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LECTURE OUTLINE

ƒ Ray OPTICS: INTRODUCTION & ƒ optics FUNDAMENTALS ƒ Statistical optics ƒ Fourier optics presented by ƒ Dr. K. C. Toussaint, Jr. 2009 Nano-Biophotonics Summer School University of Illinois at Urbana-Champaign

6/1/09 2009 Nano-Biophotonics Summer School 2

WHAT IS OPTICS? SOURCE MATERIAL

• Deals with the generation, propagation, and detection of light

Optics ~ Photonics ~ Light

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Quantum optics Quantum • Ray optics-Limit of wave optics when optics Electromagnetic is infinitesimally small Electromagnetic optics optics • Wave optics-provides a description of optical Wave optics phenomena using scalar wave theory

Ray optics • Electromagnetic optics- provides most complete treatment of light within classical optics Wave optics • -provides a quantum mechanical Ray optics description of the electromagnetic theory

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EM SPECTRUM INDEX OF REFRACTION

• Light is really, really An optical medium is characterized by a fast! quantity n >=1 8 Speed of light in vacuum • co = 3.0 x 10 m/s c = 30 cm/ns n = o Speed of light in a medium = 0.3 mm/ps c Optical pathlength Travel time d nd t = = c co

Image Source: Fundamentals of Photonics 6/1/09 2009 Nano-Biophotonics Summer School 7 6/1/09 2009 Nano-Biophotonics Summer School 8 RAY OPTICS FERMAT’S PRINCIPLE

• • Rays traveling between two points follow a path such that the optical pathlength is an extremum (point of inflection) relative to neighboring points

• Describes light as rays that travel in maximum accordance with a set of geometrical rules

minimum • Deals with location and direction of rays • Usually minimum, so Light rays travel along the path of least time

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HERO’S PRINCIPLE REFLECTION

• n is the same everywhere in a homogeneous medium Incident ray θθ ’ Reflected ray • Law of reflection: • Path of min time = Path of min distance Reflected ray lies in plane Mirror of incidence Plane of incidence • Principle of path of min distance states that the path of min distance between points is a straight line

• Light rays travel in straight lines

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n sinθ = n sinθ • Occurs at boundary 1 1 2 2

between two different n1 n2 n1 n2 Incident ray media θ1 θ1 Reflected ray

n1 θ2 θ 1 θc n2 • Reflected ray follows θ θ2 1 θ θ2 law of reflection 1 Plane of Refracted ray incidence • Refracted ray obeys

Snell’s law External refraction Internal refraction

n1 < n2 n > n n1 sinθ1 = n2 sinθ2 1 2 θ < θ 2 1 θ2 > θ1 -ray bends away -ray bends towards from boundary boundary

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TOTAL INTERNAL REFLECTION SPHERICAL

n1 n2 n1 > n2 • Bounded by 2 spherical surfaces

θ with radii R1 and R2 θ n • Thin assumption-ray at output of lens is ~ same height as ray at input to lens

• Paraxial approximation [rays Total Internal reflection travel close to, and make small • Refraction does not occur! thickness angles (sinθ~θ) with optical axis]

• Light is completely reflected Optical axis – center of symmetry for optical components • Used in fiber optics communications Image Source: Fundamentals of Photonics Image Source: Fundamentals of Photonics 6/1/09 2009 Nano-Biophotonics Summer School 15 6/1/09 2009 Nano-Biophotonics Summer School 16 THIN LENS (FOCUSING) THIN LENS (IMAGING)

• Rays coming from P1 meet at P2 1  1 1  = ()n −1  −  • Imaging equation (lens f  R1 R2  law): 1 1 1 + = z1 z2 f

• For biconvex lens (very popular), R1 +, and R2 -, leading to + f (focusing lens) • Magnification − z • For biconcave lens, R -, and R +, leading to - f conjugate planes M = 2 1 2 z (diverging lens) Real image 1

Image Source: Fundamentals of Photonics Image Source: Fundamentals of Photonics 6/1/09 2009 Nano-Biophotonics Summer School 17 6/1/09 2009 Nano-Biophotonics Summer School 18

F-NUMBER AND DEPTH OF THIN LENS (IMAGING) FIELD

• Virtual image – outgoing rays do not intersect at a point; • The f-number of a lens is the ratio of its focal length to its “effective” diameter can be viewed as a “virtual” image creating diverging f F /# = rays D

• Collect more light with small F/# lens but are more sensitive to aberrations • Occurs for negative lenses (f < 0); positive lenses if object is less than a focal length away • Depth of field (DOF) = range of distances where the object (subject) is in “acceptable” focus (often in µm for microscopes)

• DOF increases with increasing #

• A lens with F/# = f/64 has larger DOF than F/# = f/1.4 [F/# Æ f/0.5 for microscopes] virtual image

