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## Modern Optics I – wave properties of light

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**Modern Optics I – wave properties of light**Special topics course in IAMS Lecture speaker: Wang-Yau Cheng 2006/4**Outline**• Wave properties of light • Polarization of light • Coherence of light • Special issues on quantum optics**Properties of wave**Propagation • Phase • Wave equation • Phase velocity • Group velocity • Refraction of wave • Interference of wave • Electro-magnetic (EM) wave • Spectrum of EM wave**Waves, the Wave Equation, and Phase Velocity**What is a wave? Forward [f(x-vt)] vs. backward [f(x+vt)] propagating waves The one-dimensional wave equation Phase velocity Complex numbers**What is a wave?**A wave is anything that moves. To displace any function f(x) to the right, just change its argument from x to x-a, where a is a positive number. If we let a = v t, where v is positive and t is time, then the displacement will increase with time. So f(x-vt) represents a rightward, or forward, propagating wave. Similarly, f(x+vt) represents a leftward, or backward, propagating wave. v will be the velocity of the wave.**The one-dimensional wave equation**We’ll derive the wave equation from Maxwell’s equations. Here it is in its one-dimensional form for scalar (i.e., non-vector) functions, f: Light waves (actually the electric fields of light waves) will be a solution to this equation. And v will be the velocity of light.**Electromagnetism is linear: The principle of**“Superposition” holds. If f1(x,t) and f2(x,t) are solutions to the wave equation, then f1(x,t) + f2(x,t) is also a solution. Proof: and This means that light beams can pass through each other. It also means that waves can constructively or destructively interfere.**The solution to the one-dimensional wave equation**where f (u) can be any twice-differentiable function. The wave equation has the simple solution:**The 1D wave equation for light waves**where E is the light electric field We’ll use cosine- and sine-wave solutions: or where:**Waves using complex numbers**The electric field of a light wave can be written: E(x,t) = A cos(kx – wt – q) Since exp(ij) = cos(j) + i sin(j), E(x,t) can also be written: E(x,t) = Re { A exp[i(kx – wt – q)] } or E(x,t) = 1/2 A exp[i(kx – wt – q)] + c.c. where "+ c.c." means "plus the complex conjugate of everything before the plus sign." We often write these expressions without the ½, Re, or +c.c.**Waves using complex amplitudes**We can let the amplitude be complex: where we've separated the constant stuff from the rapidly changing stuff. The resulting "complex amplitude" is: So: How do you know if E0 is real or complex? Sometimes people use the "~", but not always. So always assume it's complex.**Properties of wave**• Propagation Phase Wave equation Phase velocity Group velocity • Refraction of wave • Interference of wave • Electro-magnetic (EM) wave • Spectrum of EM wave**Definitions: Amplitude and Absolute phase**E(x,t) = A cos[(k x – w t ) – q ] A = Amplitude q = Absolute phase (or initial phase)**Definitions**Spatial quantities: Temporal quantities:**The Phase Velocity**How to measure the velocity of the moving wave? First of all, measures the wavelength, secondly, count for how many wave peaks go through per second. The phase velocity is the wavelength / period: v = l / t In terms of the k-vector, k = 2p / l, and the angular frequency, w = 2p / t, this is: v = w / k**The Phase of a Wave**The phase is everything inside the cosine. E(t) = A cos(j), where j = kx – wt – q In terms of the phase, w = –¶j/¶t k = ¶j/¶x and –¶j/¶t v = ––––––– ¶j/¶x This formula is useful when the wave is really complicated.**When two waves of different frequency interfere, they**produce "beats." Indiv- idual waves Sum Envel- ope Irrad- iance:**When two light waves of different frequency interfere, they**produce beats.**Group velocity**Light wave beats (continued): Etot(x,t) = 2E0 cos(kavex–wavet) cos(Dkx–Dwt) This is a rapidly oscillating wave [cos(kavex–wavet)] with a slowly varying amplitude [2E0 cos(Dkx–Dwt)] The phase velocity comes from the rapidly varying part: v = wave / kave What about the other velocity? Define the "group velocity:" vg ºDw /Dk In general, we define the group velocity as: vgº dw /dk**Group velocity is not equal to phase velocity**if the medium is dispersive (i.e., n varies).**Calculating the Group velocity**vgº dw /dk Now, w is the same in or out of the medium, but k = k0n, where k0 is the k-vector in vacuum, and n is what depends on the medium. So it's easier to think of w as the independent variable: Using k = w n(w) / c0, calculate: dk /dw = ( n + w dn/dw ) / c0 vg = c0 / ( n + w dn/dw ) = (c0/n) / (1 + w/n dn/dw ) Finally: So the group velocity equals the phase velocity when dn/dw = 0, such as in vacuum. Otherwise, since n increases with w, dn/dw > 0, and: vg < vphase.**Calculating Group Velocity vs. Wavelength**We more often think of the refractive index in terms of wavelength,so let's write the group velocity in terms of the vacuum wavelength l0.