICAL COMPONENTS We now proceed to examine the transmission of optical waves through transparent optical components such as plates and lenses. The effect of reflection at the surfaces of these components will be ignored, since it cannot be properl described using the scalar wave optics model of light. The effect of absorption in the material is also ignored. The main emphasis here is on the phase shift introduced b these components and on the associated wavefront bending. 4.1 Transparent Plate of constant thickness Consider first the transmission of a plane wave through a transparent plate of refractive index n and thickness d surrounded b free space. The surfaces of the plate are the planes z = 0 and z = d and the incident wave travels in the z direction. Let U( x,, z ) be the complex amplitude of the wave. Since external and internal reflections are ignored, U( x,, z ) is assumed to be continuous at the boundaries. The ratio t( x, ) U( x,, d)/ U( x,,0) therefore represents the complex amplitude transmittance of the plate. The incident plane wave continues to propagate inside the plate as a plane wave with wavenumber nk 0, so that U( x,, z ) is proportional to exp jnk0d. Thus U( x,, d)/ U( x,,0) exp jnk0d, so that t( x, ) exp jnk d 0 k n i.e., the plate introduces a phase shift nk0d d /. 0 d z 13

17 4. Thin Transparent Plate of Varing Thickness We consider now a thin transparent plate whose thickness d( x, ) varies smoothl as a function of x and. The incident wave is a paraxial wave. The plate lies between the planes z = 0 and z = d 0, which are regarded as the boundaries of the optical component. The wave crosses a thin plate of width d( x, ) surrounded b thin laers of air of total width d0 d( x, ). The local transmittance is the product of the transmittances of a thin laer of air of thickness d0 d( x, ) and a thin laer of material of thickness d( x, ) so that t( x, ) exp jnk d( x, ) exp jk d d( x, ), from which where h0 exp jk0d0 t( x, ) h0exp j n 1 k0d( x, ) is a constant phase factor. This relation is valid in the paraxial approximation (all angles are small) and when the thickness d 0 is sufficientl small. A transparent plate of varing thickness. 4.3 Thin Lens The general expression for the complex amplitude transmittance of a thin transparent plate of variable thickness is now applied to the plano-convex thin lens. 14

18 Since the lens is the cap of a sphere of radius R, the thickness at the point ( x, ) is d( x, ) = d 0 - PQ= d 0 - (R QC) or d( x, ) d0 R R x This expression can be simplified b considering onl points for which x and are sufficientl small in comparison with R, so that x R. In that case and 1/ 1/ x x R x R 1 R 1 1/ R R We obtain then x d( x, ) d0 R t( x, ) h exp jk x f 0 0 where f R n 1 is the focal length of the lens and h0 exp jnk0d0 no significance. is a constant phase factor that is usuall of The lens imparts to the incident wave a phase proportional to x : it bends the planar wavefronts of a plane wave, transforming it into a paraboloidal wave centered at a distance f from the lens. 15

19 Double-Convex Lens: the complex amplitude transmittance of the double-convex lens is given b x x t( x, ) exp jk0( n 1) exp jk0( n 1) R 1 R t( x, ) h exp jk x f f R1 R with n 1 Recall that, b convention, the radius of a convex/concave surface is positive/negative, i.e., R 1 is positive and R is negative for the lens in Figure. The parameter f is recognized as the focal length of the lens. General case: In the Fresnel approximation, the transmittance of a thin lens can be simpl written: t( x, ) exp x f jk0 where f is the focal length of the lens. A lens is therefore characterized b its focal length onl, regardless of its geometrical shape. EXERCISE : Focusing of a plane wave b a thin lens. Show that when a plane wave is transmitted through a thin lens of focal length f in a direction parallel to the axis of the lens, it is converted into a paraboloidal wave (the Fresnel approximation of a spherical wave) centered about a point at a distance f from the lens. EXERCISE 3: Imaging propert of a Lens. We consider a point source P 1 emitting a paraboloidal wave. What is the wave just after the lens of focal length f? Where is the image point P? Check the Descartes s law of Ra Optics. 16

20 EXERCISE 4: Show that a thin plate of uniform thickness d 0 and quadraticall graded refractive index 1 n( x, ) n 1 x 0, where d0 1, acts as a converging lens. Express the focal length f. 17

21 Chapter : FOURIER OPTICS 1. INTRODUCTION Fourier optics provides a description of the propagation of light waves based on harmonic analsis and linear sstems. Harmonic analsis is based on the expansion of an arbitrar function of time as a sum of harmonic functions of time of different frequencies and different amplitudes: f ( t) f ( )exp j t d An arbitrar function f() t can be analzed as a sum of harmonic functions of different frequencies and complex amplitudes. The function ( )exp f ( ). f j t is a harmonic function with frequenc and complex amplitude The complex amplitude f ( ), as a function of frequenc, is called the Fourier transform of f() t : f ( ) f ( t)exp j t dt This approach is useful for the description of linear sstems. If the response of the sstem to each harmonic function is known, the response to an arbitrar input function can be determined (principle of superposition). The response of a linear sstem can therefore be determined b the use of harmonic analsis at the input and superposition at the output. 18

