EM waves and interference Review of EM wave equation and plane waves Energy and intensity in EM waves Interference
Maxwell's Equations to wave eqn The induced polarization, P, contains the effect of the medium: E = 0 E = B B = 0 Take the curl: E ( ) = t Use the vector ID: t B = E c t + µ 0 B = t ( ) = B( A C) C( A B) A B C P t E c t + µ 0 E ( ) = E ( ) ( )E = E E c E t = µ 0 P t P t Inhomogeneous Wave Equation
Wave equation in a medium The induced polarization, P, contains the effect of the medium: E c E t = µ 0 P t Sinusoidal waves of all frequencies are solutions to the wave equation The polarization (P) is sometimes a driving term for the waves (esp in nonlinear optics). In this case the polarization determines which frequencies will occur. For linear response, P will oscillate at the same frequency as the input. P( E) = ε 0 χe In nonlinear optics, the induced polarization is more complicated: ( ) P( E) = ε 0 χ () E + χ () E + χ (3) E 3 +... The extra nonlinear terms can lead to new frequencies.
Solving the wave equation: linear induced polarization For low irradiances, the polarization is proportional to the incident field: P( E) = ε 0 χe The displacement vector combines the effect of E and P D = ε 0 E + P = ε ( 0 + χ )E = ε E = n E In this simple (and most common) case, the wave equation becomes: E c E t = c χ E t E n c E t = 0 Using: ε 0 µ 0 = / c ε 0 ( + χ ) = ε = n D does not always point in the same direction as E: Birefringence: n is a function of linear polarization direction Optically activity: n is a function of circular polarization state In this case, χ and ε are tensors operating on E to change its direction
Vector wave equation The EM wave equation in vector form E is a vector that depends on r and t Wave equation is actually a set of 3 equations: E x E y E z! E n c E t = 0 ( r,t) n c t E x ( r,t) n c t E y ( r,t) n c t E z ( r,t) = 0 ( r,t) = 0 ( r,t) = 0
Plane wave solutions for the wave equation If we assume the solution has no dependence on x or y: E( z,t) = x E ( z,t ) + y E ( z,t ) + z E ( z,t ) = z E ( z,t ) E z n c E t = 0 The solutions are oscillating functions, for example ( ) = ˆx E x cos k z z ωt E z,t ( ) Where ω = k c, k = πn / λ, v ph = c / n This is a linearly polarized wave.
Complex notation for waves Write cosine in terms of exponential E( z,t) = ˆx E x cos( kz ωt +φ) = ˆx E x Note E-field is a real quantity. It is convenient to work with just one part We will use Svelto: E 0 e Then take the real part. ( ) +i kz ωt E 0 e ( ) i kz ωt ( ( ) + e i( kz ωt+φ ) ) ei kz ωt+φ No factor of In nonlinear optics, we have to explicitly include conjugate term
Wave energy and intensity Both E and H fields have a corresponding energy density (J/m 3 ) For static fields (e.g. in capacitors) the energy density can be calculated through the work done to set up the field ρ = εe + µh Some work is required to polarize the medium Energy is contained in both fields, but H field can be calculated from E field
Calculating H from E in a plane wave Assume a non-magnetic medium ( ) = ˆx E x cos( kz ωt) E z,t! E = B t = µ 0 H t Can see H is perpendicular to E H µ 0 =! ˆx ŷ ẑ E = x y z t E x 0 0 Integrate on time to get H-field: H = ŷ k E x sin( k z ω t)dt = ŷ k E cos k z ωt x µ 0 µ 0 ω = ŷ z E x = ŷke x sin( kz ωt) ( ) = ŷ k E x cos k z ωt ω µ 0 ( )
