OPTICS (Ray Optics & Wave Optics)

 

PHYSICS

Chapter 9 & 10

OPTICS

Ray Optics  ◆  Wave Optics

 

PART A: RAY OPTICS

Ray Optics, also called Geometrical Optics, is the branch of physics that studies the behaviour of light using the concept of rays. A ray of light is an imaginary line that represents the path along which light energy travels. Ray optics is valid when the dimensions of obstacles or apertures encountered by light are much larger than the wavelength of light. In this model, light travels in straight lines, and its interaction with mirrors, lenses, and prisms can be analysed using simple geometric principles.

This branch forms the foundation for understanding everyday optical devices such as mirrors, magnifying glasses, microscopes, telescopes, cameras, and the human eye.

1. Nature of Light

Light is a form of electromagnetic radiation that is visible to the human eye. It occupies the wavelength range of approximately 400 nm (violet) to 700 nm (red) in the electromagnetic spectrum. Light possesses a dual nature — it exhibits wave properties (interference, diffraction, polarisation) and particle properties (photoelectric effect).

Speed of Light: In vacuum (free space), the speed of light is c = 3 × 10⁸ m/s. In any medium with refractive index n, the speed becomes v = c/n.

 

2. Reflection of Light

Reflection is the phenomenon by which a ray of light bounces back into the same medium after striking a polished surface. The reflecting surface is called a mirror or reflector.

2.1 Laws of Reflection

◆        The incident ray, the reflected ray, and the normal at the point of incidence all lie in the same plane.

◆        The angle of incidence (∠i) is always equal to the angle of reflection (∠r):  ∠i = ∠r

 

Figure 1: Diagram illustrating the Laws of Reflection at a plane mirror

The normal is a perpendicular line drawn at the point of incidence on the reflecting surface. Both angles are measured from this normal.

2.2 Types of Reflection

◆        Regular (Specular) Reflection: Occurs on smooth, polished surfaces like mirrors. Reflected rays are parallel and form a clear image.

◆        Irregular (Diffuse) Reflection: Occurs on rough surfaces like paper or walls. Reflected rays scatter in all directions — no clear image is formed.

 

3. Spherical Mirrors

A spherical mirror is a curved mirror whose reflecting surface is a part of a hollow sphere. They are of two types:

◆        Concave Mirror: The reflecting surface is curved inward (like the inside of a bowl). It converges light rays.

◆        Convex Mirror: The reflecting surface is curved outward (like the outside of a sphere). It diverges light rays.

3.1 Key Terminology

Term	Definition
Pole (P)	The centre point of the reflecting surface of the mirror.
Centre of Curvature (C)	The centre of the sphere of which the mirror is a part. Distance PC = Radius of curvature (R).
Principal Axis	The straight line passing through the pole P and the centre of curvature C.
Principal Focus (F)	The point on the principal axis where rays parallel to the axis converge (concave) or appear to diverge from (convex) after reflection.
Focal Length (f)	The distance between the pole P and the principal focus F. Relation: f = R/2
Aperture	The diameter of the circular boundary of the mirror's reflecting surface.

 

Figure 2: Ray diagram for a concave mirror showing key points P, F, C and image formation

3.2 Mirror Formula

The relationship between object distance (u), image distance (v), and focal length (f) for a spherical mirror is given by the Mirror Formula:

Mirror Formula

1/v  +  1/u  =  1/f

 

Focal Length – Radius

f  =  R / 2

 

Linear Magnification

m  =  −v / u  =  h₂ / h₁

 

Sign Convention (New Cartesian): All distances are measured from the pole P. Distances in the direction of incident light are positive; distances opposite to incident light are negative.

3.3 Uses of Spherical Mirrors

◆        Concave mirrors: Used in torches, headlights, solar furnaces, shaving mirrors, and as a reflector in telescopes.

◆        Convex mirrors: Used as rear-view mirrors in vehicles as they provide a wider field of view and always form erect, diminished, virtual images.

 

4. Refraction of Light

Refraction is the phenomenon of bending of light when it passes obliquely from one transparent medium into another. This bending occurs because the speed of light changes as it moves from one medium to another.

Figure 3: Bending of light as it passes from a rarer to a denser medium (Refraction)

4.1 Laws of Refraction (Snell's Law)

◆        The incident ray, the refracted ray, and the normal at the point of incidence all lie in the same plane.

◆        The ratio of the sine of angle of incidence to the sine of angle of refraction is constant for a given pair of media. This constant is called the Refractive Index (n).