**The group velocity is the velocity of the envelope or**irradiance: the math. The carrier wave propagates at the phase velocity. And the envelope propagates at the group velocity: Or, equivalently, the irradiance propagates at the group velocity:**The group velocity can exceed c0 whendispersion is**anomalous. vg = c0 / (n + w dn/dw ) dn/dw is negative in regions of anomalous dispersion, that is, near a resonance. So vg can exceed c0 for these frequencies! One problem is that absorption is strong in these regions. Also, dn/dw is only steep when the resonance is narrow, so only a narrow range of frequencies has vg > c0. Frequencies outside this range have vg < c0. Pulses of light (which are broadband) therefore break up into a mess.**Beating the speed of light**To exceed c, we need a region of negative dn/dw over a fairly large range of frequencies. And the slope should not vary much—to avoid pulse break-up. And absorption should be minimal. One trick is to excite the medium in advance with a laser pulse, which creates gain (instead of absorption), which inverts the curve. Then two nearby resonances have a region in between with minimal absorption and near-linear negative slope:**Negative dispersion (vg = c0 / (n + wdn/dw) and dn/dw <0)**Naturally Artificially Grating pair Optical fiber (or, some special designed waveguide) Photonic crystal Prisms in mode-locked lasers EIT**Properties of wave**• Propagation • Phase • Wave equation • Phase velocity • Group velocity Refraction of wave Interference of wave • Electro-magnetic (EM) wave • Spectrum of EM wave**An interesting question is what happensto wave when it**encounters a surface. At an oblique angle, light can be completely transmitted or completely reflected. "Total internal reflection" is the basis of optical fibers, a billion dollar industry.**Standing Waves, Beats, and Group Velocity**Superposition again Standing waves: the sum of two oppositely traveling waves Beats: the sum of two different frequencies Group velocity: the speed of information Going faster than light...**Superposition allows waves to pass through each other.**Otherwise they'd get screwed up while overlapping**Adding waves of the same frequency, but different initial**phase, yields a wave of the same frequency. This isn't so obvious using trigonometric functions, but it's easy with complex exponentials: where all initial phases are lumped into E1, E2, and E3.**Adding waves of the same frequency, but opposite direction,**yields a "standing wave." Waves propagating in opposite directions: Since we must take the real part of the field, this becomes: (taking E0 to be real) Standing waves are important inside lasers, where beams are constantly bouncing back and forth.**A Standing Wave: Experiment**Mirror 3.9 GHz microwaves Input beam The same effect occurs in lasers. Note the node at the reflector at left.**Interfering spherical waves also yield a standing wave**Antinodes**Two Point Sources**Different separations. Note the different node patterns.**When two waves of different frequency interfere, they**produce beats. Take E0 to be real.**Young’s Two-Slit Experiment**What happens when light passes through two slits? Light pattern that emerges “fringes” The idea is central to many laser techniques, such as holography, ultrafast photography, and acousto-optic modulators. Tests of quantum mechanics also use it.**Diffraction**Light bends around corners. This is called diffraction. Light patterns after passing through rectangular slit(s): One slit: Two slits: The diffraction pattern far away is the Fourier transform of the slit transmission vs. position.**Fourier decomposing functions plays a big role in optics.**Here, we write a square wave as a sum of sine waves of different frequency.**The Fourier transform is perhaps one of the most important**equation in optics. It converts a function of time to one of frequency: and converting back uses almost the same formula:**Light electric field**Time What do we hope to achieve with theFourier Transform? We desire a measure of the frequencies present in a wave. This will lead to a definition of the term, the “spectrum.” Plane waves have only one frequency, w. This light wave has many frequencies. And the frequency increases in time (from red to blue). It will be nice if our measure also tells us when each frequency occurs.**Properties of wave**• Propagation • Phase • Wave equation • Phase velocity • Group velocity • Refraction of wave • Interference of wave Electro-magnetic (EM) wave Spectrum of EM wave**“Light is just electromagnetic wave”**• Review of Maxwell equations • The solutions which is convenient for optics • EM wave spectrum**The equations of optics are Maxwell’s equations.**where is the electric field, is the magnetic field, r is the charge density, e is the permittivity, and m is the permeability of the medium.**Longitudinal vs. Transverse waves**Motion is along the direction of Propagation Longitudinal: Motion is transverse to the direction of Propagation Transverse: Space has 3 dimensions, of which 2 directions are transverse to the propagation direction, so there are 2 transverse waves in ad- dition to the potential longitudinal one.**Vector fields**Light is a 3D vector field. A 3D vector field assigns a 3D vector (i.e., an arrow having both direction and length) to each point in 3D space.