22 Consider now an arbitrar function f ( x, ) of the two variables x and. We suppose that x and represent the spatial coordinates in a plane. f ( x, ) can also be written as a sum of harmonic functions of x and : f ( x, ) f ( x, )exp j xx d xd f (, ) is the complex amplitude of the harmonic functions, x and are the spatial x frequencies in the x and directions, respectivel. The harmonic function f ( x, )exp j xx is the building block of the Fourier Optics theor. It can be used to generate an arbitrar function of two variables f ( x, ), as illustrated below. The weights f (, ) of the different harmonic functions are determined b use of the two-dimensional Fourier transform x f ( x, ) f ( x, )exp j xx dxd An arbitrar function f ( x, ) can be analzed as a sum of harmonic functions of different frequencies and complex amplitudes. Consider a plane wave U( x,, z) Aexp jkxx k kzz. In an arbitrar plane z, U( x,, z ) is a harmonic function. In the z = 0 plane, for example, U( x,,0) is the harmonic function Aexp j xx frequencies. where k / and k / are the spatial x x Principle of Fourier optics: an arbitrar wave can be analzed as a superposition of planes waves. 19

23 Since an arbitrar function f ( x, ) can be analzed as a sum of harmonic functions, an arbitrar optical wave U( x,, z ) ma be analzed as a sum of plane waves. If it is known how a linear optical sstem modifies plane waves, the principle of superposition can be used to determine the effect of the sstem on an arbitrar wave. It is therefore useful to describe the propagation of light through linear optical components, including free space (free space is a linear sstem because the wave equation is linear). The complex amplitudes in two planes normal to the optic (z) axis are regarded as the input and output of the sstem. The transmission of an optical wave U( x,, z ) through an optical sstem between an input plane z = 0 and an output plane z = d. This is regarded as a linear sstem whose input and output are the functions f ( x, ) U( x,,0) and g( x, ) U( x,, d), respectivel.. PROPAGATION OF LIGHT IN FREE SPACE.1 Correspondence Spatial Harmonic Function / Plane Wave Consider a plane wave of complex amplitude U( x,, z) Aexp jkxx k kzz wavevector k kx, k, kz, wavelength, wavenumber k 1 x z k k k 1 1 complex envelope A. The vector k makes angles x sin kx k and sin k k -z and x-z planes, respectivel, as illustrated below. with, and with the 0

24 The complex amplitude in the z = 0 plane, U( x,,0), is a spatial harmonic function f ( x, ) Aexp j xx k with spatial frequencies k and (ccles/mm). The angles of the wavevector are therefore related to the spatial frequencies of the harmonic function b x x 1 and 1 are the periods of the harmonic function in the x and x x 1 1 directions. We see that the angles x sin x and sin are governed b the ratios of the wavelength of light to the period of the harmonic function in each direction. The wavefronts of the wave match to the periodic pattern of the harmonic function in the z = 0 plane. If k x << k and k << k, so that the wavevector k is paraxial, the angles x, and are small and Thus the angles of inclination of the wavevector are directl proportional to the spatial frequencies of the corresponding harmonic function.. Transmission through optical components Consider a thin optical element with complex amplitude transmittance f ( x, ) exp j xx. When a plane wave of unit amplitude traveling in the z direction is transmitted through this optical element, the wave is modulated b the harmonic function, so that U( x,,0) f ( x, ). The incident wave is then converted into a plane wave 1 1 with a wavevector at angles sin, and sin (see Figure). The optical element is a diffraction grating which acts like a prism. x x 1

25 An arbitrar transmittance of an optical element f ( x, ) can be decomposed as a sum of several harmonic functions of different spatial frequencies. The transmitted optical wave is therefore also the sum of plane waves dispersed into different directions; each spatial frequenc is mapped into a corresponding direction. The amplitude of each wave is proportional to the amplitude of the corresponding harmonic component of f ( x, ). More generall, if f ( x, ) is a superposition integral of harmonic functions, f ( x, ) F( x, )exp j xx d xd with frequencies (, ) and amplitudes F(, ), the transmitted wave U( x,, z ) is the superposition of plane waves, x x U( x,, z) F( x, )exp j xx exp jkzz d xd with complex envelopes (, ) F x, where 1/ k 1/ 1/ z k kx k x that F(, ) is the Fourier transform of f ( x, ). x. Note Since an arbitrar function ma be Fourier analzed as a superposition integral, the light transmitted through an optical element of arbitrar transmittance ma be written as a superposition of plane waves, provided that EXERCICE 5: Diffraction grating 1. x 1. Consider a diffraction grating with a sinusoidal amplitude transmittance : t( x, ) 1 cos x / p Show that an incident plane wave (normal incidence) of wavelength is converted into 3 plane waves. In which directions?