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• Light propagates in the form of Quantum Ray optics-Limit of wave optics when wavelength is optics Electromagnetic infinitesimally small optics • Can be described by a scalar (wave) function Wave optics-provides a description of optical phenomena using scalar wave theory

Electromagnetic optics- provides most complete • Does not completely describe some phenomena treatment of light within classical optics such as reflection and refraction, and Wave optics Quantum optics-provides a quantum mechanical polarization effects Ray optics description of the electromagnetic theory

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POSTULATES OF WAVE OPTICS POSTULATES OF WAVE OPTICS

• The • Optical intensity [Watts/cm2]

2 1 ∂ u()r,t 2 ∇2u()r,t − = 0 I(r,t)()= 2 u r,t c2 ∂t 2

measurable ∂ 2 ∂ 2 ∂2 wavefunction (real) ∇2 = + + • Optical power [Watts] • Optical energy [joules] ∂x2 ∂y 2 ∂z 2 • Equation is linear->principle of superposition P(t) = I(r,t)dA E = P(t)dt ∫A ∫t • Describes how a wave behaves in space and time area normal to direction of • Approximately applicable if n(r) and c(r) time interval propagation

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generally, position dependent • More convenient for studying optics 1 u(r,t )= a ()r cos[]2πυt +ϕ ()r u()r,t = Re{}U ()r,t = [U ()r,t +U * ()r,t ] 2

single (3x1011-1016 Hz) U (r,t) = a(r)()exp[]jϕ r exp()j2πυt Complex wavefunction Complex amplitude,U ()r

• U(r,t) must also satisfy wave equation

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OPTICAL INTENSITY (MONOCHROMATIC WAVE) I()r = U (r) 2 ∇2U + k 2U = 0

2πυ ω , k = = • averaged over a time longer than an c c optical period • Describes the wave’s behavior in space only • I is time independent for monochromatic • Convenient for studying many concepts waves

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2 2 U r = Aexp − jk ⋅r = Aexp − j k x + k y + k z ∇ U + k U = 0 ( ) ( ) [ ( x y z )] complex envelope

• What are some solutions? wavevector, k = (kx ,k y ,kz ) •Plugging U(r) into Helmholtz equation gives 2 2 2 2 • Simplest solutions are plane wave, k = k = kx + k y + kz spherical wave • of U(r) is given by arg{U (r)} = arg{A}− k ⋅r

• The wavefronts (surfaces of constant phase) obey

k ⋅r = kx x + k y y + kz z = 2πq + arg{A}

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THE PLANE WAVE WHAT DOES IT ALL MEAN?

k ⋅r = k x + k y + k z = 2πq + arg{}A x y z • Eq. describes parallel planes • Given ν, and remembering that co is that are perpendicular to k constant air n • Wavelength, λ = 2π/k =c/ν [nm]

• Intensity (I=|A|2) is constant everywhere---not realistic!

λ λ c λ c = o , λ = o , k = nk n n o Image Source: Fundamentals of Photonics Image Source: Fundamentals of Photonics 6/1/09 2009 Nano-Biophotonics Summer School 31 6/1/09 2009 Nano-Biophotonics Summer School 32 PARAXIAL HELMHOLTZ THE SPHERICAL WAVE EQUATION ∂A • Solution to Helmholtz equation in spherical ∇2 A − j2k = 0 T ∂z coordinates 2 2 2 ∂ ∂ A transverse Laplacian, ∇T = + U ()r = o exp()− jkr ∂x2 ∂y 2 r • Paraxial approximation of the Helmholtz radial distance from origin equation 2 Ao I()r = 2 • Applies to paraxial waves, i.e., wavefront r normals make small angles with z-axis

• Some solutions are the paraboloidal wave and the Gaussian beam Image Source: Fundamentals of Photonics 6/1/09 2009 Nano-Biophotonics Summer School 33 6/1/09 2009 Nano-Biophotonics Summer School 34

THE PARABOLOIDAL WAVE THE PARABOLOIDAL WAVE

• Consider spherical wave at points close to z-axis • Far from origin spherical wave and far from origin ~ paraboloidal wave

Ao U ()r = exp()− jkr • Very far ~ plane wave r ()x2 + y 2 r ≈ z + r ≈ z 2z • Approximation is valid when 2 N Fθm Fresnel approximation of a spherical wave << 1 4 2 2 Radius of circle Ao  x + y  2 U ()r ≈ exp()− jkz exp− jk  a within which approx where, N = is valid z  2z  F λz Fresnel number Image Source: Fundamentals of Photonics 6/1/09 2009 Nano-Biophotonics Summer School 35 6/1/09 2009 Nano-Biophotonics Summer School 36 TRANSMISSION THROUGH A THIN TRANSPARENT PLATE TRANSPARENT PLATE (VARYING THICKNESS) • For arbitrary paraxial wave • Thus, transmittance at output is input t(x, y) = exp(− jnkod) t(x, y) ≈ exp(− jk d)exp[]− j(n −1)k d(x, y) • Plate only introduces a phase shift o o

