## Diffraction limit and near field imaging ### Light as an electromagnetic waves Like all other waves, we care about the wave properties of electromagnetic waves. And directly from Maxwell equations, we may obtain in vacuum, electric and magnetic fields are orthogonal and oscillate in phase. Also as for our interest, light is a transverse wave and propagates at the speed of light $c$. Also when we speak of light, we care about $\omega$ and $k$, i.e., angular frequency and wavevector. This is because $\omega$ gives energy by $E = \hbar \omega$, while $k$ gives momentum by $p=\hbar k$. And energy and momentum are two typically conserved quantities. ### Velocity of electromagnetic waves When talking about the velocity of waves, we should first review the concept of **dispersion relation**. It describe the relation of $\omega$ and $k$ in a medium, therefore, one may obtain the frequency dependent quantities inside certain medium. For electromagnetic waves, we have two types of waves in general. >[!Note] >It is quite fun that if we perform Fourier transform for the Helmholtz equation in free space, both time and space, we will have > $\nabla^2 \mathbf{E} = \frac{1}{c^{2}}\frac{\partial^2 \mathbf{E}}{\partial t^2}$ > change into > $-\mathbf{k^{2}} \tilde{\mathbf{E}}+\frac{\omega^{2}}{c^{2}} \tilde{ \mathbf{E}} = 0$ > This is exactly the dispersion relation. In other words, any wave satisfy the dispersion (in vacuum) will be a possible form of solution to the wave equation, which is reasonable. **Phase velocity**: $v_{p} = \frac{\omega}{k}$, this is the velocity for single sine wave at $\omega$. ![[Drawing 2024-03-13 16.58.55.excalidraw.svg]] **Group velocity**: $v_{g}=\frac{\mathrm{d} v}{\mathrm{d} k}$, this is the velocity for actual signal carried by the waves. In other words, the velocity of wave packet, or envelope. >[!Note] >A good illustration is give in https://en.wikipedia.org/wiki/Group_velocity. > [!Example] > While group velocity in wave optics describes the envelope speed of modulated light waves, the same concept applies in quantum mechanics to matter waves. > For a free particle, the wavefunction takes the form of a plane wave: $\psi(x,t) = e^{i(kx - \omega t)}$, with the dispersion relation $\omega = \frac{\hbar k^2}{2m}$. > Constructing a narrow-band wave packet centered at $k_0$, the group velocity is given by: > $ > v_g = \frac{d\omega}{dk} = \frac{\hbar k_0}{m} = \frac{p}{m} > $ > This exactly matches the classical velocity of a particle with momentum $p$. > Physically, the group velocity represents the propagation speed of the wave packet envelope, corresponding to the particle's motion in classical mechanics. ### Wavevector as photon momentum ![[Drawing 2024-03-13 17.46.30.excalidraw.svg]] Because we have the momentum being $\hbar k$, due to uncertainty principle, we have $\Delta x \cdot \Delta p_{x} \geq \frac{\hbar}{2}$ $\Delta x \cdot \Delta k_{x} \geq \frac{1}{2}$ This means if we wanna limit $\Delta x$ under a wavelength (as an example), the $\Delta k_{x}$ should be $\frac{1}{2\lambda}$, and $k$ being $\pm \frac{1}{4\lambda}$ in each directions, this can be converted into a diffraction angle of $\pm 2.3 ^\circ$. And if we consider the smallest possible focus, we have $\Delta x = \frac{1}{2\Delta k_{x}}$. And because the minimum range for diffraction angle $\theta$ is $\pm 90^{0}$, we have $k_{x} = \pm \frac{2\pi}{\lambda}$, $\Delta k_{x} = \frac{4\pi}{ \lambda}$. This gives the the diffraction limit as $\Delta x = \frac{\lambda}{8\pi}$. This can also be interpreted as $k_{x}$ as spatial frequency, if we consider this in Fourier space, then smaller pattern would only be possible to investigate with small wavelength, i.e., high spatial frequency or large $k$. Because of this diffraction limit, with a given specific numerical aperture $NA$, the maximum resolution would be determined under *Rayleigh criterion*, which is critical in all types of microscopes, including [[Index (Characterization)#EM related characterizations|EMs]]. This also limits the number of modes an optical fiber could carry in maximum, if the size is determined. And since it affect the focusing of the light beam, such limit makes having characterization of nano-sized object very difficult due to inefficient light-matter interact due to focus mismatch, and causing focusing intense light beam onto material at a spot (which is critical for nonlinear optics study) very hard. ### Diffraction limit and near field But the good news is, this limit can be overcome if we only consider the near field case. In previous cases, we have a propagating light along $z$ direction, and therefore limited by $k_{x}$ and $k_{y}$. A larger $k_{x}, k_{y}$ could make the focal spot smaller. Since the wavelength is fixed, as $|\mathbf{k}| = \sqrt{k_{x}^{2} +k_{y}^{2} +k_{z}^{2}} = \frac{2\pi}{\lambda}$ We only have to make the $k_{z}^{2}$ term smaller than 0. This would cause the wave no longer propagating and being evanescent, as $z$ direction would have an exponentially decay. >[!Note] >The full form can be see from the Helmholtz equation, consider we have a plane wave electric field $\mathbf{E} = \mathbf{E_{0}} U(r)$, and plug it into the Helmholtz equation we had > $\nabla^2 \mathbf{E} - \frac{1}{c^{2}}\frac{\partial^2 \mathbf{E}}{\partial t^2} = 0$ > $\nabla^2 U(r) - |\mathbf{k}|^{2}\frac{\partial^2 \mathbf{U}}{\partial t^2} = 0$ > Do a Fourier transform in $x, y$ directions, we have > $-(k_{x}^{2} +k_{y}^{2}) V + \frac{\partial^{2}V}{\partial z^{2}} + |\mathbf{k}|^{2} V = 0$ > $ \frac{\partial^{2}V}{\partial z^{2}} + (|\mathbf{k}|^{2}-k_{x}^{2} +k_{y}^{2}) V = 0$ > $V(k_{x}, k_{y}, z) = V(k_{x}, k_{y}, 0) e^{i\gamma z} $ > Here $\gamma$ is a propagation factor, we have > $\gamma = \begin{cases} \sqrt{\frac{\omega^2}{c^2} - k_x^2 - k_y^2} & \text{for } \frac{\omega^2}{c^2} \geq k_x^2 + k_y^2, \\[10pt] i\sqrt{k_x^2 + k_y^2 - \frac{\omega^2}{c^2}} & \text{for } \frac{\omega^2}{c^2} < k_x^2 + k_y^2. \end{cases}$ > So for near field, we have an exponentially decay, and could not propagate to far field. >[!Info] >- For Fourier transform, see [[Frequency domain and time domain]], [[Real space and reciprocal space]] as an introduction. And [[Index (Fourier Analysis)]] for more information. >- As for books, see *Greffet, Jean-Jacques, 'Introduction to near-field optics and plasmonics', in Claude Fabre, and others (eds), Quantum Optics and Nanophotonics (Oxford, 2017; online edn, Oxford Academic, 24 Aug. 2017)* >- One application for near field imaging is NSOM, Near-field scanning optical microscope https://en.wikipedia.org/wiki/Near-field_scanning_optical_microscope.