26 .3 Transfer function of free space As seen before, the complex amplitude of a plane wave can be written as where A plane wave at z Aexp j xx x z x exp z x z U( x,, z) Aexp j k x k k z Aexp j k x k jk z Aexp j x exp jk z k, k. x x d is equal to this plane wave at z 0 (harmonic function ) multiplied b the phase factor exp jkzd. Since k kx k kz (Helmholtz) 1/ 1/ 1/ k k k k z x x 1 If x,then k z is real. The sign of k z depends on the direction of propagation. Here we consider a propagation in the direction of increasing z (+ sign). We can then write U( x,, d) U( x,,0) H(, ) x where 1/ 1 H( x, ) exp j x d H is called the transfer function of free space. The magnitude of the transfer function H(, ) 1. The intensit of a plane wave is not modified during propagation, but it undergoes a phase change. x If x 1,then 1/ k j 1/. The transfert function is now z x 1/ 1 real: H( x, ) exp x d. The + sign gives an exponentiall growing function, which is phsicall unacceptable. We have therefore 3

27 1/ 1 H( x, ) exp x d, which represents an attenuation factor. The wave is then called an evanescent wave. We ma therefore regard 1/ as the cutoff frequenc of the sstem. Features contained in spatial frequencies greater than 1/ (details smaller than ) cannot be transmitted b an optical wave over distances much greater than the wavelength. The spatial resolution of imaging sstem in the far field is limited b the wavelength. Near-filed optical imaging was demonstrated recentl to achieve super-resolution (not limited b the wavelength). Fresnel approximation The expression for the transfer function ma be simplified if the spatial frequencies are such that 1/. x 1/ 1 1/ 1 x 1 x We can approximate the transfer function b x x H( x, ) exp jkd exp jd x In this approximation, the phase is a quadratic function of the spatial frequencies. This approximation is known as the Fresnel approximation. The condition of validit of the Fresnel approximation is that 1/ 3 8 be rewritten as : x d. Noting x x, the condition of validit can 4 d 4 1 If a is the largest radial distance in the output plane, the largest angle is a/ d. The condition of validit of the Fresnel approximation ma be written in the form m N F 4 m 1 where N F a / d is the Fresnel number. 4

28 Input-output relation Given an arbitrar monochromatic wave with complex amplitude U( x,, z ) at z 0. It can be decomposed into a sum of plane waves: U( x,,0) U( x,,0)exp j xx d xd x U(,,0) is the Fourier transform of U( x,,0). It represents the complex envelop of planewave components at z 0 : U ( x,,0) FT U ( x, ) ( x,,0) (, )exp x The complex envelop of plane-wave components at z U U x j x dxd d is : U(,, d) U(,,0) H(, ) x x x The complex amplitude of the wave U( x,, z ) at z d is the sum of the contribution of these plane waves : U( x,, d) U( x,, d)exp j xx d xd 1 U( d) FT FT U(0) H U ( x,,0) U ( x ', ', d ) FT FT -1 H ( x, ) U (,,0) U (,, d ) x x H ( x, ) exp jkd exp j d x 5

29 .4 Impulse-response function of free space From U( d) FT 1 FT U (0) H 1, we see that U( d) U(0) FT ( H) (convolution). The inverse Fourier transform of the Transfer function is called the impulse-response function * : x j 0 0 h( x, ) h exp jk with h exp jkd d d The impulse-response function of free space is proportional to the complex amplitude at z d of a paraboloidal wave centered about the origin (0, 0). An alternative procedure for relating the complex amplitude of an arbitrar monochromatic wave between z 0 and z d is to regard the wave at z 0 as a superposition of points, each producing a paraboloidal wave. The wave originating at the point ( x', ',0) has an amplitude U( x', ',0) and is centered about ( x', ',0) so that it generates a wave with amplitude U( x', ',0) h( x x', '). The sum of these contributions is the twodimensional convolution U( x,, d) U( x', ',0) h( x x', ') dx ' d ' ( x x') ( ') U( x,, d) h0 U( x', ',0)exp j dx ' d ' d This relation is also called the Fresnel integral. Note that it is possible to obtain this relation b simplifing the Raleigh and Sommerfeld integral (using the Fresnel approximation), this integral being derived from the Helmholtz and Kirchhoff theorem based on the Helmholtz equation and the Green theorem (see appendix) 1 a a * Using the following Fourier transform relation: exp a x exp x 6

30 This result is consistent with the Hugens-Fresnel principle which states that each point on a wavefront generates an elementar spherical wave. The envelope of these elementar waves constitutes a new wavefront. Their superposition constitutes the wave in another plane. In the Fresnel approximation, a spherical wave centered at (0, 0) is approximated b the paraboloidal wave : x h( x, ) exp jk d.5 Summar In summar, within the Fresnel approximation, there are two approaches to determining the complex amplitude of an arbitrar wave at z d, given the amplitude at z 0. In the space domain approach, in which the wave is expanded in terms of paraboloidal elementar waves, and in the frequenc-domain approach in which the wave is expanded as a sum of plane waves. The procedure of calculation is described in the following table: U ( x,,0) h ( x, ) U ( x ', ', d ) FT FT -1 H ( x, ) U (,,0) U (,, d ) x x jkd x j exp h ( x, ) exp j d d H ( x, ) exp jkd exp j d x 7

