Title: All Decks / ETH_Sem6_old / QE

Charge density and current decomposition

Flux density/ displacement field

Polarisation density <=> Charge density

Magnetic field

Magnetisation density <=> Current density

Approximations for Fields in Matter:
1) Free charge/current = average over cube
2) Bound charge/current = boundry effects
3) Make  \(P(E)\) and \(M(B)\)
4) Make dependence local
5) Make dependence relative (homogeneity)
6) Make dependence immediate (dispersionless)
7) Make dependence linear
8) Make dependence colinear (isotropy)

Simplified relation Flux Density, Magnetic field to E and B

Refractive index

Optical impedance

Q: Polarisation convention:
A: (Right hand rule)

Header: Types of optics

Geometrical Optics:
• Restriction: λ ≪ size of objects
• Parameters: intensity, direction
Wave Optics:
• Restriction: λ ≈ size of objects
• Parameters: intensity, direction, coherence, phase
Electromagnetic Optics:
• Parameters: intensity, direction, coherence, phase, polarization, photons

Poynting Theorem in Materials:

Lorentz polarisation model

Q: Quality factor resonnator:
A: \(Q = \omega_0/ \gamma\)
- The 𝑄 factor equals 2𝜋 times the number of oscillation periods required for the stored energy to decay to 1/𝑒 (≈ 37%) of its initial value.

Q: Plasma frequency, susceptibility and permitivity:
[...]
[...]
[...]
A: \(\omega_p = \sqrt{Ne^2/m\varepsilon_0}\); \(\chi^{(1)} = - \frac{\omega_p^2}{\omega^2 - i\gamma \omega}\); \(\varepsilon_r \approx 1-\frac{\omega_p^2}{\omega^2}\)

Q: Propagation of light in metals:
\(\omega < \omega_p\): [...]
 \(\omega > \omega_p\): [...]
\(\omega= \omega_p\): [...]
A: Attenuation; Propagation according to \(\varepsilon_r\); No propagation, pure electrical oscillation: Plasmon

Dielectric material interaction with light:

Q: Normal vs anomalous dispersion:
[...]
A: \(\dd{n'}{\omega} > 0\) (Normal)

Q: Idea: Sellmeier equation
A: Index of refraction is given by sum of Lorentz oscillators at different frequencies:

Helmholtz Equation:

Q: Group dispersion + meaning of positive:
A: positive (red faster than blue) :
Below \(\omega_0\) positive, above negative

Group velocity and index:

Q: How to quantify the optimality of laser pulses?
A: Compare time bandwith product with niquist limit.
(time bandwidth product = FWHM\((\omega)\) * FWHM(t)

Dispersion parameter:

Brewster Angle:

Q: Longitudinal coherence:
A: For stationary fields:
Complex degree of temporal coherence:
Note that \(g\) decays with a characteristic time \(\tau\) the coherence time.

Q: Power spectral density:
A: Intuition:
\(V_T\): Fourier coefficient for \(T\) periodic signal
\(S\): Power/ Frequency Area (via parseval)

Wiener Kinchin Theorem:

Relation coherence time and linewidth:

Q: Optical coherence tomography intuition:
A: - Use incoherent source
- Split into sample and reference
- Lot's of light gets scattered, but only phase matching light interferes
- We only 'see' parts which have exactly controlled distance (by refrence)

Optical coherence tomography setup:

Q: Die geometrische Summenformel \[[...] = [...]\]
A: \sum_{k=0}^n q^k; \begin{cases} n+1 & \text{falls } q=1 \\  \frac{q^{n+1}-1}{q-1} & \text{falls } q \neq 1 \end{cases}

Q: Spectrum of a Fabry-Perot Etalon:
A: with \(\nu_F\): Free spectral range
and \(\delta \nu \) : FWHM
and \(F\): Finesse

Transmission of Fabri-Perot Etalon:

Free spectral range of Fabri Perrot Etalon

Q: Intuition Finesse:
A: how narrow the resonances are in relation to their frequency distance

Q: Explain the image generated by this setup:
A: We get peaks depending on the angle the beam enters the first lense. The less orthogonal the light enters the setup, the closer the peaks.

