Ray matrices The Gaussian beam Complex q and its propagation Ray-pulse “Kosten- bauder” matrices The prism pulse compressor Gaussian beam in space and time and the complex Q matrix Spatio-temporal characteristics of light and how to model them Optical system ↔ 4x4 Ray-pulse matrix

Ray Optics 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 the axis. axis x in,  in x out,  out

Ray Optics A light ray can be defined by two co-ordinates: x in,  in x out,  out its position, x its slope,  Optical axis optical ray x  These parameters will change with distance and as the ray propagates through optics.

Ray Matrices & Ray Vectors For many optical components, we can define 2x2 "Ray Matrices." The effect on a ray is determined by multiplying its "Ray Vector." Ray matrices can describe simple and com- plex systems. These matrices are often called "ABCD Matrices." Optical system ↔ 2x2 Ray matrix

Ray matrices as derivatives Since the displacements and angles are assumed to be small, we can think in terms of partial derivatives. It’s easy to evaluate these derivatives for most optical components. angular magnification spatial magnification

Rays in free space or a medium If x in and  in be the position and slope upon entering, let x out and  out be the position and slope after propagating from z = 0 to z. x in,  in z = 0 x out  out z Rewriting this expression in matrix notation:

Rays at a lens x out = x in  out = (–1/f) x in +  in x in,  in z = z in x out  out z = z out f If the lens is thin, then only the ray slope changes. Rewriting this expression in matrix notation:

Ray Matrices for Space and Lenses Free space: f z Lens:

Ray Matrix for an Interface At the interface, clearly: x out = x in. Now calculate  out. Snell's Law says: n 1 sin(  in ) = n 2 sin(  out ) which becomes for small angles: n 1  in = n 2  out   out = [n 1 / n 2 ]  in  in n1n1  out n2n2 x in x out

Ray Matrices for Curved Mirrors Consider a mirror with radius of curvature, R : On axis: Off axis: where R e = R cos  if in plane of incidence ("tangential") and R e = R / cos  if  to plane of incidence ("sagittal")

For cascaded elements, multiply ray matrices Notice that the order looks opposite to what it should be. O1O1 O3O3 O2O2

A system images an object when B = 0. When B = 0, all rays from a point x in arrive at a point x out, independent of angle. x out = A x in

The Lens Law From the object to the image, we have: 1) A distance d 0 2) A lens of focal length f 3) A distance d i

So And this arrangement maps position to angle: 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

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 w 0. 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. But lasers are Gaussian Beams, not rays.

Gaussian Beam Math The expression for a real laser beam's electric field is given by: where: w(z) is the spot size vs. distance from the waist, R(z) is the beam radius of curvature, and  (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, Radius, and Phase The expressions for the spot size, radius of curvature, and phase shift: where z R is the Rayleigh Range (the distance over which the beam remains about the same diameter), and it's given by:

Twice the Rayleigh range is the distance over which the beam remains about the same size, that is, remains “collimated.” _____________________________________________.225 cm km km 2.25 cm 0.3 km 5 km 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 Collimation Collimation Collimation Waist spot Distance Distance size w 0 = 10.6 µm = µm Longer wavelengths expand faster than shorter ones.

Gaussian Beam Divergence Far away from the waist, the spot size of a Gaussian beam will be: The beam 1/e divergence half angle is then w(z) / z as z  : The smaller the waist and the larger the wavelength, the larger the divergence angle.

Focusing a Gaussian Beam A lens will focus a collimated Gaussian beam to a new spot size: w 0  f /  w 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. Phase relative to a plane wave: Irradiance (for reference):

The Gaussian-Beam Complex q Parameter We can combine these two factors (they’re both Gaussians): where: 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. This relation holds for all systems for which ray matrices hold: Nice, eh? Optical system

Propagating q: an example Free-space propagation through a distance z : Then: The ray matrix for free-space propagation is:

Propagating q: an example (cont’d) So: Now: RHS: Does q(z) = q 0 + z ? This is equivalent to: 1/q(z) = 1/(q 0 + z). so which is just this. LHS:

Propagating q: another example Focusing a collimated beam (i.e., a lens, f, followed by a distance, f ): A collimated beam has big spot size ( w ) and Rayleigh range ( z R ), and an infinite radius of curvature ( R ), so: q in = i z R After some algebra, we find: w input w focus f f A well-known result for the focusing of a Gaussian beam But:

Now consider the time and frequency of a light pulse in addition We’d like a matrix formalism to predict such effects as: angular dispersion ∂  /∂  group-delay dispersion ∂t/∂  spatial chirp ∂x/∂  pulse-front tilt ∂t/∂x time vs. angle ∂t/∂ . This pulse has all of these distortions!