2 • For oblique incident angle, • This is true if (do/λο)θ /2n<<1 • Complex amplitude transmittance (paraxial approximation) U (x, y,d ) t(x, y) = U ()x, y,0

Thin transparent plate •Inside plate, wave continues as a plane wave t(x, y) = exp(− jnkod cosθ1) Image Source: Fundamentals of Photonics Image Source: Fundamentals of Photonics 6/1/09 2009 Nano-Biophotonics Summer School 37 6/1/09 2009 Nano-Biophotonics Summer School 38

TRANSMISSION THROUGH A EXAMPLE THIN LENS • At the point (x, y)  x2 + y 2  U 2 (x, y) = exp(− jkz)ho exp jko  2 2 2  2 f  d()x, y = do − [R − R − ()x + y ] • Thus, lens transforms incoming wavefronts x2 + y2 ≈ d − into paraboloidal wavefronts centered o 2R • t(x,y) for a thin lens is at (0,0,f)  x2 + y 2   x2 + y 2  t(x, y) ≈ exp(− jkod)exp jko  t(x, y) ≈ exp(− jkod)exp jko  2 f  2 f    • What does lens do to incident R plane wave? f = U (x, y) n −1 t(x, y) = 2 U1(x, y) • Transforms planar wavefronts to paraboloidal wavefronts centered at f exp(-jkz) Image Source: Fundamentals of Photonics Image Source: Fundamentals of Photonics 6/1/09 2009 Nano-Biophotonics Summer School 39 6/1/09 2009 Nano-Biophotonics Summer School 40 EXAMPLE DIFFRACTION GRATINGS

x •For small input angle • Periodically modulates amplitude or phase of input fθ U1(x, y) = exp[]− jk()z +θx wave θ z  (x − fθ )2 + y2  U 2 (x, y) = exp(− jkz)ho exp jko  • Here incident plane wave is 2 f f   split into multiple plane waves traveling in different directions In paraxial approximation, •Transforms input to paraboloidal wave centered at (f θ,0,f) λ θ = θ + q Grating equation q i Λ

incident angle grating period

Image Source: Fundamentals of Photonics diffraction order Image Source: Fundamentals of Photonics 6/1/09 2009 Nano-Biophotonics Summer School 41 6/1/09 2009 Nano-Biophotonics Summer School 42

DIFFRACTION GRATINGS GRADED-INDEX COMPONENTS

• More general grating Transverse Variation in equation Optical Component thickness of material λ sinθ = sinθ + q q i Λ Prism Linear Lens Quadratic • λ dependent Diffraction grating Periodic

• Used as filters and spectrum analyzers • GRIN components can be designed to produce the same effects

t(x, y) = exp(− jn()x, y kodo ) Image Source: Fundamentals of Photonics 6/1/09 2009 Nano-Biophotonics Summer School 43 6/1/09 2009 Nano-Biophotonics Summer School 44 INTERFERENCE

• Let’s consider the superposition of two monochromatic waves INTERFERENCE U (r) = U1(r) +U 2 (r)

2 2 2 2 * * I = U = U1 +U 2 = U1 + U 2 +U1U 2 +U1U 2

U1 = I1 exp( jϕ1 ) , U 2 = I 2 exp()jϕ2

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INTERFERENCE INTERFEROMETERS

  d  I = 2Io 1+ cos2π    λ  Interference equation

I = I1 + I2 + 2 I1I2 cosϕ

ϕ = ϕ2 −ϕ1

2 ϕ  I = 2Io ()1+ cosϕ = 4Io cos    2 

Image Source: Fundamentals of Photonics Image Source: Fundamentals of Photonics 6/1/09 2009 Nano-Biophotonics Summer School 47 6/1/09 2009 Nano-Biophotonics Summer School 48 INTERFERENCE OF TWO BREAK OBLIQUE PLANE WAVES

STATISTICAL OPTICS I = 2Io []1+ cos()k sinθx

Image Source: Fundamentals of Photonics 6/1/09 2009 Nano-Biophotonics Summer School 49 6/1/09 2009 Nano-Biophotonics Summer School 50

DEFINITIONS DEFINITIONS

• Study of random light

• Occurs b/c of fluctuations of the light either due to the source or the medium that it propagates through

• Examples: • Deterministic light is called “coherent” 1) Light scattered from rough surfaces (imparts random variations in the wavefront) • Ex: monochromatic plane wave

2) Natural light radiated by a hot object is made up of A • Random light, the behavior of the wavefunction U(r,t) with respect to LOT of atoms, each radiating at different position and time is not (completely) predictable and phases • Statistical methods are used to describe random light Image Source: Fundamentals of Photonics 6/1/09 2009 Nano-Biophotonics Summer School 51 6/1/09 2009 Nano-Biophotonics Summer School 52 WHAT CAN WE DO? OPTICAL INTENSITY

• Recall that for deterministic (coherent) light I(r,t)()= U r,t 2

• For random light, U(r,t) fluctuates with position and time 2 Average intensity I()r,t = U ()r,t

Ensemble average over • Which is more “intense”? Instantaneous intensity many instances

• Which has a “faster” fluctuating envelope?

