Document 6503875
Transcription
Document 6503875
Spatio-temporal characteristics of light and how to model them Spatio-temporal distortions Ray matrices The Gaussian beam Complex q and its propagation Ray-pulse “Kostenbauder” matrices The prism pulse compressor Optical system ↔ 4x4 Ray-pulse matrix ⎡ xin ⎤ ⎢θ ⎥ ⎢ in ⎥ ⎢ tin ⎥ ⎢ ⎥ ⎢⎣ν in ⎥⎦ ⎡A ⎢C ⎢ ⎢G ⎢ ⎣0 B 0 D 0 H 1 0 0 E⎤ F ⎥⎥ I⎥ ⎥ 1⎦ Gaussian beam in space and time and the complex Q matrix ⎡ xout ⎤ ⎢θ ⎥ ⎢ out ⎥ ⎢ tout ⎥ ⎢ ⎥ ⎣⎢ν out ⎦⎥ Spatio-temporal distortions Ordinarily, we assume that the pulse-field spatial and temporal factors (or their Fourier-domain equivalents) separate: E ( x, y, z , t ) = Exyz ( x, y, z ) Et (t ) Eˆ% (k x , k y , k z , ω ) = Eˆ xyz (k x , k y , k z ) E% t (ω ) where the tilde and hat mean FTs with respect to t and x, y, z In pulses with spatio-temporal distortions, they don’t: We’ll use dt0 coordinates xt : E ( x, y, z, t ) → E[ x, y, z, t + ( x − x0 )] such as x – x0, dx that are (small) dx0 deviations from % % xω : E ( x, y, z , ω ) → E[ x + (ω − ω0 ), y, z, ω ] their means. dω dk x 0 ˆ% ˆ% kω : E (k x , k y , k z , ω ) → E0 [k x + (ω − ω0 ), k y , k z , ω ] dω Spatio-temporal distortions can be useful or inconvenient. Good: They allow pulse compression. They help to measure pulses (tilted pulse fronts). They allow pulse shaping. They can increase bandwidth in nonlinear-optical processes. Bad: They usually increase the pulse length. They reduce intensity. They can be hard to measure. Angular dispersion is an example of a spatio-temporal distortion. dk x 0 dω In the presence of angular dispersion, the off-axis k-vector component kx depends on ω: dk x 0 ˆ ˆ % % (ω − ω0 ), k y , k z , ω ] E (k x , k y , k z , ω ) = E0 [k x + dω where kx0(ω) is the mean kx vs. frequency ω. x z Input pulse Prism Angularly dispersed output pulse Note that the mean off-axis k-vector component, kx0, depends on ω. Spatial chirp is a spatio-temporal distortion in which the color varies spatially across the beam. Propagation through a prism pair produces a beam with no angular dispersion, but with spatial dispersion, often called spatial chirp. Prism pair Input pulse Spatially chirped output pulse Prism pairs are inside nearly every ultrafast laser. A third and fourth prism undo this distortion, but must be aligned carefully. Spatial chirp is difficult to avoid. Simply propagating through a tilted window causes spatial chirp! Tilted window Input pulse Spatially chirped output pulse Because ultrashort pulses are so broadband, this distortion is very noticeable—and often problematic! How to think about ∂x0 spatial chirp ∂ω x x0(ω9) x0(ω8) x0(ω7) x0(ω6) x0(ω5) x0(ω4) x0(ω3) x0(ω2) x0(ω1) z Suppose we send the pulse through a set of monochromatic filters and find the beam center position, x0, for each frequency, ω. E% ( x, y, z, ω ) → dx0 % E[ x + (ω − ω0 ), y, z , ω ] dω where x0 is the center of the beam component of frequency ω. Pulse-front tilt is another common spatio-temporal distortion. Phase fronts are perpendicular to the direction of propagation. Because the group velocity is usually less than phase velocity, pulse fronts tilt when light traverses a prism. Undistorted input pulse Angularly dispersed pulse with pulse-front tilt Prism Angular dispersion causes pulse-front tilt. Angular dispersion causes pulse-front tilt even when group velocity is not involved. Diffraction gratings also yield pulse-front tilt. Angularly dispersed pulse with pulsefront tilt Undistorted input pulse Diffraction grating The path is simply shorter for rays that impinge on the near side of the grating. Of course, angular dispersion and spatial chirp occur, too. Gratings have about ten times the dispersion of prisms, and they yield about ten times the tilt. Modeling pulse-front tilt dt0 dx Pulse-front tilt