docs: Adding basic lensing example to docs (#134)

* Add minimal lensing example, update basic intro tutorial

* Add minimal example to home page of website

* Update minimal example to include vmap and jacobian

---------

Co-authored-by: pre-commit-ci[bot] <66853113+pre-commit-ci[bot]@users.noreply.github.com>

README.md

@@ -25,6 +25,65 @@ Simply install caustics from PyPI:
 pip install caustics
 ```
 
+## Minimal Example
+
+```python
+import matplotlib.pyplot as plt
+import caustics
+import torch
+
+cosmology = caustics.cosmology.FlatLambdaCDM()
+sie = caustics.lenses.SIE(cosmology=cosmology, name="lens")
+src = caustics.light.Sersic(name="source")
+lnslt = caustics.light.Sersic(name="lenslight")
+
+x = torch.tensor([
+#   z_s  z_l   x0   y0   q    phi     b    x0   y0   q     phi    n    Re
+    1.5, 0.5, -0.2, 0.0, 0.4, 1.5708, 1.7, 0.0, 0.0, 0.5, -0.985, 1.3, 1.0,
+#   Ie    x0   y0   q    phi  n   Re   Ie
+    5.0, -0.2, 0.0, 0.8, 0.0, 1., 1.0, 10.0
+])  # fmt: skip
+
+minisim = caustics.sims.Lens_Source(
+    lens=sie, source=src, lens_light=lnslt, pixelscale=0.05, pixels_x=100
+)
+plt.imshow(minisim(x, quad_level=3), origin="lower")
+plt.axis("off")
+plt.show()
+```
+
+![Caustics lensed image](./media/minimal_example.png)
+
+### Batched simulator
+
+```python
+newx = x.repeat(20, 1)
+newx += torch.normal(mean=0, std=0.1 * torch.ones_like(newx))
+
+images = torch.vmap(minisim)(newx)
+
+fig, axarr = plt.subplots(4, 5, figsize=(20, 16))
+for ax, im in zip(axarr.flatten(), images):
+    ax.imshow(im, origin="lower")
+plt.show()
+```
+
+![Batched Caustics lensed images](../../media/minisim_vmap.png)
+
+### Automatic Differentiation
+
+```python
+J = torch.func.jacfwd(minisim)(x)
+
+# Plot the new images
+fig, axarr = plt.subplots(3, 7, figsize=(20, 9))
+for i, ax in enumerate(axarr.flatten()):
+    ax.imshow(J[..., i], origin="lower")
+plt.show()
+```
+
+![Jacobian Caustics lensed image](../../media/minisim_jacobian.png)
+
 ## Documentation
 
 Please see our [documentation page](https://caustics.readthedocs.io/en/latest/)

docs/source/intro.md

@@ -20,7 +20,65 @@ are ready to go! You can check out the tutorials afterwards to see some of
 caustics' capabilities. If you want to help out with building the caustics code
 base check out the developer installation instructions instead.
 
-### Contents
+## Minimal Example
+
+```python
+import matplotlib.pyplot as plt
+import caustics
+import torch
+
+cosmology = caustics.cosmology.FlatLambdaCDM()
+sie = caustics.lenses.SIE(cosmology=cosmology, name="lens")
+src = caustics.light.Sersic(name="source")
+lnslt = caustics.light.Sersic(name="lenslight")
+
+x = torch.tensor([
+#   z_s  z_l   x0   y0   q    phi     b    x0   y0   q     phi    n    Re
+    1.5, 0.5, -0.2, 0.0, 0.4, 1.5708, 1.7, 0.0, 0.0, 0.5, -0.985, 1.3, 1.0,
+#   Ie    x0   y0   q    phi  n   Re   Ie
+    5.0, -0.2, 0.0, 0.8, 0.0, 1., 1.0, 10.0
+])  # fmt: skip
+
+minisim = caustics.sims.Lens_Source(
+    lens=sie, source=src, lens_light=lnslt, pixelscale=0.05, pixels_x=100
+)
+plt.imshow(minisim(x, quad_level=3), origin="lower")
+plt.show()
+```
+
+![Caustics lensed image](../../media/minimal_example.png)
+
+### Batched simulator
+
+```python
+newx = x.repeat(20, 1)
+newx += torch.normal(mean=0, std=0.1 * torch.ones_like(newx))
+
+images = torch.vmap(minisim)(newx)
+
+fig, axarr = plt.subplots(4, 5, figsize=(20, 16))
+for ax, im in zip(axarr.flatten(), images):
+    ax.imshow(im, origin="lower")
+plt.show()
+```
+
+![Batched Caustics lensed images](../../media/minisim_vmap.png)
+
+### Automatic Differentiation
+
+```python
+J = torch.func.jacfwd(minisim)(x)
+
+# Plot the new images
+fig, axarr = plt.subplots(3, 7, figsize=(20, 9))
+for i, ax in enumerate(axarr.flatten()):
+    ax.imshow(J[..., i], origin="lower")
+plt.show()
+```
+
+![Jacobian Caustics lensed image](../../media/minisim_jacobian.png)
+
+## Contents
 
 ```{tableofcontents}