Comprehensive Study Notes on Geometrical Optics and Reflection

Fundamentals and Nature of Light

• Acoustics refers to the science of sound, whereas optics is the term used for the science of light.

• Light was historically defined as a form of radiant energy that makes objects visible due to the stimulation of the retina of the eye. It propagates in both the presence and absence of a material medium.

• At the beginning of the 20th century, light was proved to consist of electromagnetic (EM) waves.

• Quantum theory subsequently established the particle nature of light, introducing photons as energy carrier particles.

• Experiments using a countable number of photons established that light possesses a dual nature, consisting of energy carrier photons guided by the rules of EM waves.

• Speed of Light in Vacuum:

  • The maximum possible speed for any physical object in the universe, according to Einstein's special theory of relativity, is the speed of light in vacuum $c$.
  • Exact value: $c = 299792458\,\text{m/s}$.
  • Standard practical value: $c = 3 \times 10^8\,\text{m/s}$.

• Speed of Light in a Material Medium:

  • In a material medium, the speed of electromagnetic waves is given by:     $c = \frac{1}{\sqrt{\varepsilon \mu}}$
  • The parameters permittivity ($\varepsilon$) and permeability ($\mu$) are constants that depend directly on the electric and magnetic properties of the medium.
  • The ratio of the speed of light in vacuum to the speed of light in a medium ($\frac{c}{v}$) is defined as the absolute refractive index ($n$), which is an intrinsic property of the medium.

Categories of Optical Phenomena

Light phenomena are broadly split into three distinct categories based on physical models:

• Ray Optics or Geometrical Optics:

  • Defines a light ray as a particular direction of energy propagation from a light source.
  • Used to explain and analyze phenomena such as reflection, refraction, double refraction, and total internal reflection.

• Wave Optics or Physical Optics:

  • Treats light energy as propagating in the form of electromagnetic waves.
  • Required to explain wave-based phenomena such as interference, diffraction, polarization, and the Doppler effect.

• Particle Nature of Light:

  • Treats light as quanta (photons) interacting with matter.
  • Required for phenomena that cannot be explained by classical wave theory, including the photoelectric effect, emission of spectral lines, and the Compton effect.

Fundamental Laws of Geometrical Optics

Geometrical optics examines image formation by optical devices such as mirrors, lenses, and prisms. It relies on four fundamental laws:

1. Rectilinear Propagation in Isotropic Homogeneous Media:

   • Light travels in a straight line in a medium that is both homogeneous and isotropic.
   • Homogeneous: The physical properties of the medium are identical at every location throughout the medium.
   • Isotropic: The physical properties of the medium are identical in all directions.

2. Independence of Light Rays:

   • Two or more rays can intersect at a point without affecting or altering their respective paths beyond that point of intersection.

3. Laws of Reflection:

   • The reflected ray lies in the same plane formed by the incident ray and the normal drawn at the point of incidence; the incident and reflected rays lie on opposite sides of the normal.
   • The angle of incidence ($i$) is strictly equal to the angle of reflection ($r$):      $i = r$

4. Laws of Refraction:

   • Applied at the boundary separating two distinct optical media.
   • The refracted ray lies in the plane formed by the incident ray and the normal drawn at the point of incidence; both rays lie on opposite sides of the normal.
   • Snell's Law: The angle of incidence ($\theta_1$) in a medium of refractive index $n_1$ and the angle of refraction ($\theta_2$) in a medium of refractive index $n_2$ satisfy the relationship:      $(n_1)\sin(\theta_1) = (n_2)\sin(\theta_2)$

Fermat's Principle and Worked Examples

• Fermat's Principle:

  • All four fundamental laws of geometrical optics can be derived from a single unified concept known as Fermat's Principle.
  • Definition: While travelling from one point to another through one or more reflections or refractions, a ray of light always chooses the path of least time (strictly speaking, an extreme path corresponding to minimum or maximum time).

