Ray Optics(Geometrical Optics)

Ray Optics, also known as Geometrical Optics, is a branch of physics that explains how light travels in straight lines and interacts with different optical surfaces. It focuses on concepts like reflection, refraction, image formation, and the behavior of lenses and mirrors. By treating light as rays rather than waves, ray optics provides simple and powerful tools for understanding everyday optical phenomena such as how mirrors form images, how lenses focus light, and how devices like cameras, telescopes, and microscopes work. Ideal for beginners, ray optics forms the foundation for studying more advanced optical systems and technologies.

1.0Introduction

Optics: Optics is the branch of physics in which we study the behaviour and properties of light, including its interactions with matter.

Light is that part of electromagnetic waves which gives sensation to our eyes.

2.0Properties of Light

(1) Speed of light in vacuum, denoted by $c=3\times 10^{8}m/s$ approximately.

(2) Light is an electromagnetic wave .It consists of varying electric fields and magnetic fields.

(3) Light carries energy and momentum.

(4) The formula $v=f\lambda$ is applicable to light.

3.0Reflection of Light

When light rays hit the boundary between two media, such as air and glass, some of the light is reflected back into the original medium.

(a) Regular Reflection: Regular reflection occurs when light reflects off a smooth, flat surface.

(b)Diffused Reflection: When light reflects off a rough surface, scattering in multiple directions.

Laws of Reflection

(a) The incident ray, reflected ray, and the normal at the point of incidence all lie in the same plane, known as the plane of incidence (or plane of reflection).

This condition can be expressed mathematically as $(\hat{n}\times \hat{r})=0$

(b)The angle of incidence (the angle between the normal and the incident ray) is equal to the angle of reflection (the angle between the reflected ray and the normal), i.e., i.e.

$\angle i=\angle r$

 In Vector form  $\hat{r}=\hat{e}-2(\hat{e}\cdot \hat{n})\hat{n}$

Plane Mirror

• A plane mirror is formed by polishing one surface of a plane thin glass plate. It is also said to be silver on one side.
• A beam of parallel light rays incident on a plane mirror will reflect as a parallel beam of rays.

Properties of Image formed by a Plane Mirror

Point object

1. The object distance equals the image distance from the mirror.
2. Incident rays from a point object converge at a single point after reflection, forming an image.
3. The line joining the object and its image is normal to the reflecting surface.
4. A real object forms a virtual image, and a virtual object forms a real image.
5. The field of view is the area where the observer must be to see the image.

Extended object:

• An extended object refers to an object that occupies a finite amount of space and consists of multiple points, each of which can produce its own image. This contrasts with a point object, which has no dimensions.

4.0Reflection from Spherical Surface (Spherical Mirror)

Spherical Mirrors

A curved mirror is part of a hollow sphere. If reflection takes place from the inner surface then the mirror is called concave and if its outer surface acts as reflector it is convex. 

Important terms related with spherical mirrors

(a) Center of Curvature (C) : The center of curvature is the center of the sphere from which a spherical mirror is made.

(b) Pole (P) :The center of the mirror, known as the pole, is represented by point P on the mirror APB.

(c) Principal Axis :The principal axis is a line perpendicular to the mirror's plane, passing through the pole.

(d) Aperture (A) :The aperture is the part of the mirror that reflects light; in the figure, APB represents the aperture.

(e) Focal Length: When a parallel beam of light strikes a concave mirror, reflected rays converge at the principal focus (F). For a convex mirror, the rays appear to diverge from F. If the light strikes at an angle, the rays converge or diverge from a point in the focal plane, perpendicular to the principal axis.

Mirror formulae

$\frac{1}{v}+\frac{1}{u}=\frac{1}{f}$

Magnification

Lateral Magnification

$m=\frac{h_i}{h_0}==-\frac{v}{u}$

5.0Refraction of Light

Refraction occurs when light changes mediums, causing a shift in speed and direction. If light strikes at an angle (0° < i < 90°), it bends due to this speed change. Light incident normally passes straight but is still refracted. Refraction without reflection is impossible, and as the angle of incidence increases, more energy is reflected. The refractive index is the ratio of light speed in vacuum to its speed in the medium.

• The refractive index of a medium is the ratio by which the speed of light decreases compared to its speed in a vacuum.

  $\mu =\frac{c}{v}=\frac{\text{Speed of light in vacuum}}{\text{Speed of light in medium}}$

• A higher refractive index indicates slower light speed in the medium, making it optically denser, while a lower refractive index means the medium is optically rarer.