31 3. OPTICAL FOURIER TRANSFORM 3.1 Amplitude in the far-field Consider the relation between U( x,, d ) and have The phase in the exponent is: U( x', ',0) in the space-domain approach. We ( x x') ( ') U( x,, d) h0 U( x', ',0)exp j dx ' d ' d Suppose that ( / d) ( x x') ( ') ( / d) ( x ) ( x' ' ) ( xx' ') U( x', ',0) is confined to a circle of radius b, and that the distance d is sufficientl large so that the Fresnel number N ' b / d 1. F We have then ( / d) ( x x') ( ') ( / d)( x ) ( / d)( xx' ') x xx' ' U( x,, d) h0 exp j U( x', ',0)exp j dx ' d ' d d Suppose that U( x,, d) is confined to a circle of radius a, and that the distance d is sufficientl large so that the Fresnel number N a / d 1. We have then F We can recognize a Fourier transform: xx' ' U( x,, d) h0 U( x', ',0)exp j dx' d ' d U ( x,, d x ) U (,,0) d d This approximation is called the Fraunhofer approximation. It is valid when the Fresnel numbers N F and N F ' are small. The Fraunhofer approximation is more difficult to satisf than the Fresnel approximation, which requires that is possible to satisf the Fresnel condition 1. N F m /4 1. Since 1 in the paraxial approximation, it m N F m /4 1 for Fresnel numbers N F not necessaril 8

32 In the Fraunhofer approximation, the complex amplitude U( x,, d) of a monochromatic wave of wavelength, at z d, is proportional to the Fourier transform of the complex amplitude U( x,,0) at z 0, evaluated at the spatial frequencies x x/ d and / d. The approximation is valid if U( x,,0) is confined to a circle of radius b satisfing b / d 1 and if U( x,, d ) is confined to a circle of radius a satisfing a / d 1. EXERCICE 6: Conditions of validit of the Fresnel and Fraunhofer approximations. Demonstrate that the Fraunhofer approximation is more restrictive than the Fresnel approximation b taking = 0.5 µm; assuming that the objects points (x, ) lie within a circle of radius a = 1 cm, and determining the range of distances d for which the two approximations are applicable. The wave is observed on the axis z. 3. Amplitude in the back focal plane of a lens An arbitrar wave can be decomposed into a sum of plane waves. The plane wave-components ma be separated b use of a converging lens as illustrated in the Figure below. The spherical lens transforms each plane wave into a paraboloidal (spherical) wave focused in the back focal plane of the lens (see Exercice ). If a plane wave arrives at small angles x,, it is transformed into a paraboloidal wave centered about the point x xf, f, where f is the focal length of the lens. The lens therefore maps each direction x,, determined b the spatial frequencies x x /, / into a single point x, in the focal plane and thus separates the different plane waves components. Reference : Fundamentals of Photonics (Wile Series in Pure and Applied Optics). Bahaa E. A. Saleh, Malvin Carl Teich. John Wile & Sons, 1st edition (August 15, 1991) Let f, x be the complex amplitude of the optical wave in the plane z = 0, located at an arbitrar distance d from the lens. Light is decomposed into plane waves with complex amplitudes proportional to f x,. The planes waves are focused in the focal plane at points x, defined b x xf xf and f f. The complex amplitudes at points 9

33 x, in the focal plane are therefore proportional to the Fourier transform of f x, evaluated at x x/ f, / f. x g( x, ) f, f f To determine the proportionalit factor, we analse the incident wave into its Fourier components (plane wave decomposition) and propagate them into the optical sstem. We then superimpose the contributions of these plane waves in the focal plane. We assume that the Fresnel approximation is valid. In the plane 0 z ; the plane wave with angles x, U( x,,0) f ( x, )exp j xx has a complex amplitude. In the plane z d (just before crossing the lens), the amplitude of the plane wave is U( x,, d ) f ( x, )exp j xx Hd x, with Hd x, exp jkd exp jd x., Upon crossing the lens, the complex amplitude is multiplied b the transmittance of the lens. We suppose that the lens is ver thin ( = 0). Within the Fresnel approximation, the complex transmittance of the lens is x x 0 t( x, ) exp jk exp j f f. Upon crossing the lens, the amplitude of the plane wave is therefore B noting U( x,, d ) f ( x, )exp j xx Hd x, t( x, ) xx x / f x x fx / f ( x x0) x0 / f / f f / f ( 0) 0 / f with x0 x f and 0 f, the amplitude of the wave can be rewritten where x x0 0 U( x,, d ) A x, exp j f A x, exp jkd exp j d f x f ( x, ) 30