Image Source: Fundamentals of Photonics 6/1/09 2009 Nano-Biophotonics Summer School 53 6/1/09 2009 Nano-Biophotonics Summer School 54

STATIONARY VS. TEMPORAL COHERENCE NONSTATIONARY • Consider stationary light at fixed position r : U(r,t) ÆU(t) and I(r)ÆI

- We characterize U(t) by a time scale that

• (Statistically) Stationary - I(r,t) = I(r) is time independent represents the memory of the random fluctuation Ex: light bulb driven by constant electric current T 1 2 I()r = lim U (r,t) dt T →∞ ∫ - Quantitatively this is represented by the 2T −T autocorrelation function G(τ) • (Statistically) Nonstationary – I(r,t) is not time independent Ex: light bulb driven by pulse of electric current Image Source: Fundamentals of Photonics 6/1/09 2009 Nano-Biophotonics Summer School 55 6/1/09 2009 Nano-Biophotonics Summer School 56 TEMPORAL COHERENCE DEGREE OF TEMPORAL FUNCTION COHERENCE • Autocorrelation(temporal coherence function) • Complex degree of temporal coherence

∗ ∗ G()τ = U ()(t U t +τ ) G()τ U ()t U (t +τ ) g τ = = 0 ≤ g(τ ) ≤1 () ∗ 1 T G()0 U ()t U ()t G()τ = lim U ∗ ()t U (t +τ )dt T →∞ 2T ∫ −T Normalized autocorrelation I = G()0 • For monochromatic wave, U (t) = Aexp( jωot)

• Contains info about intensity and coherence g()τ = Aexp (jωoτ ) g(τ ) =1 (degree of correlation) • Tells you how much two points in time are correlated (for a fixed position)

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COHERENCE TIME COHERENCE LENGTH

• Light can be considered coherent if distance lc is >> than all optical path-length differences encountered in an optical system

lc = cτ c • Within τc, wave is predictable; At times greater than τc, one cannot predict amplitude and phase of wave

∞ τ = g τ 2dτ Commonly used definition c ∫ () −∞

τ c = ∞ Monochromatic wave

Image Source: Fundamentals of Photonics Image Source: Fundamentals of Photonics 6/1/09 2009 Nano-Biophotonics Summer School 59 6/1/09 2009 Nano-Biophotonics Summer School 60 COMPLEX DEGREE OF MUTUAL INTENSITY COHERENCE • To examine the spatial correlation of light • Spatial and temporal fluctuations of U(r,t) can be described by cross-correlation between U(r ,t) and ∗ 1 G(r1,r2 ,0) = G(r1,r2 )()()= U r1,t U r2 ,t U(r2,t) ∗ Mutual coherence G()()()r1,r2 ,τ = U r1,t U r2 ,t +τ function G()r ,r 1 2 Normalized mutual g()r1,r2, 0 = g()r1,r2 = intensity I()()r1 I r2 G(r1,r2,τ ) g()r1,r2,τ = 0 ≤ g()r1,r2 ,τ ≤1 I()()r1 I r2 • Measure of spatial coherence between 2 • Measure of coherence between 2 positions at 2 different positions for 0 time delay times

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COHERENCE AREA PARTIAL COHERENCE

• Represents the spatial extent of g ()r 1 , r 2 as

a function of r1 for a fixed r2

• If size of aperture in optical system is <

than coherence area ( g () r 1 , r 2 ≈ 1 ) then light may be considered coherent; if vice-versa, light is considered incoherent Plane wave Spherical wave

Image Source: Fundamentals of Photonics 6/1/09 2009 Nano-Biophotonics Summer School 63 6/1/09 2009 Nano-Biophotonics Summer School 64 INTERFERENCE OF PARTIALLY BREAK COHERENT LIGHT partially coherent waves 1 and 2

2 I = U1 +U 2 = I1 + I2 + 2 I1I2 g12 cosϕ

• If both waves are completely FOURIER OPTICS correlated w/ g = exp () j ϕ then: ∗ 12 U1U 2 g g = Phase of 12 g12 =1 12 I1I2 and I is given by usual expression 2 I I we know Imax − I min 1 2 V = = g12 Imax + Imin I1 + I2 • If both waves are completely uncorrelated w/ g 12 = 0 then:

