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--- courses:ast201:6 [2023/08/13 00:39] – [Optical materials] asad
+++ courses:ast201:6 [2023/11/25 23:43] (current) – asad
@@ Line 23: / Line 23: @@
 A **coherent** source emits all light in phase, and in this case the geometrical wavefronts also correspond to surfaces of constant phase called **phase fronts**.
+{{:courses:ast201:n-lambda.png?nolink|}}
 ==== - Reflection and refraction ====
@@ Line 63: / Line 65: @@
 reflectance  $R=0$  for TM polarization and only the TE polarization is reflected.
-=== Spherical surface and interface ===
+==== - Spherical interfaces ====
 {{:courses:ast201:reflection-spherical.png?nolink&650|}}
@@ Line 84: / Line 86: @@
 where  $P_{12}$  is the power in diopters. Positive  $P$  tells you how strongly converging a lens is, and vice versa.
-===== - Mirrors, lenses and antennae =====
+===== - Mirrors and lenses =====
 {{:courses:ast201:thick-lens.png?nolink|}}
@@ Line 119: / Line 121: @@
 Optical fibres are used in astronomy for transferring light from the focal plane to somewhere else for further analysis. Sometimes putting a large spectrometer or detector or sensor at the focal plane is not practical as it obstructs the optical path, so optical fiber openings are placed at the focal plane instead.
-==== - Optical materials ====
-{{:courses:ast201:n-lambda.png?nolink|}}
-Surface coating can increase or decrease reflectance. A coating with 1/4 wavelength thick film will create two reflected waves that are exactly 1/2 wavelength out of phase with each other; they will destruct each other if they have the same amplitude. If the glass and the coating have indices  $n_s$  and  $n_c$ , then the amplitude of the two reflected beams will be equal if
- $n_c = \sqrt{n_s}$ .
 ==== Prisms ====
@@ Line 211: / Line 207: @@
 ===== - Aberrations =====
+==== - Monochromatic wavefront aberrations ====
+For reflecting systems, chromatic aberration does not occur. But even monochromatic light (light of just one wavelength) can have differing focus after getting reflected from a mirror.
+{{:courses:ast201:optical-planes.webp?nolink&800|}}
+The plane of the diagram (a) is called the **tangential** plane and the plane perpendicular to it is called the **sagittal** plane. The chief ray  $V'VF$  and the ray  $CFBFC$  are shown on the tangential plane. The test ray  $P'PF$  could be on a different plane than the tangential and sagittal.
+Diagram (b) shows the plane of the aperture and a point here can be described by the spherical polar coorcinates  $\phi$  and  $\rho$  (radius) and the distance  $b=R\sin\theta$  that gives the distance of the source from the optical axis,  $R$  being the radius of curvature.
+{{:courses:ast201:perfect-actual-rays.png?nolink|}}
+The focal plane for the perfect focus is called the **Gaussian image plane**, and the distance from this image plane of the point where the actual ray meets the optical axis can be measured using the quantity  $\Delta w$  which is located by the coordinates  $\rho,\phi$  and the distance  $b$ .
+In **third-order aberration theory** (where  $\sin\theta \approx \theta - \theta^3/3!$ ), it has been proved that
+ $\Delta w(\rho,\phi,b) = C_1 \rho^4 + C_2 \rho^3 b \cos\phi + C_3 \rho^2 b^2 \cos^2\phi + C_4 \rho^2 b^2 + C_5 \rho b^3 \cos\phi$
+where the coefficients  $C_i$  depend on the detailed shape of the reflecting surface. The five terms in this equation are responsible for five different types of aberrations called the **Seidel aberrations** as given below.
+^ Aberration ^ Functional dependence ^
+| Spherical aberration |  $\rho^4$  |
+| Coma |  $\rho^3 b \cos\phi$  |
+| Astigmatism |  $\rho^2 b^2 \cos^2\phi$  |
+| Curvature of field |  $\rho^2 b^2$  |
+| Distortion |  $\rho b^3 \cos\phi$  |
+The list is given in order of importance. Because spherical aberration (SA) is the most prominent, it is corrected for first, and then the effect of 'coma' is corrected, and so on.
+==== - Surface shapes ====
+Spherical mirrors are the easiest to make, but other shapes are created by rotating a conic section around its axis of symmetry.
+{{:courses:ast201:surface-shapes.png?nolink&700|}}
+Panel (a) shows the  $y-z$  coordinates of the system, where  $z$ -axis is along the optical axis and also the axis of symmetry of a conic section. The cross-section of a mirror surface satisfies the equation
+ $y^2 = 2Rz - (1-e^2)z^2$
+where  $e$  is the eccentricity of the conic and  $R$  the radius of curvature at the vertex.
