PHYS 408 Applied Optics (Lecture 12) Jan-April 2017 Edition Jeff Young AMPEL Rm 113

Average =71%, Median =74%

Plan The next assignment will involve redoing the multiple choice part, with explanations, plus a couple of additional problems, one associated with Q4 on the exam. Due Monday Feb. 27 at 5 pm, along with a question that will be posted today.

General Reminder Optics is nothing but: Understanding what light is, and what parameters characterize light, together with Understanding how light interacts with materials Fundamentally it always boils down to knowing how to solve the Maxwell Equations in some environment. If stuck, can always go back to expanding solutions in uniform regions and using boundary conditions, but in many cases, there are tricks or approximations that greatly simplify the analysis/engineering. Ask them to give examples in each case

Quick review of key points from last lecture Electromagnetic modes, or more properly, normal modes, are stationary, non-static solutions of the Maxwell equations in lossless environments. Plane waves are normal modes of the vacuum. The dielectric environment, specified by , in general imposes boundary conditions that “mix” plane waves via scattering, leading to more complicated, spatially varying normal modes. Cavity modes, or cavity normal modes, are the discrete modes supported in some localized region defined by the boundary conditions, with the property that no light can escape from that region, at least for some range of frequencies. In practice, real cavities must couple to the outside world to be useful, so the idealized cavity normal modes are either a) used in perturbative treatments of scattering problems that couple the cavity normal modes to radiation (continuum) modes by taking account of the imperfect localizing nature of the real dielectric environment, or b) replaced by quasi-normal modes that have complex eigen frequencies w, and represent the solutions of the Maxwell equations obtained using outward-propagating boundary conditions. For “good quality” cavity modes these quasi- normal modes will typically have almost identical form as the corresponding normal modes in and near the cavity region.

Gaussian Beams (relevant to a large class of cavities)

From here Develop a mathematical and intuitive description of the simplest form of a scalar Gaussian beam The defining mathematical formula cast in various forms Remarkably simple method for following the propagation of a paraxial Gaussian beam as it interacts with optical elements that maintain its paraxial nature (using the “ABCD matrix”) Regression to understand the ABCD matrix that is most easily derived using Fermat’s principle (ray optics)

The defining equation What is A? (Everything except the e^(ikz) term) What is r? sqrt(x^2+y^2) Why would it be called a Gaussian beam? Transverse amplitude dependence. CRUCIAL Point: How many independent parameters define the Gaussian mode? (2, lambda and w_0 assuming you don’t care about the origin, otherwise 3, including where z=0 is in some a particular reference frame) Qualitativley sketch the z dependence of R

Re-arrange What labels would you give these separate terms?

z dependence of Radius of curvature

z dependence of the beam waist What is the total angular spread of the Gaussian beam at large distances? This is the type of “physics”, heuristic knowledge like the conditions for anti-reflection on exam, that expect you to know.

The q parameterization

The real and imaginary parts of 1/q What are R and w at z=z_0? If you knew R and W at some z value (in a fixed reference frame), do you know everything about the beam, everywhere else? Yes, because other than the reference frame, there are only two parameters…algorithm for finding lambda and z_0? (find z_0 from R(z) equation, then lambda from w_0 equation).

Follow q through a paraxial optical system

The ABCD Matrix: Very simple! Fundamentally based on Fermat’s Principle, which is?

LIGO: Gravitational Wave Detector How is this related to what we have covered in this course?