Nanophotonics Lec 2

Derivations: ω ∝ k (Photon) and ω ∝ k² (Electron)

Derivations in Free Space

(1) Photon: \( \omega \propto k \)   |   (2) Electron (non-relativistic): \( \omega \propto k^2 \)

A. Photon in Free Space: Why \( \omega \propto k \)

Massless particle + Maxwell/Relativity
Key idea: \(E = pc\)

For light in free space, energy is proportional to momentum. Using Planck and de Broglie relations gives a linear dispersion relation.

  • Step 1: Photon energy–frequency relation (Planck) \( E = \hbar \omega \)
  • Step 2: Photon momentum–wavevector relation (de Broglie) \( p = \hbar k \)
  • Step 3: Relativistic relation for a massless particle (photon) \( E = pc \)
    More general: \(E^2 = (pc)^2 + (mc^2)^2\). For photon, \(m=0\) → \(E=pc\).
  • Step 4: Combine Step 1, 2 and 3 \( \hbar\omega = (\hbar k)c \) Cancel \( \hbar \): \( \omega = ck \)
Result: \( \boxed{\omega = ck \;\Rightarrow\; \omega \propto k} \)
Phase velocity: \(v_p=\omega/k=c\) Group velocity: \(v_g=d\omega/dk=c\)

B. Free Electron (Non-Relativistic): Why \( \omega \propto k^2 \)

Massive particle + Schrödinger equation
Key idea: \(E = p^2/2m\)

A free electron (no potential) is described by the Schrödinger equation. A plane-wave solution leads to a quadratic dispersion relation.

  • Step 1: Free-particle Schrödinger equation \[ i\hbar \frac{\partial \psi}{\partial t} = -\frac{\hbar^2}{2m}\nabla^2 \psi \]
  • Step 2: Try a plane-wave solution \[ \psi(\mathbf{r},t)=A e^{i(\mathbf{k}\cdot \mathbf{r}-\omega t)} \]
  • Step 3: Compute derivatives
    \[ \frac{\partial \psi}{\partial t} = -i\omega \psi \quad\Rightarrow\quad i\hbar \frac{\partial \psi}{\partial t} = \hbar\omega \psi \]
    \[ \nabla^2 \psi = -k^2 \psi \quad\Rightarrow\quad -\frac{\hbar^2}{2m}\nabla^2\psi = \frac{\hbar^2 k^2}{2m}\psi \]
  • Step 4: Substitute back into Schrödinger equation
    \[ \hbar\omega\psi = \frac{\hbar^2 k^2}{2m}\psi \]
    Cancel \( \psi \) and one \( \hbar \): \(\;\; \omega = \frac{\hbar k^2}{2m}\)
Result: \( \boxed{\omega = \frac{\hbar k^2}{2m} \;\Rightarrow\; \omega \propto k^2} \)
Group velocity: \(v_g=d\omega/dk=\hbar k/m\) Curved dispersion → wavepacket spreads

Quick Alternative (Short Derivation)

Using \(E=\hbar\omega\) and \(p=\hbar k\)
\[ E=\frac{p^2}{2m},\quad p=\hbar k,\quad E=\hbar\omega \]
\[ \hbar\omega = \frac{(\hbar k)^2}{2m} \Rightarrow \omega = \frac{\hbar k^2}{2m} \]

C. One-Line Nanophotonics Connection

Dispersion engineering
Why this matters

In nanophotonics we try to modify the photon dispersion (normally linear \( \omega=ck \)) using waveguides, cavities, photonic crystals, or plasmonic structures, so that light can be confined, slowed, or made to have bandgaps (electron-like behavior).

Comments

Post a Comment

Popular posts from this blog

Diagonalization of Matrix

Diagonalization of 2×2 Matrix using Similarity (SAS⁻¹ / S⁻¹AS) Diagonalization of a 2×2 Matrix using Similarity Transformation Method: Choose eigenvectors as columns of \(S\), then compute \(D = S^{-1}AS\) (equivalently \(A = SDS^{-1}\)). Given \[ A=\begin{pmatrix}5&4\\[2pt]1&2\end{pmatrix} \] We will diagonalize \(A\) using \(D=S^{-1}AS\), where \(S\) is formed from eigenvectors. Step 1 Find the eigenvalues Characteristic equation: \(\det(A-\lambda I)=0\) \[ \det(A-\lambda I)= \begin{vmatrix} 5-\lambda & 4\\ 1 & 2-\lambda \end{vmatrix} =(5-\lambda)(2-\lambda)-4 \] \[ =(10-7\lambda+\lambda^2)-4=\lambda^2-7\lambda+6 \] ...
Read more

Short Question Asso Leg poly_Orthogonal ppt

3-Mark Question: Orthogonality of Associated Legendre Polynomials 3 Marks Short Question: Orthogonality of Associated Legendre Polynomials Question (3 Marks) State the orthogonality property of associated Legendre polynomials \(P_\ell^{\,m}(x)\) for fixed \(m\) on the interval \([-1,1]\). \[ \text{(Write the standard relation and define } \delta_{\ell n}\text{.)} \] Expected Answer (Brief) \[ \boxed{ \int_{-1}^{1} P_\ell^{\,m}(x)\,P_n^{\,m}(x)\,dx = \frac{2}{2\ell+1}\,\frac{(\ell+m)!}{(\ell-m)!}\,\delta_{\ell n} } \] where \(\delta_{\ell n}=1\) if \(\ell=n\) and \(0\) if \(\ell\neq n\). Scheme of Evaluation Marking Scheme (Total: 3 Marks) ...
Read more

MOF trial

Lab Handout: Zn–Histidine Bio-MOF Synthesis (Beginner SOP) LAB HANDOUT (Beginner SOP) Synthesis of Zn–Histidine Bio-MOF (Real Metal–Organic Framework / Coordination Framework) Target output: ≥ 4 g (by combining 2 identical batches)  |  Recommended level: MSc Nanoscience project  |  Balance: 4-digit (0.0001 g) OK 1) Objective To synthesize a real MOF / coordination framework using Zn 2+ (metal node) and L-histidine (organic linker) under controlled pH. The key is to keep pH around 6.5 so that we form a coordination network (MOF) and avoid Zn(OH) 2 / ZnO formation . 2) Safety (Mandatory) Wear lab coat, gloves, and goggles throughout. NaOH is corrosive —add slowly; wash spills immediately with plenty of water. Autoclave safety: do not fill above 80%; do not open while hot/pressurized; cool naturally. Dispose washings as instructed (d...
Read more