Lecture 19 Optical MEMS (1)

Transcription

1 EEL6935 Advanced MEMS (Spring 5) Instructor: Dr. Huikai Xie Lecture 19 Optical MEMS (1) Agenda: Optics Review EEL6935 Advanced MEMS 5 H. Xie 3/8/5 1 Optics Review Nature of Light Reflection and Refraction Total Internal Reflection Lenses Numerical Aperture Diffraction Polarization Interference Doppler Effect Coherence Optic Fibers: Basics EEL6935 Advanced MEMS 5 H. Xie

2 The Nature of Light Particle Light consists of photons Wave Light travels as a transverse electromagnetic wave Ray Theory Light travels along a straight line and obeys laws of geometrical optics. Ray theory is valid when the objects are much larger than the wavelength EEL6935 Advanced MEMS 5 H. Xie 3 Spherical and plane wave fronts A wave front is the locus all points in the wave train which have the same phase EEL6935 Advanced MEMS 5 H. Xie 4

3 Field Distribution of Plane Wave Electric and magnetic fields are orthogonal to each other and to the direction of propagation Z EEL6935 Advanced MEMS 5 H. Xie 5 Snell s Law Reflection and Refraction n 1 n θ 1 θ sinθ n sinθ = n 1 1 EEL6935 Advanced MEMS 5 H. Xie 6

4 Light incident to a less dense medium (i.e., n 1 >n ) Total Internal Reflection Critical angle (i.e., when θ =9º) θ n 1 c = sin n1 For glass/air interface, n1=1.5 and n=1, then θ c = 41.8º n 1 θ 1 n 1 θ 1 n 1 θ 1 Evanescent wave n n n θ θ θ θ θ > θ 1 < θc θ1 = θc 1 c Total Internal Reflection EEL6935 Advanced MEMS 5 H. Xie 7 Optical Fiber n 1 n Step Index Fiber n 1 >n Core has greater refractive index for total internal reflection Size varies Large cores for imaging and high-power illumination Small cores for optical communications Standard fibers for telecom: 15µm cladding/9µm core EEL6935 Advanced MEMS 5 H. Xie 8

5 Optical Fiber φ n 1 c = sin n1 θ = 9 φ c c nsinθ = n sinθ = n cosφ = n n,max 1 c 1 c 1 Maximum acceptance angle: θ ( ) 1/ = sin n n 1,max 1 NA = sinθ = n n Numerical Aperture: ( ) 1/,max 1 EEL6935 Advanced MEMS 5 H. Xie 9 Principle of Linear Superposition Two plane waves of the same frequency ω E(1) E1 exp = i( k1 i r ωt+ φ1 ) E() E exp = i( k i r ωt+ φ ) Irradiance: I = E = E E = E + E + E E cosθ * i 1 1i 1 1i cosθ = I + I + E E θ = k ir k ir + φ φ 1 1 φ 1 -φ is constant mutually coherent φ 1 -φ is randomly changing with time mutually incoherent no fringes The interference term depends on polarization. If the polarizations are mutually orthogonal, then E 1 E =. No interference fringes. EEL6935 Advanced MEMS 5 H. Xie 1

6 Theory of Partial Coherence Two plane waves of the same frequency ω but with varying phases I = EiE = E + E E + E ( 1 ) i( 1 ) * * * * ( i ) = E + E + Re E E 1 1 = I + I + Re E E * 1 1 Time average f 1 T lim T T f() t dt S 1 Usually E 1 and E originate from a common source. They differ due to traveling different optical paths. Path 1 S Path S P EEL6935 Advanced MEMS 5 H. Xie 11 Theory of Partial Coherence Mutual Coherence Function (or Correlation Function) * ( τ ) E ( t) E ( t τ ) Γ = Self-Coherence Function (or Autocorrelation Function) * ( τ ) E ( t) E ( t τ ) Γ = Degree of Partial Coherence γ 1 ( τ) ( τ ) ( ) Γ ( ) 11 ( τ ) Γ1 Γ1 = = Γ I I 1 γ 1 1 = 1 (complete coherence) < γ < 1 (partial coherence) γ 1 = (complete incoherence) EEL6935 Advanced MEMS 5 H. Xie 1

7 Theory of Partial Coherence Fringes ( ) I = I + I + I I Reγ τ I = I + I + I I γ max I = I + I I I γ min Fringe Visibility V I II max I γ min = I + I I + I 1 1 max min 1 V = γ 1 If I 1 = I EEL6935 Advanced MEMS 5 H. Xie 13 Theory of Partial Coherence Coherence Time τ and Coherence Length l c lc = cτ For a monochromatic wave with a finite duration τ, i.e., () f t iωt τ τ e for - < t < = otherwise Taking Fourier transform yields ( ) τ 1 1 sin ω ω τ / iωt g( ω) = e dt = τ - π π ω ω EEL6935 Advanced MEMS 5 H. Xie 14

8 Theory of Partial Coherence Coherence Time τ and Coherence Length l c Power spectrum: sin ω ω τ / G( ω) = g( ω) = π ω ω ( ) ( ) π The first minima occur at ω = ω ± τ π Spectral width ω = τ 1 v = τ l c c λ = c τ = = v λ EEL6935 Advanced MEMS 5 H. Xie 15 Doppler Effect Observed frequency f c v v f ' = f f 1 c v = + + c c v f 1 c Fixed Mirror Beamsplitter v(t) Laser Photodetector f(t) ~ v(t) Moving Mirror EEL6935 Advanced MEMS 5 H. Xie 16

9 Lenses Spherical lens, cylindrical lens, GRIN (gradient-index) rod lens Numerical Aperture NA = nisinα D α D α NA f For small NA f F-number f f # = D Resolution.61λ r = NA EEL6935 Advanced MEMS 5 H. Xie 17 Gaussian Beam D w θ f z R z R Beam waist (or spot size) Raleigh Range Beam radius at z Divergence angle w.44λ f = D π w z R = λ z w( z) = w 1 + z R w z λ w θ = = z π w z ( ) R EEL6935 Advanced MEMS 5 H. Xie 18 1/

10 Polarization If the electric field is oscillating along a straight line, it is called a linearly polarized (LP) or plane polarized wave If the E field rotates in a circle (constant magnitude) or in an ellipse then it is called a circular or elliptically polarized wave Natural light has random polarization EEL6935 Advanced MEMS 5 H. Xie 19 Polarization Adding two linearly polarized waves with zero phase shift will generate another linearly polarized wave EEL6935 Advanced MEMS 5 H. Xie

11 Polarization Adding two linearly polarized waves with a phase shift will produce an elliptically polarized light EEL6935 Advanced MEMS 5 H. Xie 1 Polarization Adding two linearly polarized waves with equal amplitude and 9 o phase shift results in circular polarized wave EEL6935 Advanced MEMS 5 H. Xie

12 Summary We discussed some basic optical phenomena: reflection, refraction, polarization, interference, diffraction, coherence and Doppler shift Lenses: numerical aperture Gaussian beam Optic Fibers: Total reflection Acceptance angle References: 1. Fowles, Introduction to Modern Optics, nd edition, Dover Publications, Inc., New York, Heavens and Ditchburn, Insight into Optics, John Wiley & Sons, Jeff Hecht, Understanding Fiber Optics, Prentice Hall, EEL6935 Advanced MEMS 5 H. Xie 3