Optics Notes PDF

# Phys 256 Exam Review ## Lecture 10 - Light has both wave-like and particle-like nature (transverse EM wave) - Wave function: $y=Asin[k(x-vt)]$, $y = a(x+vt)^2$, $y = e^x(x-vt)$. - For traveling wave, must satisfy: $\frac{\partial^2 y}{\partial x^2}$ = $\frac{1}{v^2}$ $\frac{\partial^2 y}{\parti...

# Phys 256 Exam Review ## Lecture 10 - Light has both wave-like and particle-like nature (transverse EM wave) - Wave function: $y=Asin[k(x-vt)]$, $y = a(x+vt)^2$, $y = e^x(x-vt)$. - For traveling wave, must satisfy: $\frac{\partial^2 y}{\partial x^2}$ = $\frac{1}{v^2}$ $\frac{\partial^2 y}{\partial t^2}$ or $\frac{\partial^2 y}{\partial x^2}$ = $\frac{1}{v^2}$ $\frac{\partial^2 y}{\partial t^2}$ curvature, $\frac{\partial y}{\partial t}$ acceleration. - Wave egn is a 2nd order partial differential equation, 1^st differential must be continuous. - $$y(x,t) = Asin(kx-vt)$$ – propogating right - $$y(x,t) = Asin(kx+vt)$$ – propogating left - A=amplitude, K=propagation #, v=propagation speed - $sinx = cos(x-\frac{T}{2})$, $\phi_0$: initial phase – $y(x,t) = Asin[K(x\pm vt)+\phi_0]$ - **Angular freq (ω):** how fast the wave oscillates in time. - **Wavenumber (k):** how fast it oscillates in space. - **Period (T):** duration of time of 1 cycle in a repeating event, reciprocal of the frequency. - **Frequency (f):** rate quantity that describes rate of oscillations or vibrations on per-second basis. - Eg. $Ψ(y,t) = (0.040) sin2\pi (60×10^{-7} + 20x10^{-15})$ - a) Represents a harmonic wave - b) $T=2×10^{-7}s$ – $f=\frac{1}{T}=5×10^4Hz$ - c) $x=6×10^-7m$ - d) 0.040 - e) left - f) $v=Af = 3×10^8 m/s$ - **Euler’s formula:** $e^{i\theta} = cos\theta + isin\theta$ - **Harmonic waves as complex functions:** $y = Ae^{i{(kx-wt)}}$ = $Acos(kx-wt) + i(Asin(kx-wt))$ - real – imaginary - **Wavefront:** the surface joining all points of equal phase - **Planar Wavefront:** Wave phase is constant along planar surface, invariant to propagation direction. - **Spherical Wavefront:** Wave phase constant along spherical surface, over time, wavefronts expand outwards, a spherical wave decreases in amplitude - **Wave in vector form:** $Ψ = Asin(K\cdot r-wt)$ = $(A)e^{i(kr-wt)}$ - **General harmonic wave in 3D:** $\frac{\partial^2 Ψ}{\partial x^2}$ + $\frac{\partial^2 Ψ}{\partial y^2}$ + $\frac{\partial^2 Ψ}{\partial z^2}$ = $\frac{1}{v^2}$ $\frac{\partial^2 Ψ}{\partial t^2}$ - **Rays:** Perpendicular to the wavefront, trajectories of photons, straight in homo media - **Electromagnetic Waves:** $E = E_0sin(k\cdot r-wt)$, $B=B_0sin(k\cdot r-wt)$ , $E = vB$ - **Phasor:** Complex # in polar form, consists of a magnitude (amplitude) and angle (phase). ## Lecture 12 - Eg. $E=E_0e^{i(kx-\sqrt{2}y-9.9×10^{14}t)}$ - $E=E_0e^{i(k\cdot r-wt)}$, $K=k\hat{x}+k_y\hat{y}+k_z\hat{z}$, $|k|=\sqrt{k^2}$ v = $\frac{w}{|k|}$ = 2.98×10^8 m/s - a) Angular freq. $w= 9.9×10^{14} rad/s$ - b) Wave vector, $K= 3\hat{x}-5\hat{y}$ - c) $|k|=\sqrt{Ⅱ} m^{-1}$ - d) v = $\frac{w}{|k|}$ = 2.98×10^8 m/s - **Maxwell's Equations** – tie together electric & magnetic fields, charges, & currents 1. **Gauss's Law (E):** Electric fields are produced by electric charges: ∮$E\cdot ds = \frac{Q_{enc}}{\epsilon_0}$ or $\nabla \cdot E = \rho/\epsilon_0$ 2. **Gauss's Law (B):** Magnetic fields form closed loops and have no beginning or end (no magnetic monopoles): ∮ $B\cdot ds = 0$ or $\nabla \cdot B = 0$ ## Lecture 13 - **Principle of Superposition:** $Y(r.t) = \sum_{i=1}^n C_iY_i(r,t) $ - $E_R = E_0cosαsinωt + E_0sinαcosωt = E_0sin(ωt+α)$ - $E_0^2 = E_1^2+E_2^2+2E_1E_2cos(α_2-α_1)$ - **Destructive Interference:** Active cancellation, sends signal 2λ out of phase - **Optical Path Difference:** $δ = k(x_1-x_2) = \frac{2π}{\lambda}(x_1-x_2)$, $E = \sum_{i=1}^N {E_0}_icos(α_i\pmωt)$ - **Superposition of Coherent Harmonic Waves:** $E = \sum_{i=1}^n {E_0}_icos(α_i\pmωt)$ ## Lecture 14 - **Standing Wave:** Transmit no energy. $E = 2(E_0sinkx)Cosωt$ - $sinkx=0$ or $kx=mπ$, $m=0,\pm1, \pm2$... - **Nodes:** $x=m(\frac{\lambda}{2})$ = $0,\frac{\lambda}{2},\lambda…$ - space dependent amplitude - *same frequency* - Examples: Strings on instrument, laser cavity with mirror, singing in shower - **Partial Standing Wave:** $E = E_0(x)cos(wt-k(x))$ - **Magnetic field of Standing Wave:** $B(x,t) = -2B_0coskxsinωt$ - **Addition of Waves w/ different f:** $E = 2E_0cos(k_px-w_pt)cos(k_gx-w_gt)$ - **Beat Pattern:** Superposition of 2 equal amplitude harmonic waves of different frequency producing a beat pattern. - **Beat frequency:** $w_b= |w_1-w_2|= \frac{1}{2}|w_g-w_l|$ - Example: tuning a guitar using 2 guitars, using feedback loop and doppler shift. - **Phase Velocity:** Velocity of the higher-frequency carrier wave: $v_p = \frac{ω}{k}$ - **Group Velocity:** Velocity of lower frequency envelope wave! $v_g = \frac{dω}{dk}$ = $v_p[1-(\frac{\lambda}{n}\frac{dn}{d\lambda})]=v_p[1+\frac{λ}{n}(\frac{dn}{dλ})]$ ## Lecture 15 - **Two beam Interference:** $E_p=E_1+E_2$, $E_1/E_2(r,t) = E_01/2 Cos (kr_1t - wt+\epsilon_1/2)$ - $I = <E_p^2>$ , $I_{12}=2\epsilon_0C<E_1^*E_2> = E_01.E_02<Cos(k\cdot r_1 - wt + \epsilon_1/2)>$ - **Spatial Coherence:** Uniformity of phase over points of extended source over wavefront, direction & polarization constant. Eg, Wavefronts from point source. - **Temporal Coherence:** Time interval where waves generated by source have same temporal freg, phase, polarization. Eg, laser with long coherence length - **Interference of Mutually Coherent Beams:** - $cosδ=+1$ **Constructive Int:** $I=I_1+I_2+2\sqrt{I_1I_2}$ – $δ=2mπ$ - $cosδ=-1$ **Destructive Int:** $I=I_1+I_2-2\sqrt{I_1I_2}$ – $δ=(2m+1)π$ - When $E_01=E_02$, $I_1=I_2$: $I = 2I_0(1+cosδ) = 4I_0cos^2\frac δ2$ - $I_{max}=4I_0$, for $δ=0,\pm2π,...$ - $I_{min}=0$, for $δ=\pmπ, \pm 3π,…$ - **Visibility:** A measure of fringe contrast, $visibility=\frac{I_{max}-I_{min}}{I_{max}+I_{min}}$, $m=0,\pm 1,\pm 2$ - **Spherical Sources:** $E_{12}(r_1,r_2, t) = E_01/2 ((r_1,r_2)e^{i(k\cdot r_1-wt+\epsilon_1/2)}$ - $I=4I_0cos^2\frac δ2 = 4I_0 cos^2\frac 12 [k(r_1-r_2)+(\epsilon_1-\epsilon_2)]$ - Max & Min produce interference fringes. - $δ=k(r_1-r_2)+(\epsilon_1-\epsilon_2) = \frac{2mπ}{k} = mλ$, $Max: (r_1-r_2) = \frac{2mπ}{k} = mλ$ - $Min: (r_1-r_2) = \frac{(2m+1)π}{k} = (m+\frac 12)λ$ - **Young's Double Slit Experiment:** Mono light passes through single slit, then double slits. - **Path