34 We recognize here the complex amplitude of a paraboloidal wave (spherical wave) converging toward the point x0, 0 in the lens focal plane. We now have to examine the propagation in free space between the lens and its focal plane. B using the Fresnel integral, we obtain finall where x U( x,, d f ) h f A, x x j h0 expjkf f The plane wave is focused into a single point at x0 x f, 0 f. The last step is to integrate over all the plane waves (all values of and ). We finall obtain or where The intensit in the lens focal plane is, 0 /, / g x h A x f f x d f x g( x, ) hl exp j f, f f f hl exp jkd exp j d f x f ( x, ) 1 x I( x, ) f (, ) f f f The intensit of light in the back focal plane of the lens is proportional to the squared absolute value of the Fourier Transform of the amplitude of the wave at the input plane, regardless of the distance d. If d where f (front focal plane of the lens), we have then x g( x, ) hl f, f f / exp h j f j kf. l x 31

35 The complex amplitude of light at a point ( x, ) in the back focal plane of a lens of focal length f is proportional to the Fourier transform of the complex amplitude of the complex amplitude in the front focal plane evaluated at the spatial frequencies x x/ f and / f. This relation is valid in the Fresnel approximation. That is wh the back focal plane of a lens is called the Fourier plane. Without lens, the Fourier transformation is obtained onl in the Fraunhofer approximation, which is more restrictive. 4. DIFFRACTION When an optical wave is transmitted through an aperture and travel some distances in free space, its intensit is called the diffraction pattern. If light were treated as ras, the diffraction pattern would be a shadow of the aperture. Because of the wave nature of light, however, the diffraction pattern ma deviate from the aperture shadow, depending on the distance between the aperture and the observation plane, the wavelength, and dimensions of the aperture. The aperture is characterized b a transmittance, or aperture function: 1 inside the aperture p( x, ) 0 outside the aperture The incident wave is multiplied b the aperture function and then propagates in free space. The diffraction pattern is the intensit distribution of the light at a distance d from the aperture. The diffraction pattern is known as the Fresnel diffraction or Fraunhofer diffraction, depending on whether free space propagation is described using the Fresnel approximation or the Fraunhofer diffraction, respectivel. 4.1 Fresnel diffraction The theor of Fresnel diffraction is based on the assumption that the incident wave is multiplied b the aperture function p( x, ) and propagates in free space in accordance with the Fresnel approximation (valid at an distance). Explicit analtical expression of Fresnel diffraction patterns are often difficult to establish (it was however eas for a Gaussian wave, see exercise). Numerical simulation are generall performed (see examples) 3

36 Numerical simulations of Fresnel diffraction patterns resulting from diffraction b a circular aperture (diameter = 18 µm) illuminated with a plane wave at = 63.8 nm. The plane z = 0 mm corresponds to the plane of the aperture. Numerical simulations of Fresnel diffraction patterns resulting from diffraction b a circular aperture (diameter = cm) illuminated with a plane wave at = 500 nm. The propagation distance is 1 m (left) and m (right). Numerical simulations of Fresnel diffraction. The aperture (left) is illuminated with a plane wave at = 500 nm. The diffraction pattern (right) is calculated after propagation over a distance of.56 m. 33

37 Reference : Fundamentals of Photonics (Wile Series in Pure and Applied Optics). Bahaa E. A. Saleh, Malvin Carl Teich. John Wile & Sons, 1st edition (August 15, 1991) 4. Fraunhofer diffraction The theor of Fraunhoffer diffraction is based on the assumption that the incident wave is multiplied b the aperture function p( x, ) and propagates in free space in accordance with the Fraunhoffer approximation (valid onl in far-field). Assuming that the incident wave is a plane wave of intensit I 0 travelling in the z direction, the complex amplitude in the plane of the aperture is I0 p( x, ). As seen earlier, the complex x j d d d amplitude in the far field is U( x,, d) I0h0 p(, ) with h0 exp jkd The diffraction pattern is therefore:. x I( x, ) I0 p, d d 34

38 The Fraunhofer diffraction pattern at the point (x, ) is proportional to the squared magnitude of the Fourier transform of the aperture function p( x, ) evaluated at the spatial frequencies x x/ d and / d. The Fraunhofer approximation is valid for distances d that are usuall extremel large. The are satisfied in applications of long-distance free-space optical communication such as laser radar (lidar) and satellite communication. However, if a lens of focal length f is used to focus the diffracted light, the intensit pattern in the focal plane is proportional to the squared magnitude of the Fourier transform of p( x, ) evaluated at x x/ f and / f : x I( x, ) I0 p, f f Diffraction from a rectangular aperture: The Frauhofer diffraction pattern from a rectangular aperture, of height and width D x and D, observed at a distance d is: Dx x D I( x, ) I0sinc sinc d d Fraunhofer diffraction from a rectangular aperture ( mm) (left), illuminated with a plane wave at = 500 nm. The diffraction pattern (right) is calculated at a distance of 30 m. 35

39 Diffraction from a circular aperture: the Frauhofer diffraction pattern from a circular aperture of diameter D is: 1 1/ J ( Dr / d) I( x, ) I0 with r x Dr / d The diffraction pattern consists of a central disk surrounded b rings. It is known as the Air pattern. The radius of the central disk is r0 1. d / D 0 1. / D Fraunhofer diffraction from a circular aperture (diameter = 160 µm) (left), illuminated with a plane wave at = 500 nm. The diffraction pattern (right) is calculated at a distance of 1 m. 5. SPATIAL FILTERING Consider a two-lens imaging sstem as the one illustrated below: This sstem, called 4-f imaging sstem, has a magnification of