Q: Matrix method wave optics:
A: (+ ->)
(- <-)
s and p polarisation are handeled seperatley.

Q: Definition Transfer Matrix vs Scattering matrix:
+ Change between them
A: Transfer Matrix:
Scattering Matrix:
Change of representation:

Relation scattering matrix & fresnel coefficients:

Procedure S-matrix method:

Q: Properties of S-matrix:
Lossless \(\implies\)[...]
Reciprocal \(\implies\)[...]
Lossles & Reciprocal \(\implies\)
A: unitary; \(S = S^T\)

Q: S-matrix for beamsplitter:
A: \(s \): The splitting ratio

Radial intensity for planewave, Helmholtz

Q: 2D-FT QE
A: (Sign is due to negative convention of time to space in Waveequation)

Q: Solution: Fourier Helmholz Equation:
A: The transfer function

Q: Fresnel Approximation:
A: Take only first order terms:

Q: Definition Fresnel Number:
A: \(N_{F}={\frac{a^{2}}{\lambda z}}\)

Q: Criterion for validity of Fresnel Approx:
A: \(N_{F}\frac{\theta_{m}^{2}}{4}\ll 1\)

Q: Paraxial Helmholtz equation:
A: \(\nabla_{T}^{2}A-i2k\frac{\partial A}{\partial z}=0\)

Q: Paraxial radial wavefunction:
A: \(U({\bf r})=A({\bf r})e^{-i k z}\)

Q: Gaussian beam formula:
A: \(A({\bf r})=\frac{A_{1}}{q(z)}e^{-i k(x^{2}+y^{2})/2q(z)}\)

Q: \(q\) for Gaussian Beam:
A: \(q(z) = z + iz_0\)
\(\frac{1}{q(z)}=\frac{1}{R(z)}-i\frac{\lambda}{\pi W^{2}(z)}\)

Q: Gaussian beam parameters (\(W,R,\zeta, W_0)\):
A: \(W(z)=W_{0}\sqrt{1+\left({\frac{z}{z_{0}}}\right)^{2}}\)
\(R(z)=z\left[1+\left({\frac{z_{0}}{z}}\right)^{2}\right]\)
\(\zeta(z)=\tan^{-1}\frac{z}{z_{0}}\)
\(W_{0}={\sqrt{\frac{\lambda z_{0}}{\pi}}}\)

Q: Properties \(W\) in QE
A: - Width at which intensity falls to \(1/e^2\)
- Spot size = \(2 W(z)\)
- Waist Radius \(W_0\)

Q: Half angle of divergence Gaussian Beam,
A: \(\theta_{0}=\frac{\lambda}{\pi W_{0}}\)

Q: Rayleigh range to Beamwaist:
A: \(2z_{0}=\frac{2\pi W_{0}^{2}}{\lambda}\) (depth of focus)
Note goes squared with \(W_0\)

Q: What is \(\zeta\) in Gaussian Beams?
A: Gouy Phase shift

Q: Intuitive difference Frauenhofer vs Fresnel limit:
A: Frauenhofer < Fresnel
Farfield : Nearfield
Planewaves : Parabolic Waves
FT : Complicated ( but more 'classical' beahaviour like shadows)

Q: Condition for Frauehofer Limit:
A: Input and output plane @ \(a,b\) far away:
\(N_F= \frac{(a,b)^2}{\lambda z} \ll 1\)
Largest slit^2/ smallest Distance* Wavelenght

Typical 4-f setup

Q: Intuition Fresnel zone-plate:
A: Filters destructive interference, thus creating a lense. Good for X-ray

Sketch of Hologram:

Q: Mathematical idea hologram:
A: Transparency:\(t \propto I \propto |U_0 + U_{ref}|^2 = I_r + I_0 + U_r^*U_0 + U_r U^*_0 = I_r + I_0 + 2\sqrt{I_rI_0} \cos(\arg(U_r) - \arg(U_0))\)
Using this transparency with the refereence gives:
\(U(x,y)\propto I_{r}+I_{0}(x,y)+\sqrt{I_{r}}U_{0}(x,y)+\sqrt{I_{r}}U_{0}^{*}(x,y)\)
The 3rd Term is our goal, the fourth is the ambiguity wave