 

Snell's Law

n₁ sin θ₁  =  n₂ sin θ₂

 

Refractive Index

n  =  sin i / sin r  =  c / v

 

4.2 Total Internal Reflection (TIR)

When light travels from a denser medium to a rarer medium, and the angle of incidence exceeds a critical angle (θc), the light is completely reflected back into the denser medium. This is called Total Internal Reflection. No refracted ray exits the denser medium.

Critical Angle

sin θc  =  1 / n  (for medium to air)

 

Applications of TIR: Optical fibres (communication and medical endoscopy), mirage formation in deserts, sparkling of diamonds (high refractive index ≈ 2.42), and prism-based periscopes and binoculars.

 

5. Refraction Through Lenses

A lens is a transparent refracting medium bounded by two curved surfaces, or one curved and one plane surface. Lenses work on the principle of refraction. They are broadly classified into:

◆        Convex (Converging) Lens: Thicker at the centre; converges incident parallel rays to a real focus.

◆        Concave (Diverging) Lens: Thinner at the centre; diverges incident parallel rays, appearing to come from a virtual focus.

 

Figure 4: Convex Lens — Ray diagram showing image formation by a converging lens

5.1 Lens Formula and Magnification

Lens Formula

1/v  −  1/u  =  1/f

 

Magnification

m  =  v / u  =  h₂ / h₁

 

Lens Maker's Equation

1/f  =  (n − 1) [ 1/R₁  −  1/R₂ ]

 

Power of Lens

P  =  1/f (metres)   [Unit: Dioptre, D]

 

5.2 Combination of Lenses

When two thin lenses of focal lengths f₁ and f₂ are placed in contact, the effective focal length F and total power P of the combination are:

Combined Focal Length

1/F  =  1/f₁  +  1/f₂

 

Combined Power

P  =  P₁  +  P₂

 

5.3 Refraction Through Prism

A prism is a transparent medium bounded by two plane surfaces inclined at an angle (A = angle of prism). When light passes through a prism, it undergoes refraction twice and the net bending of light is called the angle of deviation (δ).

Angle of Deviation

δ  =  (i₁ + i₂) − A

 

At Minimum Deviation

n  =  sin[(A + Dm)/2]  /  sin(A/2)

 

Dispersion: White light splits into its component colours (VIBGYOR) on passing through a prism because the refractive index of glass is different for different wavelengths (colours). Violet light is deviated the most and red the least.

 

6. Optical Instruments

6.1 The Human Eye

The human eye works like a camera. The cornea and lens together form a converging lens system. The retina acts as the screen. The ciliary muscles change the curvature (and hence the focal length) of the eye lens — a process called Accommodation. The near point of a normal eye is 25 cm (D) and far point is infinity.

6.2 Defects of Vision and Correction

Defect	Cause	Correction
Myopia (Shortsightedness)	Image forms in front of retina; eyeball too long	Concave (diverging) lens
Hypermetropia (Longsightedness)	Image forms behind retina; eyeball too short	Convex (converging) lens
Presbyopia	Loss of accommodative power with age	Bifocal lenses
Astigmatism	Non-uniform curvature of cornea	Cylindrical lenses

 

6.3 Simple Microscope

A simple microscope is a single convex lens of short focal length used to see magnified images of small objects. The object is placed between F and the optical centre of the lens, producing a virtual, erect, and magnified image at the near point.

Magnifying Power (image at D)

m  =  1  +  D/f

 

6.4 Compound Microscope

A compound microscope has two converging lenses: the objective (short focal length fo) and the eyepiece (focal length fe). The objective forms a real, magnified, inverted image of the object, which acts as an object for the eyepiece.

Magnifying Power

M  =  − L/fo × (1 + D/fe)

 

6.5 Telescope (Astronomical)

An astronomical telescope is used to view distant objects. It also has an objective lens and an eyepiece. The objective has a large aperture and long focal length to collect maximum light and form a real image at its focal plane.

Magnifying Power (normal adjustment)

M  =  − fo / fe

 

Resolving power of a telescope depends on the diameter of the objective lens — larger the aperture, better the resolution and brightness.

 

PART B: WAVE OPTICS

Wave Optics, also known as Physical Optics, treats light as a transverse electromagnetic wave characterised by its wavelength (λ), frequency (ν), and amplitude. Wave optics is essential to explain phenomena such as interference, diffraction, and polarisation — effects that cannot be explained by the ray (geometric) model of light.

The wave theory of light was proposed by Christian Huygens in the 17th century and later confirmed by Thomas Young's double-slit experiment in 1801 and further developed by Augustin-Jean Fresnel.

