PHYS 408 Applied Optics (Lecture 18) Jan-April 2017 Edition Jeff Young AMPEL Rm 113

Quick review of key points from last lecture The Fabry-Perot cavity transmission function is useful for identifying important parameters that characterize real-life cavities. The two main parameters are the cavity Finesse, and the Free Spectral Range. These are sufficient for a symmetric cavity, but if not symmetric, the maximum on-resonance transmission is also needed. All of these parameters are related to the two mirror reflectivities and the length of the cavity. Going to a Gaussian, curved-mirror cavity, the absolute value of the transmission would only match the Fabry-Perot transmission if the excitation beam transverse profile was mode-matched to the Gaussian of interest, but the mode lifetimes and free spectral range are the same. There is also an offset of the absolute cavity resonance frequencies due to the (generalized) Gouy phase.

Physical interpretation(s) of Finesse What are the two most intuitive/important physical interpretations of the Finesse of a quasi-mode? Cavity lifetime Internal intensity enhancement

Key relations for high-finesse cavities Sketch the incident and internal field profiles on resonance, and half way between resonances

Extrinsic sources of loss Finite mirror size No longer a Gaussian, how will it propagate? Know how Gaussians propagate, know how plane and spherical waves propagate, what about a more general field structure?

Fourier Optics: empirical approach Dx d Propagation through a slit a very similar problem… i.e. if mirror was very small, the reflected centre of the Gaussian would be effectively like the laser passing through a small hole.

What does this suggest? Why does this sort of make sense, based on what we know about plane waves? Hint: what is a good guess for what the electric field distribution is in the plane of the slit? Want a functional form, E(x). Pause, let them work on it

? Analyze kx x Eslit -Dx/2 Dx/2 Dx/2 -Dx/2 ? Let them work on it, emphasize what the argument is

Essence of Diffraction The 2D Fourier transform of the field in a plane illuminated by a plane wave at a fixed frequency: Together with the knowledge that any valid solution of the Helmholtz Eqn. can be expressed as a linear superposition of plane waves, lead to the powerful result that the field at any location on the outgoing side of the illuminated plane can be expressed as where

? So have: x kx d Eslit Dx x At screen, if screen far from slit Dx/2 Dx/2 -Dx/2 kx ? What seems to be happening here? How can these be related? Spend awhile on this…see if they can get to relating to 3D plane waves and association of kx/omega with a direction. Don’t forget pi in definition of sinc! Relate this to expansion in terms of plane waves

Approximation? If put the screen right after the slit, would just see a shadow of the slit, so this mapping of the Fourier transform only applies in the limit that the screen is far away from the illuminated plane. This means that the 3D plane waves contributing to the image on the screen at large separations must be propagating almost normal to the planes defined by the screen and the aperture, so then we can write: and Taylor expand the square root.

Fill in the blanks When illuminated by a plane wave of harmonic radiation, an aperture with a transmission function that can be represented via ___1___ in terms of a function T(___2___), generates a distribution of light ___3___ on a screen located ___4___ from the aperture, that has a functional form T(___5___)2. This implies that under these conditions, there must be ___6___. 2 dimensional Fourier transformation k_x,k_y Intensity A long distance away x,y a one to one correspondence between the in-plane Fourier components, k_x,k_y of the field pattern at the aperture, and the x,y coordinates, x(k_x), and y(k_y) on the screen where the electric field is proportional to the corresponding Fourier component of the field at the aperture. i.e. propagation over a large distance produces a pattern of light intensity that spatially maps out the square of the Fourier transform of the field distribution at the aperture.