H field from E field H field for a propagating wave is in phase with E- field E( z,t) = ˆx E x cos( k z ω t) H( z,t) = ŷ k E x cos( k z ωt) ω µ 0 Amplitudes are not independent H y = H y = k ω µ 0 E x k = n ω c n cµ 0 E x = nε 0 ce x c = µ 0 ε 0 µ 0 c = ε 0 c
Energy density in a traveling EM wave Back to energy density, non-magnetic ρ = εe + µ 0 H ε = ε 0 n ρ = ε 0 n E + µ 0 n ε 0 c E µ 0 ε 0 c = ρ = ε 0 n E + ε 0 n E = ε 0 n E 0 cos Equal energy in both components of wave H = nε 0 ce ( kz ωt)
Cycle-averaged energy density Optical oscillations are faster than detectors Average over one cycle: ρ = ε 0 n T E 0 cos ( k z z ωt)dt T 0 Graphically, we can see this should = ½.0 0.8 0.6 0.4 k z = 0 k z = π/4 0. 0.5.0.5.0 Regardless of position z t/t ρ = ε 0 n E 0
Intensity and the Poynting vector Intensity is an energy flux (J/s/cm ) In EM the Poynting vector give energy flux S = E H For our plane wave, S = E H = E 0 cos( k z z ωt)nε 0 ce 0 cos k z z ωt S = nε 0 ce 0 cos ( k z z ωt)ẑ S is along k Time average: S = nε 0 ce 0ẑ Intensity is the magnitude of S ( ) ˆx ŷ I = nε ce = c 0 0 n ρ = V ρ phase Photon flux: F = I hν
General 3D plane wave solution Assume separable function E(x, y,z,t) ~ f ( x) f y E( z,t) = x E z,t ( ) f 3 z ( ) + ( )g t y E z,t E(x, y,z,t) = E 0 e i( k r ωt ) Now k-vector can point in arbitrary direction With this solution in W.E.: ( ) ( ) + z E z,t ( ) = n ( ) c t E z,t Solution takes the form: E(x, y,z,t) = E 0 e ikxx e ikyy e ikzz e iωt = E 0 e i ( k xx+k y y+k z z) e iωt n ω c = k x + k y + k z = k k Valid even in waveguides and resonators
Grad and curl of 3D plane waves Simple trick: E = x E x + y E y + z E z For a plane wave, E = i k x E x + k y E y + k z E z Similarly, Consequence: since ( ) = i k E ( ) E = i k E E = 0, ( ) k E For a given k direction, E lies in a plane E.g. x and y linear polarization for a wave propagating in z direction
Writing electric field expressions Write an expression for a complex E-field as shown: y E 0 θ z E( y,z,t ) = E [ 0 ŷcosθ ẑsinθ ]e i( kysinθ+kzcosθ ωt)
Interference: ray and wave pictures Interference results from the sum of two waves with different phase: E tot ( Δφ) = E e ikz + E e ikz+δφ We measure intensity, which leads to interference ( Δφ) E e ikz + E e ikz+δφ = E + E e iδφ I tot = I + I + I I e iδφ + I I e iδφ = I + I + I I cos( Δφ) For the case where I = I, I tot ( Δφ) = I + cos Δφ ( ) ( ) ( ) = 4I cos Δφ / How to generate, calculate phase difference?
The Fizeau Wedge Interferometer The Fizeau wedge yields a complex pattern of variable-width fringes, but it can be used to measure the wavelength of a laser beam.
Fizeau wedge calculation Interference between reflections from internal surfaces α L Angle is very small, neglect change in direction Path difference: Δl = L sinα Lα Phase difference: Δφ = ω L n Δl π c λ α n = Interference: I tot ( Δφ) = I + I + I I cos Δφ ( ) One fringe from one max to the next, so maxima are at Δφ = π m In this interferometer, minimum path = 0, we can measure absolute wavelength: Δφ = π m = π L λ α λ = L m α
Newton's Rings
Newton's Rings Get constructive interference when an integral number of half wavelengths occur between the two surfaces (that is, when an integral number of full wavelengths occur between the path of the transmitted beam and the twice reflected beam). This effect also causes the colors in bubbles and oil films on puddles.