involves coupling between the space and time domains: dt0 E ( x, y, z, t ) → E[ x, y, z, t + ( x − x0 )] dx For a given transverse position in the beam, x, the pulse mean time, t0, varies in the presence of pulse-front tilt. Pulse-front tilt occurs after pulse compressors that aren’t aligned properly. Angular dispersion always causes pulsefront tilt! Angular dispersion means that the off-axis k-vector depends on ω: ˆ ˆ E% (k x , k y , k z , ω ) = E% 0 [k x + γ (ω − ω0 ), k y , k z , ω ] where γ = dkx0 /dω Inverse Fourier-transforming with respect to kx, ky, and kz yields: i γ (ω −ω0 ) x % % E ( x, y, z , ω ) = E0 ( x, y, z , ω ) e using the shift theorem Inverse Fourier-transforming with respect to ω−ω0 yields: ⇒ E ( x, y, z , t ) = E0 ( x, y, z , t + γ x) using the shift theorem again which is just pulse-front tilt! The combination of spatial and temporal chirp also causes pulse-front tilt. The theorem we just proved assumed no spatial chirp, however. So it neglects another contribution to the pulse-front tilt. Dispersive medium Spatially chirped input pulse Spatially chirped pulse with pulse-front tilt, but no angular dispersion vg(red) > vg(blue) The total pulse-front tilt is the sum of that due to dispersion and that due to this effect. Xun Gu, Selcuk Akturk, and Erik Zeek A pulse with temporal chirp, spatial chirp, and pulse-front tilt. Suppressing the y-dependence, we can plot such a pulse: 4.5 mrad ψ = 11.3 mrad 797 803 nm nm where the pulsefront tilt angle is: ∂t 0 ψ ≡c ∂x xx [[m mm m] ] ] tt [[ffss] 775 nm 777 nm We’ll need a nice formalism for calculating these distortions! Ray Optics axis We'll define light rays as directions in space, corresponding, roughly, to k-vectors of light waves. Each optical system will have an axis, and all light rays will be assumed to propagate at small angles to it. This is called the Paraxial Approximation. The Optic Axis A mirror deflects the optic axis into a new direction. This “ring laser” has an optic axis that scans out a rectangle. Optic axis A ray propagating through this system We define all rays relative to the relevant optic axis. The Ray Vector xin, θin xout, θout A light ray can be defined by two co-ordinates: its position, x ray l a c i t op its slope, θ θ x Optical axis These parameters define a ray vector, which will change with distance and as the ray propagates through optics. ⎡x⎤ ⎢θ ⎥ ⎣ ⎦ Ray Matrices For many optical components, we can define 2 x 2 ray matrices. An element’s effect on a ray is found by multiplying its ray vector. Ray matrices can describe simple and complex systems. Optical system ↔ 2 x 2 Ray matrix ⎡ xin ⎤ ⎢θ ⎥ ⎣ in ⎦ ⎡A ⎢C ⎣ B⎤ D ⎥⎦ ⎡ xout ⎤ ⎢θ ⎥ ⎣ out ⎦ These matrices are often called "ABCD Matrices." Ray matrices as derivatives Since the displacements and angles are assumed to be small, we can think in terms of partial derivatives. spatial magnification ∂xout ∂xin xout ∂xout ∂xout = xin + θin ∂xin ∂θin θout ∂θout ∂θout = xin + θin ∂xin ∂θin ∂xout ∂θin ⎡ xout ⎤ ⎡ A B ⎤ ⎡ xin ⎤ ⎢θ ⎥ = ⎢C D ⎥ ⎢θ ⎥ ⎦ ⎣ in ⎦ ⎣ out ⎦ ⎣ ∂θout ∂xin ∂θout ∂θin angular magnification We can write these equations in matrix form. For cascaded elements, we simply multiply ray matrices. ⎡ xin ⎤ ⎢θ ⎥ ⎣ in ⎦ O2 O1 ⎡ xout ⎤ ⎪⎧ ⎢θ ⎥ = O3 ⎨O2 ⎣ out ⎦ ⎪⎩ ⎛ ⎜ O1 ⎝ O3 ⎡ xout ⎤ ⎢θ ⎥ ⎣ out ⎦ ⎡ xin ⎤ ⎞ ⎪⎫ ⎡ xin ⎤ ⎢θ ⎥ ⎟ ⎬ = O3 O2 O1 ⎢θ ⎥ ⎣ in ⎦ ⎠ ⎪⎭ ⎣ in ⎦ Notice that the order looks opposite to what it should be, but it makes sense when you think about it. Ray matrix for free space or a medium If xin and θin are the position and slope upon entering, let xout and θout be the position and slope after propagating from z = 0 to z. xout θout xin, θin xout = xin + z θin θ out = θin Rewriting these expressions in matrix notation: z z=0 Ospace ⎡1 z ⎤ = ⎢ ⎥ 0 1 ⎣ ⎦ ⎡xout ⎤ ⎡1 z⎤ ⎡xin ⎤ ⎢θ ⎥ = ⎢0 1⎥ ⎢θ ⎥ ⎣ ⎦ ⎣ in ⎦ ⎣ out ⎦ Ray Matrix for an Interface At the interface, clearly: θin xout = xin. xin xout n1 Now calculate θout. Snell's Law says: θout n2 n1 sin(θin) = n2 sin(θout) which becomes for small angles: n1 θin = n2 θout ⇒ θout = [n1 / n2] θin Ointerface 0 ⎤ ⎡1 =⎢ ⎥ n n 0 / ⎣ 1 2⎦ Ray matrix for a curved interface At the interface, again: xout = xin. To calculate θout, we must calculate θ1 and θ2. If θs is the surface-normal slope at the height xin, then θs θin R θ2 θ1 xin = xout n1 n2 θout θs z θ1 = θin+ θs and θ2 = θout+ θs If R is the surface radius of curvature, the surface z coordinate will be: z = R − R 2 − xin2 = R − R 1 − ( xin / R) 2 ≈ R − R ⎡⎣1 − 12 ( xin / R) 2 ⎤⎦ = 12 ( xin2 / R) dz ⇒ θs ≈ ≈ xin / R dxin Ray matrix for a curved interface (cont’d) θ1 = θin+ xin / R and θ2 = θout+ xin / R Snell's Law: n1 θ1 = n2 θ2 θ1 θ2 ⇒ n1 (θin + xin / R) ≈ n2 (θ out + xin / R) ⇒ θ out ≈ (n1 / n2 )(θin + xin / R) − xin / R n1 ⇒ θ out ≈ (n1 / n2 )θin + (n1 / n2 − 1) xin / R Ocurved interface 1 0 ⎤ ⎡ =⎢ ⎥ n n R n n − ( / 1) / / ⎣ 1 2 1 2⎦ Now the output angle depends on the input position, too. n2 A thin lens is just two curved interfaces. We’ll neglect the glass in between (it’s a really thin lens!), and we’ll take n1 = 1. Ocurved interface 1 0 ⎤ ⎡ =⎢ ⎥ ( / 1) / / n n − R n n 1 2⎦ ⎣ 1 2 R1 n=1 R2 n≠1 n=1 1 0⎤ ⎡ 1 0 ⎤ ⎡ Othin lens = Ocurved Ocurved = ⎢ ⎥ ⎢ (1/ n − 1) / R 1/ n ⎥ ( 1) / − n R n interface 2 interface 1 2 1 ⎣ ⎦⎣ ⎦ 1 0 ⎤ ⎡ 1 0⎤ ⎡ =⎢ =⎢ ⎥ ⎥ ( n 1) / R n (1/ n 1) / R n (1/ n ) ( n 1) / R (1 n ) / R 1 − + − − + − ⎣ 2 1 ⎦ ⎣ 2 1 ⎦ 1 0⎤ ⎡ =⎢ ⎥ ( n 1)(1/ R 1/ R ) 1 − − ⎣ 2 1 ⎦ This can be written: where: 1/ f = (n − 1)(1/ R1 − 1/ R2 ) ⎡ 1 ⎢ −1/ f ⎣ 0⎤ 1 ⎥⎦ The Lens-Maker’s Formula Ray matrix for a lens Olens 1/ f = (n − 1)(1/ R1 − 1/ R2 ) ⎡ 1 = ⎢ ⎣-1/f 0⎤ ⎥ 1⎦ The quantity, f, is the focal length of the lens. It’s the single most important parameter of a lens. It can be positive or negative. In a homework problem, you’ll extend the Lens Maker’s Formula to lenses of greater thickness. R1 > 0 R2 < 0 f>0 If f > 0, the lens deflects rays toward the axis. R1 < 0 R2 > 0 f<0 If f < 0, the lens deflects rays away from the axis. A lens focuses parallel rays to a point one focal length away. For all rays A lens followed by propagation by one focal length: xout = 0! f ⎤⎡xin ⎤ ⎡ 0 ⎤ ⎡xout ⎤ ⎡1 f ⎤⎡ 1 0⎤⎡xin ⎤ ⎡ 0 ⎢θ ⎥ = ⎢0 1⎥⎢−1/ f 1⎥⎢ 0 ⎥ = ⎢−1/ f 1⎥⎢ 0 ⎥ = ⎢−x / f ⎥ ⎦⎣ ⎦⎣ ⎦ ⎣ ⎦⎣ ⎦ ⎣ in ⎦ ⎣ out ⎦ ⎣ f f Assume all input rays have θin = 0 At the focal plane, all rays converge to the z axis (xout = 0) independent of input position. Parallel rays at a different angle focus at a different xout. Types of lenses Lens nomenclature Which type of lens to use (and how to orient it) depends on the aberrations and application. Ray Matrix for a Curved Mirror Consider a mirror with radius of curvature, R, with its optic axis perpendicular to the mirror: θ1 = θin − θ s θ s ≈ xin / R R θout θs θin θ out = θ1 − θ s = (θin − θ s ) − θ s ≈ θin − 2 xin / R θ1 θ1 xin = xout z 0⎤ ⎡ 1 ⇒ Omirror = ⎢ ⎥ − 2 / 1 R ⎣ ⎦ Like a lens, a curved mirror will focus a beam. Its focal length is R/2. Note that a flat mirror has R = ∞ and hence an identity ray matrix. Laser Cavities Mirror curvatures matter in lasers. Two flat mirrors, the “flat-flat” laser cavity, is difficult to align and maintain aligned. Two concave curved mirrors, the “stable” laser cavity, is easy to align and maintain aligned. Two convex mirrors, the “unstable” laser cavity, is impossible to align! A system images an object when B = 0. When B = 0, all rays from a point xin arrive at a point xout, independent of angle. xout = A xin A is the magnification. The Lens Law From the object to the image, we have: 1) A distance d0 2) A lens of focal length f 3) A distance di ⎡1 di ⎤ ⎡ 1 O=⎢ ⎥ ⎢ −1/ f 0 1 ⎣ ⎦⎣ ⎡1 d i ⎤ ⎡ 1 =⎢ ⎥ ⎢ −1/ f 0 1 ⎣ ⎦⎣ B ⎡ A =⎢ ⎣ −1/ f 1 − d 0 / 0 ⎤ ⎡1 d 0 ⎤ 1 ⎥⎦ ⎢⎣ 0 1 ⎥⎦ d0 ⎤ 1 − d 0 / f ⎥⎦ ⎤ f ⎥⎦ B = d 0 + di − d 0 di / f = d 0 di [1/ d 0 + 1/ di − 1/ f ] = 0 if 1 1 1 + = d 0 di f ⎡1 1⎤ A = 1 − di / f = 1 − di ⎢ + ⎥ ⎣ d 0 di ⎦ d ⇒ Mag = − i d0 Lenses can also map angle to position. From the object to the image, we have: 1) A distance f 2) A lens of focal length f 3) A distance f 0⎤ ⎡1 f ⎤ ⎡xin ⎤ ⎡xout ⎤ ⎡1 f ⎤ ⎡ 1 ⎢θ ⎥ = ⎢0 1 ⎥ ⎢−1/ f 1⎥ ⎢0 1 ⎥ ⎢θ ⎥ ⎦⎣ ⎦⎣ ⎦ ⎣ in ⎦ ⎣ out ⎦ ⎣ f ⎤ ⎡xin ⎤ ⎡1 f ⎤ ⎡ 1 =⎢ ⎥ ⎢−1/ f 1 ⎥ ⎢θ ⎥ 0 1 ⎣ ⎦⎣ ⎦ ⎣ in ⎦ f ⎤ ⎡xin ⎤ ⎡ f θin ⎤ ⎡ 0 =⎢ =⎢ ⎢ ⎥ ⎥ ⎥ x f θ / − f 1/ 0 − ⎣ ⎦ ⎣ in ⎦ ⎣ in ⎦ So xout ∝ θ in And this arrangement maps position to angle: θ out ∝ xin If an optical system lacks cylindrical symmetry, we must analyze its x- and ydirections separately: Cylindrical lenses A "spherical lens" focuses in both transverse directions. A "cylindrical lens" focuses in only one transverse direction. When using cylindrical lenses, we must perform two separate ray-matrix analyses, one for each transverse direction. Large-angle reflection off a curved mirror also destroys cylindrical symmetry. The optic axis makes a large angle with the mirror normal, and rays make an angle with respect to it. Optic axis before reflection tangential ray Optic axis after reflection Rays that deviate from the optic axis in the plane of incidence are called "tangential.” Rays that deviate from the optic axis ⊥ to the plane of incidence are called "sagittal.“ (We need a 3D display to show one of these.) Ray Matrix for Off-Axis Reflection from a Curved Mirror If the beam is incident at a large angle, θ, on a mirror with radius of curvature, R: tangential ray Optic axis θ ⇔ R where Re = R cosθ for tangential rays and Re = R / cosθ for sagittal rays ⎡ 1 ⎢ −2 / R e ⎣ 0⎤ 1 ⎥⎦ But lasers are Gaussian beams, not rays. Real laser beams are localized in space at the laser and hence must diffract as they propagate away from the laser. The beam has a waist at z = 0, where the spot size is w0. It then expands to w = w(z) with distance z away from the laser. The beam radius of curvature, R(z), also increases with distance far away. Gaussian beam math The expression for a real laser beam's electric field is given by: E ( x, y , z ) ∝ % exp [ −ikz − iψ ( z ) ] w( z ) ⎡ x2 + y 2 π x2 + y 2 ⎤ exp ⎢ − 2 −i ⎥ λ w ( z ) R ( z ) ⎣ ⎦ where: w(z) is the spot size vs. distance from the waist, R(z) is the beam radius of curvature, and ψ(z) is a phase shift. This equation is the solution to the wave equation when we require that the beam be well localized at some point (i.e., its waist). Gaussian beam spot size, radius, and phase The expressions for the spot size, radius of curvature, and phase shift: w( z ) = w0 1 + ( z / z R ) 2 R ( z ) = z + z R2 / z ψ ( z ) = arctan( z / z R ) where zR is the Rayleigh Range (the distance over which the beam remains about the same diameter), and it's given by: z R ≡ π w0 / λ 2 Gaussian beam collimation Twice the Rayleigh range is the distance over which the beam remains about the same size, that is, remains “collimated.” 2 z R = 2π w02 / λ Collimation Collimation Waist spot Distance Distance λ = 10.6 µm λ = 0.633 µm size w0 _____________________________________________ .225 cm 0.003 km 0.045 km 2.25 cm 0.3 km 5 km Longer wavelengths expand faster than shorter ones. 22.5 cm 30 km 500 km _____________________________________________ Tightly focused laser beams expand quickly. Weakly focused beams expand less quickly, but still expand. As a result, it's very difficult to shoot down a missile with a laser. Gaussian beam divergence Far away from the waist, the spot size of a Gaussian beam will be: w( z ) = w0 1 + ( z / z R ) ≈ w0 2 ( z / zR ) 2 = w0 z / z R The beam 1/e divergence half angle is then w(z) / z as z → ∞ : θ1/ e ⇒ w0 z w0 w0 = = = z R z z R π w02 / λ θ1/ e = λ / (π w0 ) The smaller the waist and the larger the wavelength, the larger the divergence angle. Focusing a Gaussian beam winput f wfocus f A lens will focus a collimated Gaussian beam to a new spot size: wfocus ≈ λ f / πwinput So the smaller the desired focus, the BIGGER the input beam should be! The Guoy phase shift The phase factor yields a phase shift relative to the phase of a plane wave when a Gaussian beam goes through a focus. ⎧ π / 2 when z → +∞ ψ ( z ) = arctan( z / z R ) → ⎨ ⎩−π / 2 when z → −∞ ψ(z) Phase relative to a plane wave: π/2 -zR zR −π/2 Recall the i in front of the Fresnel integral, which is a result of the Guoy phase shift. The Gaussian-beam complex-q parameter ⎡ π x2 + y2 ⎤ ⎡ x2 + y2 ⎤ E ( x, y, z ) ∝ exp ⎢ −i exp ⎢ − 2 ⎥ ⎥ λ R z w z ( ) ( ) ⎣ ⎦ ⎣ ⎦ We can combine these two factors (they’re both Gaussians): ⎡ π x2 + y 2 ⎤ E ( x, y, z ) ∝ exp ⎢ −i ⎥ ( ) λ q z ⎣ ⎦ where: 1 1 λ ≡ −i q( z ) R ( z ) π w2 ( z ) q completely determines the Gaussian beam. Ray matrices and the propagation of q We’d like to be able to follow Gaussian beams through optical systems. Remarkably, ray matrices can be used to propagate the q-parameter. ⎡A B⎤ O=⎢ ⎥ C D ⎣ ⎦ Optical system This relation holds for all systems for which ray matrices hold: qout Aqin + B = Cqin + D Just multiply all the matrices first and use this result to obtain qout for the relevant qin! Important point about propagating q Use qout Aqin + B = Cqin + D to compute qout. But use matrix multiplication for the various components to compute the total system ray matrix. Don’t compute qout Aqin + B = Cqin + D for each component. You’d get the right answer, but you’d work much harder than you need to! Propagating q: an example qout Aqin + B = Cqin + D Free-space propagation through a distance z: The ray matrix for free-space propagation is: Ospace 1q(0) + z = q(0) + z Then: q(z) = 0 q(0) + 1 ⎡1 z ⎤ =⎢ ⎥ 0 1 ⎣ ⎦ Propagating q: an example (cont’d) Does q(z) = q0 + z? LHS: This is equivalent to: 1/q(z) = 1/(q0 + z). 1 1 λ ≡ − i 2 πw (z) R(z) q(z) 1 1 1 1 1 ≡ − i = − i 2 2 2 2 2 q(z) z + zR / z zR (1+ z / zR ) z + zR / z zR + z /zR Now: RHS: π w20 q(0) = i = i zR λ so q(0) + z = izR + z 1 z − iz z 1 iz R = = 2 R2 = 2 − q(0) + z z + zR z + z2R z 2 + zR2 z + izR = 1 1 − i z + z2R / z z2 / zR + zR 1 = q(z) So: which is just this. q(z) = q(0) + z Propagating q: another example winput Focusing a collimated beam (i.e., a lens, f, followed by a distance, f ): wfocus f f A collimated beam has a big spot size (w) and Rayleigh range (zR), and an infinite radius of curvature (R), so: qin = i zR OspaceOlens ⎡1 =⎢ ⎣0 Cqin + D 1 = qout Aqin + B So: [ Im 1/ q focus But: [ ] Im 1/ q focus ≡ − ⇒ ] f ⎤⎡ 1 1 ⎥⎦ ⎢⎣ −1/ f 1 qafter lens = 0⎤ ⎡ 0 =⎢ ⎥ 1 ⎦ ⎣ −1/ f f⎤ 1 ⎥⎦ (−1/ f )(iz R ) + 1 1 − iz R / f = 0 (izR ) + f f π winput zR = − 2 =− f λf 2 2 λ πw 2 focus ⇒ w focus = λf π winput The well-known result for the focusing of a Gaussian beam Now consider the time and frequency of a light pulse in addition We’d like a matrix formalism to predict such effects as the: group-delay dispersion ∂t/∂ω angular dispersion ∂kx /∂ω or ∂θ /∂ω spatial chirp ∂x/∂ω pulse-front tilt ∂t/∂x time vs. angle ∂t/∂θ. This pulse has all of these distortions! where we’ve dropped “0” subscripts for simplicity. We’ll need to consider, not only the position (x) and slope (θ ) of the ray, but also the time (t) and frequency (ω ) of the pulse. Propagation in space and time: Ray-pulse “Kostenbauder” matrices Kostenbauder matrices are 4x4 matrices that multiply 4-vectors comprising the position, slope, time (group delay), and frequency. Optical system ↔ 4x4 Ray-pulse matrix ⎡ xin ⎤ ⎢θ ⎥ ⎢ in ⎥ ⎢ tin ⎥ ⎢ ⎥ ⎣⎢ν in ⎦⎥ ⎡A B 0 E⎤ ⎢C D 0 F ⎥ ⎢ ⎥ ⎢G H 1 I ⎥ ⎢ ⎥ ⎣0 0 0 1⎦ ⎡ xout ⎤ ⎢θ ⎥ ⎢ out ⎥ ⎢ tout ⎥ ⎢ ⎥ ⎣⎢ν out ⎦⎥ where each vector