I = I1 + I2

Image Source: Fundamentals of Photonics 6/1/09 2009 Nano-Biophotonics Summer School 65 6/1/09 2009 Nano-Biophotonics Summer School 66

BACKGROUND: FOURIER ANALYSIS (TIME-DOMAIN APPROACH)

Linear time-invariant

LTI x(t) h(t) y(t)

output signal • Arbitrary function can be represented as a Arbitrary input signal described as described as weighted superposition of harmonic functions impulse response superposition of superposition of weighted and shifted weighted and shifted impulse responses, impulses • Frequencies are orthogonal h(t) y(t) = x(t)∗h(t)

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Linear time-invariant

LTI • Y(f) is the of y(t) X(f) H(f) Y(f)

output signal Arbitrary input signal • y(t) is the Inverse Fourier transform of Y(f) described as described as Frequency response superposition of superposition of or weighted and shifted weighted sinusoids Transfer function response H(f) (harmonic signals)

Y ( f ) = X ( f )⋅ H ( f )

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SOME COMMON TRANSFORM INTRODUCTION PAIRS

• Image can be decomposed into weighted sum of harmonic functions • Orthogonal spatial frequencies • Different complex amplitudes

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Arbitrary optical wave = Superposition of plane waves (building blocks) Spatial angular frequency [radians/mm] k k v = x v = x x 2π y 2π

[cycles/mm]

Composed of various • Tells us how often structure (e.g., in an spatial frequencies Each plane wave has unique spatial frequency ν image) repeats per unit distance

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THE EFFECT OF SPATIAL LINEAR-SYSTEMS APPROACH FREQUENCIES

Linear system can be described in either of two pictures

HPF LPF

Impulse response function Transfer function -response to impulse or point at input -response to spatial harmonic function [courtesy: http://www.icaen.uiowa.edu/~dip/LECTURE/LinTransforms.html]

6/1/09 2009 Nano-Biophotonics Summer School 75 6/1/09 2009 Nano-Biophotonics Summer School 76 SPATIAL HARMONIC FUNCTION ANGLES • Consider plane wave at arbitrary plane U (x, y,0) = Aexp[− j2π (ν x x +ν y y)]= f (x, y) U ()x, y, z = Aexp[− j(kx x + k y y + kz z)]

• At z=0 plane consistent w/ plane wave traveling at angles −1 −1 θ x = sin λν x θ y = sin λν y U ()x, y,0 = Aexp[− j(kx x + k y y)]= f (x, y) in paraxial approximation  k   kx y  θ ≈ λν , θ ≈ λν = Aexp− j2π  x + y x x y y   2π 2π 

= Aexp[]− j2π ()ν x x +ν y y notice wavelength dependence

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SPATIAL SPECTRAL ANALYSIS EXAMPLES

• Plane wave is deflected up

f(x,y)=exp[-j2π(νxx+νyy)] f(x,y) Response

cos(2πν x x) • Incident plane wave is transformed • Plane wave is deflected down 1 into components bent both upward for f(x,y)=exp[+j2π(ν x+ν y)] = []exp()()− j2πν x + exp + j2πν x and downward x y 2 x x −1 ± sin λν x • Interference phenomenon where two points separated by • Incident plane wave is transformed Λ interfere constructively for 1+ cos(2πν y y) into components bent left and right as pathlength difference λ well as a portion that goes through

Image Source: Fundamentals of Photonics 6/1/09 2009 Nano-Biophotonics Summer School 79 6/1/09 2009 Nano-Biophotonics Summer School 80 EFFECT OF THIN OPTICAL MATHEMATICAL DESCRIPTION ELEMENT

f x, y = F ν ,ν exp − j2π ν x +ν y dν dν ()∫∫ ()x y [] (x y )x y −∞

superposition of harmonic functions • Transmitted light can be written as superposition of f(x,y) plane ways provided that 2 2 −2 ν x +ν y ≤ λ U (x, y, z) = F ν ,ν exp − j 2πν x + 2πν y exp − jk z dν dν ∫∫ ( x y ) []( x y ) ( z ) x y −∞

superposition of plane waves

2 2 2 −2 2 2 Thin optical element kz = ± k − kx − kz = 2π λ −ν x −ν y FT f ()x, y ↔ F(ν x ,ν y ) Image Source: Fundamentals of Photonics 6/1/09 2009 Nano-Biophotonics Summer School 81 6/1/09 2009 Nano-Biophotonics Summer School 82

EXAMPLE EXAMPLE

• Illuminate transparency from • Phase shift 2πφ(x,y) is previous example w/ plane introduced wave

