Optical Microscopy: Lecture 4. Interpretation of the 3-Dimensional Images and Fourier Optics. Pavel Zinin HIGP, University of Hawaii, Honolulu, USA

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1 GG 711: Advanced Techniques in Geophsics and Materials Science Optical Microscop: Lecture 4 Interpretation of the 3-Dimensional Images and Fourier Optics Pavel Zinin HIGP, Universit of Hawaii, Honolulu, USA

2 Lecture Overview Introduction to Fourier Transform Fourier Spectrum Approach of Image in 3-D Contrast in Reflection and Transmission Microscop Emulated Transmission Confocal Raman Microscop Using scanning optical or acoustical microscopes it is possible to reconstruct structure of three-dimensional objects. How the image we obtain is related to the structure of the real object? Is it possible to simulate images of simple-shaped bodies obtained b optical or acoustical microscopes?

3 Interpretation of the Acoustical Images in Medicine Acoustical images are difficult for direct interpretation: Ultrasound images of a fetus during seventeen of development (left) and an artist s rendering of the image. (after Med. Encclopedia, 2005) B striving to do the impossible, man has alwas achieved what is possible. Bakunin

4 The Fourier Transform What is the Fourier Transform? The continuous limit: the Fourier transform (and its inverse) F( ) f ( t) ep( it) dt f 1 ( t ) ( ) ep( ) 2 F i t d Fourier analsis is named after Joseph Fourier, who showed that representing a function b a trigonometric series greatl simplifies the stud of heat propagation. Jean Baptiste Joseph Fourier ( )

5 The Fourier Transform Consider the Fourier coefficients. Let s define a function F() that incorporates both cosine and sine series coefficients, with the sine series distinguished b making it the imaginar component: F( ) f ( t) cos( t) dt i f ( t) sin( t) dt where t is the time and is the angular frequenc; = 2 f, where f is the frequenc F( ) f ( t) ep( it) dt The Fourier Transform F() is called the Fourier Transform of f(t). It contains equivalent information to that in f(t). We sa that f(t) lives in the time domain, and F() lives in the frequenc domain. F() is just another wa of looking at a function or wave.

6 The Fourier Transform and its Inverse F( ) f ( t) ep( it) dt FourierTransform f 1 ( t ) ( ) ep( ) 2 F i t d Inverse Fourier Transform So we can transform to the frequenc domain and back. Interestingl, these functions are ver similar. There are different definitions of these transforms. The 2π can occur in several places, but the idea is generall the same.

7 What do we hope to achieve with the Fourier Transform? We desire a measure of the frequencies present in a wave. This will lead to a definition of the term, the spectrum. F E( t)cos( t) E( t)cos( t) ep( i t) dt E( t) ep( i0t) ep( i0t) ep( i t) dt E( t)ep( i[ 0] t) dt E( t)ep( i[ 0] t) dt 2 2 F 1 1 E( t)cos( 0t) E( 0) E( 0) 2 2 Eample: E(t) = ep(-t 2 ) E( t)cos( t) 0 t F E( t)cos( t) From: Prof. Rick Trebino, Georgia Tech

8 Fourier Transform of a rectangle function: rect(t) F() f(t) F( ) ep( it) dt [ep( )] it i 1 [ep( i ) ep( i i 2 ep( i ) ep( i 2i sin( = 1 t = 1 F( 2 sinc( Asinc(

9 Sinc() and wh it's important f(t) F() Sinc( 0 o o 1 o ; fo = 1 Sinc() is the Fourier transform of a rectangle function. Sinc 2 () is the Fourier transform of a triangle function. Sinc() describes the aial field distribution of a lens. Sinc 2 (a) is the diffraction pattern from a slit. It just crops up everwhere = t

10 F() f(t) Fourier Transform of a rectangle function: rect(t) F() F() = t o 1 o ; fo = = 5 = = 10 =

11 Eample: the Fourier Transform of a Gaussian function F 2 2 {ep( )} ep( )ep( ) at at i t dt 2 ep( / 4 a) 2 ep( at ) 2 ep( / 4 a) 0 t 0