40 This optical sstem can be recognized as an association of two Fourier-transforming subsstems. The first subsstem performs a Fourier Transform, and the second another Fourier transform. As a result, in the absence of an aperture the image is the perfect replica of the object (with a magnification -1). Let f ( x, ) be the complex amplitude in the object plane and g(x, ) the complex amplitude in the image plane. The first lens analzes f ( x, ) into its spatial Fourier transform and separates its Fourier components so that each point in the Fourier plane corresponds to a single spatial frequenc. These components are then recombined b the second lens sstem and the object distribution is perfectl reconstructed. The 4-f imaging sstem can be used as a spatial filter in which the image g(x, ) is a filtered version of the object f ( x, ). Since the Fourier components of f ( x, ) are available in the Fourier plane, a mask ma be used to adjust them selectivel, blocking some components and transmitting others, as illustrated. Reference : Fundamentals of Photonics (Wile Series in Pure and Applied Optics). Bahaa E. A. Saleh, Malvin Carl Teich. John Wile & Sons, 1st edition (August 15, 1991) 37

41 Examples of spatial filters The ideal low-pass filter has a transmittance represented b the Disk function. For example, if the radius of the disk is r 0 and the optical wavelength, the filter eliminates the spatial frequencies greater than c r0 / f. The high-pass filter is the complement of the low-pass filter. It blocks low frequencies and transmits high frequencies. The mask is a clear transparenc with an opaque central disk. The filter output is high at regions of large rate of change and small at regions of smooth or slow variation of the object. This filter is therefore useful for edge enhancement in image-processing applications. The vertical-pass filter blocks horizontal frequencies and transmits vertical frequencies. Onl variations in the x direction are transmitted. If the mask is a vertical slit of width D, the highest transmitted frequenc is ( D/)/ f. 38

42 Composit photo of the moon (a). Filtered photo without horizontal lines (b). Fourier transform (c). Filtered Fourier transform (d). Reference : Fundamentals of Photonics (Wile Series in Pure and Applied Optics). Bahaa E. A. Saleh, Malvin Carl Teich. John Wile & Sons, 1st edition (August 15, 1991) 6. Image formation We consider image formation using the single lens imaging sstem shown below: Reference : Fundamentals of Photonics (Wile Series in Pure and Applied Optics). Bahaa E. A. Saleh, Malvin Carl Teich. John Wile & Sons, 1st edition (August 15, 1991) 39

43 6.1 Impulse response function B definition, the impulse response of the complex amplitude in the object plane of the wave emitted b the point (an impulse) on the optical axis in the object plane. This point emits a spherical wave, whose complex amplitude in the aperture plane (just before the lens) is x U( x, ) h1 exp jk d1 j d 1 where h1 expjkd1 Upon crossing the aperture and the lens, the complex amplitude becomes x U1( x, ) U( x, )exp jk px, f The resultant field U1( x, ) then propagates in free space a distance d. The complex amplitude in the image plane is then obtained b a convolution, according to ( x x') ( ') h( x, ) h U1( x', ')exp j dx ' d ' d We have finall: where h exp jkd j d x x 1 1 d d d h( x, ) h h exp j p,, where p 1 is the Fourier transform of the function d d f 1 x p1( x, ) p( x, )exp j, and If the sstem is focused ( = 0), p 1 = p. We have then where h0 h1h. x x 0 d d d h( x, ) h exp j p, The function h( x, ) is called the impulse response function of the imaging sstem. x If the phase factor d is much smaller than 1, it can be neglected, so that x h( x, ), h0 p d d The sstem s impulse response is proportional to the Fourier transform of the pupil function p. 40

44 6. Image formation with coherent illumination In the ra optics theor, each point A of the object, located at x0, 0, has an image which is a point A located at x0', 0'. In the wave optics theor, the image of the point A is proportional to the impulse response function centered on A : hx' x0', ' 0' U image ( waveoptics ) in the image plane is therefore:. The complex amplitude U ( x', ') U ( x ', ') h x' x ', ' ' dx ' d ' image ( wave optics ) image ( raoptics ) where Uimage ( raoptics ) is the image obtained b ra tracing. This result can be written in a condensed manner as using the convolution operator: U U h image ( waveoptics ) image ( raoptics ) * The image in amplitude is the perfect image (in amplitude, according to the ra optics theor) convoluted b the impulse response function. The spatial frequencies of the wave in the image plane are given b U U H image ( waveoptics ) image ( raoptics ) where H(, ) is the transfert function of the imaging sstem, defined as the Fourier x transform of the impulse response function. where px, is the pupil function. H(, ) p d, d x x The image in intensit (which is what we can observe on a screen or a camera) is then: image ( waveoptics ) image ( raoptics ) * U U h The description of image formation considered here is valid if we consider coherent light. The concept of coherence will be presented in the next chapter 6.3 Image formation with incoherent illumination In the previous studies, the light was supposed to be spatiall coherent. This means that when several optical waves are superimposed, the amplitude of the resulting wave is the sum of the amplitude of each wave. If the light is supposed to be spatiall incoherent, this result is not true. The superposition of several incoherent waves gives a wave whose intensit (and not amplitude) is the sum of the intensit (and not amplitude) of each wave. 41

