Geometric Optics, Optical Instruments, and Interference & Diffraction: Study Notes

Geometric Optics

Plane Mirrors and Image Formation

Geometric optics studies the behavior of light as it interacts with mirrors and lenses. A plane mirror forms images by reflection, and understanding the properties of these images is fundamental.

• Object Distance (s): The distance from the object to the mirror.

• Image Distance (s'): The distance from the image to the mirror. For a plane mirror, $s = -s'$ or $\frac{1}{s} + \frac{1}{s'} = 0$.

• Magnification (m): $m = \frac{y'}{y}$; for a plane mirror, $m = 1$ (image is same size as object).

• Image Properties: Images are virtual (cannot be projected), erect (upright), and reversed (left-right inversion).

Example: A person standing 2 m in front of a plane mirror sees their image 2 m behind the mirror, for a total distance of 4 m between object and image.

Spherical Mirrors: Concave and Convex

Spherical mirrors can be concave (inward-curving) or convex (outward-curving). Images are formed by reflection, and the sign conventions are important.

• Mirror Equation: $\frac{1}{s} + \frac{1}{s'} = \frac{1}{f}$

• Focal Length (f): $f = \frac{R}{2}$, where $R$ is the radius of curvature. $f > 0$ for concave, $f < 0$ for convex mirrors.

• Magnification: $m = \frac{y'}{y} = -\frac{s'}{s}$

• Sign of R: Positive if the center of curvature is on the same side as outgoing light; negative otherwise.

Graphical Methods: Four principal rays are used to locate images. Any two suffice to determine image position and size.

Refraction at Spherical Surfaces

When light passes through a spherical interface between two media, images are formed by refraction.

• Refraction Equation: $n_a \frac{1}{s} + n_b \frac{1}{s'} = \frac{n_b - n_a}{R}$

• Magnification: $m = \frac{y'}{y} = -\frac{n_a s'}{n_b s}$

• Applications: Objects in water or glass appear shifted due to refraction (e.g., a fish in a bowl).

Thin Lenses: Converging and Diverging

Thin lenses form images by refraction. Lenses are classified as converging (convex) or diverging (concave).

• Thin Lens Equation: $\frac{1}{s} + \frac{1}{s'} = \frac{1}{f}$

• Magnification: $m = \frac{y'}{y} = -\frac{s'}{s}$

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

• Sign Conventions: $R_1$ is positive if the first surface is convex as seen from the object side; $R_2$ is positive if the second surface is convex as seen from the image side.

Graphical Methods: Principal rays are used to construct images for lenses, similar to mirrors.

Optical Instruments

The Human Eye

The eye is a natural optical instrument. Its far point (maximum clear distance) is infinity for a normal eye, and the near point (minimum clear distance) is 25 cm.

Vision Defects and Correction

• Myopia (Nearsightedness): Difficulty focusing on distant objects. Corrected with a diverging (concave) lens.

• Hyperopia (Farsightedness): Difficulty focusing on near objects. Corrected with a converging (convex) lens.

• Lens Power: $\text{Power} = \frac{1}{f}$ (in meters), measured in diopters (D).

Magnifying Glass (Simple Magnifier)

• Angular Size: $\theta \approx \frac{y}{s}$ (in radians), where $y$ is object height and $s$ is distance to eye.

• Angular Magnification: $M = \frac{\theta'}{\theta} = \frac{25\ \text{cm}}{f}$

• For a near point of 25 cm, $\theta = \frac{y}{25\ \text{cm}}$, $\theta' = \frac{y}{f}$

Compound Microscope

• Principle: Uses two lenses (objective and eyepiece) to achieve high magnification.

• Angular Magnification: $M = m_1 M_2 = \frac{(25\ \text{cm}) s_1'}{f_1 f_2}$

• Where $f_1$ and $f_2$ are focal lengths of objective and eyepiece, $s_1'$ is image distance from objective.

Interference and Diffraction

Interference of Light

Interference occurs when two or more coherent light waves overlap, producing regions of constructive and destructive interference.

• Constructive Interference: $r_2 - r_1 = m \lambda$

• Destructive Interference: $r_2 - r_1 = (m + \frac{1}{2}) \lambda$

• $m$ is an integer (order of the fringe).

Young's Double-Slit Experiment

• Bright Fringes (Constructive): $d \sin \theta = m \lambda$

• Dark Fringes (Destructive): $d \sin \theta = (m + \frac{1}{2}) \lambda$

• Position of m-th Bright Band: $y_m = R \frac{m \lambda}{d}$

• Position of m-th Dark Band: $y_m = R \frac{(m + 1/2) \lambda}{d}$

• Fringe Pattern: Alternating bright and dark bands on the screen.

Thin Film Interference

Thin film interference arises from the superposition of light reflected from the top and bottom surfaces of a thin layer.

• Wavelength in Film: $\lambda = \frac{\lambda_0}{n_{film}}$

• No Phase Shift: $2t = m \lambda$ (constructive), $2t = (m + \frac{1}{2}) \lambda$ (destructive)

• With Phase Shift: $2t = m \lambda$ (destructive), $2t = (m + \frac{1}{2}) \lambda$ (constructive)

• Nonreflective Coating Thickness: $t = \frac{1}{4} \lambda$

Example: Soap bubbles and oil films show colorful patterns due to thin film interference.

Diffraction

Diffraction is the bending and spreading of waves when they encounter an obstacle or aperture. Both interference and diffraction result from the superposition principle and Huygens’s principle.

• Single Slit Diffraction: Most power is in the central bright band (~85%).

• Dark Bands: $\sin \theta = \frac{m \lambda}{a}$, $m = \pm1, \pm2, \pm3, ...$

• Small Angle Approximation: $\theta = \frac{m \lambda}{a}$

• Position of Dark Fringes: $y_m = x \frac{m \lambda}{a}$

• Intensity Distribution: Central maximum is much brighter and wider than side maxima.

Multiple Slit Diffraction (Diffraction Grating)

• Constructive Interference: $d \sin \theta = m \lambda$, $m = 0, \pm1, \pm2, ...$

• Intensity Distribution: Sharp, bright maxima at specific angles; used for precise wavelength measurements.

Summary Table: Key Equations in Geometric Optics and Interference

Phenomenon	Key Equation	Notes
Plane Mirror	$s = -s'$, $m = 1$	Virtual, erect image
Spherical Mirror	$\frac{1}{s} + \frac{1}{s'} = \frac{1}{f}$, $f = \frac{R}{2}$	Sign of $f$ and $R$ depends on mirror type
Thin Lens	$\frac{1}{s} + \frac{1}{s'} = \frac{1}{f}$, $\frac{1}{f} = (n-1)(\frac{1}{R_1} - \frac{1}{R_2})$	Converging or diverging
Young's Double Slit	$d \sin \theta = m \lambda$	Bright fringes
Single Slit Diffraction	$\sin \theta = \frac{m \lambda}{a}$	Dark fringes
Thin Film Interference	$2t = m \lambda$ or $2t = (m + \frac{1}{2}) \lambda$	Depends on phase shift

Additional info: The notes above expand on the brief review points by providing definitions, sign conventions, and context for each formula, as well as examples and applications relevant to college-level physics.