7. Huygens' Principle

Huygens' Principle provides a geometric method to determine the new position of a wavefront at a later time. It states:

Huygens' Principle

1. Every point on a wavefront acts as a source of secondary wavelets, which spread out in all directions with the speed of light in that medium. 2. The new wavefront at any later instant is the forward envelope (tangential surface) of all these secondary wavelets.

 

A wavefront is the locus of all points of a medium that are in the same phase of oscillation. For a point source, wavefronts are spherical. For a distant source, wavefronts become planar (plane wavefronts).

Using Huygens' principle, one can derive the laws of reflection and refraction of light mathematically and show that they are natural consequences of the wave theory.

 

8. Interference of Light

Interference is the phenomenon of redistribution of light energy (intensity) when two or more coherent light waves superpose (overlap). At certain points the waves add constructively (bright fringes) and at other points they cancel destructively (dark fringes), producing an interference pattern.

8.1 Conditions for Sustained Interference

◆        The two sources must be coherent — they must emit light waves of the same frequency and have a constant phase difference.

◆        The amplitudes of the two waves should be equal (or nearly equal) for maximum contrast in the fringe pattern.

◆        The two sources must be very close to each other, and the screen should be far away.

◆        The light used should ideally be monochromatic (single wavelength).

 

8.2 Young's Double Slit Experiment (YDSE)

Thomas Young (1801) demonstrated interference by allowing light from a single source to pass through two narrow parallel slits S₁ and S₂, which act as two coherent sources. The light from these slits overlaps on a screen and produces alternating bright and dark bands called fringes.

Figure 5: Young's Double Slit Experiment — Interference pattern showing bright and dark fringes

8.3 Mathematical Analysis of YDSE

Let:  d = distance between the two slits,  D = distance between slits and screen,  λ = wavelength of light used,  y = position of fringe on screen.

Fringe Width (β)

β  =  λD / d

 

Condition for Bright Fringe

y_n  =  nλD/d      (n = 0, ±1, ±2, ...)

 

Condition for Dark Fringe

y_n  =  (2n−1)λD / 2d

 

Resultant Intensity

I  =  I₁ + I₂ + 2√(I₁I₂) cos δ

 

Here δ is the phase difference between the two waves at the point of superposition. At bright fringes δ = 2nπ and at dark fringes δ = (2n−1)π.

The central fringe (n = 0) is always a bright fringe. Fringes are equally spaced with fringe width β = λD/d. Decreasing d or increasing D increases the fringe width.

 

9. Diffraction of Light

Diffraction is the phenomenon of bending and spreading of light waves around the edges of an obstacle or through a narrow aperture (slit). It is a consequence of the wave nature of light and is most pronounced when the size of the obstacle or slit is comparable to the wavelength of light.

Figure 6: Single Slit Diffraction — Intensity distribution showing central maximum and minima

9.1 Single Slit Diffraction

When monochromatic light of wavelength λ passes through a narrow slit of width 'a', a diffraction pattern is observed on a screen. The pattern consists of a wide, bright central maximum flanked by alternating dark and bright bands of decreasing intensity.

Condition for Dark Fringes (Minima)

a sin θ  =  nλ      (n = ±1, ±2, ...)

 

Angular Width of Central Maximum

2θ  =  2λ / a

 

The central maximum is twice as wide as secondary maxima. As the slit width 'a' decreases, the diffraction pattern spreads wider, and vice versa. This inverse relationship between slit width and diffraction spread is a fundamental wave property.

9.2 Difference: Interference vs Diffraction

Interference	Diffraction
Superposition of waves from two or more coherent sources	Superposition of waves from different parts of the same wavefront
All bright fringes have the same intensity	Intensity decreases from the central maximum outward
Fringes are equally spaced	Central maximum is twice as wide as secondary maxima
Minima are perfectly dark (if amplitudes are equal)	Minima are not perfectly dark

 

 

10. Polarisation of Light

Light is a transverse electromagnetic wave. An ordinary (unpolarised) light wave has electric field vibrations in all planes perpendicular to its direction of propagation. Polarisation is the phenomenon by which the vibrations of the electric field vector are confined to a single plane. Such light is called Plane Polarised Light or Linearly Polarised Light.

10.1 Methods of Polarisation

◆        Polarisation by Reflection (Brewster's Law): When light is reflected from a surface at a specific angle called Brewster's angle (iB), the reflected ray is completely plane polarised. Brewster's Law: tan iB = n (refractive index of the medium).