Relative refractive index

${}^1{\mu }_2=\frac{{\mu }_2}{{\mu }_1}=\frac{\left(\frac{c}{v_2}\right)}{\left(\frac{c}{v_1}\right)}=\frac{v_1}{v_2}$

Laws of Refraction

(a) "The incident ray, the normal at the point of incidence, and the refracted ray all lie in the same plane, known as the plane of incidence."

(b) $\frac{\sin i}{\sin r}=\text{Constant}\Rightarrow \text{Snell’s Law}$$\Rightarrow \frac{\sin i}{\sin r}=\frac{n_2}{n_1}=\frac{v_1}{v_2}=\frac{{\lambda }_1}{{\lambda }_2}$

$n_1\sin i=n_2\sin r$

$\frac{n_2}{n_1}={}^1{n}_2$= Refractive Index of the second medium with respect to the first medium.

Special Cases

1. When Normal incidence : i = 0 ; from Snell’s law : r = 0

2. When light moves from an optically denser to an optically rarer medium, it bends away from the normal.

3. When light moves from an optically rarer to an optically denser medium, it bends towards the normal.

6.0Critical Angle and Total Internal Reflection ( T. I. R.)

The critical angle is the angle in the denser medium where the refraction in the rarer medium is 90°. If the angle exceeds the critical angle, total internal reflection occurs, and the interface acts like a mirror.

${\theta }_c={\sin }^{-1}\left(\frac{n_r}{n_d}\right)$

Conditions of T. I. R.

(a) Light is incident on the interface from denser medium.

(b) Angle of incidence should be greater than the critical angle $(i>{\theta }_c)$

7.0Refraction Through Single Curved Surfaces

Law of refraction at spherical surface:

When light passes from a medium of refractive index 1 to a medium of refractive index 2 by a spherical surface of radius of curvature R then the relation between object distance u and image distance v is given by

$\frac{{\mu }_2}{v}-\frac{{\mu }_1}{u}=\frac{{\mu }_2-{\mu }_1}{R}$

Sign convention

• All distances are quantified from the pole (P).
• Distances along the course of the incident rays are considered positive.
• Distances above the principal axis are taken as positive.

Terms related to refraction at spherical surfaces

(A) Centre of curvature (C): It is the centre of a sphere of which the surface is a part.

(B) Radius of curvature (R): It is the radius of the sphere of which the surface is a part.

(C) Pole (P) :It is the geometrical centre of the spherical refracting surface.

(D) Principal Axis : Straight line joining the centre of curvature to the pole.

(E) Focus :"When a parallel beam of paraxial rays hits a spherical refracting surface, the rays converge or diverge depending on the surface's curvature and refractive indices. The point of convergence or apparent divergence on the principal axis is called the focus."

Note:

(1) It is not always necessary that for a convex boundary the parallel rays always converge. Similarly, for concave boundaries the incident parallel ray may converge or diverge depending upon the refractive index of two media.

(2) Laws of refraction are valid for spherical surfaces also.

(3) Pole, centre of curvature, Radius of curvature, Principal axis etc. are defined as spherical mirrors except for the focus.

Lateral Magnification

$m=\frac{h_i}{h_o}=\frac{\text{height of image}}{\text{height of object}}$

$m=-\frac{\frac{v}{{\mu }_2}}{\frac{u}{{\mu }_1}}=-\frac{{\mu }_1}{{\mu }_2}\cdot \frac{v}{u}$

8.0Lens

A lens is a transparent object with two refracting surfaces, at least one of which is curved, and the refractive index of the material differs from that of its surrounding medium.

Types of lenses

Depending upon the shape of the refracting surfaces following types of lenses can be formed:

Lens-Maker’s formula

It relates the focal length of the lens to the relative refractive index  of the lens material and the radii of curvature of the two surfaces.

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

$\mu =\frac{{\mu }_2}{{\mu }_1}=\frac{\text{Refractive index of lens}}{\text{Refractive index of surrounding}}$

$R_1$ is the radius of curvature of first surface and $R_2$ is the radius of curvature of the second surface from where light emerges out in the first medium.

Note:

(1) The Lensmaker’s formula is applicable for thin lenses only. The values of $R_1$ and are to be put in accordance with the cartesian sign convention.

(2) Position of object and image are interchangeable. These positions are called conjugate positions.