• φ(x,y) =-x2/2λf • Each part of wave is deflected by a different angle • Wave is deflected at position • Transparency acts as a (x,y) by  jπx2   λ∂φ   − x  f ()x, y = exp  cylindrical lens θ = sin −1 = sin −1   λf  2 x        jπx   ∂x   f  f ()x, y = exp  ν  λf  x • Transparency acts as a • In paraxial limit x spherical lens for 2 2 θ ≈ − different x-values yield  jπ (x + y ) x f ()x, y = exp  f different angles  λf  Image Source: Fundamentals of Photonics Image Source: Fundamentals of Photonics 6/1/09 2009 Nano-Biophotonics Summer School 83 6/1/09 2009 Nano-Biophotonics Summer School 84 TRANSFER FUNCTION OF FREE TRANSFER FUNCTION OF FREE SPACE SPACE

• We use systems analysis to determine the output g(x, y) = f (x, y)⋅ H (ν x ,ν y ) g(x,y)=U(x,y,d), given input f(x,y)=U(x,y,0) −2 2 2 H (ν x ,ν y )= exp[− j2πd λ −ν x −ν y ] • System is linear shift-invariant (LSI) because of Helhmoltz equation and invariance of free space to displacement

Image Source: Fundamentals of Photonics Image Source: Fundamentals of Photonics 6/1/09 2009 Nano-Biophotonics Summer School 85 6/1/09 2009 Nano-Biophotonics Summer School 86

TRANSFER FUNCTION OF FREE TRANSFER FUNCTION OF FREE SPACE SPACE •For d>>λ, attenuation factor drops sharply for magnitude spatial frequencies that slightly exceed λ-1

• λ-1 [cycles/mm] is ~ the spatial bandwidth of free space

• Harmonic function with spatial • For any image, features with details finer than 2 2 −2 -1 frequencies ν x + ν y ≤ λ will have its phase λ (spatial frequencies greater than λ ) cannot magnitude go unaltered, but it will be transmitted by light of wavelength λ for undergo a spatial phase shift d>>λ

2 2 −2 • For ν x + ν y > λ , term under square • Conceptually, this is the difference between root leads to a real exponential near field and far field microscopy exp − 2πd ν 2 +ν 2 − λ−2 [ x y ] attenuation factorÆ evanescent wave Image Source: Fundamentals of Photonics Image Source: Fundamentals of Photonics 6/1/09 2009 Nano-Biophotonics Summer School 87 6/1/09 2009 Nano-Biophotonics Summer School 88 FRESNEL APPROXIMATION FRESNEL APPROXIMATION

2 2 −2 2 2 • For ν x + ν y << λ , we use paraxial H (ν x ,ν y )≈ H o exp[jπλd(ν x +ν y )] θ ≈ λν θ ≈ λν approximation such that x x and y y exp(-jkd)

• Phase is a quadratic function of the spatial 2 2 2 2 2 2 frequencies • Let θ = θ x + θ y ≈ λ ( ν x + ν y ) , then phase factor θ 2 becomes • Approximation holds if N m <<1 F 4 −2 2 2 d 2 Taylor series expansion a2 2πd λ −ν x −ν y = 2π 1−θ N = λ F λd d  θ 2 θ 4  = 2π 1− + −⋅⋅⋅ λ  2 2  •For, a=1 cm, d=100 cm, λ=500 nm, 2 NFθ /4=.005<<1

Image Source: Fundamentals of Photonics 6/1/09 2009 Nano-Biophotonics Summer School 89 6/1/09 2009 Nano-Biophotonics Summer School 90

INPUT-OUTPUT RELATION INPUT-OUTPUT RELATION • Given input f(x,y) we can determine output using Fourier transform (spatial-frequency) approach

1. Determine complex envelope of input plane-wave • Using Fresnel approximation for H(νx,νy) components FT F ν ,ν = f x, y exp j2π ν x +ν y dxdy ↔ f x, y ()x y ∫∫ ()[]()x y () −∞ g x, y = H F ν ,ν exp jπλd ν 2 +ν 2 exp − j2π ν x +ν y dν dν ( ) o ∫∫ ( x y ) [ ( x y )] []()x y x y 2. Determine complex envelope of output plane-wave −∞

components by H (ν x ,ν y )F(ν x ,ν y )

3. Determine complex amplitude g(x,y) by summing individual contributions inverse FT g x, y = H ν ,ν F ν ,ν exp − j2π ν x +ν y dν dν ()∫∫ ()()x y x y [ (x y )] x y −∞