45 The image of a point given b an optical sstem (the impulse response) has an intensit ' 0', ' 0' h x x. The image formation with incoherent illumination is therefore given b I ( x', ') I ( x ', ') h x' x ', ' ' dx ' d ' image ( wave optics ) image ( raoptics ) 0 0 inc Using condensed notation, we write :, with h x', ' h x', ' inc I I h. image ( waveoptics ) image ( raoptics ) * inc B taking the Fourier transform of the previous relation, we have I I H, image ( waveoptics ) image ( raoptics ) inc where Hinc hinc is referred to as the incoherent transfer function of the optical sstem. The modulation transfert function is defined as H inc. 4

46 Summar 43

47 Example: Imaging sstem with a circular aperture If the pupil is a circle of radius a, the pupil function px, 1 for points inside the circle and zero elsewhere. The pupil function is described b px, The coherent impulse response function of the imaging sstem is then h x, The incoherent impulse response function is J1 sr a where s r d s r dis a where r x. J1 sr hinc x, hx, sr (Air function) d The first zeros of h and h inc are located at rs 1,. a With incoherent illumination, each point in the object plane has an image that is proportional to h inc. When two points of the object plane are close enough so that their images are distant of d the quantit r s, then it becomes impossible to resolve them. The quantit rs 1, is a a measure of the resolution of the imaging sstem. In the case of coherent illumination, the coherent transfert function of the imaging sstem is H, x dis s where x In the case of incoherent illumination, the incoherent transfert function of the imaging sstem is H 1/ 1 cos 1, s s s if 0 otherwise inc x s a d The cutoff frequenc is c a d with coherent illumination with incoherent illumination The impulse response functions and tranfert functions with coherent or incoherent illumination are shown below: 44

48 Impulse-response functions and transfer functions of a diffraction-limited single lens under coherent (a) and incoherent (b) illumination. The pupil is supposed to be a circle of radius a. 45

49 Chapter 3: COHERENCE In the previous chapters, light was assumed to be deterministic. The dependence of the wave function on time and position was perfectl predictable. In realit, optical waves have random fluctuations. For example, light radiated b a hot object (thermal light) is random because it is a superposition of emissions from a large number of atoms radiating independentl and at different frequencies, amplitudes and phases. Randomness in light ma also be a result of scattering from rough surfaces, diffused glass or turbulent fluids. The theor of the random fluctuations of light is known as statistical optics or also as the theor of optical coherence. This theor is based on statistical averaging to define a number of nonrandom quantitative measures. 1. Statistical properties of random light An arbitrar optical wave is defined b a wavefunction u(, t) Re U (, t) r r, where U( r, t) is the complex wavefunction. For random light, U( r, t) is a random function that can be characterized b a number of statistical averages introduced in this section. 1.1 Optical intensit The intensit of coherent (deterministic) light is defined as I( r, t) U( r, t) 46

50 For monochromatic deterministic light, the intensit is independent of time. In the case of random light, the intensit It is then called the instantaneous intensit. The average intensit is defined as U( r, t) is also a random function of position and time. I( r, t) U( r, t) The smbol denotes an ensemble average over man realizations. The average intensit ma be time independent or not (see figure below). When the optical wave is statisticall stationar, its statistical average does not depend on time. Stationar does not mean constanc; it means constanc of the average properties (example = halogen lamp with constant electric current suppl). (a) Statisticall stationar wave. The average intensit does not var with time. (b) A statisticall non-stationar wave has a time-varing intensit. When the light is stationar, the statistical averaging operation can be determined b time averaging over a long duration (instead of averaging over man realizations), so that: 1 T I( r) lim U(, t) dt T T r T 47

51 1. Temporal coherence and spectrum We consider the fluctuation of stationar light as a function of time, at a fixed position. The random fluctuation of Ut () can be characterized b a memor time. Fluctuations separated b a time longer than the memor time are independent, so that the process forgets itself. The function Ut () appears to be smooth within the memor time, but rough when examined over a longer time scale. A quantitative measure of this temporal behaviour can be established b defining a statistical behaviour known as the autocorrelation function. Temporal coherence function The autocorrelation function of a stationar complex random function Ut () is G U t U t * ( ) ( ) ( ) or 1 T * G( ) lim U ( t) U( t ) dt T T T In the theor of optical coherence, the autocorrelation function G( ) is called the temporal * coherence function. This function has a Hermitian smmetr: G( ) G ( ). The intensit of the wave is equal to G( ) when 0 : G(0) I. The phase of the function G( ) is equal to the average of the phase difference between Ut () and Ut ( ). If there is no correlation between Ut () and Ut ( ), G( ) 0. Degree of temporal coherence The temporal coherence function G( ) carries information about the degree of correlation (coherence) of stationar light and the intensit. A measure of the coherence that is insensitive to the intensit is given b the normalized autocorrelation function, known as the degree of temporal coherence: * G( ) U ( t) U( t ) g( ) * G(0) U ( t) U( t) g(0) 1. The absolute value of g( ) cannot be greater than one. 0 g( ) 1 The value of g( ) is a measure of the degree of correlation between Ut () and Ut ( ) case of deterministic and monochromatic light, i.e. U( t) Aexp j 0t Ut ( ) are completel correlated for all time delas. Usuall, g( ) drops from 1 for 0 to 0 for sufficientl large value of.. In the, g( ) 1. Ut () and 48