Curved wavefronts Rays are directed normal to surfaces of constant phase These surfaces are the wavefronts Radius of curvature is approximately at the focal point Spherical waves are solutions to the wave equation (away from r = 0) E + n ω E = 0 E Scalar r ( c r ei ± kr ωt ) + outward I - inward r
Paraxial approximations For rays, paraxial = small angle to optical axis Ray slope: tanθ θ For spherical waves where power is directed forward: e i k r = exp ik x + y + z k x + y + z = k z + x + y k z + x + y z z e i k r ω t ( ) e i k z exp i k x + y z ω t Wavefront = surface of constant phase For x, y >0, t must increase. Wave is diverging: Expanding to st order z is radius of curvature = R φ = 0 k x + y R = ω t
Newton s rings: interfere plane and spherical waves Add two fields: E ( r ) = E0 + E0 e i k r R Assume equal amplitude For Newton s rings, x phase shift Calculate intensity: I ( r ) E0 + E0 e k r i R k r = E0 + E0 cos R Local k increases with r kr cos r = cos ( klocal r ) R
Shearing interferometer Combine two waves with a lateral offset ( shear ) x s E tot ( x) = E 0 exp i +E 0 exp i k R k R (( x x s ) + y ) (( x + x s ) + y ) I tot ( x) = I 0 + exp i ( x x s ) x + x s k( x x s ) R k ( x + x s ) R + c.c. ( ) = x xx s + x s x + xx s + x s ( ) = 4xx s I tot ( x) = I 0 + cos kx s R x Fringes are straight, equally spaced Combine with constant tilt in y direction: leads to fringe rotation with divergence
Tilted window: ray propagation Calculate phase shift caused by the insertion of the window into an interferometer. Ray optics: Add up optical path for each segment Subtract optical path w/o window Δd = nl AB + L BC L A B L L AB = B C L w L A B cosθ = L w cosθ L BC = L B C + L B B sinθ Use Snell s Law to reduce to: θ A L w θ C B C Reference line B z Δd = n L w cosθ L w cosθ n n n
Tilted window: wave propagation Write expression for tilted plane wave ( ) = E 0 exp i k x x + k z z E x,z ( ) = E exp i ω 0 c n xsinθ + z cosθ ( ) Snell s Law: phase across surfaces is conserved k x x = ω c nsinθ is constant Δφ = ( k cosθ )L w ( k cosθ )L w B C L w C This approach can be used to calculate phase of prism A θ B z Reference line pairs and grating pairs θ n n n
Multiple-beam interference: The Fabry-Perot Interferometer or Etalon A Fabry-Perot interferometer is a pair of parallel surfaces that reflect beams back and forth. An etalon is a type of Fabry-Perot etalon, and is a piece of glass with parallel sides. The transmitted wave is an infinite series of multiply reflected beams.
Multiple-beam interference: general formulation Incident wave: E 0 Reflected wave: E 0r Reflected wave: n = t t e iδ / E 0 Transmitted wave: E t t 0t eiδ / r n n = ( ) e iδ E 0 r, t = reflection, transmission coefficients from air to glass r, t = from glass to air δ = round-trip phase delay inside medium = k 0 ( n L cos θ t ) Transmitted wave: E 0t = t t e iδ / E 0 + r E 0r = re 0 + t t r e iδ E 0 + t t r r ( ( ) e iδ + ( ( r ) e iδ ) + ( ( r ) e iδ ) 3 ) + (( ) e iδ ) E 0 + t t e iδ / r t t e iδ / r (( ) e iδ (( ) e iδ ) E 0 )3 E 0
Stokes Relations for reflection and transmission = Time reversal: Same amplitudes, reversed propagation direction E oi = ( E oi r)r + ( E oi t) t t t = r ( E oi t) r + ( E oi r)t = 0 r = r Notes: relations apply to angles connected by Snell s Law true for any polarization, but not TIR convention for which interface experiences a sign change can vary
Fabry-Perot transmission The transmitted wave field is: E 0t = t t e iδ / E 0 + r r ( ( ) e iδ + ( ( ) e iδ ) + ( ( r ) e iδ ) 3 ) + Stokes relations r = r r = r t t = r = t t e iδ / E 0 E 0t = t t / eiδ r e E iδ 0 Power transmittance: ( + r e iδ + ( r e iδ ) + ( r e iδ ) 3 ) + = t t eiδ / E 0 ( t t ) T E 0t E 0 = t t / eiδ r e iδ = ( t t ) ( r e +iδ )( r e iδ ) = {+ r 4 r cosδ} = ( r ) {+ r 4 r [ sin (δ / )]} = ( r ) { r + r 4 + 4r sin (δ / )]} Dividing numerator and denominator by T = ( ) + F sin δ / Where: ( r ) n=0 ( x) = + x + x + x 3 +! where: ( r e iδ ) n r F = r
Multiple-beam interference: simple limits Reflected waves Full transmission: sin( ) = 0, d = π m Minimum transmission: sin( ) =, d = π (m+/) T = Destructive interference for reflected wave ( ) + F sin δ / } internal st reflection reflections Constructive interference for reflected wave
Etalon transmittance vs. thickness, wavelength, or angle T = The transmittance varies significantly with thickness or wavelength. We can also vary the incidence angle, which also affects δ. As the reflectance of each surface (R=r ) approaches, the widths of the high-transmission regions become very narrow. ( ) + F sin δ / Transmission max: sin( ) = 0, d = π m δ = ω c n L cos [ θ t ] = πm At normal incidence: λ m = n L n L = m λ m or m
The Etalon Free Spectral Range The Free Spectral Range is the wavelength range between transmission maxima. λ FSR = Free Spectral Range λ FSR For neighboring orders: 4πnL λ 4πnL λ = π λ λ = nl = λ λ λ λ λ λ = λ λ λ λ FSR λ FSR λ nl λ FSR λ = λ nl = ν FSR ν ν FSR c nl /(round trip time)
Etalon Linewidth The Linewidth δlw is a transmittance peak's full-width-half-max (FWHM). T = A maximum is where δ / mπ + δ / and sin δ / Under these conditions (near resonance), T = + F δ / 4 This is a Lorentzian profile, with FWHM at: F 4 δ LW ( ) + F sin δ / = δ LW 4 / F ( ) δ / This transmission linewidth corresponds to the minimum resolvable wavelength.
