LECTURE OUTLINE Ray optics OPTICS: INTRODUCTION & Wave optics FUNDAMENTALS Statistical optics Fourier optics presented by Diffraction 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 6/1/09 2009 Nano-Biophotonics Summer School 3 6/1/09 2009 Nano-Biophotonics Summer School 4 OPTICS THEORIES OPTICS: THEORIES Quantum optics Quantum • Ray optics-Limit of wave optics when optics Electromagnetic wavelength 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 • Quantum optics-provides a quantum mechanical Ray optics description of the electromagnetic theory 6/1/09 2009 Nano-Biophotonics Summer School 5 6/1/09 2009 Nano-Biophotonics Summer School 6 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 • Geometrical optics • 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 6/1/09 2009 Nano-Biophotonics Summer School 9 6/1/09 2009 Nano-Biophotonics Summer School 10 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 6/1/09 2009 Nano-Biophotonics Summer School 11 6/1/09 2009 Nano-Biophotonics Summer School 12 REFRACTION PLANAR BOUNDARIES 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 6/1/09 2009 Nano-Biophotonics Summer School 13 6/1/09 2009 Nano-Biophotonics Summer School 14 TOTAL INTERNAL REFLECTION SPHERICAL LENSES n1 n2 n1 > n2 • Bounded by 2 spherical surfaces θ with radii R1 and R2 θ n • Thin lens 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 6/1/09 2009 Nano-Biophotonics Summer School 19 6/1/09 2009 Nano-Biophotonics Summer School 20 OPTICS: THEORIES WAVE OPTICS • Light propagates in the form of waves 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 6/1/09 2009 Nano-Biophotonics Summer School 21 6/1/09 2009 Nano-Biophotonics Summer School 22 POSTULATES OF WAVE OPTICS POSTULATES OF WAVE OPTICS • The wave equation • 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 6/1/09 2009 Nano-Biophotonics Summer School 23 6/1/09 2009 Nano-Biophotonics Summer School 24 MONOCHROMATIC WAVES COMPLEX REPRESENTATION 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 frequency (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 6/1/09 2009 Nano-Biophotonics Summer School 25 6/1/09 2009 Nano-Biophotonics Summer School 26 OPTICAL INTENSITY HELMHOLTZ EQUATION (MONOCHROMATIC WAVE) I()r = U (r) 2 ∇2U + k 2U = 0 2πυ ω wavenumber, 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 6/1/09 2009 Nano-Biophotonics Summer School 27 6/1/09 2009 Nano-Biophotonics Summer School 28 HELMHOLTZ EQUATION THE PLANE WAVE 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 • Phase 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} 6/1/09 2009 Nano-Biophotonics Summer School 29 6/1/09 2009 Nano-Biophotonics Summer School 30 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(−