Added CTF tableu and interactive line plot.

TEM.md

@@ -49,7 +49,7 @@ One example of the importance of aberrations and wavfunction modification is sho
 ```{figure} #app:tem_contrast
 :name: fig_tem_phase
 :placeholder: ./static/tem_contrast.png
-**TEM imaging of covid spike protein**: (left) In focus, (middle) defocued, and (right) Zernike phase plate TEM images. [Covid structure](https://www.rcsb.org/structure/3jcl) from {cite}`walls2016cryo`
+**TEM imaging of COVID spike protein**: (left) In focus, (middle) defocued, and (right) Zernike phase plate TEM images. [Covid structure](https://www.rcsb.org/structure/3jcl) from {cite}`walls2016cryo`
 ```
 
 For the same, dose the image contrast can be increased by modfying the wavefunctions through the introduction of aberrations. This is most often done with defocus, which is illustrated in the middle pannel. The resulting image has improved contrast, making it easier to visualize the covid spike protein. The middle pannel shows the defocused image, but the aberrations can often be corrected for later in post-processing. 

aberrations.md

@@ -5,10 +5,15 @@ numbering:
 label : CTF_page
 ---
 
+(wave-aberrations)=
+## Wave aberrations
+
 An ideal lens forms a spherical wave converging on or emerging from a single point. This means that a plane wave will be converted to a spherical wave converging at the focal point of the lens, and the image of a point source is also a point. In STEM we want the objective lens to produce the smallest possible probe and in HRTEM we want the objective lens to produce a perfect magnified image of the sample. In most cases, both of these objectives require that we minimize all aberrations as much as possible. 
 
 However, while the last decades has seen enormous improvements in the optics of electron microscopes, they are far from ideal optical system. Imperfections causes the focused wave front to deviate from the ideal spherical surface. 
 
+### Contrast transfer and the point-spread function
+
 This deviation is typically expressed as a phase error or the aberration function, $\chi(\bm{k})$. Given a Fourier space wavefunction $\Psi_0(\bm{k})$ entering the lens, the wavefunction after passing through that lens can thus be expressed as 
 ```{math}
     \Psi(\bm{k}) = \Psi_0(\bm{k}) \mathrm{e} ^ {-i \chi(\bm{k})}.
@@ -18,22 +23,82 @@ Another, possibly more intuitive way of framing the phase error is the point spr
 ```{math}
     \mathrm{PSF}(\bm{r}) = \mathscr{F}^{-1}_{\bm{k} \rightarrow \bm{r}}\left\{ \mathrm{e} ^ {-i \chi(\bm{k})} \right\} ,
 ```
-the point spread function describes how an imaging system responds to a point source. In STEM, the PSF would be the image of the probe, given an infinite objective aperture, in HRTEM the PSF would be how a perfect point source is imaged by an objective lens with an infinite collection angle.  
+the point spread function describes how an imaging system responds to a point source. In STEM, the PSF would be the image of the probe, given an infinite objective aperture, in HRTEM the PSF would be how a perfect point source is imaged by an objective lens with an infinite collection angle.
 
 The phase error can be expressed in different ways, however, it is traditionally written as a series expansion. One popular choice is to write it as an expansion in polar coordinates
 ```{math}
     \chi(k, \phi) = \frac{2 \pi}{\lambda} \sum_{n,m} \frac{1}{n + 1} C_{n,m} (k \lambda)^{n+1} \cos\left[m (\phi - \phi_{n,m}) \right] ,
 ```
 where $k$ is the Fourier space radial coordinate and $\phi$ is the corresponding azimuthal coordinate. The coefficient $C_{n,m}$ represents the magnitude of an aberration and $\phi_{n,m}$ gives a direction to that aberration. There are other ways of representing the aberration function, you might have seen the Cartesian representation where some parameters has an "a" and "b" version.
 
-For an uncorrected microscope the dominant aberration is ther third order spherical aberration. Assuming the other non-symmetric components have been well aligned by the user, the contrast transfer function simplifies to:
-```{math}
-    \chi(k) \approx \frac{2\pi}{\lambda}\left( \frac{\lambda^2 k^2}{2} \Delta f + \frac{\lambda^4 k^4}{4} C_s \right) \quad .
+#### Visualizing the CTF and PSF
+
+Below we visualize each aberration by mapping the phase to different colors. We can also visualize each phase aberration by showing its effect on a small probe wave function using the PSF. Note that the magnitude of the aberrations are scaled by a power law are scaled by the radial order ($n$) according to a power law.
+
+````{tab-set}
+```{tab-item} CTF
+![ctf](./static/ctf.png)
+```
+```{tab-item} PSF
+![ctf](./static/psf.png)
 ```
-Here we used the common aliases of the aberration coefficients, so $C10 = -\Delta f$ is the negative defocus and $C_{30} = C_s$ is the third order spherical aberration. 
+````
+
+We can also visualize this interactively as shown in [](#fig_ctf_psf).
 
 ```{figure} #app:ctf_psf
 :name: fig_ctf_psf
 :placeholder: ./figures/ctf_psf.png
-**Interactive widget showing the influence of common aberrations on the point spread function (PSF) and contact transfer function (CTF) **: 
+**Interactive widget showing the influence of common aberrations on the point spread function (PSF) and contact transfer function (CTF).**
+```
+
+### Aperture
+
+In practice, the maximum transferred frequency is limited by the aperture of the objective lens. This is conventionally described as a multiplication with the aperture function:
+
+$$
+    \psi_{\mathrm{image}}(k, \phi) = A(k) \exp[-i \chi(k, \phi)] \psi_{\mathrm{exit}}(k, \phi),
+$$
+
+where $A(k)$ is the aperture function
+
+$$
+    A(k) = \begin{cases} 1 & \text{if } x \leq k_{cut} \\ 0 & \text{if } x > k_{cut} \end{cases} .
+$$
+
+We typically cut off the `CTF` at the angle corresponding to the Scherzer [point resolution](https://en.wikipedia.org/wiki/High-resolution_transmission_electron_microscopy#Scherzer_defocus), which is defined as the angle where the phase of the `CTF` crosses the abscissa for the first time (`crossover_angle`).
+
+### Partial coherence (quasi-coherent)
+Partial coherence acts similarly to the aperture function to dampen the high spatial frequencies of the signal. Partial coherence may be approximated by multiplying the `CTF` by envelope functions:
+
+$$
+    \psi_{\mathrm{image}}(k, \phi) = \mathrm{CTF}(k, \phi) \psi_{\mathrm{exit}}(k, \phi),
+$$
+
+where the $\mathrm{CTF}$ is now given as
+
+$$
+    \mathrm{CTF}(k, \phi) = E_t(k) E_s(k) A(k) \exp[-i \chi(k, \phi)]
+$$
+
+and $E_t(k)$ and $E_s(k)$ are the temporal and spatial envelopes, respectively (see [abTEM documentation](https://abtem.github.io/doc/user_guide/walkthrough/contrast_transfer_function.html#partial-coherence-quasi-coherent) for more detail).
+
+### Spherical-aberration limited contrast transfer
+
+For an uncorrected microscope the dominant aberration is the third order spherical aberration ($C_{3,0}$). Assuming the other non-symmetric components have been well aligned by the user, the contrast transfer function simplifies to:
+```{math}
+    \chi(k) \approx \frac{2\pi}{\lambda}\left( \frac{\lambda^2 k^2}{2} \Delta f + \frac{\lambda^4 k^4}{4} C_s \right) \quad .
 ```
+Here we used the common aliases of the aberration coefficients, so $C10 = -\Delta f$ is the negative defocus and $C_{30} = C_s$ is the third order spherical aberration.
+
+#### Visualizing the contrast transfer function
+
+The contrast transfer function can also be plotted as a line profile, which *by convention* display the imaginary part of complex exponential of the phase error. The horizontal axis denotes a specific spatial frequency, while the vertical axis how the contrast at that spatial frequency is transferred.
+
+Below in [](#fig_ctf_line), we visualize the CTF interactively for different primary beam energies, defocus values, spherical aberration, aperture cutoff, and temporal and spatial incoherences.
+
+```{figure} #app:ctf_line
+:name: fig_ctf_line
+:placeholder: ./static/ctf_line.png
+**Interactive widget showing the influence of common coherent and incoherent parameters on the contact transfer function (CTF).**
+```