Difference:** Length difference of total path of 2 waves of light - **Constructive Interference** occurs when path diff is I integer x λ or mλ - **Destructive Interference** occurs when path diff is integer +0.5 x λ or (m+½)λ - Examples of Interference: Soap film, oil film. ## Lecture 16 - **Diffraction:** The bending of a wave around the edges of an opening or obstacle - - If diffraction is observed for something, it is evidence that it's a wave. - **Huygen's Principle:** Every point on a wave front serves as a source of secondary wavelets, and the new wave front is the tangetial surface to all the secondary wavelets. - **Young's Double Slit Experiment:** - **Path Difference:** $r_2-r_1 = ∆ = mλ = asinθ$ - **Small angle approximation:** $sine=tanθ = \frac{y}{L}$ or, $∆ = (m+\frac 12)λ = asinθ$ - **The bright fringe position:** $y = m\frac{λL}{a}$ - **The dark fringe position:** $y = (m+\frac 12)\frac{λL}{a}$ - **Irradiance of fringes on screens:** $I = 4I_0cos^2(\frac{πd}{λ}) $, $E_0=2E_0i cos(\frac{πd}{λ})$ - ↑ slit seperation → fringe size ↑; also ↑ fringe size - Constant seperation between irradiance maxima corresponding to successive values m. $△y=Y_{m+1}-Y_m=\frac{λL}{a}$ - **Far field:** see a series of straight, parallel, bright and dark fringes - **Near field:** 2 waves arrive at point with phase angle and amplitude difference. - **Interference of White Light:** 0^th order fringe is white, higher order shows multiple colours. - **Fresnel's Double Mirror:** Line of intersection parallel to source slit. 2 mirrors bend light. - **Lloyd's Mirror:** Fringes formed from superposition of light at screen from 2 sources. - **Frensel's Biprism:** Sources are virtual images formed by refraction of 2 halfs of prism. - **Wavefront Split** - Young's double slit experiment - **Amplitude Split** - Thin film interference ## Lecture 17 - **Thin Film:** Permits both reflection & transmission of light that interfere. - **External Reflection:** Reflection at a denser surface produces a reflected wave 180° out of phase wrt the incident wave (n_e>n_i) - **Internal Reflection:** Reflection at lower density produces "in-phase" reflected wave (n_f<n_i) - **Optical Path Difference Condition:** $∆ = 2n_tdcosθ$ & **Amplitude Condition** - **Single Film Interference:** $∆ = 2n_tdcosθ$ $δ = k∆$ Path change: $Ar = \frac{\lambda}{2}$ - **Net Path difference:** $∆ = Ar$ - **Max:** $tcosθ = \frac{(2m+1)λ}{4n_t}$ - **Min:** $tcosθ = \frac{2mλ}{4n_t}$ - **Intermal Reflection:** Constructive: t = $\frac{λ}{4}$ Destructive: t = $\frac{3λ}{4}$ - **Extenal Reflection:** Con: t = $\frac{λ}{4}$ Des: t = $\frac{3λ}{4}$ - **Anti-Reflective Coating:** - $r = \frac{1-n_1}{1+n_1}$, $n_1=√n_2n_3$, $n_3=√n_1n_4$ - **Enhancement of Reflected Light:** Multilayer dielectric mirror, alternating high & low index. $t = \frac{λ}{2}$ - **Michelson Interferometer:** $I = 4I_0cos^2(δ)$ – $δ = 2dcosθ$ - **For bright fringes:** $2dcosθ= (m+\frac{1}{2})λ$, $m_{max}=\frac{2d}{\lambda}$ , $I=4I_0$ - **The center fringe is dark:** $2dcosθ= mλ$, $m_{max}=\frac{2d}{\lambda}$ , $I=0$ - **For dark frnges:** $2dcosθ= mλ$ - $p = m_{max} - m = \frac{2dcosθ}{\lambda}-(m)$ = $2d(1-cosθ)$ - **As d↓ :** # of