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Camera Obscura The Wave Nature of Light Maxwell’s equations ⟼ waves Plane and spherical waves Energy flow / Intensity The Eikonal and Fermat’s principle optimum pinhole size From geometrical to wave optics contradicts expectations from geometrical optics Huygen’s principle from Hecht, Optics lecture 1 lecture 2 Maxwell’s Equations ~ ·B ~ =0 r ~ ·D ~ =0 r Plane and Spherical Waves ~ ~ ⇥H ~ = dD r dt ~ ~ ⇥E ~ = dB r dt ~ = r2 E ⇣ n ⌘ 2 d2 ~ E c dt2 and ~ = r2 H ⇣ n ⌘ 2 d2 ~ H c dt2 plane wave ~ D ~ B ~ ✏r ✏0 E ~ = µr µ0 H ⇣ ⌘ 2 d2 ~ = n ~ r2 E E c dt2 lecture 2 linear isotropic medium ⇢=0 J~ = 0 = and ~ r, t) = E ~ 0 ei(~k~r E(~ !t) spherical wave ⇣ ⌘2 d2 ~ = n ~ r2 H H c dt2 E(~r, t) = E0 ei(kr lecture 2 !t) /r p ✏ rµ r p ✏ 0µ 0 = n ⌫ 1/ /k = = ! = c c/n = ~ vp ~ ⇥H E ~ = S 2⇡ = nk0 k= Scalar Approach ⟼ Eikonal E(~r, t) = E0 (~r)e Fresnel Number i!t (r2 + n2 k02 )E0 (~r) = 0 E(~r, t) = A0 (~r)ei(k0 L(~r) F = a !t) b (r · L)2 = n2 Eikonal equation a2 b F L3 L2 L1 lecture 2 L0 1 Geometrical optics F =1 Fresnel diffraction F ⌧1 Fraunhofer diffraction wave optics lecture 2 Wave Nature of Light Maxwell’s wave equations Wave Nature of Light + k 2 )ur = 0 ( Maxwell’s wave equations Plane and spherical transverse waves ( + k 2 )ur = 0 Plane and spherical transverse waves Wave propagation single-frequency wave: plane or spherical waves: wave vector: angular frequency: intensity: lecture 3 u(~r, t) = ur (~r)e u(~r, t) = u0 e i!t i(~ k~ r !t) u0 i(kr e or r Energy flow & Poynting vector Eikonal equation Fermat’s principle (geom. Optics) !t) k = 2⇡/ ! = 2⇡/T = c/ I / |u(~r, t)|2 Fresnel number 0 ⇥ c/n F = lecture 3 a2 b a b Huygens’ wavelet Huygens’ wavelet ebcid:com.britannica.oec2.identifier.AssemblyIdentifier?assemblyId=... ebcid:com.britannica.oec2.identifier.AssemblyIdentifier?assemblyId=... Huygens’ wavelet Huygens’ wavelet Wave Nature of Light Huygen’s Principle Wave propagation Every point on a wave front can be considered as a source of secondary spherical waves Huygen’s principle Kirchhoff integral Historic disputes Fraunhofer diffraction Figure 2: Huygens' wavelets. Originating along the fronts of (A) circular waves and (B) plane waves, wavelets recombine to produce the propagating wave front. (C) The diffraction of sound around a corner Interference and diffraction arising from Huygens' wavelets. Z ~ ik|R ~ r | waves and (B) plane waves, Figure 2: Huygens' wavelets. Originating along the fronts ofe(A) circular ~ /wave front. wavelets recombine to produce the propagating u(R) u(~r)(C) The diffraction dS of sound around a corner ~ ~r| arising from Huygens' wavelets. |R Superposition of waves Fourier methods lecture 3 lecture 3 Huygen’s Principle Huygens’ wavelet Wave Propagation ebcid:com.britannica.oec2.identifier.AssemblyIdentifier?assemblyId=... print articles Huygens’ wavelet Fresnel’s Theory of wave propagation From Maxwell’s equations: ~ / u(R) Z ⇣ ~ G(R G(~r) = ~ / u(R) Z ~ r u(~r) ~r)r ~ r G(R ~ u(~r)r ⌘ ~ ~r) dS print articles 1 ik|~r| e |~r| ~ 1 of 1 ⌘(✓i , ✓o )u(~r) eik|R ~r| dS ~ ~r| |R 23.11.2008 19:56 Uhr Figure 2: Huygens' wavelets. Originating along the fronts of (A) circular waves and (B) plane waves, wavelets recombine to produce the propagating wave front. (C) The diffraction of sound around a corner arising from Huygens' wavelets. Huygen‘s wavelets recombine to produce the propagating