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. A Kostenbauder matrix requires five additional parameters, E, F, G, H, I. Optical system ↔ 4x4 Ray-pulse matrix where each vector component corresponds to the deviation from a mean value for the ray or pulse.

Kostenbauder matrix elements As with 2x2 ray matrices, consider each element to correspond to a small deviation from its mean value. So we can think in terms of partial derivatives. spatial chirp time vs. angle GDD pulse-front tilt angular dispersion the usual 2x2 ray matrix

Some Kostenbauder matrix elements are always zero or one.

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: I = ∂t out /∂ in where L is the thickness of the medium, n is its refractive index, and k” is the GVD: So: The 2  is due to the definition of K-matrices in terms of, not .

Example: Using the Kostenbauder matrix for propagation through free space Apply the free-space propagation matrix to an input vector: This approach works in much more complex situations, too. The position varies in the usual way, and the beam angle remains the same. The group delay increases by k”L  in The frequency remains the same.

Kostenbauder matrix for a lens The ABCD elements are always the same as the ray matrix. Everything else is a zero or one. 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 wavelength- dependent focal length, other lens imperfections cannot be modeled using Kostenbauder matrices. So:

no spatial chirp (yet) Kostenbauder matrix for a diffraction grating Gratings introduce magnification, angular dispersion and pulse-front tilt: where  is the incidence angle, and  ’ is the diffraction angle. The zero elements (E, H, I ) will become nonzero when propagation follows. So: time is independent of angle no GDD (yet) pulse-front tilt angular dispersion spatial magnification angular magnification

Kostenbauder matrix for a general prism All new elements are nonzero. is the GVD, spatial chirp time vs. angleGDDpulse-front tilt angular dispersion where L and spatial magnification angular magnification

Kostenbauder matrix for a Brewster Prism If the beam passes through the apex of the prism: (this simplifies the calculation a lot!) where Use + if the prism is oriented as above; use – if it’s inverted. Just dispersion and pulse-front tilt. No GDD or spatial chirp—yet. Brewster angle incidence and exit

Using the Kostenbauder matrix for a Brewster Prism This matrix takes into account all that we need to know for pulse compression. 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). Pulse-front tilt yields GDD. Brewster angle incidence and exit Dispersion changes the beam angle.

Modeling a prism pulse compressor using Kostenbauder matrices K prism K air K = K 7 K 6 K 5 K 4 K 3 K 2 K 1 Use only Brewster prisms

Free space propagation in a pulse compressor There are three distances in this problem. n = 1 in free space L1L1 L2L2 L3L3

K-Matrix for a prism pulse compressor Spatial chirp unless L 1 = L 3. K = K 7 K 6 K 5 K 4 K 3 K 2 K 1 Negative GDD! The GDD is negative and can be tuned by changing the amount of extra glass in the beam.

Propagating spot size, radius of curvature, pulse length, and chirp! We’d like to be able to follow beams that are Gaussian in both space and time through optical systems. What allowed us to propagate Gaussian beams in space was the fact that they’re quadratic in space (x and y): A Gaussian pulse is quadratic in time. And the real and imaginary parts also have important meanings (pulse length and chirp):

The complex Q matrix We define the complex Q-matrix so that the space and time dependence of the pulse can be written: the complex q parameter for Gaussian beams When Q 12 = -Q 21 = 0 : pulse length and chirp parameter for Gaussian pulses (It gets considerably more complicated when they’re not. For example, what is the pulse length of a tilted pulse?)

K-matrices and the propagation of Q This relation holds for all systems: This is actually more elegant than it looks. Kostenbauder matrices can be used to propagate the Q-matrix. Division means multiplication by the inverse.

Propagating the Q-matrix Notice the symmetry in the 2x2 matrices in the Q- propagation equation. In terms of these 2x2 matrices: [A][A] [C][C] [D][D] [B][B]

A case for complex Q and K-matrices… K-matrices can model even very complex situations. Here is a case with a little of everything… The tendency of different colors to propagate differently can cause the pulse to have severe spatio-temporal distortions. Beam divergence angle  depends on :  = 2 /  w, where w = beam spot size. So if ranges from 400 nm to 1600 nm,  varies by a factor of 4. The lens focal length will also depend on. Now send this light into a pulse compressor… We’re actually doing this experiment!