Geometrical Optics

The formula for refraction at a spherical interface is:

$\frac{\mu_1}{u} + \frac{\mu_2}{v} = \frac{\mu_2 - \mu_1}{R}$

Derivation steps:

1. Consider a paraxial ray from object striking surface at angle i
2. Apply Snell's law: μ₁sin(i) = μ₂sin(r)
3. For small angles: sin(i) ≈ i and sin(r) ≈ r
4. Use geometric relationships between angles and distances
5. Combine and simplify equations

Sign convention:

• Distances measured along the direction of incident light are positive
• Distances measured opposite to incident light are negative
• R positive when center of curvature is in second medium
• R negative when center of curvature is in first medium

This equation works for all spherical interfaces when proper signs are used and is fundamental to lens design and optical systems analysis.

The exterior angle property:

• When looking at triangles formed in ray diagrams, the exterior angle equals the sum of two opposite interior angles.
• This is a fundamental property from Euclidean geometry.

Application to mirror equation derivation:

1. From the triangle containing the object point:

   • β = α + θ (where θ is the angle of incidence)

2. From the triangle containing the image point:

   • γ = α + 2θ (using the law of reflection)

3. Eliminating θ gives: 2β = α + γ

4. Under paraxial approximation:

   • α ≈ height/object distance = h/u
   • β = height/radius = h/R (exact)
   • γ ≈ height/image distance = h/v

5. Substituting: 2(h/R) = h/u + h/v

6. Simplifying: 1/u + 1/v = 2/R

This elegant application of geometric principles to optical systems allows us to predict image formation without complex calculations.

For a convex mirror:

$\frac{1}{u} + \frac{1}{v} = \frac{2}{R}$

Where:

• u = object distance (negative for real objects)
• v = image distance (positive for virtual images)
• R = radius of curvature (positive for convex mirrors)

This can also be written in terms of focal length: $\frac{1}{u} + \frac{1}{v} = \frac{1}{f}$

Where:

• f = focal length = R/2 (positive for convex mirrors)

The radius of curvature is twice the focal length: R = 2f

Application: A convex security mirror with R = 4m will form an image at v = +0.8m when an object is placed 4m away (u = -4m).

Concave Mirror (Object beyond focal point):

• Forms a real, inverted image
• Image is formed on the same side as the incident light
• Ray tracing rules:
  1. Rays parallel to principal axis reflect through the focus
  2. Rays through center of curvature reflect back along the same line
  3. Rays through focus reflect parallel to principal axis

Convex Mirror:

• Always forms a virtual, erect image
• Image is formed behind the mirror
• Image is smaller than the object
• Formula: 1/f = 1/u + 1/v (where f is negative for convex mirrors)

For both mirrors, the magnification is given by: m = -v/u

When viewing an object through a medium with refractive index μ:

• The apparent depth is different from the real depth

• For a medium with refractive index μ > 1 (like water or glass):

  • Apparent depth < Real depth
  • Specifically: Apparent depth = Real depth / μ

• The shift in position (Δt) toward the observer is:

  • Δt = (1 - 1/μ) × t
  • Where t is the thickness of the medium above the object

Example: A coin at the bottom of a 6 cm thick glass cube (μ = 1.5) appears to be only 4 cm deep because apparent depth = 6/1.5 = 4 cm.

The angle of deviation (δ) is the angular difference between the original direction of the incident ray and the final direction of the emergent ray after passing through a prism.

It is calculated using:

• δ = (i + i′) - A

Where:

• i = angle of incidence at the first surface
• i′ = angle of emergence at the second surface
• A = angle of the prism (the angle between the two refracting surfaces)

This relationship derives from the fact that the sum of internal angles r + r′ = A, where r and r′ are the angles of refraction at the first and second surfaces respectively.

To minimize spherical aberration:

1. For distant objects (parallel rays):
   • Place the surface with greater curvature (smaller radius) facing the incident light
2. For close objects:
   • Follow the "aplanatic points" rule - each surface should contribute to refraction proportional to its curvature

Key principle: Distribute the total deviation between the two surfaces rather than having most refraction occur at a single surface.

Example: For a planoconvex lens imaging a distant object, the curved surface should face the incident light, not the plane surface.

Five monochromatic aberrations:

1. Spherical aberration

   • Cause: Rays through different zones of lens focus at different points
   • Effect: Blurred disc image instead of point
   • Fix: Use stops, parabolic surfaces, or lens combinations

2. Coma

   • Cause: Off-axis points form asymmetric ray patterns through lens zones
   • Effect: Comet-shaped blurring for off-axis points
   • Fix: Special curvature designs, proper stops

3. Astigmatism

   • Cause: Off-axis rays in different planes focus at different distances
   • Effect: Image spread along principal axis
   • Fix: Proper lens design, stops

4. Curvature of field

   • Cause: Best image forms on curved surface, not plane
   • Effect: Cannot focus entire field on flat surface
   • Fix: Field flattener lenses, lens combinations

5. Distortion

   • Cause: Variation of magnification with distance from axis
   • Effect: Straight lines appear curved (barrel or pincushion)
   • Fix: Symmetric lens design, aperture placement

Lens Maker's Formula: $\frac{1}{f} = (\mu - 1)\left(\frac{1}{R_1} - \frac{1}{R_2}\right)$

Where:

• $f$ = focal length of lens
• $\mu$ = refractive index of lens material (in air)
• $R_1$ = radius of curvature of first surface
• $R_2$ = radius of curvature of second surface

Lens Equation: $\frac{1}{v} - \frac{1}{u} = \frac{1}{f}$

Where:

• $u$ = object distance
• $v$ = image distance
• $f$ = focal length

Relationship:

• The lens maker's formula determines the focal length of a lens based on its physical properties
• The lens equation then uses this focal length to relate object and image positions
• Together they form a complete system for predicting image formation

• Convergent (convex) lens:
  • Parallel rays incident on the lens converge at a point called the second focus (F₂)
  • The second focus is a real point where light actually passes through
  • Produces real images when object is beyond focal length
• Divergent (concave) lens:
  • Parallel rays incident on the lens diverge after refraction
  • The rays appear to originate from a virtual point called the second focus (F₂)
  • The second focus is on the same side as the incident light
  • Always produces virtual images

Note: The focal length is positive for convergent lenses and negative for divergent lenses.