6/1/09 2009 Nano-Biophotonics Summer School 91 6/1/09 2009 Nano-Biophotonics Summer School 92 IMPULSE RESPONSE FUNCTION FREE-SPACE PROPAGATION (FREE SPACE) () • In the Fresnel approximation we define • Input-output can be related using space-domain approach  x2 + y 2  h()x, y ≈ ho exp− jk   2d  • Think of f(x,y) as superposition of different points (delta functions) (j /λd)exp(-jkd) Impulse response function Æ response g(x,y) when input is a g(x, y) = ∫∫ f (x', y')(h x − x', y − y' )dx'dy' (point-spread function) point at the origin −∞ • In Fresnel approximation,  ()()x − x' 2 + y − y' 2  • Each input point generates a paraboloidal wave that is g x, y = h f x', y' exp − jπ dx'dy' ()o ∫∫ ( )  summed at output −∞  λd 

6/1/09 2009 Nano-Biophotonics Summer School 93 6/1/09 2009 Nano-Biophotonics Summer School 94

FOURIER TRANSFORM OPTICAL FOURIER TRANSFORM (FAR FIELD) • Light can be used to determine FT of 2-D • This is valid under Fraunhofer approximation function f(x,y) which requires that ' N F <<1 and N F <<1

2 2 • Design a transparency with amplitude a ' b N F = and N F = transmittance f(x,y) and then illuminate with a λd λd •Can spatially resolve each plane wave plane wave contribution if d is long enough describes largest describes largest circular region in circular region where  x y  observation plane points lie in object plane g()x, y ≈ ho F ,  •FT of f(x,y) is then determined either after light  λd λd  propagates a really long distance d or by using a • More difficult to satisfy than lens Fresnel approximation

Image Source: Fundamentals of Photonics 6/1/09 2009 Nano-Biophotonics Summer School 95 6/1/09 2009 Nano-Biophotonics Summer School 96 FOURIER TRANSFORM FOURIER TRANSFORM (USING LENS) (USING LENS) (front focal plane) (back focal plane)

• Lens maps each direction (θx,θy) to a single point • Complex amplitude g(x,y) at point (x,y) in output (θxf, θyf) plane is (using Fresnel approximation)  (x2 + y2 )()d − f   x y  g x, y = h exp jπ F ,  ()l  2    • Allows us to separate different plane wave  λf   λf λf  contributions Fourier transform of (x,y) evaluated H h o o at spatial frequencies νx and νy Image Source: Fundamentals of Photonics Image Source: Fundamentals of Photonics 6/1/09 2009 Nano-Biophotonics Summer School 97 6/1/09 2009 Nano-Biophotonics Summer School 98

FOURIER TRANSFORM BREAK (USING LENS)

• Intensity of light at back focal plane (independent of

input distance d) 2 1  x y  I x, y = F ,  () 2   ()λf  λf λf  DIFFRACTION •For d=f, we get 2-f system

 x y  g()x, y = hl F ,   λf λf 

(j /λf )exp(-j2kf)

Image Source: Fundamentals of Photonics 6/1/09 2009 Nano-Biophotonics Summer School 99 6/1/09 2009 Nano-Biophotonics Summer School 100 DIFFRACTION OF A WAVE DIFFRACTION BY A SLIT ∞ • Relates to the obstruction of a 0 wave by an obstacle •Occurs for any type of • Obstruction could be bending of wave phenomenon an optical wave by edges or when passing through an aperture (of appropriate size) • Effect is more pronounced a λ as the size of the slit • Amplitude or phase of wave is approaches the λ of the altered wave

Diffraction patterns of the teeth of a saw

-would be perfect geometric • Interference phenomenon shadow if λÆ 0 (as in ray optics) 0 ∞

Image Source: Fundamentals of Photonics Image Source: Optics 6/1/09 2009 Nano-Biophotonics Summer School 101 6/1/09 2009 Nano-Biophotonics Summer School 102

BASIC APPROACH

• Used when light (free-space) propagation 1, inside the aperture p()x, y = (beyond aperture) is described by Fraunhofer 0, outside the aperture approximation Largest radial distance aperture (pupil) function 2 within aperture ' b • Valid when N F = <<1 • Assumption: Incident wave passes • Diffraction pattern is λd through unaltered within aperture; intensity at the observation reduces to 0 on opaque part of plane • For U ( x , y ) = I i , f (x, y) = Ii p()x, y aperture 2 I()x, y = g(x, y) f ()x, y = U ()()x, y p x, y  x y  Fraunhofer (far-field) diffraction g()x, y ≈ Ii ho P ,  or  λd λd  Fresnel (near-field) diffraction FT of p(x,y) Image Source: Fundamentals of Photonics 6/1/09 2009 Nano-Biophotonics Summer School 103 6/1/09 2009 Nano-Biophotonics Summer School 104 FRAUNHOFER DIFFRACTION

• Used when light (free-space) propagation

FT (beyond aperture) is described by Fresnel P ν ,ν = p x, y exp j2π ν x +ν y dxdy ↔ p x, y ()x y ∫∫ ( ) []()x y ( ) approximation −∞