12 The Fourier Transform Properties Linearit Shifting Modulation Convolution af (, ) bg(, ) af( u, v) bg( u, v) f e F u v j 2 ( u0v 0 (, ) ) (, ) 0 0 j2 ( u0v0 ) e f F u u v v (, ) (, ) f (, )* g(, ) F( u, v) G( u, v) 0 0 Multiplication f (, ) g(, ) F( u, v)* G( u, v) Separable functions f (, ) f ( ) f ( ) F( u, v) F( u) F( v)

13 Scale Theorem The Fourier transform of a scaled function, f(at): F { f ( at)} F( / a) / a Proof: F F { f ( at)} f ( at) ep( i t) dt Assuming a > 0, change variables: u = at { f ( at)} f ( u) ep( i [ u/ a]) du / a f ( u) ep( i [ / a] u) du / a F( / a) / a If a < 0, the limits flip when we change variables, introducing a minus sign, hence the absolute value. (From. Prof. Rick Trebino, Georgia Tech. )

14 f(t) F() The Scale Theorem in action Short pulse t The shorter the pulse, the broader the spectrum! Mediumlength pulse t This is the essence of the Uncertaint Principle! Long pulse t (From. Prof. Rick Trebino, Georgia Tech. )

15 Sampling and Nquist Theorem Nquist theorem: to correctl identif a frequenc ou must sample twice a period. So, if is the sampling, then π/ is the maimum spatial frequenc

16 Sampling and Nquist Theorem The Nquist Shannon sampling theorem is a fundamental result in the field of information theor, in particular telecommunications and signal processing. Sampling is the process of converting a signal (for eample, a function of continuous time or space) into a numeric sequence (a function of discrete time or space). The theorem states: Theorem: If a function (t) contains no frequencies higher than B hertz, it is completel determined b giving its ordinates at a series of points spaced 1/(2B) seconds apart. OR In order to correctl determine the frequenc spectrum of a signal, the signal must be measured at least twice per period. In order for a band-limited (i.e., one with a zero power spectrum for frequencies f > B) baseband ( f > 0) signal to be reconstructed full, it must be sampled at a rate f 2B. A signal sampled at f = 2B is said to be Nquist sampled, and f =2B is called the Nquist frequenc. No information is lost if a signal is sampled at the Nquist frequenc, and no additional information is gained b sampling faster than this rate. The Nquist Sampling Theorem states that to avoid aliasing occuring in the sampling of a signal the samping rate should be greater than or equal to twice the highest frequenc present in the signal. This is refered to as the Nquist sampling rate.

17 The Fourier Transform of 2-D objects Scanning of the intensit along a line Fourier Transform of the intensit of the contrast of the image along a line Intensit of the contrast of the image along a line

18 The Fourier Transform of 2-D objects I log{ F{I} 2 +1} [F{I}] FT of an Image (Magnitude + Phase) Peters, Richard Alan, II, "The Fourier Transform", Lectures on Image Processing, Vanderbilt Universit, Nashville, TN, April 2008, Available on the web at the Internet Archive,

19 The Fourier Transform of 2-D objects I Re[F{I}] Im[F{I}] FT of an Image (Real + Imaginar) Peters, Richard Alan, II, "The Fourier Transform", Lectures on Image Processing, Vanderbilt Universit, Nashville, TN, April 2008, Available on the web at the Internet Archive,

20 The Fourier Transform in Space Gallanger et al., AJR

21 Eample of the Fourier Transform in Space Bahadir Gunturk

22 Light Waves: Plane Wave Definition: A plane wave is a wave in which the wavefront is a plane surface; a wave whose equiphase surfaces form a famil of parallel surfaces (MCGraw-Hill Dict. Of Phs., 1985). Definition: A plane wave in two or three dimensions is like a sine wave in one dimension ecept that crests and troughs aren't points, but form lines (2-D) or planes (3-D) perpendicular to the direction of wave propagation (Wikipedia, 2009). Ae i k k k zt z The large arrow is a vector called the wave vector, which defines (a) the direction of wave propagation b its orientation perpendicular to the wave fronts, and (b) the wavenumber b its length. We can think of a wave front as a line along the crest of the wave.