wavefront lecture 3 lecture 3 1 of 1 23.11.2008 19:56 Uhr Fresnel‘s Theory of Wave Propagation Fresnel‘s Theory of Wave Propagation Fresnel-Kirchhoff diffraction integral i ⇤ up = (⇥in , ⇥out ) plane-to-plane u0 ikr e dS r obliquity factor (⇥in , ⇥out ) = 1 (cos ⇥in + cos ⇥out ) 2 Fraunhofer (far field) diffraction is a special case lecture 3 eikr ⇤ eikr0 · ei( Huygens secondary sources on wavefront at -z radiate to point P on new wavefront at z = 0 x x+ y y) Fresnel‘s Theory of Wave Propagation Fresnel‘s Theory of Wave Propagation plane-to-plane r= ! up = u0 up = ⇥(⇤in , ⇤out ) Huygens secondary sources on wavefront at -z radiate to point P on new wavefront at z = 0 u0 ikr e dS r lecture 3 q2 + ⇥2 ⇥2 q⇥ 2q Fresnel Diffraction → Talbot Effect Complementary Models Near-field diffraction of an optical grating Geometric Optics Fermat’s principle Light rays Corpuscular explanation (Newton) self-imaging at zT = 2d2 / Wave Optics Huygen’s principle Fresnel-Kirchhoff integral Interference and diffraction lecture 3 lecture 3 Wave or Particle? (17th century) Isaac Newton Christiaan Huygens Wave or Particle? ✤ Wave theory of light Christiaan Huygens (1690) ✤ Explains: reflection, refraction, colours, diffraction, interference lecture 3 lecture 3 Wave or Particle? ✤ Wave or Particle? Corpuscular theory of light ✤ Isaac Newton (1704) ✤ Isaac Newton (1704) Explains: ✤ reflection, refraction, colours ✤ Explains: reflection, refraction, colours but not: ✤ diffraction, interference but not: diffraction, interference lecture 3 lecture 3 Wave or Particle? ✤ Corpuscular theory of light Young‘s Double Slit Evidence for light waves intensity Thomas Young (1803) ✤ light behaves like a wave Explains: diffraction interference bending light around corners lecture 3 lit eS oubl D lecture 3 position Young‘s Double Slit Young‘s Double Slit intensity light behaves like a wave intensity position position Interference: crest meets crest through meets through crest meets through lecture 3 ➙ annihilation lecture 3 Poisson versus Fresnel particles } ➙ amplification Poisson versus Fresnel waves particles waves Poisson versus Fresnel Fraunhofer Diffraction çois Fran o g Ara y aperture θ Poisson Spot lecture 4 Diffraction in the far field = (k sin ⇥) · y Fraunhofer Diffraction Fraunhofer Diffraction A diffraction pattern for which the phase of the light at the observation point is a linear function of the position for all points in the diffracting aperture is Fraunhofer diffraction A diffraction pattern for which the phase of the light at the observation point is a linear function of the position for all points in the diffracting aperture is Fraunhofer diffraction eikr ⇤ eikr0 · ei( x x+ y y) How linear is linear? lecture 4 Fraunhofer Diffraction Fraunhofer Diffraction d r = |rmax a2 ⇥ ⇥/8 d| ⇤ 8d lecture 4 r= 1 + 2 a2 ⇥ 8 1 1 + ds dp lecture 4 Fraunhofer Diffraction Fraunhofer Diffraction y y ⇥ ⇥/8 f aperture θ aperture θ = (k sin ⇥) · y = (k sin ⇥) · y Fraunhofer Diffraction Fraunhofer Diffraction y f A diffraction pattern formed in the image plane of an optical system is subject to Fraunhofer diffraction illumination aperture θ Diffraction in the image plane what is being imaged? = (k sin ⇥) · y lecture 4 Fraunhofer Diffraction Fraunhofer Diffraction Fraunhofer diffraction: in the image plane lecture 4 Equivalent lens system: Fraunhofer diffraction independent on aperture position lecture 4