fringes ↓, contract towards center, $θ_d$ for first fringe - **As d↑ :** # of fringes ↑, expands outwards, $θ_d$ for first fringe - Examples, Keck twin telescopes, makes planets visible, cancels light from stars - **Haidinger fringes:** Fringes of equal inclination. $δ = \frac{4π_tn_tdcosθ}{λ}$ - **Fizeau fringes:** Fringes of equal thickness. $α = \frac{λ}{2n_t}$, $△x = \frac{λ}{2π}$, $△x = \frac{λ}{2}$ - **Newton's Rings:** An interference pattem is created by reflection of light between 2 surfaces; spherical & adjacent flat surface. $2n_ft+Ar = (m+\frac 12)λ$ - $n_1=1$, $m=0$, $t=0$ at center. $Ar=black$ $x=\sqrt{Rmλ}$ (distance from m^th ring to center) - Applications: Irregularities of lens surface, measure the radius of curvature of lens. ## Lecture 19 - **Fraunhofer Diffraction:** far field diffraction, the source of light and observation screen are effectively far enough from the diffraction aperture → plane wavefronts. - - Moving screen changes the size of the pattern but not it's shape. - **Fresnel Diffraction:** Near field diffraction, the source of light and observation screen is close → spherical or cylindrical wavefront undergoes diffraction - Both shape & size of pattern depends on distance between aperture & screen. - Fraunhofer Diffraction will occur at an aperture of greatest width b when R > $\frac{b^2}{λ}$ - Wavefronts in and out from the aperture are planar. - Path difference to P → phase shift - These path differences describable by a linear function of the 2 aperture variables. - **Single Slit Diffraction** - Long thin slit is = to point source with a pattem in the horizontal plane - Large aperture leaves plane wave unchanged, small apertures show more diffraction - $dEp=(Eds)e^{i{kr-wt}}$ $E_p= \frac{E_0b}{iλ}e^{i{kr-wt}} \int_{-b/2}^{b/2}e^{i{kbsinθ}} d\xi$ - $B=kbsinθ$, $∆ = bsinθ$ - $|B|$ = magnitude of phase difference, Total path diff is $bsinθ$, b=slit width - $I=ZE_p^*E_p=EC |\frac{E_0bsinθ}{iλ}|^2 = I_0 (\frac{sinß}{ß})^2 = I_0sinc^2(ß)$ - **Phase difference across the slit:** $δ = 2πbsinθ$ - **Secondary Maxima Pattern:** $ß = tanß$ - **The Angular Width of the central maximum is the angle 40:** $40 =\frac{2λ}{b}$ - **Width of the central maximum is W = LAO = 2λ\frac{L}{b}$** , $L_{min}=\frac{2λ}{b}$ - **Arbitrary Aperture:** $E=E_0e^{i(wt-kR)}$ - **2D Pattern-Rectangular Aperture:** $y_m=\frac{mλL}{a}$ or $x_n=\frac{nλL}{b}$, $I = I_0 (sinc^2ß) (sinc^2α)$ - $I = I_0 (sinα)^2$, where $α = (\frac{πa}{λ})sinθ$ - **Circular Aperture:** $I_0 = (2J_1(z))²$ where $z = kDsinθ$ - **Airy disc:** Far-field angular radius (angular half-width) of the Airy disk is $40/2 = \frac{1.22λ}{D}$ - **Resolution:** Ability to provide distinct images for distant object points, either physically close together or seperated by a small angle at the lens. To improve: ↑ lens diameter, ↓ λ - **Rayleigh's Criterion:** Requires that the angular seperation of the centers of the image patterns be no less than the angular radius of the Airy disc. - **Limit of Resolution:** $(40)_{min} = 1.22\frac λD$ - **Resolution Power:** $RP = (40)_{min} = 1.22\frac λD$ - **Resolution for a Microscope:** $X_{min} = f40_{min} = f(1.22\frac λD)$, $X_{min} = \frac {0.61λ}{NA}$ - Eg. ultraviolet, x-ray, and electron microscopes - **Resolution of the human eye:** Diffraction by the eye with pupil as aperture limits the resolution of objects subtending angle (40)_min - $E_{theoretical}(40)_{min} = 29×10^{-5}rad$, $RP_{eyes}=\frac {1}{40}=\frac{1}{50×10^{-5}}= 2×10^3$ - $E_{experimental}(40)_{min} = 50×10^{-5}rad$ ## Lecture 20 - **Double Slit Diffraction:** Each aperture generates the same single slit diffraction pattern. Primary wave excites secondary sources at each slit, resultant wavelets are coherent and interference must occur. - $B = \frac{1}{2}kbsinθ$, $kbsin θ = \frac{2π}{λ}bsinθ = πasinθ$ - $E_0= 2E_0sinβcosα$ - $I = +I_0(\frac{sinβ}{β})^2cos^2α$ – $α = kbsinθ = πasinθ$, $sinα = \frac{sin(πasinθ)}{πasinθ}$ - **Diffraction Minima:** $bsinθ = mλ$ - **Interference Minima:** $asinθ = (p+\frac{1}{2})λ$ - **Interference Maxima:** $asinθ = pλ$ - **Pattern depends on slit width (b) and slit seperation (a).** - **Missing Orders:** $a = m(\frac{λ}{2})$ ## Lecture 21 - **Multiple Slit Diffraction:** $E_p=e^{i{kr-wt}}(\frac {bsinß}{ß}) \frac{sinNα}{sinα}$ , $I = I_0(\frac{sinß}{ß})^2(sinNα/sinα)^2$ - Interference factor describes series of irradiance peaks (principal maxima). - **Conditions for Principal Maxima:** For a = 10, principal maxima occur for p=0, ±N, ±2N or m = m or α=Mπ . Recall β = πasinθ. $asinθ = mπ$ ⇒ Diffraction Grating Egn. - **Secondary Minima** occur for p = all other integer values. - **Principal Maxima:** When all waves arrive in phase, resulting phasor diagram is formed by adding N phasors all in the same "direction" giving a maximum resultant. - **Secondary Maxima:** Uniform phase difference between waves from adjoining slit causes the phase diagram to curl up, with a smaller resultant. - **Minima:** Phasor diagram forms closed figure, cancellation is complete. - **Diffraction Grating:** Repetitive array of diffracting elements (apertures) that has the effect of producing periodic alterations in the phase, amplitude, of emergent wave. - Dispersion of prism is non-linear white gratings offer linear dispersions - Prisms refract input light; gratings diffract. - Gratings are more efficient, no absorption effects that limit wavelength range of prism. ## Lecture 22 - **Polarization:** The direction of the electric field vector È (EM wave). - **Linear Polarization:** Orientation of the E field is constant, magnitude & sign vary in time. - **E** resides in plane of vibration, contains **E** and **K** - **Removes most È vibrations in a direction, allowing I ubrations to be transmitted** - **E** = $Ex(z,t) + Ey(z,t) = (\hat{x}E_{0x} + \hat{y}E_{0y})cos(kz-wt)$ - $tanΦ = \frac{E_{0y}}{E_{0x}}$ - If the relative phase difference $δ$ = 0, ±π, ±2π, … the light is linearly polarized. - **Circular Polarization** - Perpendicular waves have equal amplitudes $E_{0x} = E_{0y} = E_0$, phase difference $δ$ =-2π + 2πm - $E_x^2 + E_y^2 = E_0^2[cos^2 (kz-wt)+sin^2 (kz-wt)] = E_0^2 = constant$ - ← it’s a circle - Rotates clockwise with the same frequency which it oscillates. - 2 – antennas radiating w/ 90° phase difference produces circularly polarized EM waves - If É rotates CCW, wave is