Geometrical Optics

Geometrical Optics

The Basic Law of Geometrical Optics

Index of Refraction

For most materials, \(\kappa_m \approx 1\), so \(\(v = \frac{1}{\sqrt{\kappa_e\kappa_m\epsilon_0\mu_0}} \approx \frac{1}{\sqrt{\mu_0\epsilon_0\kappa_e}} = \frac{c}{\sqrt{\kappa_e}}\)\)

Let's define the index of refraction of a material as \(\(v = \frac{c}{n}\)\)

It's obvious that \(n \approx \sqrt{\kappa_e} > 1\).

Total Internal Reflection

When the light incident into a material with a with a lower index of refraction, total internal reflection will occur if the incident angle satisfies \(\(\theta \ge \sin^{-1} \frac{n_2}{n_1}\)\)

Fermat's Principle

The time of a light ray traveling from point \(Q\) to point \(P\) is given by \(\(t = \sum \frac{\Delta l_i}{v_i} = \sum \frac{n_i \Delta l_i}{c} = \frac{\sum n_i \Delta l_i}{c}\)\)

Define the optical path length as \(\((QP) = \sum n_i \Delta l_i = \int_Q^P ndl\)\)

By Fermat's Principle, the actual path taken by the light ray is the one that minimizes the optical path length.

To form an image, the rays of light emitted from a point on an object must converge, so the optical path between the object point and the image point is equivalent.

Image Formation

For the refraction on a spherical surface, we have \(\(\frac{o^2}{n^2(o + r)^2} - \frac{i^2}{{n'}^2(i - r)^2} = -4r\sin^2 \frac{\phi}{2} \left[ \frac{1}{n^2(o + r)} + \frac{1}{{n'}^2(i - r)} \right]\)\)

\(o\) and \(i\) are determined at same time. For one spherical surface, there is only one group points that satisfy the equation for all \(\phi\).

For paraxial rays, suppose \(\phi \rightarrow 0\), then we have

\[ \begin{aligned} \frac{o^2}{n^2(o + r)^2} = \frac{i^2}{{n'}^2(i - r)^2} \\\\ \Rightarrow \frac{n(o + r)}{o} = \frac{{n'}(i - r)}{i} \\\\ \Rightarrow \frac{n}{o} + \frac{n'}{i} = \frac{n' - n}{r} \\ \end{aligned} \]

For the first focal point, \(i \rightarrow \infty\), \(o = f = \dfrac{n}{n' - n} r\).

For the second focal point, \(o \rightarrow \infty\), \(i = f' = \dfrac{n'}{n' - n} r\).

Therefore, we have the image formation equation: \(\(\frac{f}{f'} = \frac{n}{n'}, \quad \frac{f}{o} + \frac{f'}{i} = 1\)\)

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If we suggest that the incident light ray from left to right, then:

When \(Q, Q', C\) is at the left of the surface, \(o > 0, i < 0, r < 0\).

When \(Q, Q', C\) is at the right of the surface, \(o < 0, i > 0, r > 0\).

For example, for the reflection at the surface of a mirror, \(\(n = -n', \quad f = - \frac{r}{2}, \quad f' = \frac{r}{2}\)\)

So the image formation equation becomes \(\(\frac{(- \dfrac{r}{2})}{o} + \frac{\dfrac{r}{2}}{(-i)} = 1 \quad \Rightarrow \quad \frac{1}{o} + \frac{1}{i} = - \frac{2}{r}\)\)

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For the refraction, the lateral magnification is \(\(m = \frac{y'}{y} = - \frac{i\theta'}{o\theta} = - \frac{ni}{n'o}\)\)

For the reflection, the lateral magnification is \(\(m = - \frac{i}{o}\)\)

Thin Lens

For two closely adjacent surfaces, we have

\[ \begin{gathered} \begin{cases} \dfrac{f_1}{o_1} + \dfrac{f'_1}{i_1} = 1 \\ -o_2 \approx i_1 \\ \dfrac{f_2}{o_2} + \dfrac{f'_2}{i_2} = 1 \end{cases} \Rightarrow \frac{f}{o} + \frac{f'}{i} = 1 \end{gathered} \]

where

\[ \begin{gathered} f' = \frac{f'_1f'_2}{f'_1 + f_2} = \frac{\dfrac{n_L}{n_L - n}\cdot \dfrac{n'}{n' - n_L} \cdot r_1r_2}{\dfrac{n_L}{n_L - n}r_1 + \dfrac{n'}{n' - n_L}r_2} = \frac{n'}{\dfrac{n_L - n}{r_1} + \dfrac{n' - n_L}{r_2}} \\\\ f = \frac{f_1f_2}{f'_1 + f_2} = \frac{\dfrac{n}{n_L - n}\cdot \dfrac{n_L}{n' - n_L} \cdot r_1r_2}{\dfrac{n_L}{n_L - n}r_1 + \dfrac{n'}{n' - n_L}r_2} = \frac{n}{\dfrac{n_L - n}{r_1} + \dfrac{n' - n_L}{r_2}} \\\\ \frac{f'}{f} = \frac{n'}{n} \end{gathered} \]

If \(n = n' = 1\), then

\[ \begin{gathered} f = f' = \frac{1}{(n_L - 1)(\dfrac{1}{r_1} - \dfrac{1}{r_2})} \\\\ \frac{1}{i} + \frac{1}{o} = \frac{1}{f} \end{gathered} \]

For converging lens, \(r_1 > 0, r_2 < 0\), so \(f > 0, f' > 0\).

For diverging lens, \(r_1 < 0, r_2 > 0\), so \(f < 0, f' < 0\).

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Sign conventions: If \(Q\) is at the left of \(F\), \(x > 0\); If \(Q'\) is at the right of \(F'\), \(x' > 0\).

Define \(o = f + x\), \(i = f' + x'\), then we have the Newton's Form:

\[ \begin{gathered} \frac{1}{f + x} + \frac{1}{f' + x'} = \frac{1}{f} \\\\ \Rightarrow xx' = f^2 = ff' \end{gathered} \]

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The Lateral Magnification of thin lens is

\[ \begin{gathered} m_1 = - \frac{ni_1}{n_Lo_1}, \quad m_2 = - \frac{n_Li_2}{n'o_2} \\\\ m = m_1m_2 = \frac{ni_1}{n_Lo_1} \cdot \frac{n_Li_2}{n'(-i_1)} = - \frac{ni}{n'o} = - \frac{fi}{f'o} \end{gathered} \]

Shorthands

• Image formation equation on a spherical surface
• Image formation equation for a mirror
• Lateral magnification
• Lens maker’s equation
• Focal of lens
• Newton's Form