Paraxial ray-optics,lense transfer matrix:

Q: Paraxial ray-optics, examples of elements:
A: Propagation:
Interface from \(n_1 \to n_2\)
Spherical  (concave) mirror with Radius \(R\)
Lense with focal length \(f\)

Q: How to model a lense in gaussian optics and how is the beam transformed?
A: Transmission of: \(e^{i k\rho^{2}/2f}\)
Beam transforms via \({\frac{1}{R^{\prime}}}={\frac{1}{R}}-{\frac{1}{f}}\)
To find the rest of the beam determine \(z_0'\)

Q: ABCD Law:
A: \(q_{2}=\frac{A q_{1}+B}{C q_{1}+D}\)
Note that the ABCD Matrix is the same as for paraxial ray optics

Q: \(k^2\) in dispersive medium
A: \(k^2 = \mu \varepsilon \omega^2 + i \mu \sigma \omega \)

Q: How do we obtain Hermite Gaussian beams from Gaussian beams (mathematically)?
A: \(A(x,y,z)=\mathcal X[\frac{\sqrt{2}x}{ W({z})}]\mathcal{Y}[\frac{\sqrt{2}y}{W(z)}]e^{i\mathcal{Z}(z)}A_{G}(x,y,z)\)
With
 \(\mathcal X(\sqrt 2 x / W(z))= \mathcal X(u) = H_l (u)\)
\(\mathcal Z = (l+m)\tan^{-1}(z/z_0) + C\)
And \(A_G = \frac{A_{1}}{q(z)}e^{-i{k}\bigl(x^{2}+y^{2}\bigr)/2q(z)}\)

Draw the first few hermite gaussian functions:

Q: Recurrence relation Hermite polynomials
A: \(H_{l+1}(u)=2u H_{l}(u)-2l H_{l-1}\)

Q: Intuition topological charge QE:
A: Number of separate rotating 'lobes' in a laguerre gaussian beam

Q: What is special about bessel beams?
A: Bessel beams are exact solutions to Helmholtz!

Q: Bessel beam formula:
A: \(A(\rho,\phi)=A_{m}J_{m}(k_{T}\rho)e^{i m\phi}\)

Q: The paraxial ray matrix for one round-trip through a spherical resonator
A: \(\left(\begin{array}{c c}{{A}}&{{B}}\\ {{C}}&{{D}}\end{array}\right)=\left(\begin{array}{c c}{{1}}&{{0}}\\ {{-\frac2{R_{1}}}}&{{1}}\end{array}\right)\left(\begin{array}{c c}{{1}}&{{d}}\\ {{0}}&{{1}}\end{array}\right)\left(\begin{array}{c c}{{1}}&{{0}}\\ {{-\frac2{R_{2}}}}&{{1}}\end{array}\right)\left(\begin{array}{c c}{{1}}&{{d}}\\ {{0}}&{{1}}\end{array}\right)\)

Q: Stability of resonator - formula:
A: \(|A+D|\lt 2\)
or
\(0\lt {\bigg(}1-{\frac{d}{R_{1}}}{\bigg)}\left(1-{\frac{d}{R_{2}}}\right)\lt 1\)
Alternativeley with the \(g\) parameters:
\(g_i = 1-\frac{d}{R_i}\)

Complex beam parameter from cavity ABCD matrix

Q: Frequency domain: \(k(\omega)=\)
A: \(n(\omega)\omega/c\)

What is the exact resonance conditions for two spherical mirrors?

