Reflection and Refraction Study Notes

Reflection and Refraction\n

Introduction

Optics is a crucial branch of physics with applications in entertainment, remote sensing, space exploration, and communication (optical fibers). Reflection, refraction, and dispersion are fundamental processes explained using ray or wave concepts. This chapter focuses on the ray concept to explain optical systems and instruments.

Light Rays

Light rays are narrow streams of radiation traveling in a straight line, used to explain the rectilinear propagation of light. A broad beam is a bundle of parallel rays. The ray model is useful for studying reflection, refraction, and image formation by mirrors and lenses.

Light rays do not exist physically but are theoretical constructs. Isolating a ray leads to pencils of light, and very small holes cause light to behave as a wave.

Reflection at Plane Surfaces (Mirrors)

Any smooth surface can act as a mirror. Mirrors are made by coating glass with silver or depositing aluminum in a vacuum. A protective layer of silicon monoxide or magnesium fluoride is added. Mirrors redirect light rays, forming an image of an object.

Object and Image

An object emits or reflects light rays. Objects can be point objects (no physical extent) or extended objects (length, width, height). Reflected rays converge or appear to diverge from a position, creating the impression of an object. This apparent object is the image.

Images can be real (containing light energy, can be seen on a screen, essential for photography) or virtual (cannot be received on screen, light rays do not pass through it, photographed with a converging optical system).

Image of a Point Object

Image formation follows the law of reflection. In Fig. 3.1, object O is at distance u in front of the mirror. Ray OA is normal, reflected along its path. Ray OB makes angle i and is reflected at the same angle. Reflected rays extended backward intersect at I, distance v behind the mirror. I is the virtual image of O.

u is the object distance.
v is the image distance.

Triangles OAB and IAB are congruent, so O and I are equidistant from the mirror. Object distance u is positive; image distance v is negative.

The object and image distances are related by:

(3.1) \quad u = -v

The incident ray is deviated through an angle \theta = 180^\circ - 2i.

Image of an Extended Object

Consider a linear object (arrow) of height h parallel to the mirror (Fig. 3.2). The image is an extended image. Ray BD is normal, reflected along DB. Ray BC, at angle i, is reflected along CQ. Rays from B appear to diverge from image point B'. The image is A'B' with height h'. Triangles ABC and A'B'C are congruent, so h = h'. The image is erect.

Lateral magnification m is the ratio of image height to object height:

(3.2) \quad m = \frac{h'}{h}

For a plane mirror, m = +1.

Image Reversal

Mirrors form three-dimensional images (Fig. 3.3). Fingers parallel to the mirror are not reversed; the finger pointing toward the mirror is reversed front to back. A mirror reverses front to back, not right to left.

Effect of Rotation of Mirror on the Reflected Ray

If a mirror rotates through an angle \alpha, the reflected beam moves through 2\alpha (Fig. 3.4).

\angle CBC' = \angle ABC' - \angle ABC = 2(i + \alpha) - 2i = 2\alpha

Multiple Plane Mirrors

Multiple images are formed with two or more plane mirrors due to multiple reflections (Fig. 3.5). The number of images, N, depends on the angle \alpha between the mirrors:

(3.3) \quad N = \frac{360^\circ}{\alpha} - 1

For parallel mirrors (\alpha = 0^\circ), N is infinite.

Properties of Images in Plane Mirrors

Image is virtual and erect.
Image is as far behind as the object is in front.
Right side transforms into the left side.
Magnification is unity (image size equals object size).
Rotation of the mirror by an angle causes the reflected ray to turn twice the angle.
Multiple images are formed in multiple mirrors, given by N = \frac{360^\circ}{\alpha} - 1.

Applications

Plane mirrors are used as choppers, beam deflectors, image rotators, scanners, and to amplify/measure rotations in laboratory apparatus.

Caution: Reflection occurs if the average depth of surface irregularities is much less than the wavelength of incident light. Reflector size must be larger than the wavelength; otherwise, light scatters.

Reflection at Spherical Mirrors

Mirrors can be curved. Spherical mirrors are segments of a spherical surface with circular edges. Types: concave (inner surface reflects) and convex (outer surface reflects).

Basic Terms

Centre of curvature, C: Center of the sphere.
Vertex (pole), P: Midpoint of the mirror.
Radius of curvature, R: Radius of the sphere.
Principal axis: Line through C and P.
Principal focus, F: Point where rays parallel to the principal axis converge (concave) or appear to emerge from (convex).
Focal plane: Plane through F, perpendicular to the principal axis.
Focal length, f: Distance PF; f = R/2.
Power of the mirror: P = 1/f.
Aperture: Diameter MN of the circular outline (Fig. 3.6); determines the light gathering capability.

Paraxial Rays and Paraxial Approximation

A paraxial ray has a small angle \theta with the principal axis (less than 10°). Small angle approximation: \cos \theta = 1, \sin \theta \approx \theta, \tan \theta \approx \theta. This is the first-order theory (Gaussian optics). Good image formation with monochromatic light occurs when rays are paraxial.

Sign Convention

Incident light travels from left to right.
Distances are measured from the vertex.
Distances in the incident light direction are positive; opposite are negative.
Upward heights are positive; downward are negative.
Clockwise angles are negative; counterclockwise are positive.

To derive formulas, treat distances and angles as positive, deduce an equation, then apply the sign convention.

Spherical Mirror Equation

Consider a concave mirror (Fig. 3.8). Point O is on the optic axis, distance from O to P is greater than R. A ray from O strikes the mirror at Q, making equal angles with the normal CQ. A second ray through C is normal. Reflected rays intersect at I, forming a real image.

Exterior angle of a triangle equals the sum of opposite interior angles. From triangle OQC, \gamma = \alpha + \theta. From triangle OQI, \beta = \alpha + 2\theta. Eliminate \theta to get \alpha + \beta = 2\gamma (3.6).

Let h be the height of Q above the axis, and \delta be the distance BP.

\tan \alpha = \frac{h}{u-\delta}
\tan \beta = \frac{h}{v-\delta}
\tan \gamma = \frac{h}{R-\delta}

In paraxial approximation, angles are small, so \tan \alpha \approx \alpha etc., and \delta is negligible.

$$\alpha = \frac{h}{u}