component corresponds to the deviation from a mean value for the ray or pulse. A Kostenbauder matrix requires five additional parameters, E, F, G, H, I. Kostenbauder matrix elements As with 2x2 ray matrices, consider each element to correspond to a small deviation from its mean value (xin = x – x0 ). So we can think in terms of partial derivatives. the usual 2x2 ray matrix angular dispersion ⎡ xout ⎤ ⎡A ⎢θ ⎥ ⎢C ⎢ out ⎥ = ⎢ ⎢ tout ⎥ ⎢G ⎢ ⎥ ⎢ ⎢⎣ν out ⎥⎦ ⎣0 pulsefront tilt ∂tout ∂xin B D H 0 ∂θ out ∂ν in E ⎤ ⎡ xin ⎤ ∂xout ∂ν in ⎢ ⎥ ⎥ 0 F ⎥ ⎢θ in ⎥ 1 I ⎥ ⎢ tin ⎥ ⎥⎢ ⎥ 0 1 ⎦ ⎢⎣ν in ⎥⎦ 0 time vs. angle ∂tout ∂θ in spatial chirp GDD ∂tout ∂ν in Some Kostenbauder matrix elements are always zero or one. ⎡ xout ⎤ ⎡A B ⎢θ ⎥ ⎢C D out ⎢ ⎥ = ⎢ ⎢ tout ⎥ ⎢G H ⎢ ⎥ ⎢ ⎢⎣ν out ⎥⎦ ⎣0 0 0 E ⎤ ⎡ xin ⎤ 0 F ⎥⎥ ⎢⎢θ in ⎥⎥ 1 I ⎥ ⎢ tin ⎥ ⎥⎢ ⎥ 0 1 ⎦ ⎢⎣ν in ⎥⎦ Kostenbauder matrix for propagation through free space or material The ABCD elements are always the same as the ray matrix. Here, the only other interesting element is the GDD: So: ^ material ⎡1 L / n ⎢0 1 ⎢ = ⎢0 0 ⎢ 0 ⎣0 ⎤ ⎥ ⎥ 1 2π Lk ′′⎥ ⎥ 0 1 ⎦ 0 0 0 0 I = ∂tout/∂νin The 2π is due to the definition of K-matrices in terms of ν, not ω. where L is the thickness of the medium, n is its refractive index, and k” is the GVD: d 2k k ′′ ≡ = 2 dω ω 0 λ3 d 2n 2πc 2 dλ 2 Example: Using the Kostenbauder matrix for propagation through free space Apply the free-space propagation matrix to an input vector: ⎡1 ⎢0 ⎢ ⎢0 ⎢ ⎣0 ⎤ ⎡ xin ⎤ ⎡ xin + Lθ in ⎤ ⎥ ⎥ ⎢θ ⎥ ⎢ θ in ⎥ ⎥ ⎢ in ⎥ = ⎢ 0 1 2π Lk ′′⎥ ⎢ tin ⎥ ⎢tin + 2π Lk ′′ν in ⎥ ⎥ ⎥⎢ ⎥ ⎢ ν in 0 0 1 ⎦ ⎢⎣ν in ⎥⎦ ⎢⎣ ⎥⎦ L 0 1 0 0 0 The position varies in the usual way, and the beam angle remains the same. The group delay increases by k”Lωin The frequency remains the same. Because the group delay depends on frequency, the pulse broadens. This approach works in much more complex situations, too. Kostenbauder matrix for a lens The ABCD elements are always the same as the ray matrix. Everything else is a zero or one. So: ^ lens ⎡ 1 ⎢ −1/ f = ⎢ ⎢ 0 ⎢ ⎣ 0 0 1 0 0 0 0 1 0 0⎤ 0 ⎥⎥ 0⎥ ⎥ 1⎦ where f is the lens focal length. The same holds for a curved mirror, as with ray matrices. While chromatic aberrations can be modeled using a wavelengthdependent focal length, other lens imperfections cannot be modeled using Kostenbauder matrices. Kostenbauder matrix for a diffraction grating Gratings introduce magnification, angular dispersion and pulse-front tilt: So: angular magnification spatial magnification cos( β ′) ⎡ − ⎢ cos( β ) ⎢ ⎢ 0 ⎢ ^ grating = ⎢ ⎢ sin( β ) − sin( β ′) ⎢ c cos( β ) pulse⎢ front ⎢ 0 ⎣ tilt 0 − cos( β ) cos( β ′) 0 0 0 1 0 0 no spatial chirp (yet) ⎤ ⎥ ⎥ λ[sin( β ') − sin( β )] ⎥ ⎥ c cos( β ') ⎥ ⎥ 0 ⎥ ⎥ ⎥ 1 ⎦ 0 time is independent of angle angular dispersion no GDD (yet) where β is the incidence angle, and β’ is the diffraction angle. The zero elements (E, H, I) will become nonzero when propagation follows. Kostenbauder matrix for a general prism ϑin ϑout L ψ in ψ out angular dispersion angular magnification ^ = spatial magnification ⎡ ⎢ ⎢ ⎢ −2π ⎢ ⎢⎣ min mout Lmout / nmin 0 1/ min mout dn dω min (tanψ in + tanψ out ) / λ0 0 −2π dn dω L tanψ out / nmin λ0 0 pulse-front tilt d 2k where k ′′ ≡ dω 2 All new elements are nonzero. ω0 Lmout tanψ in / n ⎤ ⎥ 0 −2π ddnω (tanψ in + tanψ out ) / mout ⎥ 2 dn 2 1 4π ( dω ) L tanψ in tanψ out / nλ0 + 2π Lk ′′⎥ ⎥ 0 1 ⎥⎦ 0 time vs. angle is the GVD, min ≡ −2π spatial chirp dn dω GDD cosψ in cos ϑin and mout cos ϑout ≡ cosψ out Kostenbauder matrix for a Brewster prism Brewster angle incidence and exit If the beam passes through the apex of the prism: ψ in = ψ out ≡ ψ min = 1/ mout L→0 (this simplifies the calculation a lot!) ^ prism where W = − 4π (dn /d ω ) tanψ / mout = −4π min (dn /d ω ) tanψ 0 1 0 0 0 0 ⎤ 0 ±W ⎥⎥ 1 0 ⎥ ⎥ 0 1 ⎦ Use + if the prism is oriented