left-circularly polarized (RHR) - **Elliptical Polarization:** - $tan2α =\frac{2E_{0x}E_{0y}cosδ}{E_{0x}^2+E_{0y}^2-2E_{0x}E_{0y}cosδ} = \frac{sin^2δ}{cos^2δ}$ - Size of bounding box is determined by $E_{0x}$ and $E_{0y}$ - **Randomly Polarized (Simply Unpolarized):** Polarizations of the individual fields produced by oscillators are randomly distributed in direction. - **Partially Polarized:** EM field consists of the superposition of fields w/ many polarizations where I is predominant. - A perfectly monochromatic plane wave is always polarized. - **Polarizer:** An optical device whose input is natural light and output is some form of polarized light. The transmitted light is linearly polarized in direction of transmission. - - **Perfect Polarizer:** Lets 1/2 of randomly oriented light through - **Analyzer:** Transmission axis is vertical. Amplitude of light: $E_{0z} = E_{0A}cosθ$ - **Malus’s Law:** The intensity of a plane-polarized light that passes through an analyzer is a $cos^2θ$ of the angle between the plane of the polarizer and the transmission axis of the analyzer: $I(θ) = I_0cos^2θ$ - **Phase Retarder:** Doesn’t remove any I components of E vibrations, but introduces a phase difference between them. - **Rotator:** Effect of rotating direction of linearly polarized light incident by some angle. - **Dichroism:** Selective absorption of I of the 2 – P-state components of incident beam. - **Anisotropy of Absorption:** Dichroic polarizer is physically anisotropic, producing strong asymmetrical absorption of I field component, being transparent' to the other. - Egs: Wire-Grid Polarizer, Polaroid, Dichroic Crystals. - $κ$ - absorption coefficient - **Efficiency of dichroic absorber is a function of thickness:** $I=I_0e^{-κx}$ - **Birefringence - Double Refraction:** Cause double refraction - appearance of 2 refracted beams due to existence of 2 different indices of refraction for a single material. - E-ray does not generally obey Snelli’s law - **Birefringence Materials:** Crystals belonging to hexagonal, tetragonal, triagonal systems. Optically anisotropic and birefringement. Calcite, Quartz, Ice, Mica. - **Complex Refractive Index:** $n = n + ik$ - • ideal dichroic material: $n_x=n_y$, $κ_x=κ_y$ - • ideal birefringent material: $n_x≠n_y$, $κ_x = κ_y$ - where κd absorption coefficient - **Scattering:** Removal of energy from an incident wave by a scattering medium and the reemission of some portion of energy in many directions. - **Rayleigh Scattering:** Elastic scattering, particles with dimensions < wavelengths light, induce oscillating dipoles reradiate or scatter energy in all directions except along the dipole axis itself. Reason why the sky is blue. - **Mie Scattering:** Scattering particles has a size comparable to or > λ light. - **Polarization by Reflection:** Light that is specularly reflected from dielectric Surfaces is at least partially polarized. - $E_s$ – to plone of incidence - $E_p$ – in plane of incidence - **Brewster’s Angle:** Angle of incidence that produces a linearly polarized beam: E_s reflection is 0^o. É_p component is missing from reflected beam. - $tanθ_B = n_2/n_1$ = $tan(Φ_t)$ - Eg of Polarization: 3D glasses.