Intuition Quality factor

Mode density at a given frequency

Einstein coefficients to observables

Einstein relation

Q: What is g in the context of lasers?
A: Lineshape function

Finesse of laser cavity

Energy density in laser mode given photon density

Q: Definition gain coefficient
A: How much is the laser amplified pro distance?
Note that \(\sigma^* \) is the transition cross-section (adapted to the cavity)

Q: Intuition threshold gain coefficient \(\gamma_{th}\)
A: The gain coefficient that barely compensates the round trip attenuation \( \alpha \)

Q: Photon cavity lifetime
A: With \(c_n= c/n \) and \(\alpha_r \) the threshold gain coefficient

Q: Stability criterion for symmetric cavity
A: \(0< d/R < 2\)

Q: Transition cross-section, relation to gain coefficient?
A: It's the (cavity adjusted) crossection times the population inversion.

Q: Photon flux conversion:
A: \(\phi = \frac{I}{h\nu} = \frac{nc}{V}\)
Number of photons per \(m^2\) per \(s\)

Threshold gain coefficient:

Q: What can we read from this graph?
A: We see the relation between the effective population inversion \(N\) and the absorbtion/emission probability density \(W_i\) as related to the saturation time \(\tau_s\)
\(W_i\): is a stand in for the photon flux density \(W_i = \phi \sigma\).
We therefore see how the population inversion is depleted from it's equilibrium if we have higher mode density.
We also see how shorter saturation times lead to higher possible photon flux. (for this we immagine that the model stabilizes at \(W_i = 1/\tau_s\) (this is wrong! but helpfull)

Q: Formula for gain clamping:
A: WIth \(\phi_s\) the saturation photon-flux density. (The pfd where gain drops by 2)

Q: Explain the following graphic
A: We see on the y axis the effective population inversion (at 1 we have 50:50).
On the x axis we have the small signal population inversion \(N_0\).
We see that below threshold (when we have no lasing yet) that \(N\) and \(N_0\) agree.
(This is because there is negligible emission probability density \(W\))
After we reach the threshold we start getting real roundttrip gain. This leads to a saturation of \(N\).
But now we are in a regime where we have competition between pumping and lasing. This is nearly linear.

Q: Intuition \(N_0\)
A: The population inversion we would have if we had no feedback. (Pureley through the kinetics of pumping and decaying)

Jones vectors for different polarisations:

Q: page 200 in qe
A: do more pages

Q: Explain the following graphic:
A: On the left we see the finite number of possible constructive interferences (modes) for a planar mirro waveguide. We get this distribution from the condition \(\sin(\theta) = \lambda/2d\).
On the right we see the relation between the transverse component (\(k_y\)) and the longitudinal component \(\beta_m\) (the propagation constant) of possible modes.
We see that the transversal condition leads to 'particle in a box' like quantisations, whereas the longitudinal vector is non linearly distributed.
The relation between the wavevectors along the direction of effective propagation \(z\) and the transverse \(y\) is given by pythagoras.
The boundry condition is the same as for braggs law!

Q: Transversal modes in mirror waveguide
A: Note these are TEM modes. for propagation in z

Plane mirror waveguide cuttof frequency and mode number

Q: Effective refractive index for mirror waveguide
A: The \(z\) component of the wave vector, normalized to the free space wavevector. (note this also includes \(n_{medium}\) via \(\beta_m\)

Q: Explain this figure:
A: Relation of frequency to number of modes. This makes sense, as higher frequencies allow for more 'numbers of wavelengths' to fit the bragg condition.
Frequency to effective propagation wave vector (phase velocity). Low order modes approximate free space dispersion.
Frequency to group velocity. The low order modes approximate the speed of light. For ultra small group velocities we can have transversal standing waves.

Q: Phase shift on total reflection
A: This is not analytically solvable...
The condition for periodicity is then

Q: What is the cutoff frequency for a dielectric slab waveguide?
A: This is the same as for a mirror with \(n \to \sqrt{n_1^2 -n_2^2}\)

Q: Explain this image:
A: The allowed modes need to be between the light line for the core \(c_1\) and the cladding \(c_2\)

Definition of \(NA\)

\(NA\) for dielectric planar waveguide?

Q: Types of fibers, and advantages
A: GRIN fibers have lower mode dispersion (see path lenght difference)