This is a list of commonly used formulas and diagrams used in Geometrical Optics. The sign convention follows the one adopted by the College of Optical Sciences at the University of Arizona. For a more complete list and guide please refer to “Field Guide to Geometrical Optics” by J.E. Greivenkamp.

  • Snell’s law of refraction:

    \[n_1 \sin \theta_1 = n_2 \sin \theta_2\]

  • Total internal reflection (TIR):

    \[\sin \theta_c = \frac{n_2}{n_1}\]

    subject to

    \[n_2 < n_1\]

  • Power of an optical surface:

    \[\phi = (n' - n) C = \frac{n' - n}{R}\]

  • Curvature:

    \[C = \frac{1}{R}\]

  • Effective Focal Length:

    \[f = f_E = \frac{1}{\phi}\]

  • The front and rear focal lengths are related to the EFL:

    \[f_F = -n f_E\] \[f'_R = n' f_E\] \[\frac{f'_R}{f_F} = - \frac{n'}{n}\]

  • Reflective surfaces:

    In a reflection $$ n’ = -n$$

    \[\phi = -2nC = -\frac{2n}{R}\] \[f_F = f'_R = -\frac{n}{\phi} = -n f_E = \frac{R}{2} = \frac{1}{2C}\]

  • Gaussian Equations:

    \[m = \frac{z' / n'}{z / n} = - \frac{z'}{z} \frac{f_F}{f'_R}\] \[\frac{n'}{z'} = \frac{n}{z} + \frac{1}{f_E}\]

  • Longitudinal magification:

    \[\Delta z = z_2 - z_1\] \[\Delta z' = z'_2 - z'_1\] \[m_1 = \frac{h'_1}{h_1}\] \[m_2 = \frac{h'_2}{h_2}\] \[\frac{\Delta z'}{\Delta z} = - \left( \frac{f'_R}{f_F} \right) m_1 m_2\] \[\frac{\Delta z' / n' }{\Delta z / n} = m_1 m_2\]

    As the plane seperation goes to zero, the local longitudinal magnification is obtained:

    \[\overline{m} = \left( \frac{n'}{n} \right) m^2\] \[\frac{ \Delta z' / n'}{ \Delta z / n} = m^2\]

  • Gaussian reduction:

    Gaussian reduction is the process of combining elements two at a time into a single equivalent focal systems

    \[\phi = \phi_1 + \phi_2 - \phi_1 \phi_2 \tau\] \[\tau = \frac{t}{n_2}\] \[\frac{d}{n} = \frac{\phi_2}{\phi} \tau\] \[\frac{d'}{n'} = -\frac{\phi_1}{\phi} \tau\]

    \(n_2\) is the index of refraction of the space between the front and rear principal planes

    \(\phi\) is the power of the combined system

    \(\phi_1\) is the power of the first element or system

    \(\phi_2\) is the power of the second element or system

    \(d\) is the shift in object space of the front system principal plane

    \(d'\) is the shift in image space of the rear system principal plane

    \(t\) is the directed distance in the intermediate optical space from the rear principal place of the first system to the front principal plane of the second system.

  • Power of a thin lens in air:

    \[\phi = \left( n - 1 \right) \left( C_1 - C_2 \right)\] \[d = d' = 0\]

    The principal planes and nodal points are located at the lens.

  • Vertex distances:

    Back focal distances is the distance from the rear vectex to the rear focal point.

    \[\text{BFD} = f'_R + d'\]

    Front focal distance is the distance from the front vertex to the front focal point.

    \[\text{FFD} = f_F + d\] \[s = z + d\] \[s' = z' + d'\]

  • Paraxial optics:

    \[\omega = n u\]

    Refraction (and reflection):

    \[n' u' = n u - y \phi\] \[\omega' = \omega - y \phi\]

    Transfer:

    \[y' = y + u' t'\] \[y' = y + \omega' \tau'\]

    • Rear Cardinal Points:

      Trace a ray parallel to the axis in object space ( \(u = \omega = 0\) ). This ray must go through the rear focal point \(F'\)

      \[\phi = - \frac{n' u'_k }{y_1} = - \frac{\omega'_k}{y_k}\] \[\text{BFD} = - \frac{n' y_k}{\omega'_k} = -\frac{y_k}{u'_k}\] \[d' = \text{BFD} - f'_R\]

    • Front Cardinal Points:

      Trace a ray from the system front focal point \(F\) that emerges parallel to the axis in image space. The reverse raytrace equations are used to work from image space back to object space.

      \[\phi = \frac{n u_1}{y_k} = \frac{\omega_1}{y_k}\] \[\text{FFD} = - \frac{n y_1}{\omega_1} = -\frac{y_1}{u_1}\] \[d = \text{FFD} - f_F\]