• Example 9.1: Light Propagation through Spectacle Glass:

  • Problem: A spectacle lens has a thickness of $2\,\text{mm}$ and a refractive index of $1.5$. Calculate the time taken by light to cross this thickness, expressing the answer with the most convenient prefix attached to seconds.
  • Given Data:
  • Speed of light in vacuum $c = 3 \times 10^8\,\text{m/s}$
  • Refractive index $n_{\text{glass}} = 1.5$
  • Thickness $s = 2\,\text{mm} = 2 \times 10^{-3}\,\text{m}$
  • Step-by-Step Calculation:
  1. Calculate the speed of light in glass ($v_{\text{glass}}$):        $v_{\text{glass}} = \frac{c}{n_{\text{glass}}} = \frac{3 \times 10^8\,\text{m/s}}{1.5} = 2 \times 10^8\,\text{m/s}$
  2. Calculate travel time ($t$):        $t = \frac{s}{v_{\text{glass}}} = \frac{2 \times 10^{-3}\,\text{m}}{2 \times 10^8\,\text{m/s}} = 10^{-11}\,\text{s}$
  3. Convert to convenient SI prefix (pico- = $10^{-12}$):        $t = 10 \times 10^{-12}\,\text{s} = 10\,\text{ps}$

Cartesian Sign Convention

To ensure consistent algebraic equations across all positions of objects and optical elements, the Cartesian sign convention is applied (analogous to standard coordinate geometry):

• All linear distances are measured directly from the optical center or pole ($P$) of the optical element. For thin lenses and spherical mirrors, the optical center coincides with the geometrical center.
• Optical diagrams must be drawn such that incident light rays travel strictly from left to right towards the reflecting or refracting surface.
• Object Beam Classifications:
  • Diverging Beam: Originates from a real point object located to the left of the pole ($u$ is negative).
  • Converging Beam: Directed toward a virtual object located to the right of the pole ($u$ is positive).
  • Parallel Beam: Represents an object positioned at infinity ($u = \infty$).
• The principal axis corresponds to the x-axis, with the origin placed at the pole ($P$).
• Distance Directions:
  • Distances measured to the left of the pole (against the direction of incident light) are negative.
  • Distances measured to the right of the pole (along the direction of incident light) are positive.
  • Distances measured vertically above the principal axis are positive.
  • Distances measured vertically below the principal axis are negative.
• Unless explicitly specified otherwise, all objects are assumed to be real objects.

Reflection from Plane Surfaces and Multiple Image Formation

• Single Plane Reflecting Surface:

  • Images formed by plane mirrors are virtual, erect, laterally inverted, identical in size to the object, and located at an equal perpendicular distance behind the mirror surface as the real object is in front of it.
  • If an observer stands on the bank of still water or on a plane mirror, the reflected image seen is laterally reversed, equal in size, and formed on the opposite side of the interface.

• Multiple Images from Inclined Plane Mirrors:

  • When an object is placed between two plane mirrors inclined at an angle $\theta$, multiple reflections create several distinct images.
  • The total number of visible images ($N$) is calculated by first evaluating the parameter $n$:     $n = \frac{360^\circ}{\theta}$
  • Rules for determining $N$:
  1. If $n$ is an even integer: $N = n - 1$, regardless of whether the object lies on or off the angle bisector.
  2. If $n$ is an odd integer and the object is placed on the angle bisector (symmetrically): N = n - 1$.\n 3. If n$is an **odd integer** and the object is placed **off the angle bisector** (asymmetrically):$N = n$.
  3. If $n$ is not an integer: $N = m$, where $m$ is the integer part of $n$ ($m = \lfloor n \rfloor$).