23 The Fourier Transform in Space Time Fourier Transform F( ) f ( t) ep( it) dt Fourier Transform in Space? U( k, k, Z) (,, Z) ep Ae i k i k k k zt z As must satisf the Helmholtz equation, ever solution can be decomposed into plane waves ep(ik + ik + ik z z) with wave vector k = (k, k, k z ), k =/c, and c being the velocit of sound in the coupling liquid. If k 2 + k 2 k², then. Such waves are denoted as homogeneous waves. k dd (,, Z) U( k, k, Z)ep i k k dk dk

24 The Fourier Transform in Space: Angular Spectrum Conversel, the potential can then be written as the inverse Fourier transform of the angular spectrum. dk dk k k i Z k k U Z ep ),, ( 2 1 ),, ( 2 The inverse Fourier transform represents the wavefield in the plane Z as a superposition of plane waves ep(ik + ik ). dd k k i Z Z k k U ep ),, ( ),, (

25 Properties of the Fourier Transform The Fourier spectrum of the field at the plane z = Z, U(k,k,Z) can be epressed through the Fourier spectrum of the field at the plane z = 0, U(k,k,0). U ( k, k, Z) U ( k, k,0)epik Z i i z

26 Properties of the Fourier Transform () U( k, k ) (, )ep i k k dd The Fourier spectrum of the circular Function. 1, r 1 Pr () 0, r 1 Is the jinc function: U J 1 ( kr ) A ( k k r r ) k r k k k 2 2 r J. W. Goodman. Introduction to Fourier Optics,, McGraw-Hill, (1996).

27 Aial Intensit Distribution Debe integral can be simplified for = 0. ( z) u f ep( ikf ) ep( ikz cos )sin d u f o o () z B 0 ep( ikf ) ep( ikz ) ep( ikz cos ) kz sin 0.5 kz(1 cos ) 0.5 kz(1 cos ) For small angles cos ~ sin 2, then 2 2 sin kz sin / 4) sin kz NA / 4) () z B B 2 2 kz sin / 4 kz NA / 4 PSF: Point Spread Function

28 Field in the Focus and Fourier Spectrum Approach 2 ( r) A P(, )ep( ikr(cos cos sin sin cos( )) sin d d 0 0 P P P In the limit f (Debe approimation) this equation is valid throughout the whole space. It epresses the field in the focal region as a superposition of plane waves whose propagation vectors fall inside a geometrical cone formed b drawing straight lines from the edge of the aperture through the focal point, which, in contrast to the usual Debe integral, are weighted with P(,). Where P(,) is the Pupil Function. We ma sa that each point on P(,) is responsible for the emission (and, b reciprocit, for the detection) of the plane wave component emitted along the line from the point on P(,) through the focal point

29 Field in the Focus and Fourier Spectrum Approach To find the angular spectrum of the emitted field in the focal plane, we set the third coordinate of r equal to zero and substitute Cartesian coordinates for the angular integration variables. For the vector components of k hold: k = k sinθ cosφ, k = k sinθ sinφ, kz = cosφ = k k k. The Jacobi determinant corresponding ields. 1 sindd dk dk 2 k cos 1 kk z dk dk Denoting the Cartesian coordinates of r b,, z, Equation can be rewritten: i 1 (,, z. 0) 2 2 P( k, k ) ep i k k dk kk dk z (1) Up to a constant P(k,k ) is defined as P( k, k ) 2 2 P(, ), k k 0, elsewhere k 2

30 Field in the Focus and Fourier Spectrum Approach (,,0) U( k, k,0)ep i k k dk dk The Pupil Function of an ideal lens can be described b a circle function Pk ( ) r 1, kr ksin 0, otherwise Fourier transform of the circle function. ( ) J 1 ksin A k sin J. W. Goodman. Introduction to Fourier Optics,, McGraw-Hill, (1996).