Etalon Finesse resolution The Finesse, I, is the ratio of the Free Spectral Range and the Linewidth: δ = π corresponds to one FSR I δ δ FSR FW = π π F = 4 / F Using: r F = r I = π r taking r The Finesse is the number of wavelengths the interferometer can resolve.
Tools: scanning Fabry-Perot Resonator with piezo control over mirror separation http://www.thorlabs.us/newgrouppage9.cfm?objectgroup_id=859 Wavelength range: 535-80nm (ours) SA00 (ours) FSR.5 GHz Finesse > 00 Resolution 7.5MHz SA0 FSR 0 GHz Finesse > 50 Resolution 67MHz
Tools: fixed plate Fabry-Perot
Multilayer coatings Typical laser mirrors and camera lenses use many layers. The reflectance and transmittance can be custom designed
Multilayer thin-films: wave/matrix treatment Use boundary conditions to relate fields at the boundaries Phase shifts connect fields just after I to fields just before II Express this relation as a transfer matrix Multiply matrices for multiple layers
High-reflector design 0 periods periods Reflectivity can reach > 99.99% at a specific wavelength > 99.5% for over 50nm Bandwidth and reflectivity are better for S polarization.
Interference filter design Transmission % A thin layer is sandwiched between two high reflector coatings - very large free spectral range, high finesse - typically 5-0nm bandwidth, available throughout UV to IR
Geometric optics Approaches: Paraxial calculations (assume small angle to optical axis) Imaging equations: simple lens, Lensmakers for thick lens ABCD matrices for optical system Non-paraxial, general case Analytic calculation of aberrations: Zernike polynomials, Taylor-type expansions of wavefront error Computer tracing (no approximations). Example: Zemax, Oslo Design procedure Find existing design close to what could work for application Paraxial trace with ray diagram Calculate magnification, limiting apertures Optimize with ABCD matrices or computer program Analyze aberrations
Geometric optics: lenses For a nice summary, see Lens (optics) Wikipedia page https://en.wikipedia.org/wiki/lens_(optics)
Geometric optics: imaging equation f = s + s s, s are positive as shown s, s = infinity: rays are collimated M = s s Magnification calculate with similar triangles M < 0 inverted image M = - when s = s = f, : imaging
Geometric optics: virtual images s, s < 0 if on the opposite side of lens virtual image: position where image seems to come from Concave lens: f < 0 Same convention for s, s
Raytracing: single curved interface Snell: n sin( θ i ) = n sin( θ r ) θ i y p = tan(θ i φ) θ i φ y q = tan(φ θ r ) φ θ r y R = sin(φ) φ p θ i φ 0 y θ r φ R φ θ r q n n n = sin θ i n sin θ r ( ) ( ) θ i = φ + y p θ r φ y q = y R + y p y R y q = R + p R q n R ( q ) = n R + p ( ) R n n In paraxial appx, y s cancel ( ) = n q + n p
Raytracing: two curved interfaces Eqn from st: - add second interface: R > 0 if center is to right - assume y =y ( n n ) = n R q + n p Adapt to nd interface: n n q q p q ( ) = n R n n q n q y p 0 R q q Solve eqn for image distance n q = R n n R n n ( ) = n ( ) n q ( n n )+ n R p n = R R n q + p p n n n n = R R n f f = s o + s i Focal length (lensmaker s eqn) Imaging equation
Aberrations For non-paraxial rays errors in how rays focus Spherical aberration Coma: tilted lens, off-axis points
Lens systems It is possible to cascade the lens imaging equation for multiple lenses: Image from lens is object for lens s s s 3 =d s s 4 For more complicated systems, use a matrix method: ABCD matrices Also good for resonators Does not account for aberrations d
ABCD ray matrices Formalism to propagate rays through optical systems Keep track of ray height r and ray angle θ = dr/dz = r Treat this pair as a vector: r r Optical system will modify both the ray height and angle, e.g. r A B r = r C D r Successive ABCD matrices multiply from the left Translation r = r + L r L r = r 0 r r
Translation: Flat interface r = r Window: calculate matrix L Refraction in ABCD L 0 n sinθ = n sinθ n θ n θ r = n r n 0 0 n / n 0 0 n = 0 0 n L 0 Special case: n =, n = n L / n 0 / n n n 0 0 / n 0 0 / n = L / n 0 Effective thickness reduced by n
Curved surfaces in ABCD Thin lens: matrix computes transition from one side of lens to other r = r r = r / p p 0 q r = r / q n n n q = f p r = r f p = r f + r Spherical interface: radius R lens A C B D 0 / f 0 n n n R n n
3D wave propagation E n j c t E = z E + E n(r) c t E = 0 Note: = x + y = r r( r r ) + r φ All linear propagation effects are included in LHS: diffraction, interference, focusing Previously, we assumed plane waves where transverse derivatives are zero. More general examples: Gaussian beams (including high-order) Waveguides Arbitrary propagation Can determine discrete solutions to linear equation (e.g. Gaussian modes, waveguide modes), then express fields in terms of those solutions.