• Summary Table for Inclined Plane Mirror Images (Table 9.1):

  • Angle $\theta = 0^\circ$: $n = \infty$, $N = \infty$
  • Angle $\theta = 120^\circ$ ($n = 3$):
  • On angle bisector: $N = 2$
  • Off angle bisector: $N = 3$
  • Angle $\theta = 110^\circ$ ($n = 3.28$): Position = Anywhere $\rightarrow N = 3$
  • Angle $\theta = 90^\circ$ ($n = 4$): Position = Anywhere $\rightarrow N = 3$
  • Angle $\theta = 80^\circ$ ($n = 4.5$): Position = Anywhere $\rightarrow N = 4$
  • Angle $\theta = 72^\circ$ ($n = 5$):
  • On angle bisector: $N = 4$
  • Off angle bisector: $N = 5$
  • Angle $\theta = 60^\circ$ ($n = 6$): Position = Anywhere $\rightarrow N = 5$
  • Angle $\theta = 50^\circ$ ($n = 7.2$): Position = Anywhere $\rightarrow N = 7$

• Example 9.2: Continuously Increasing Mirror Angle:

  • Problem: A small object is kept symmetrically between two plane mirrors inclined at $38^\circ$. The angle is gradually increased to $41^\circ$ while maintaining symmetrical placement. Determine the number of visible images throughout the process.
  • Calculation:
  • At $\theta = 38^\circ$:       $n = \frac{360^\circ}{38^\circ} = 9.47$       Since $n$ is non-integer, N = \text{integral part} = 9$.\n - As \theta$increases up to$40^\circ$,$N = 9$remains valid because the integer portion stays 9 (at$40^\circ$,$n = 9$, and being symmetric,$N = 9 - 1 = 8$right at transition beyond$40^\circ).\n - Beyond 40^\circ$up to$\theta = 41^\circ:\n      n = \frac{360^\circ}{41^\circ} = 8.78\n      Since n < 9$, the integral part becomes 8. Thus, 8 images are visible from above$40^\circ$up to$41^\circ\n\n# Reflection from Curved Spherical Mirrors\n\n- Curved mirrors are required to converge or diverge parallel or divergent light beams. Torches, headlights, and vehicle rear-view mirrors use spherical surfaces, whereas searchlights utilize parabolic surfaces.\n- Spherical mirrors are polished spherical caps: convex mirrors are polished on the outside, and concave mirrors are polished on the inside.\n- **Radius of Curvature (R)**: The radius of the sphere of which the curved mirror forms a part.\n- **Focal Length (f)**: For spherical mirrors, focal length is exactly half of the radius of curvature:\n  f = \frac{R}{2}\n - For a concave mirror, f$is the distance at which incident parallel rays converge ($f$is negative:$-f = -\frac{R}{2}).\n - For a convex mirror, f$is the distance from which reflected parallel rays appear to diverge ($f$is positive:$+f = +\frac{R}{2}).\n- **Aperture Condition**: A spherical mirror is classified as a small mirror if its aperture (diameter) is much smaller (at least one-tenth) relative to the values of u$,$v$, and$f$.

• Focal Power ($P$):

  • Measures the converging or diverging capacity of a lens or mirror:     $P = \frac{1}{f}$
  • Expressed in the SI unit of diopters (D), where:     $1\,\text{diopter (D)} = 1\,\text{m}^{-1}$

• Lateral Magnification ($m$):

  • Ratio of the linear size of the image to that of the object measured perpendicular to the principal axis:     $m = -\frac{v}{u}$
  • For a convex mirror, $m < 1$ for all real object positions (always forms virtual, erect, and diminished images).