52 Coherence time / coherence length The width of g( ) denoted c is a measure of the memor time of the fluctuations known as the coherence time. For c, the fluctutations are strongl correlated, whereas for c the are weakl correlated. Although a definition of the width of a function is arbitrar, the usual definition of the coherence time is the power-equivalent width: c g( ) d The coherence time of monochromatic light is infinite. The coherence length is defined as l c c c Power spectral densit The power spectrale densit is defined as S( ) U( ) The power spectral densit is the Fourier Transform of the autocorrelation function. This relation is known as the Wiener-Khinchin theorem S( ) G( )exp j d Demonstration of the Wiener-Khinchin theorem * G( ) U ( t) U( t ) dt * * et ( ) ( )exp U( t) U( )exp j t d U t U j t d *,,,, ( ) ( )exp exp *,,,, U ( ) U( ) exp j d d *,,, G( ) U ( )exp j t d U( )exp j t d dt U U j t j d d dt * U ( ) U( )exp j d U( ) exp j d 49

53 We have ( ) ( )exp G S j d Spectral width The spectral width of light is defined as the width of the spectral densit S( ). Because of the Fourier-transform relation between S( ) and G( ), their widths are inversel related. A light source of broad spectrum has a short coherence time (and short coherence length). 1 c c l c c Source Filtered sunlight ( 0 = µm) LED ( 0 = 1 µm, = 50 nm) Low-pressure Sodium lamp Multimode He-Ne laser ( 0 = 633 nm) Single-mode He-Ne laser ( 0 = 633 nm) c l c.67 fs 800 nm 67 fs 0 µm ps 600 µm 0.67 ns 0 cm 1 µs 300 m 1.3 Spatial coherence Mutual coherence function A descriptor of the spatial and temporal fluctuations of the random function U( r, t) is the crosscorrelation function of U( r 1, t) and U( r, t) : G( r, r, ) U ( r, t) U( r, t ) * 1 1 This function of the time dela is known as the mutual coherence function. Its normalized form is called the complex degree of mutual coherence: g( r, r, ) I( r1) I( r) G( r, r, ). 50

54 When the two points coincide ( r 1 r r ), g( rr,, ) is the degree of temporal coherence. The absolute value of the degree of coherence is bounded between zero and unit. It is a measure of the degree of correlation between the fluctuations at r 1 and those at r at a time later. Degree of spatial coherence The spatial correlation of light ma be assessed b examining the dependence of the mutual function on position for a fixed time dela ( 0 usuall). The mutual coherence function at 0 is G( r, r,0) U ( r, t) U( r, t) * 1 1 This function is denoted G( r1, r ) for simplicit. Its normalized form called the degree of spatial coherence: g( r, r ) I( r1) I( r) G( r, r ) Coherence area For a given point at r, the area scanned b the point r 1 within which g( r1, r ) 1/ (for example) is called the coherence area around the point r. In the case of perfectl spatiall coherent light the coherence area is infinite. In the case of totall spatiall incoherent light the coherence area is reduced to a point. It is important to consider the coherence area when using an optical sstem. If the area of coherence is greater than the size of the aperture through which light is transmitted, the light ma be regarded as coherent. If the coherence area is smaller than the resolution of the optical sstem (size of the sstem s incoherent impulse response), the light can be considered as incoherent. Light emitted b a hot object (thermal light) with an extended surface has an area of coherence on the order of, where is the central wavelength. It can therefore be considered as spatiall incoherent is most cases. However, complete coherence or incoherence are onl idealizations (two limits of partial coherence). In a microscope using Köhler illumination, the spatial coherence of illumination can be adjusted with the aperture diaphragm (D1). The primar advantage of Köhler illumination is to provide even illumination of the sample. 51

55 Principle of Köhler illumination in a microscope (in transmission here) 1.4 Gain of spatial coherence b propagation: Zernike and Van Cittert theorem We consider a spatiall incoherent light source in the plane z = 0. We propose to calculate the degree of, spatial coherence in an arbitrar plane z after propagation over a distance d., ss, and 1, r r represent two arbitra points in the plane of the source and in the plane z respectivel. Within the Fresnel approximation, the complex amplitude in the plane z can be obtained b evaluating the Fresnel integral. We therefore have:, * j *, ( r1 s), U ( r1 ) exp jkd U ( )exp j d d s d s j ( r s) U( r ) exp jkd U( )exp j d d s d s The (spatial) mutual coherence function in the plane z is 5