◆        Polarisation by Refraction: The refracted beam is partially polarised. Both the reflected and refracted beams are perpendicular to each other at Brewster's angle.

◆        Polarisation by Selective Absorption (Polaroids): Polaroid films (like those in sunglasses and LCD screens) transmit only those vibrations parallel to the polarisation axis and absorb the rest.

◆        Polarisation by Scattering: Sunlight scattered by air molecules (Rayleigh scattering) is partially polarised. The blue colour of the sky is due to this scattering effect.

 

Brewster's Law

tan iB  =  n₂ / n₁

 

Malus's Law

I  =  I₀ cos² θ

 

10.2 Applications of Polarisation

◆        Polaroid sunglasses: Reduce glare from reflected light.

◆        LCD screens and 3D cinema glasses use polaroids.

◆        Stress analysis in engineering using photoelasticity.

◆        Used in photography to reduce reflections and enhance contrast.

 

11. Resolving Power and Rayleigh's Criterion

Resolving Power is the ability of an optical instrument to distinguish between two closely spaced point objects and show them as separate images. According to Rayleigh's Criterion, two point sources are just resolved when the central maximum of the diffraction pattern of one source falls on the first minimum of the other.

Limit of Resolution (Telescope)

θ_min  =  1.22 λ / D

 

Resolving Power of Telescope

RP  =  D / 1.22 λ

 

Resolving Power of Microscope

RP  =  2n sin α / 1.22 λ

 

Here D is the diameter of the objective lens of the telescope, n is the refractive index of the medium between object and lens, and α is the half-angle of the cone of light from the object. The quantity n sin α is called the Numerical Aperture (NA).

To increase resolving power: use shorter wavelength light (e.g., UV or electron beams in electron microscopes), use objective lenses with a large aperture, or use oil-immersion objectives that increase the effective refractive index.

 

12. Quick Reference — Important Formulae

Concept	Formula	Symbols
Mirror Formula	1/v + 1/u = 1/f	u=object dist, v=image dist, f=focal length
Magnification (Mirror)	m = −v/u	Negative → inverted image
Snell's Law	n₁ sin i = n₂ sin r	n = refractive index
Critical Angle	sin θc = 1/n	n = n_denser / n_rarer
Lens Formula	1/v − 1/u = 1/f	Same sign convention as mirrors
Lens Power	P = 1/f (m)  [D]	f in metres, P in Dioptre
Lens Maker's Eq.	1/f = (n−1)[1/R₁−1/R₂]	R₁,R₂ = radii of curvature
YDSE Fringe Width	β = λD/d	D=slit-screen dist, d=slit separation
Single Slit Minima	a sin θ = nλ	a = slit width, n = 1,2,3...
Malus's Law	I = I₀ cos² θ	θ = angle between polaroids
Brewster's Law	tan iB = n	iB = Brewster's angle
Resolving Power	θ = 1.22λ/D	D = aperture diameter

 

 

13. Key Points to Remember

Ray Optics — Key Points

• Reflection: ∠i = ∠r always. Mirror formula: 1/v + 1/u = 1/f. • Concave mirror: can form real or virtual images depending on object position. • Convex mirror: always forms virtual, erect, and diminished images — used as rear-view mirror. • Refractive index n = c/v = sin i / sin r. Higher n → lower speed → more bending toward normal. • TIR occurs when light goes from denser to rarer medium and ∠i > θc. Used in optical fibres. • Convex lens: converging; concave lens: diverging. Lens formula: 1/v − 1/u = 1/f. • Power of lens (P) = 1/f(m). Convex lens: +ve power. Concave lens: −ve power. • Dispersion in prism: violet deviates most (highest n), red deviates least (lowest n).

 

Wave Optics — Key Points

• Huygens' principle: every point on a wavefront is a source of secondary wavelets. • Coherent sources: same frequency, constant phase difference — necessary for sustained interference. • YDSE: Bright fringe at path diff = nλ; dark fringe at path diff = (2n−1)λ/2. Fringe width β = λD/d. • Single slit diffraction: minima at a sinθ = nλ. Central maximum width = 2λ/a. • Diffraction shows light is a wave; polarisation proves light is a TRANSVERSE wave. • Malus's Law: I = I₀ cos²θ. At θ=0° → max intensity; at θ=90° → zero intensity. • Resolving power increases with larger aperture and shorter wavelength. • Rayleigh's criterion: θ_min = 1.22 λ/D for just resolved images.

 

— End of Chapter: Optics (Ray Optics & Wave Optics) —