• Mirror Formula:

  • Relates object distance $u$, image distance $v$, and focal length $f$ for point objects or small finite objects:     $\frac{1}{f} = \frac{1}{v} + \frac{1}{u}$

Image Formation by Concave Mirrors and Calculation Example

• Concave Mirror Image Characteristics (Table 9.2):

  • Object at $u = \infty$: Image at $v = f$, Real, Lateral Magnification $m = 0$
  • Object at $u > 2f$: Image at $2f > v > f$, Real, Lateral Magnification $m < 1$
  • Object at $u = 2f$: Image at $v = 2f$, Real, Lateral Magnification $m = 1$
  • Object at $2f > u > f$: Image at $v > 2f$, Real, Lateral Magnification $m > 1$
  • Object at $u = f$: Image at $v = \infty$, Real, Lateral Magnification $m = \infty$
  • Object at $u < f$: Image at $v > u$ (behind mirror), Virtual, Lateral Magnification $m > 1$

• Example 9.3: Image Size of an Extended Longitudinal Object:

  • Problem: A thin pencil of length $20\,\text{cm}$ is aligned along the principal axis of a concave mirror with a radius of curvature of $30\,\text{cm}$. The nearest end of the pencil is located $20\,\text{cm}$ from the pole of the mirror. Find the length of the image of the pencil.
  • Given Data:
  • Concave mirror radius $R = 30\,\text{cm} \implies f = -\frac{R}{2} = -15\,\text{cm}$
  • Pencil length = $20\,\text{cm}$
  • Nearest end distance $u_1 = -20\,\text{cm}$
  • Farthest end distance $u_2 = -(20 + 20) = -40\,\text{cm}$
  • Step 1: Locate image of nearest end ($v_1$):     $\frac{1}{f} = \frac{1}{v_1} + \frac{1}{u_1} \implies -\frac{1}{15} = \frac{1}{v_1} - \frac{1}{20}$$\frac{1}{v_1} = -\frac{1}{15} + \frac{1}{20} = \frac{-4 + 3}{60} = -\frac{1}{60} \implies v_1 = -60\,\text{cm}$
  • Step 2: Locate image of farthest end ($v_2$):     $\frac{1}{f} = \frac{1}{v_2} + \frac{1}{u_2} \implies -\frac{1}{15} = \frac{1}{v_2} - \frac{1}{40}$$\frac{1}{v_2} = -\frac{1}{15} + \frac{1}{40} = \frac{-8 + 3}{120} = -\frac{5}{120} = -\frac{1}{24} \implies v_2 = -24\,\text{cm}$
  • Step 3: Calculate length of the image:     $\text{Length} = |v_1| - |v_2| = 60\,\text{cm} - 24\,\text{cm} = 36\,\text{cm}$

Optical Aberrations and Spherical Aberration

• Theoretical derivations of simplified optical formulas (such as $f = \frac{R}{2}$ and \frac{1}{f} = \frac{1}{v} + \n\frac{1}{u}) depend strictly on three fundamental assumptions:

  1. Objects and images are situated very close to the principal axis.
  2. Rays diverging from objects are confined to a narrow cone with a small angle.
  3. Incident parallel beams are strictly paraxial (parallel and extremely close to the principal axis).

• When these conditions are violated, image distortions called aberrations occur. Commonly occurring defects include spherical aberration, coma, astigmatism, curvature, and distortion.

• All image defects except spherical aberration arise specifically from rays inclined at angles to the principal axis.

• Spherical Aberration Mechanics:

  • Spherical aberration arises purely due to the spherical surface geometry of a mirror or lens.
  • The relation $f = \frac{R}{2}$ yields a single focal point only for small aperture mirrors receiving paraxial rays.
  • For rays farther from the principal axis (marginal rays), reflected rays intersect the axis closer to the pole, causing the point focus to shift progressively toward the pole.
  • This results in a blurred, unsharp image with unclear, fuzzy boundaries.

• Quantitative Measures of Spherical Aberration:

  • Longitudinal Spherical Aberration: Defined as the axial distance along the principal axis between the marginal focus ($F_m$) and the paraxial focus ($F_p$).
  • Circle of Least Confusion: In the presence of spherical aberration, a point object does not produce a point focus on a screen at any location; instead, the focused image appears circular. The specific axial plane where the diameter of this circular cross-section is minimized (marked across line $AB$) is called the circle of least confusion.
  • Transverse Spherical Aberration: Defined as the radius of the circle of least confusion.