Lens formula

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

$\frac{1}{f}=\left(\frac{{\mu }_2}{{\mu }_1}-1\right)\left(\frac{1}{R_1}-\frac{1}{R_2}\right)$

Lateral magnification

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

Note:

(1) If converging rays fall, the focus is on the other side of the direction of incidence and for diverging rays focus is on the same region of the direction of incidence.

(2) m has negative and positive values for real virtual pairs.

(3) Use cartesian sign convention with optical centre of lens as origin.

Power of a lens

When focal length is written in metre then $P=\frac{\mu }{f}D$ is known as the power of the lens. Where D is (diopter) unit of power.

Combination of lenses

(A) Net magnification, $m=m_1\times m_2\times m_3\times \cdots$

(B) If thin lenses are kept close together with their principal axis coincide then, $\frac{1}{f}=\frac{1}{f_1}+\frac{1}{f_2}+\frac{1}{f_3}+\cdots$

f = (+) ve gives equivalent converging lens

= (–) ve gives equivalent diverging lens

9.0 Prism

"A prism is a homogeneous, solid, transparent refracting medium with two plane surfaces inclined at an angle."

A=Refracting angle or the Angle of Prism( Apex angle)

$\delta$== Angle of Deviation

Angle of Deviation ($\delta$)

It is the angle between the emergent and the incident ray. In other words, it is the angle through which the incident ray turns while passing through a prism.

$\delta =i+e-A$

Condition of No Emergence

For face AB is

$r_i=\max =90^{\circ },\text{at face AC }{r}_2>{\theta }_c$

• A ray of light will not emerge out of a prism (what ever be the angle of incidence) if $A>2{\theta }_c$
• $\mu >Cosec(\frac{A}{2})$

Critical Angle

• It is the angle of prism above which incident light ray on first surface will not emerge out from the second surface for all possible values of angle of incidence$A={\theta }_{2}c$

Condition of Grazing Emergence

If a ray can emerge out of a prism, the value of angle of incidence i for which angle of emergence e=90° is called the condition of grazing emergence.

$r_2={\theta }_c$

$r_1=A-r_2=A-{\theta }_c$

$i={\sin }^{-1}\left[\sqrt{{\mu }^2-1}\sin A-\cos A\right]$

Note: The light will emerge out of a given prism only if the angle of incidence is greater than the condition of grazing emergence.

Condition for Minimum Deviation

Minimum deviation happens when the angle of incidence equals the angle of emergence.

$i=e,r_1=r_2=r,{\delta }_{\min }=2i-A,r=\frac{A}{2}$

$\mu =\frac{\sin i}{\sin r}=\frac{\sin \left(\frac{{\delta }_{\min }+A}{2}\right)}{\sin \left(\frac{A}{2}\right)}$

Note: In the condition of minimum deviation the light ray passes through the prism symmetrically, i.e., the light. The ray in the prism becomes parallel to its base.

10.0Graphical Representation of Angle of Deviation

11.0Optical Instruments

1. Simple microscope

(1) It is a single convex lens of lesser focal length.

(2) Also called magnifying glass or reading lens.

(3) Magnification’s, when final image is formed at D and ∞ i.e $m_{0}and{m}_e$

$m_o={\left(1+\frac{D}{f}\right)}_{\max }\text{and}{m}_o={\left(\frac{D}{f}\right)}_{\min }$

2. Compound microscope

(1) Consist of two converging lenses called objective and eye lens.

(2) $f_{eyelens}>f_{objective}$ and $(Diameter{)}_{eyelens}>(Diameter{)}_{objective}$

(3) The final image is magnified, virtual and inverted.

Magnification

$m=m_{\text{objective}}\times m_{\text{eye lens}}$

$m_D=-\left(\frac{v_o-f_o}{f_o}\right)\left(1+\frac{D}{f_e}\right)$

$m_{\infty }=-\left(\frac{v_o-f_o}{f_o}\right)\frac{D}{f_e}$

3. Astronomical Telescope

(1) Used to see heavenly bodies.

(2) $f_{\text{objective}}>f_{\text{eye lens}}and{d}_{\text{objective}}>d_{\text{eye lens}}$

(3) The intermediate image is real, inverted and small.

(4) The final image is virtual, inverted and small.

(5) Magnification $m_D=\frac{f_o}{f_e}\left(1+\frac{f_e}{D}\right)\text{and}{m}_{\infty }=-\frac{f_o}{f_e}$

(6)  Length  $L_D=f_o+u_e=f_o+\frac{f_{e}D}{f_e+D}\text{and}{L}_{\infty }=f_o+f_e$