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_psets/7.md

+---
+title: Problem Set 7
+---
+
+## (9.1)
+
+{:.question}
+Optics (as well as most of physics) can be derived from a global law as well as a local one, in this
+case Fermat’s Principle: a light ray chooses the path between two points that minimizes the time to
+travel between them. Apply this to two points on either side of a dielectric interface to derive
+Snell’s Law.
+
+
+## (9.2)
+
+### (a)
+
+{:.question}
+Use Fresnel’s equations and the Poynting vectors to find the reflectivity and transmissivity of a
+dielectric interface, defined by the ratios of incoming and outgoing energy.
+
+### (b)
+
+{:.question}
+For a glass–air interface (n = 1.5) what is the reflectivity at normal incidence?
+
+### (c)
+
+{:.question}
+What is the Brewster angle?
+
+### (d)
+
+{:.question}
+What is the critical angle?
+
+
+## (9.3)
+
+{:.question}
+Consider a wave at normal incidence to a dielectric layer with index $$n_2$$ and thickness $$d$$
+between layers with indices $$n_1$$ and $$n_3$$.
+
+### (a)
+
+{:.question}
+What is the reflectivity? Think about matching the boundary conditions, or about the multiple
+reflections.
+
+### (b)
+
+{:.question}
+Can you find values for $$n_2$$ and $$d$$ such that the reflection vanishes?
+
+
+## (9.4)
+
+{:.question}
+Consider a ray starting with a height $$r_0$$ and some slope, a distance $$d_1$$ away from a thin
+lens with focal length $$f$$. Use ray matrices to find the image plane where all rays starting at
+this point rejoin, and discuss the magnification of the height $$r_0$$.
+
+
+## (9.5)
+
+{:.question}
+Common CD players use an AlGaAs laser with a 790 nm wavelength.
+
+### (a)
+
+{:.question}
+The pits that are read on a CD have a diameter of roughly $$1 \si{\mu m}$$ and the optics are
+diffraction-limited; what is the beam divergence angle?
+
+### (b)
+
+{:.question}
+Assuming the same geometry, what wavelength laser would be needed to read $$0.1 \si{\mu m}$$ pits?
+
+### (c)
+
+{:.question}
+How large must a telescope mirror be if it is to be able to read a car’s license plate in visible
+light ($$\lambda \approx 600 \si{nm}$$) from a Low Earth Orbit (LEO) of 200 km?