Learn & Review: Optics Tutorial - 3 - Algebraic Lens Imaging

Summary of First-Order Imaging Through a Lens Using Algebraic Equations

This tutorial, presented by Scott from OpticsRealm in September 2011, aims to explain algebraic methods for understanding how lenses form images. It highlights the complexities and potential confusion arising from different sign convention systems, particularly the Cartesian and empirical methods.

1. Understanding Sign Conventions

The tutorial introduces two primary methods for sign conventions in optics:

• Cartesian Coordinate System:

  • Objects to the left of the lens are considered negative.
  • Objects to the right of the lens are considered positive.
  • This system is based purely on geometry, irrespective of whether the object or image is real or virtual.
  • The speaker indicates a preference for this system.

• Empirical System:

  • A real image to the left of the lens has a positive distance.
  • A virtual image to the left of the lens (inside the lens) has a negative distance.
  • This system is based on the nature (real or virtual) and location of the object/image relative to the lens.

The speaker acknowledges that various sources use different conventions, which can be confusing, even for experienced optical engineers.

2. Imaging Through a Single Optical Surface

• Power of an Optical Surface: Defined as the difference in refractive indices between the two media divided by the radius of curvature.
  • Power = (n' - n) / R
• Focal Length: The inverse of power.
  • Focal Length (f) = R / (n' - n)
  • If the first medium is air (n=1), then f = R / (n' - 1).
• Front and Back Focal Lengths: These relate to optical thicknesses and are given by -n * f and n' * f respectively.

3. Imaging Through a Refractive Surface

For a surface with two different refractive indices, the imaging equation (using the Cartesian system) is:

• n' / s' = (n' - n) / R + n / s
  • s' is the image distance.
  • R is the radius of curvature.
  • n' and n are the refractive indices of the second and first media, respectively.
  • s is the object distance.

If the first medium is air (n=1), the equation simplifies.

4. Imaging Through a Reflective Surface (Mirror)

This is treated as a special case where the secondary medium's index is the negative of the incident medium (n' = -n). The consequence is that the focal length is half the radius of curvature (f = R / 2).

5. Imaging Through a Lens (Thin Lens Approximation)

For a thin lens (infinitesimally small thickness) using the Cartesian coordinate system, the fundamental imaging equation is:

• 1 / s' = 1 / f + 1 / s
  • f is the focal length.
  • s is the object distance.
  • s' is the image distance.

Important Note on Sign Convention: In this equation, if the image is real, s is negative.

• Example: A lens with f = 100 mm and a real object at s = 150 mm (which is -150 mm in Cartesian) results in an image distance s' = 300 mm.

The empirical equation for a lens is similar but includes a negative sign: 1/s' = 1/f - 1/s. The speaker expresses a dislike for this version.

6. Magnification

Magnification (M) describes the ratio of image height (h') to object height (h).

• For a lens in air: M = h' / h = s' / s
• For a lens not in air: M = s' * n / (s * n')

7. Conclusion and Future Topics

The speaker emphasizes that sign conventions can be confusing and plans to introduce an imaging nomograph in a future video to help visualize lens and mirror behavior. The tutorial references "Field Guide to Geometric Optics" by John Greivenkamp as a valuable resource.