as above; use – if it’s inverted. ⎡ ⎢ = ⎢ ⎢ ±W / λ0 ⎢ ⎣ 0 1 0 Just angular dispersion and pulse-front tilt. No GDD etc. Using the Kostenbauder matrix for a Brewster prism Brewster angle incidence and exit This matrix takes into account all that we need to know for pulse compression. ⎡ 1 ⎢ 0 ⎢ ⎢ ±W / λ0 ⎢ ⎣ 0 xin 0 0 0 ⎤ ⎡ xin ⎤ ⎡ ⎤ 1 0 ±W ⎥⎥ ⎢⎢θin ⎥⎥ ⎢⎢ θ in ± W ν in ⎥⎥ = 0 1 0 ⎥ ⎢ tin ⎥ ⎢tin ± xinW / λ0 ⎥ ⎥ ⎥⎢ ⎥ ⎢ ν in 0 0 1 ⎦ ⎣ν in ⎦ ⎣ ⎦ When the pulse reaches the two inverted prisms, this effect becomes very important, yielding longer group delay for longer wavelengths (D < 0; and use the minus sign for inverted prisms). Dispersion changes the beam angle. Pulse-front tilt yields GDD. Modeling a prism pulse compressor using Kostenbauder matrices Use only Brewster prisms 1 ^prism 2 ^air 7 ^prism ^prism 6 ^prism ^air 3 ^air 4 5 K = ^7 ^6 ^5 ^4 ^3 ^2 ^1 Free space propagation in a pulse compressor There are three distances in this problem. L1 L3 L2 ^ free − space ⎡1 Li ⎢0 1 =⎢ ⎢0 0 ⎢ ⎣0 0 0 0 1 0 0⎤ 0 ⎥⎥ 0⎥ ⎥ 1⎦ n = 1 in free space K-matrix for a prism pulse compressor K = ^7 ^6 ^5 ^4 ^3 ^2 ^1 Spatial chirp unless L1 = L3. ^ = L1 + L2 + L3 ⎡1 ⎢0 1 ⎢ ⎢ W ( L1 − L3 ) ⎢0 − λ0 ⎢ ⎢⎣0 0 Time vs. angle unless L1 = L3. W ( L1 − L3 ) ⎤ ⎥ 0 0 ⎥ ⎥ W 2 1 − ( L1 + L3 ) ⎥ λ0 ⎥ ⎥⎦ 0 1 Negative GDD! 0 The GDD is negative and can be tuned by changing the amount of extra glass in the beam (which we haven’t included yet, but which is easy). Propagating spot size, radius of curvature, pulse length, and chirp To follow beams that are Gaussian in both space and time: Qout Qin We could propagate Gaussian beams in space because they’re quadratic in space (x and y): E ( x, y ) ∝ exp[ −(1/ w2 + iπ / λ R) ( x 2 + y 2 )] A Gaussian pulse is quadratic in time. And the real and imaginary parts also have important meanings (pulse length and chirp): E(t ) ∝ exp[−(1/τ G + i β )t ] 2 2 The complex-Q matrix We define the complex Q-matrix so that the space and time dependence of the pulse can be written: ⎧ π E ( x, t ) ∝ exp ⎨−i [ x, −t ] Q −1 ⎩ λ ⎡Q11−1 + Q12−1 ⎤ ⎡ x ⎤ ⎫⎪ ⎡ x⎤⎫ ⎪⎧ π ⎢ t ⎥ ⎬ = exp ⎨−i λ [ x, −t ] ⎢ −1 −1 ⎥ ⎢ ⎥ ⎬ ⎣ ⎦⎭ ⎪⎩ ⎣Q21 + Q22 ⎦ ⎣ t ⎦ ⎪⎭ ⎧⎪ π ⎡Q11−1 x + Q12−1t ⎤ ⎫⎪ = exp ⎨−i [ x, −t ] ⎢ −1 −1 ⎥ ⎬ λ + Q x Q ⎪ 22 t ⎦ ⎭ ⎣ 21 ⎩⎪ −1 −1 Note : Q21 = − Q12 ⎧ π −1 2 ⎫ = exp ⎨−i [Q11 x + 2Q12−1 xt − Q22−1t 2 ]⎬ ⎩ λ ⎭ These complex matrix elements contain all the parameters of beams/pulses that are Gaussian in space and time. The complex-Q matrix (cont’d) −1 When the off-diagonal elements, Q12 , are zero: ⎧ π −1 2 ⎫ E ( x, t ) ∝ exp ⎨−i [Q11 x − Q22−1t 2 ]⎬ ⎩ λ ⎭ Q11−1 ≡ 1/ qx = −1 22 Q 1 λ −i π w2 ( z ) R( z ) λ ≡ 1/ qt ≡ ( β + i /τ G2 ) π spatial complex-q parameter for Gaussian beams temporal complex-q parameter: the pulse length and chirp parameter for Gaussian pulses When the off-diagonal components are not zero, there is pulse-front tilt: Pulse-front tilt = Im ⎡⎣Q12−1 ⎤⎦ off-diagonal Q-1 element Im ⎡⎣Q22−1 ⎤⎦ temporal component of Q-1 K-matrices and the propagation of Q Kostenbauder matrices can be used to propagate the Q-matrix. Qout Qin This relation holds for all systems (multiply the component matrices together first and then use this complex result): Qout ⎡A ⎢G = ⎣ ⎡C ⎢0 ⎣ 0⎤ ⎡B E / λ⎤ Qin + ⎢ ⎥ ⎥ 1⎦ ⎣H I / λ ⎦ 0⎤ ⎡D F / λ⎤ Qin + ⎢ ⎥ ⎥ 0⎦ 1 ⎦ ⎣0 Division means multiplication by the inverse. This is actually more elegant than it looks... [A] Propagating the Q-matrix ⎡A ⎢C ^ =⎢ ⎢G ⎢ ⎣0 Qout B 0 D H 0 1 0 0 E /λ⎤ ⎥ F /λ⎥ I /λ ⎥ ⎥ 1 ⎦ In terms of these 2x2 matrices: 0⎤ ⎡ B E / λ⎤ Qin + ⎢ ⎥ ⎥ / 1⎦ λ H I ⎣ ⎦ 0⎤ ⎡D F / λ ⎤ Qin + ⎢ ⎥ ⎥ 0⎦ 0 1 ⎣ ⎦ ⎡A ⎢G = ⎣ ⎡C ⎢0 ⎣ [C] [D] Notice the symmetry in the 2x2 matrices in the Qpropagation equation. Qout = [ A] Qin + [ B ] [C ] Qin + [ D ] [B] Important point about propagating Q Use Qout = [ A] Qin + [ B ] [C ] Qin + [ D ] to