Optics

Optics deals with light and its behavior. It’s divided into ray optics (geometric) and wave optics (physical).

Overview

graph TD
    A[Optics] --> B[Ray Optics]
    A --> C[Wave Optics]
    B --> B1[Reflection]
    B --> B2[Refraction]
    B --> B3[Lenses & Mirrors]
    C --> C1[Interference]
    C --> C2[Diffraction]
    C --> C3[Polarization]

Interactive: Lens Ray Diagram

Adjust the object distance and focal length to see how images form through convex and concave lenses:

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Ray Optics

Reflection

Laws of Reflection:

1. Incident ray, reflected ray, and normal lie in same plane
2. Angle of incidence = Angle of reflection

Spherical Mirrors

Mirror Formula:

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

Magnification:

$$m = \frac{h_i}{h_o} = -\frac{v}{u}$$

Sign Convention (Real is positive for mirrors):

• u is negative (object on left)
• v is positive for real image, negative for virtual
• f is negative for convex, positive for concave

Mirror	Object Position	Image
Concave	Beyond C	Real, inverted, diminished
Concave	At C	Real, inverted, same size
Concave	Between F and C	Real, inverted, magnified
Concave	At F	At infinity
Concave	Between P and F	Virtual, erect, magnified
Convex	Any	Virtual, erect, diminished

Refraction

Snell’s Law:

$$\boxed{n_1 \sin i = n_2 \sin r}$$

Refractive Index:

$$n = \frac{c}{v} = \frac{\sin i}{\sin r}$$

Total Internal Reflection

Occurs when light travels from denser to rarer medium.

Critical Angle:

$$\boxed{\sin C = \frac{n_2}{n_1} = \frac{1}{_1n_2}}$$

Applications: Optical fibers, prisms, diamonds

Refraction Through Prism

Deviation:

$$\delta = i_1 + i_2 - A$$

Minimum Deviation:

$$n = \frac{\sin\left(\frac{A + \delta_m}{2}\right)}{\sin\left(\frac{A}{2}\right)}$$

At minimum deviation: $i_1 = i_2$ and $r_1 = r_2 = \frac{A}{2}$

Lenses

Interactive Demo: Lens Ray Diagram

Move the object and change focal length to see how image properties change:

Lens Maker’s Formula:

$$\frac{1}{f} = (n-1)\left(\frac{1}{R_1} - \frac{1}{R_2}\right)$$

Lens Formula:

$$\boxed{\frac{1}{v} - \frac{1}{u} = \frac{1}{f}}$$

Magnification:

$$m = \frac{v}{u}$$

Power of Lens:

$$P = \frac{1}{f} \text{ (in diopters when f in meters)}$$

Combination of Lenses

$$\frac{1}{f_{eq}} = \frac{1}{f_1} + \frac{1}{f_2}$$

(in contact)

$$P_{eq} = P_1 + P_2$$

Comparison

Property	Concave	Convex
Nature	Diverging	Converging
Focal length	Negative	Positive
Power	Negative	Positive

Optical Instruments

Simple Microscope

Magnifying Power:

$$M = 1 + \frac{D}{f}$$

(Image at D)

$$M = \frac{D}{f}$$

(Image at infinity)

Compound Microscope

$$M = m_o \times m_e = \frac{L}{f_o} \times \frac{D}{f_e}$$

Telescope

Astronomical (Normal adjustment):

$$M = \frac{f_o}{f_e}$$

Length: $L = f_o + f_e$

Wave Optics

Huygens’ Principle

Every point on a wavefront acts as a source of secondary wavelets.

Interference

Conditions for Sustained Interference:

1. Coherent sources (same frequency, constant phase difference)
2. Same amplitude (for good contrast)

Young’s Double Slit Experiment

Path Difference:

$$\Delta x = d\sin\theta \approx \frac{yd}{D}$$

Fringe Width:

$$\boxed{\beta = \frac{\lambda D}{d}}$$

Bright Fringes: $\Delta x = n\lambda$ (n = 0, 1, 2, …) Dark Fringes: $\Delta x = (2n-1)\frac{\lambda}{2}$ (n = 1, 2, 3, …)

Intensity:

$$I = I_1 + I_2 + 2\sqrt{I_1 I_2}\cos\phi$$

For equal intensities:

$$I = 4I_0\cos^2\frac{\phi}{2}$$ $$I_{max} = 4I_0, \quad I_{min} = 0$$

Diffraction

Single Slit Diffraction

Angular Position of Minima:

$$a\sin\theta = n\lambda \quad (n = 1, 2, 3, ...)$$

Width of Central Maximum:

$$\beta_0 = \frac{2\lambda D}{a}$$

Intensity:

$$I = I_0 \left(\frac{\sin\alpha}{\alpha}\right)^2$$

where $\alpha = \frac{\pi a \sin\theta}{\lambda}$

Polarization

Light is a transverse wave; polarization restricts vibrations to one plane.

Malus’ Law

$$\boxed{I = I_0 \cos^2\theta}$$

where θ is angle between polarizer and analyzer.

Brewster’s Law

When reflected and refracted rays are perpendicular:

$$\tan i_B = n$$

At Brewster’s angle, reflected light is completely polarized.

Practice Problems

1. A concave mirror has focal length 20 cm. Find image position for object at 30 cm.

2. Calculate critical angle for water (n = 4/3) to air interface.

3. In YDSE with d = 1 mm, D = 1 m, λ = 600 nm, find fringe width.

4. Light passes through two polaroids with axes at 60°. If incident intensity is $I_0$, find transmitted intensity.

Further Reading

• Electromagnetic Waves
• Dual Nature of Matter