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Optics Lecture Transparencies 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 Plane and Spherical Waves ~ ~ ~ ~ d ~ n 2 d2 n 2 d2 B =0 H = D 2E~ = E~ and 2H~ = H~ r · r⇥ dt r c dt2 r c dt2 d ⇣ ⌘ ⇣ ⌘ ~ D~ =0 ~ E~ = B~ r · r⇥ −dt plane wave ~ ~ D = ✏r✏0E linear isotropic medium ~ ✏rµr E~ (~r, t)=E~ ei(k~r !t) p ~ ~ 0 − = µ0 B = µrµ0H ⇢ =0 J~ =0 n /p ✏0 ⌫ /k = c =1 = ! c/n spherical wave = ~ vp ~ H n 2 d2 n 2 d2 E 2 ~ ~ 2 ~ ~ ~ = ⇥ E = 2 E and H = 2 H i(kr !t) S r c dt r c dt E(~r, t)=E0e − /r 2⇡ ⇣ ⌘ ⇣ ⌘ k = = nk0 λ lecture 2 lecture 2 Scalar Approach ⟼ Eikonal Fresnel Number i!t E(~r, t)=E0(~r)e− 2 2 2 2 a ( + n k0)E0(~r)=0 F = a r λb E(~r, t)=A (~r)ei(k0L(~r) !t) 0 − b ( L)2 = n2 r · F 1 Geometrical optics Eikonal L3 equation F =1 Fresnel diffraction wave optics L2 F 1 Fraunhofer diffraction ⌧ L1 lecture 2 L0 lecture 2 Wave Nature of Light Wave Nature of Light 2 2 Maxwell’s wave equations (∆ + k )ur =0 Maxwell’s wave equations (∆ + k )ur =0 Plane and spherical transverse waves Plane and spherical transverse waves Wave propagation i!t single-frequency wave: u(~r, t)=ur(~r)e− Energy flow & Poynting vector i(~k~r !t) u0 i(kr !t) plane or spherical waves: u(~r, t)=u e − or e − 0 r Eikonal equation wave vector: k =2⇡/λ Fermat’s principle (geom. Optics) angular frequency: ! =2⇡/T = c/λ0 Fresnel number intensity: I u(~r, t) 2 c/n a2 / | | ⇥ F = a λb lecture 3 lecture 3 b Huygens’ wavelet ebcid:com.britannica.oec2.identifier.AssemblyIdentifier?assemblyId=... Huygens’ wavelet 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 Huygen’s principle of secondary spherical waves Kirchhoff integral Historic disputes Fraunhofer diffraction Figure 2: Huygens' wavelets. Originating along the fronts of (A) circular waves and (B) plane waves, Interference andwavele tdiffractions recombine to produce the propagating wave front. (C) The diffraction of sound around a corner arising from Huygens' wavelets. ik R~ ~r Figure 2: Huygens' wavelets. Originating along the fronts ofe (A)| cir−cul|ar waves and (B) plane waves, Superposition of waves wavelets recombine to produce the propua(gRa~ti)ng wave furo(n~rt.) (C) The diffrdSaction of sound around a corner arising from/ Huygens' waveR~lets. ~r Fourier methods Z | − | lecture 3 lecture 3 Huygen’s Principle Wave Propagation Huygens’ wavelet ebcid:com.britannica.oec2.identifier.AssemblyIdentifier?assemblyId=... print articles Huygens’ wavelet From Maxwell’s equations: Fresnel’s Theory of wave propagation u(R~) G(R~ ~r)~ u(~r) u(~r)~ G(R~ ~r) dS~ print articles / − rr − rr − Z ⇣ ⌘ 1 ik ~r G(~r)= e | | ~r | | ik R~ ~r 1 of 1 e | − | 23.11.2008 19:56 Uhr u(R~) ⌘(✓i, ✓o)u(~r) dS / R~ ~r Figure 2: HuyHuygen‘sgens' wavelets. O rwaveletsiginating along trecombinehe fronts of (A) cir ctoula r producewaves and (B) plane waves, Z wavelets recombine to produce the propagating wave front. (C) The diffraction of sound around a corner | − | the propagatingarising from Huygens 'wavefront wavelets. lecture 3 lecture 3 1 of 1 23.11.2008 19:56 Uhr print articles 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 plane-to-plane i u u = η(⇥ , ⇥ ) 0 eikrdS p −⇤ in out r obliquity factor 1 η(⇥ , ⇥ ) = (cos ⇥ + cos ⇥ ) in out 2 in out Fraunhofer (far field) diffraction is a special case eikr eikr0 ei(βxx+βy y) Huygens secondary sources on wavefront at -z −⇤ · radiate to point P on new wavefront at z = 0 lecture 3 Fresnel‘s Theory of Wave Propagation Fresnel‘s Theory of Wave Propagation plane-to-plane ⇥2 δr = q2 + ⇥2 q − ⇥ 2q ! up = u0 u u = α ⇥(⇤ , ⇤ ) 0 eikrdS p in out r Huygens secondary sources on wavefront at -z radiate to point P on new wavefront at z = 0 lecture 3 Fresnel Diffraction → Talbot Effect Complementary Models Near-field diffraction of an optical grating Geometric Optics Fermat’s principle self-imaging at Light rays 2 Corpuscular explanation (Newton) zT = 2d /λ Wave Optics Huygen’s principle Fresnel-Kirchhoff integral Interference and diffraction lecture 3 lecture 3 Wave or Particle? (17th century) Wave or Particle? Isaac Newton Christiaan Huygens ✤ Wave theory of light Christiaan Huygens (1690) ✤ Explains: reflection, refraction, colours, diffraction, interference lecture 3 lecture 3 Wave or Wave or Particle? Particle? ✤ Corpuscular theory of light ✤ Corpuscular theory of light Isaac Newton (1704) Isaac Newton (1704) ✤ Explains: ✤ Explains: reflection, reflection, refraction, refraction, colours colours ✤ but not: ✤ but not: diffraction, diffraction, interference interference lecture 3 lecture 3 Wave or Young‘s Double Slit Particle? ✤ Evidence for light waves intensity Thomas Young (1803) light behaves like a wave position ✤ Explains: diffraction interference bending light Double Slit around corners lecture 3 lecture 3 Young‘s Double Slit Young‘s Double Slit intensity intensity light behaves like a wave position position Interference: crest meets crest through meets through } ➙ amplification crest meets through ➙ annihilation lecture 3 lecture 3 Poisson versus Fresnel Poisson versus Fresnel particles waves particles waves Poisson versus Fresnel Fraunhofer Diffraction François Arago y θ aperture Poisson Spot Diffraction in the far field δ = (k sin ⇥) y · Fraunhofer Diffraction Fraunhofer Diffraction A diffraction pattern for which the phase A diffraction pattern for which the phase of the light at the observation point is a of the light at the observation point is a linear function of the position for all linear function of the position for all points in the diffracting aperture is points in the diffracting aperture is Fraunhofer diffraction Fraunhofer diffraction eikr eikr0 ei(βxx+βy y) How linear is linear? −⇤ · lecture 4 lecture 4 Fraunhofer Diffraction Fraunhofer Diffraction d a2 a2 1 1 δr = rmax d ⇥/8 δr = δ1 + δ2 + ⇥/8 | − | ⇤ 8d ⇥ ⇥ 8 ds dp ≤ lecture 4 lecture 4 ⇥ Fraunhofer Diffraction Fraunhofer Diffraction y y f θ θ aperture aperture δ = (k sin ⇥) y δ = (k sin ⇥) y · · Fraunhofer Diffraction Fraunhofer Diffraction y f A diffraction pattern formed in the illumination image plane of an optical system is θ subject to Fraunhofer diffraction what is being imaged? aperture Diffraction in the image plane δ = (k sin ⇥) y · lecture 4 Fraunhofer Diffraction Fraunhofer Diffraction Fraunhofer diffraction: Equivalent lens system: in the image plane Fraunhofer diffraction independent on aperture position lecture 4 lecture 4.
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