inputs.md

@@ -46,7 +46,7 @@ In the widget below, we have oriented the strontium titanate structure along the
 ```{figure} #app:sto_supercell
 :name: fig_sto_supercell
 :placeholder: ./static/sto_supercell.png
-**Interactive widget showing supercell construction for the STO(110) supercell.
+**Interactive widget showing supercell construction for the STO(110) supercell**:
 ```
 
 Since the positions and atomic numbers are just `NumPy` arrays, they can be modified in-place. Below, we create an SrTiO<sub>3</sub>/LaTiO<sub>3</sub> interface by changing the atomic numbers of the Sr atoms with a $y$-coordinate less than $7.5 \ \mathrm{Å}$ in a (3,4,10) supercell oriented along the (110) zone axis. This interface created from a will be later used for [STEM image simulations](#stem-image-simulation).

myst.yml

@@ -9,15 +9,15 @@ project:
   banner: banner.png
   github: https://github.com/msa-em/guide-tem-simulation
   jupyter: true
-  #  jupyter:
-  #    binder:
-  #      url: https://y2j74k9mn0bz.curvenote.dev/services/binder/
-  #      repo: msa-em/guide-tem-simulation
-  #      ref: main
-  #  jupyter:
-  #    server:
-  #      url: 'http://localhost:8888'
-  #      token: '512ac78f14e1141db1fac17e8b4099c1e5bc7d589518b38c'
+  #jupyter:
+  #  binder:
+  #    url: https://y2j74k9mn0bz.curvenote.dev/services/binder/
+  #    repo: msa-em/guide-tem-simulation
+  #    ref: main
+  #jupyter:
+  #  server:
+  #    url: http://localhost:8891
+  #    token: toma-token-devel
   keywords:
     - Transmission electron microscopy
     - Scanning transmission electron microscopy
@@ -126,6 +126,7 @@ project:
         - file: notebooks/04_atomic_potentials.ipynb
         - file: notebooks/06.1_Atoms_STO-LTO.ipynb
         - file: notebooks/07_ctf_psf_widget.ipynb
+        - file: notebooks/07.2_ctf_lineplot_widget.ipynb
         - file: notebooks/08.1_TEM_imaging.ipynb
         - file: notebooks/08.2_TEM_diffraction.ipynb
         - file: notebooks/08.3_TEM_phase_plate.ipynb