compute Qout. But use matrix multiplication for the various components to compute the total system ray matrix. Don’t compute Qout = [ A] Qin + [ B ] [C ] Qin + [ D ] for each component. As with q, you’d get the right answer, but you’d work much harder than you need to! How to compute pulse distortions The K-matrix elements are derivatives of the output parameters with respect to the input parameters, and the spatio-temporal distortions are derivatives of means with respect to the output pulse parameters. But because νout = νin, the E = dxout/dvin, F = dθout/dvin, and I = dtout/dvin elements clearly indicate the (added) spatial chirp, angular dispersion, and group-delay dispersion. This doesn’t work for pulse-front tilt, G = dtout/dxin, however. Spatial chirp: E element of the K-matrix Angular dispersion: F element of the K-matrix Group-delay dispersion: I element of the K-matrix assuming there’s zero distortion to begin with because xout may not be xin Pulse-front tilt: You must look for an off-diagonal Q-1 element Kostenbauder matrices can model very general systems using the pulse Wigner Distribution. Okay, the complex-Q matrix tells us what happens to Gaussian pulse/beams. But what about more complex pulse/beams? Ordinarily, you’d have to numerically solve the Fresnel diffraction integral, which can yield a very complex computer computation. But there’s a simpler approach. It uses the Wigner Distribution, a different way to represent a pulse’s dependence on time: ∞ t t WEt (ω,τ ) ≡ Et (τ + ) Et *(τ − ) exp(−iω t ) dt 2 2 −∞ ∫ The Wigner Distribution converts the pulse into a plot of intensity vs. time (delay) and frequency. We can invert the Wigner Distribution to obtain the pulse field. Inverse-Fourier transform the Wigner Distribution with respect to ω : Y { -1 } 1 WEt (ω,τ ) = 2π ∞ t t Et (τ + ) Et *(τ − )exp(−iω t ) dt exp(iω t′) dω 2 2 −∞ ∫∫ −∞ ∞ t t ⎧1 = Et (τ + ) Et *(τ − ) ⎨ 2 2 ⎩ 2π −∞ ∫ ∞ ⎫ ′ exp[−iω( t − t )] dω ⎬ dt −∞ ⎭ ∫ ∞ ∞ t t = Et (τ + ) Et *(τ − ) δ ( t − t′) dt 2 2 −∞ t′ t′ = Et (τ + ) Et *(τ − ) 2 2 ∫ Setting τ = t’/2: = Et (t′) Et *(0) So we can’t determine the absolute phase, but simple inverse Fourier transforming yields the rest of E(t). Examples of Wigner Distributions The Wigner Distribution is always real, but it usually goes negative. Linearly chirped Gaussian Quadratically chirped Gaussian τ ω τ ω τ ω Double pulse Properties of the Wigner Distribution The marginals (integrals) of the Wigner Distribution yield the pulse intensity vs. time and the spectrum vs. frequency. I (τ ) = ∫ ∞ −∞ WE (ω ,τ ) dω I(τ) S (ω ) = ∫ ∞ −∞ WE (ω ,τ ) dτ S(ω) τ ω The Wigner Distribution has many other nice properties. We can define spatial and space-time Wigner Distributions, too. A spatial Wigner Distribution is in terms of x and kx: dropping the x subscript on k x′ x′ WEx (k , x) ≡ Ex ( x + ) Ex *( x − ) exp(−ikx′) dx′ 2 2 −∞ ∫ ∞ The Wigner Distribution converts the beam into a plot of intensity vs. space and spatial frequency. Then we can define a space-time Wigner Distribution: WE (k , x, ω,τ ) ≡ ∞ x′ t x′ t E ( x + ,τ + ) E *( x − ,τ − ) 2 2 2 2 −∞ exp(−ikx′ − iωt ) dx′ dt ∫∫ −∞ ∞ In terms of Wigner Distributions, Kostenbauder matrices can describe a general optical system! If the K-matrix of the system and the input-pulse Wigner Distribution are known: WEout ( xout , kout , ωout ,τ out ) = m WEin ( xin , kin , ωin ,τ in ) where the output Wigner Distribution and its parameters are determined from the input parameters, the K-matrix of the system, and the input pulse Wigner Distribution. The quantity, m, is a magnification. J. Paye and A. Migus, “Space–time Wigner functions and their application to the analysis of a pulse shaper,” J. Opt. Soc. Am. B, 12, #8, p 1480, August 1995.
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