31 Field in the Focus and Fourier Spectrum Approach Equation (1) also tells us that the lens performs Fourier transform on the incident electromagnetic. Plane wave passing the lens with a pupil function P(,) transforms into jinc function in focal plane of the lens.. h( u, v) ( u, v) P(, ) F[ P(, )] h( u, v)

32 Lens as a Fourier transform Fourier transform of letter E.

33 Lens as a Fourier transform Input laser beam Output laser beam Focused laser beam Gaussian. super-gaussian Flat top Focal πshaper 20 m a b c 1/e 2 1/e 2 r mm 6 mm Radius, mm Courtes to Vitali. Prakapenka, Universit of Chicago

34 Field in the Focus and Fourier Spectrum Approach U i ''( k', k' ) U i ( k', k' )ep ik' z Z For the reflection microscope we have, since the backscattered wave propagates opposite to the z-direction. Combining all above we obtain the output signal of the reflection microscope as dk ' dk ' dk dk V ( X, Y, Z) P( k, k) P( k ', k ' ) gs( k, k, k ', k ' )ep i k ' k X k ' k Y k ' z kz Z kk ' z

35 Image Formation of Spherical Particle in Reflection Microscope X-Z scan through a steel sphere for a reflection microscope(oxsam) at 105 MHz. The radius of the sphere was 560 μm. The semi-aperture angle of the microscope lens was Z=0 corresponds to focussing to the centre of the sphere. (b) X-Z scan through a steel sphere for reflection microscope calculated with the same parameter as in From Weise, Zinin et al., Optik 107, 45, 1997.

36 Elasticit in Medicine Acoustical images are difficult for direct interpretation: Ultrasound images of a fetus during seventeen of development (left) and an artist s rendering of the image. (after Med. Encclopedia, 2005) B striving to do the impossible, man has alwas achieved what is possible. Bakunin

37 Some Fourier transform pairs (graphical illustration) X-z scan of 3-D image of steel sphere. X-z scan of 3-D image of a liquid (water) drop. X-z scan of 3-D image of a Pleiglas sphere. Important conclusion from the theor we developed is that size of the spherical particle can be determined onl from image taken b a transmission microscope. The size of the image of the spherical particle in reflection microscope is less than the real size of the particle and is equal to a sin(α), where a is the radius of the particle, and α is semiaperure angle of the lens. This theor has found several direct application in practical microscop: surface imaging and in developing Emulated Transmission Confocal Raman Microscop.

38 Optical images of east cells Optical images of east cells on glass (100 objective) in transmission mode (a), in reflection mode (b). The red circles mark the position of the laser beam.

39 Optical images of east cells Calculated vertical scans through a transparent sphere with refraction inde of 1.05 and refraction inde of surrounding liquid of 1.33: (a) reflection microscope with aperture angle 30 o; (a) transmission microscope with aperture angle 30 o.

40 Optical images of east cells Sketch of the optical ras when cell is (a) attached to the glass substrate or (b) to mirror.

41 Counts 2933 Emulated Transmission Confocal Raman Microscop 400 (b) Raman shift (cm -1 ) Optical image of the east baker cells in the reflection confocal microscope. Rectangle shows the area of the Raman mapping. (a) Raman spectra of the cell α measured with green laser ecitation (532 nm, WiTec sstem). Map of the Raman peak intensit centered at 2933 cm-1. The intensit of the 2933 cm-1 peak is shown in a ellow color scale. (b) Map of the Raman peak intensit centered at 1590 cm-1. The intensit of the 1590 cm-1 peak is shown in a green color scale.

42 Summar Lecture 6 Sinc function is a Fourier Transform of the rectangula Function. Distribution of the field in the focal plane is the spacial Fourier Transform of the Pupil Function of the Lens Fourier Spectrum Approach of Image in 3-D Contrast in Reflection and Transmission Microscop Emulated Transmission Confocal Raman Microscop

43 Home work 1. Present a definition of the Fourier transform (SO). 2. Describe difference in image formation of transmission and reflection optical microscope of 3-D objects. (KK). 3. Formulate and eplain Nquist Sampling Theorem (SO). 4. Derive a Fourier transform of the function f(t)={1, if -1/2<t<1/2; 0 if t >1/2} (KK). 5. Derive magnification of the compound microscope (KK and SO)