Geometrical Optics

• Paraxial Rays Definition:

  • Light rays that travel close to and nearly parallel with the principal axis
  • Make small angles with the principal axis
  • Hit the mirror near its pole (center point)

• Relationship to Focus:

  • Only paraxial rays converge precisely at the focus (f = R/2)
  • Non-paraxial rays (hitting mirror far from pole) converge at slightly different points
  • This creates spherical aberration - a fundamental limitation

• Mathematical Basis:

  • The f = R/2 relationship assumes the small-angle approximation where sin θ ≈ θ
  • This approximation only holds for paraxial rays

• Practical Limitations:

  • Large aperture spherical mirrors suffer from spherical aberration
  • Image quality degrades as mirror diameter increases
  • Resolution is limited without correction

• Solutions:

  • Using small apertures (reduced light collection)
  • Implementing corrective optics
  • Using parabolic mirrors instead (which focus all parallel rays to a single point)
  • Schmidt corrector plates in some telescope designs

• Application Example: Astronomical telescopes often use parabolic rather than spherical mirrors to eliminate spherical aberration for precise star imaging

For a concave mirror with focal length f:

(a) Real and magnified image:

• Object position: between f and 2f
• Example: at 1.5f, image forms at -3f with magnification m = -2
• Magnification magnitude |m| > 1 indicates enlargement

(b) Real and same size image:

• Object position: exactly at 2f
• Image also forms at 2f
• Magnification m = -1 (same size but inverted)

(c) Real and diminished image:

• Object position: beyond 2f
• Magnification magnitude |m| < 1 indicates reduction
• As object moves beyond 2f, image size decreases

Note: The negative sign in magnification always indicates the image is inverted relative to the object.

First Focal Point (F₁):

• Point where object must be placed to produce parallel emergent rays
• For convergent lens: Real point on incident side
• For divergent lens: Virtual point on emergent side

Second Focal Point (F₂):

• Point where parallel incident rays converge after refraction
• For convergent lens: Real point on emergent side
• For divergent lens: Virtual point on incident side

Relationship between focal points:

• For thin lens in homogeneous medium: |PF₁| = |PF₂|
• Usually just referred to as "the focal length f"

Lens Power:

• P = 1/f (measured in diopters, m⁻¹)
• Positive power = converging lens
• Negative power = diverging lens
• For lenses in contact: P = P₁ + P₂

Step 1: Calculate focal length using lens maker's formula $\frac{1}{f} = (\mu - 1)\left(\frac{1}{R_1} - \frac{1}{R_2}\right)$ $\frac{1}{f} = (1.5 - 1)\left(\frac{1}{15\text{ cm}} - \frac{1}{-25\text{ cm}}\right)$ $\frac{1}{f} = 0.5\left(\frac{1}{15\text{ cm}} + \frac{1}{25\text{ cm}}\right)$ $\frac{1}{f} = 0.5\left(\frac{5}{75\text{ cm}} + \frac{3}{75\text{ cm}}\right) = 0.5 \times \frac{8}{75\text{ cm}} = \frac{4}{75\text{ cm}}$ $f = 18.75\text{ cm}$

Step 2: Use lens equation to find image position $\frac{1}{v} - \frac{1}{u} = \frac{1}{f}$ $\frac{1}{v} - \frac{1}{-10\text{ cm}} = \frac{1}{18.75\text{ cm}}$ $\frac{1}{v} + \frac{1}{10\text{ cm}} = \frac{1}{18.75\text{ cm}}$ $\frac{1}{v} = \frac{1}{18.75\text{ cm}} - \frac{1}{10\text{ cm}} = \frac{10 - 18.75}{187.5\text{ cm}} = \frac{-8.75}{187.5\text{ cm}}$ $v = -21.43\text{ cm}$

Step 3: Determine magnification and image nature $m = \frac{v}{u} = \frac{-21.43\text{ cm}}{-10\text{ cm}} = 2.14$

Image is virtual (negative v), erect (positive m), and magnified (|m| > 1)

• 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.

• Second focus (F₂):

  • Point where parallel rays incident on the lens converge (convex) or appear to diverge from (concave)
  • Related to image formation of distant objects
  • Located on the opposite side from incident light for convex lens
  • Located on the same side as incident light for concave lens

• First focus (F₁):

  • Point where an object must be placed to produce parallel emergent rays
  • For convex lens: A real point where a real object can be placed
  • For concave lens: A virtual point where only a virtual object can exist
  • When object is at F₁, image forms at infinity

Application: In camera design, when focusing on distant objects, the image sensor is placed at the second focal point F₂.

Angle of deviation (δ) calculation for a thin lens:

1. Formula: δ = h(1/v - 1/u) = h/f Where:

   • h = height of incident ray above optical axis
   • u = object distance
   • v = image distance
   • f = focal length

2. Physical meaning:

   • Represents angular change in ray direction
   • Proportional to height (h) above optical axis
   • Inversely proportional to focal length
   • Zero for rays passing through optical center

3. Applications:

   • Determines image formation characteristics
   • Used to calculate equivalent focal length of lens combinations
   • Essential for analyzing lens aberrations
   • Helps predict image magnification and orientation

Understanding deviation angles enables optical system design by showing how different rays converge to form images, and how multiple lenses work together in complex optical instruments like microscopes and telescopes.

Ray tracing procedure:

1. Draw a ray from the object tip through the optical center (P) of the lens

   • This ray passes undeviated (no angular change)

2. Draw a second ray parallel to the principal axis

   • For converging lens: This ray refracts through the second focal point (F₂)
   • For diverging lens: This ray refracts as if coming from the first focal point (F₁)

3. The intersection of these two rays locates the image tip

   • If rays actually intersect: real image
   • If backward extensions of rays intersect: virtual image

4. The base of the image lies on the principal axis directly below/above the image tip

5. The lateral magnification is given by: m = h₂/h₁ = -v/u

   • Negative m indicates inverted image
   • Positive m indicates erect image

For a ray passing at height h₁ from the axis through the first lens:

1. Deviation angle from first lens: δ₁ = h₁/f₁
2. Height at second lens: h₂ = h₁ - d·(h₁/f₁)
3. Deviation angle from second lens: δ₂ = h₂/f₂
4. Total deviation angle: δ = δ₁ + δ₂

Through substitution and algebraic manipulation, this results in: δ = h₁/F

Where F is the equivalent focal length given by: $\frac{1}{F} = \frac{1}{f_1} + \frac{1}{f_2} - \frac{d}{f_1 f_2}$

This relationship shows that the combined system acts as a single thin lens with focal length F for parallel incident rays.

Spherical aberration is an optical defect where:

• Rays passing through different zones of a lens with spherical surfaces converge at different points
• Marginal rays (passing through outer portions) are deviated more strongly than paraxial rays (near center)
• Creates a blurred disc image instead of a sharp point

For parallel rays incident on lenses:

• Convex lens: Marginal rays converge at a point F' that is closer to the lens than the ideal focus F
• Concave lens: Marginal rays appear to diverge from a point F' that is farther from the lens than the ideal focus F

This creates opposite aberration effects that can be exploited in lens combinations to reduce total aberration.