2 2 Ii  x y   2 2  I x, y = P , Ii ()()x − x' + y − y' () 2   I()x, y = p()x', y' exp − jπ dx'dy' ()λd  λd λd  2 ∫∫   ()λd −∞  λd 

Diffraction pattern

2 I  x y  I x, y = i P ,  () 2   using a lens ()λf  λf λf 

6/1/09 2009 Nano-Biophotonics Summer School 105 6/1/09 2009 Nano-Biophotonics Summer School 106

IMAGE FORMATION 4-f SYSTEM

inverted coordinate system •4-f system (a.k.a. imaging system) • In Fourier plane, individual spatial frequency • Fourier transform taken twice components are separated • Second lens performs “inverse” Fourier transform since coordinate system is inverted • Allows for spatial filtering (physical process) • Perfect replica of object obtained Image Source: Fundamentals of Photonics Image Source: Fundamentals of Photonics 6/1/09 2009 Nano-Biophotonics Summer School 107 6/1/09 2009 Nano-Biophotonics Summer School 108 SPATIAL FILTERING EXAMPLES: SPATIAL FILTERING

• Consider LPF: 2 2 2 H (ν x ,ν y )=1, ν x +ν y <ν s

• Mask is circular aperture of

diameter D=2νsλf

•For, D=2 cm, λ=1 µm, f=100 cm Transfer function Æ H ν ,ν = p λfν ,λfν ( x y ) ( x y ) νs=10 lines/mm, (same shape as pupil function) smallest feature size is .1 mm

1  x y  IFT Impulse response Æ h x, y = P ,  ↔ H ν ,ν () 2   ()x y ()λf  λf λf  Image Source: Fundamentals of Photonics Image Source: Fundamentals of Photonics 6/1/09 2009 Nano-Biophotonics Summer School 109 6/1/09 2009 Nano-Biophotonics Summer School 110

SINGLE-LENS IMAGING SYSTEM SINGLE-LENS IMAGING SYSTEM

where,

P(ν x ,ν y )↔ p1(x, y)

 x2 + y 2   j  p ()()x, y = p x, y exp− jπε    1   h1()x, y =  exp()− jkd1  λ   λd1  focusing error Immediately following aperture, • Here, p(x,y) plays same role as Upon further propagation by a distance d2, mask in 4-f system  ' 2 ' 2  generalized pupil function U1()x, y = U ()x, y ⋅t(x, y)⋅ p(x, y) ' ' (x − x ) + (y − y ) ' ' h()x, y = h2 U1()x , y exp− jπ dx dy 2 2 ∫∫ λd  x + y  −∞  2  = U x, y h exp jk p(x, y) • From object plane to aperture plane ()1    j   2 f    h2 ()x, y =  exp()− jkd2 (Fresnel approximation), for single  λd2  lens phase factor impulse input 2 2 2 2  x + y   x y   x + y  h()x, y = h h exp− jπ P  ,  U ()x, y ≈ h exp − jk 1 2   1  1    λd2   λd2 λd2   2d1  Image Source: Fundamentals of Photonics Image Source: Fundamentals of Photonics 6/1/09 2009 Nano-Biophotonics Summer School 111 6/1/09 2009 Nano-Biophotonics Summer School 112 SINGLE-LENS IMAGING SYSTEM SINGLE-LENS IMAGING SYSTEM

 x2 + y2  Now, for perfect focusing, U ()x, y ≈ h1 exp− jk   2d1  p1(x,y)=p(x,y)

 x y    h()x, y ≈ h1h2 P ,   λd2 λd2 

 x2 + y2   x y  2 2 h()x, y = h h exp− jπ P  ,   x + y  1 2   1  U x, y = U x, y h exp jk p(x, y)  λd2   λd2 λd2  1() ()1   similar result as with 4-f system  2 f  assume <<1  j     x y  h1()x, y =  exp()− jkd1   λd h()x, y = h1h2P1 ,   1   λd2 λd2  Image Source: Fundamentals of Photonics Image Source: Fundamentals of Photonics 6/1/09 2009 Nano-Biophotonics Summer School 113 6/1/09 2009 Nano-Biophotonics Summer School 114

EXAMPLE BREAK

1, ρ = x2 + y 2 ≤ D / 2 p()x, y = 0, outside the aperture MICROSCOPY

2J ()πDρ / λd h()()x, y = h 0,0 1 2 , ρ = x2 + y 2 πDρ / λd2

2 2 h()0,0 = (πD / 4λ d1d2 )

For, d 1= ∞, d2 = f Smaller F# (larger aperture) has better image quality ρs =1.22λF#

Measure of size of blur of circle Image Source: Fundamentals of Photonics 6/1/09 2009 Nano-Biophotonics Summer School 115 6/1/09 2009 Nano-Biophotonics Summer School 116