+^ Shape ^ Eccentricity ^
+| Sphere |  $0$  |
+| Oblate ellipsoid |  $0<e<1$  |
+| Prolate ellipsoid |  $e^2<0$  |
+| Paraboloid |  $e=1$  |
+| Hyperboloid |  $e>1$  |
+==== - Spherical aberration ====
+All aberrations except SA vanishes on axis when  $b=0$ , but SA remains. If the source is exactly at the center of the field of view, SA is the only aberration to take care of.
+As shown below, two rays coming from an on-axis source strike the axis at two different points  $F$  and  $G$  where G is at a distance  $0.845(R/2)$  from the vertex.
+{{:courses:ast201:spherical-aberration.png?nolink&550|}}
+The focal length of a conic of revolution
+ $f(\rho) = \frac{R}{2} - (1+K) \left[\frac{\rho^2}{4R}+\frac{(3+K)\rho^4}{16R^3}+...\right]$
+where the first term gives  $R/2$  gives the Gaussian focus,  $\rho^2$  is a third-order aberration term and  $\rho^4$  is a fifth-order aberration term. The **conic constant**  $K=-e^2$ .
+The actual image is blurred and formed not in the Gaussian plane but in the **plane of least confusion**.
+Large focal ratio minimizes both chromatic and spherical aberration of lenses. The the blur due to SA can be reduced down to the size of the seeing disk almost.
+For mirrors, a paraboloid has zero SA, because  $K=-1$ .
+A Schmidt telescope modifies a spherical reflector to have minimal SA.
+==== - Coma ====
+Before the invention of photography, astronomers did not care about off-axis sources.
+Among the four off-axis aberrations, coma and astigmatism degrade image resolution, and the curvature of field and distortion only change the position of the course.
+Both coma and SA are problematic mainly for large apertures because of the large exponent of  $\rho$ .
+Coma depends on  $\rho^3 b \cos\phi$ , so it increases with distance of the object from axis.
+{{:courses:ast201:coma.png?nolink&700|}}
+In this sagittal plane diagram, rays are coming off-axis at an angle  $\theta$ , here we are talking about a constant distance from the vertex  $\rho$ . The rays form a ring-shaped offset from the Gaussian focus. The farther the ray (its meeting point on the Gaussian plane) is from the Gaussian focus, the bigger the ring. So the overall offset creates a shape like a comet, hence the name coma.
+The angular size of the blur due to coma
+ $L = A\frac{bD^2}{f^3} = A\theta \mathcal{R}^{-2}$
+where  $D$  is the diameter of the aperture,  $\mathcal{R}$  is the focal ratio, and  $A$  depends on the shape of the surface.
+An optical system with neither SA nor coma is called **aplanatic** which can only be achieved by using multiple optical elements together.
+==== - Astigmatism ====
+Astigmatism depends on  $b^2\rho^2 \cos\phi$  and, hence, it increases more rapidly than coma for off-axis points in an image.
+In the sagittal plane,  $\phi=90^\circ$  or  $270^\circ$  and, hence, coma is zero. It is maximum in the tangential plane ( $\phi=0^\circ$  or  $180^\circ$ ).
+{{:courses:ast201:astigmatism.png?nolink&800|}}
+The tangential rays focus on the Gaussian focus on the sagittal plane, and this focus is the tangential focus. The sagittal rays focus on a sagittal focal point  $S$  farther away. The compromise focus is somewhere in between.
+The image of a star at the compromise focus has a length due to astigmatism
+ $L_{astig} = B\theta^2 \mathcal{R}^{-1}$
+where  $B$  depends on the detailed structure of the aperture surface.
+A Schmidt-Cassegrain system is anastigmatic aplanat.
+==== - Field curvature ====
+{{:courses:ast201:field-curvature.png?nolink&500|}}
+If the previous aberrations are absent, image should be formed at the focal plane at a distance  $f=R/2$  from the vertex. However, the actual imaging surface is not a //plane// but a //curved surface// called the **Petzval surface**. If the object is on-axis, there is no problem, but the off-axis sources are necessarily out-of-focus because they are not imaged on the focal plane, but on the curved Petzval surface. This is field curvature.
+The detector is usually placed on the Petzval surface. If the detector is small, the curvature can be ignored. But for large detectors, this cannot be ignored. In the past, glass photographic plates used to be bent to match the curvature of the Petzval surface. However, modern CCDs cannot be bent. So corrector plates or lenses are used.
+==== - Distortion ====
+{{:courses:ast201:distortion.png?nolink|}}
+Straight lines on the sky become curved lines in the focal plane. The **pincushion** and **barrel** distortions are shown above.
+Below you see the visual representation of all the distortions for both on-axis and off-axis objects.
+{{:courses:ast201:spot-diagram.png?nolink|}}