Paraxial, slowly-varying approximations Assume waves are forward-propagating: E( r,t) = A( r)e i ( k z ω 0 t) + c.c. Refractive index is isotropic z A + ik z A k A + A + n ω 0 A = 0 c Fast oscillating carrier terms cancel (blue) Slowly-varying envelope: compare red terms Drop nd order deriv if π λ L A L A This ignores: Changes in z as fast as the wavlength Counterpropagating waves
Gaussian beam solutions to wave equation Without any source term, paraxial equation is ik z A + A = 0 Gaussian beam solutions can be written as: A( r,z) = A 0 + iξ e r ( ) ξ = z z z R = n πw 0 R λ = k w 0 w 0 +iξ Rayleigh range w ( ) = A 0 0 w( z) ei k z ω t E r, z,t ( ) e r w z ( ) e i k r R( z ) e iη( z)
Standard form of Gaussian beam equations w ( ) = A 0 0 w( z) ei k z ω t E r, z,t w( z) = w 0 + z z R ( ) e r w z ( ) e i k r R( z ) e iη( z) ( ) = z + z R R z z 0 0-0 -5 5 0-0 R(z) z/z R 0. -0-5 5 0-0. -0-0.4.5 0.4 /R(z) z/z R.0 Gouy phase η z ( ) = arctan z z R 0.5-0 -5 5 0-0.5 -.0 z/z R -.5
Complex q form for Gaussian beam This combines beam size and radius of curvature into one complex parameter q z This form is used for for ABCD calculations A( r,z) = A 0 ( ) = R( z) i λ π w ( z) q( z) = = z + i z R = R z = R z + iξ e ( ) i w 0 z R w r ( ) w 0 +iξ z z + z R i ( z) A r,z z R z + z R ( ) i λ π w ( z) = R( z) i Z ( z) ( ) = k q( z) e i R z w ( ) = z r q z ( ) z + z R z ( ) = w 0 + z z R = z z + z R = z R ( ) w 0 z + z R
Complex q vs standard form ( ) = k q( z) e i u r,z r q z ( ) Expand exponential: r exp i k = exp i k r q( z) = exp i k r R z ( ) i π λ with ( ) = R( z) i λ π w ( z) q z R z Expand leading inverse q: q( z) = i i z z + z + z R R z + z R = i ( ) i λ π w ( z) r i λ π w ( z) = e i k z + z R z + z R = i e i arctan ( z/z R ) w z R + z = 0 / z R i z R w z ( ) eiη z r R z e i arctan z/z R ( ) ( ) e r w ( z) ( ) a + ib = a + b i arctan b/a e ( )
( ) = k q( z) e i u r,z E r, z,t Difference between Siegman s complex q and standard form r q z ( ) = w 0 i z R w z w ( ) = A 0 0 w( z) ei k z ω t ( ) eiη z ( ) e r w z ( ) e i k r R( z ) e r w ( z) ( ) e i k r R( z ) e iη( z) Siegman s form for the complex q is used almost everywhere for the ABCD calculations. He uses the exp[+ I w t] convention, which accounts for the sign difference in the complex exponentials. With exp[-i w t] convention, define q as: q z ( ) = R( z) + i λ π w ( z) = z i z R
Compare Boyd s form to standard: Boyd s complex form is consistent with standard Gaussian beam form A( r,z) = A 0 + iξ e + iξ = = + i z / z R A( r,z) = A 0 ( i z R ) r ( ) = A0 e + i z / z R w 0 +iξ z R z R + i z = i z R = i z R z i z R q z q z ( ) e+ i z R r w 0 q z ( ) ( ) = i z R A 0 q z r ( ) w 0 +i z/z R ( ) e+ i k r q z ( )