Ray Optics and Optical Instruments Notes

Optics Synopsis Ray Optics

Laws of reflection:

The incident ray, the reflected ray, and the normal to the reflecting surface are coplanar.

The angle of incidence (i) = angle of reflection (r).

Some important facts about reflection in a plane mirror:

The image and the object are equidistant from the plane mirror.

The image formed by the plane mirror is virtual and of the same size as the object but laterally inverted.

When the plane mirror is turned through an angle θ, the reflected ray is turned through 2θ in the same direction.

The number of images (N) formed due to two plane mirrors inclined at an angle θ is given by

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⇒ \(N=\frac{360^{\circ}}{\theta}-1 \text { when } \frac{360^{\circ}}{\theta} \text { is even }\)

⇒ and \(N=\frac{360^{\circ}}{\theta} \text { when } \frac{360^{\circ}}{\theta} \text { is odd }\)

For θ = 60°, N = 6-1 = 5; and for θ = 40°, N = 9.

The minimum height of the plane mirror required to view the whole image of an object is half the height of the object.

The deviation of a ray is the angle between the directions of the incident and reflected rays.

Reflection from spherical surfaces (concave and convex mirrors):

The image formed by a concave mirror is real for u ≥ f and virtual for u < f.

The image formed by a convex mirror is always virtual and diminished.

⇒ Mirror formula: \(\frac{1}{u}+\frac{1}{v}=\frac{1}{f}\)

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⇒ Lateral or transverse magnification = \(m_{\mathrm{T}}=-\frac{v}{u}\)

⇒ Longitudinal magnification = \(m_{\mathrm{L}}=-m_{\mathrm{T}}^2=-\left(\frac{v}{u}\right)^2\)

Refraction through a plane surface:

Symmetrical form of Snell’s law:

μ₁ sinθ₁ = μ₂ sinθ₂ = μ₃ sinθ₃ = … .

Here μ is the refractive index of the medium.

Note that this law is true for both plane and curved surfaces.

Refractive index (μ) and speed of light (c):

⇒ \(\mu=\frac{\text { speed of light in vacuum }}{\text { speed of light in the medium }}=\frac{c_0}{c}\)

In symmetrical form, μ₁c₁ = μ₂c₂ =c₀.

During refraction, the speed of light (c) and its wavelength (λ.) change but its frequency (f) remains invariant. Hence,

⇒ \(\mu_1 \lambda_1=\mu_2 \lambda_2=\ldots=\frac{c_0}{f}\)

Apparent depth (h):

1. Refractive index = \(\frac{\text { real depth }}{\text { apparent depth }}=\frac{H}{h}\)
2. Shift of image = \(H-h=\left(1-\frac{1}{\mu}\right) H\)

Critical angle \(\left(\theta_{\mathrm{c}}\right): \sin \theta_{\mathrm{c}}=\frac{\mu_2}{\mu_1}, \text { where } \mu_1>\mu_2\)

Refraction through a spherical surface separating two transparent media:

⇒ \(\frac{\mu_2}{v}-\frac{\mu_1}{u}=\frac{\mu_2-\mu_1}{R}\)

Magnification = \(m=\frac{\mu_1}{\mu_2} \cdot \frac{v}{u}\)

⇒ Lens-maker’s formula: \(\frac{1}{f}=(\mu-1)\left(\frac{1}{R_1}-\frac{1}{R_2}\right)\)

The power of a lens is the reciprocal of its focal length.

⇒ So, \(P(\text { in dioptres })=\frac{100}{f(\text { in centimetres })}\)

P is positive for a convex lens and negative for concave lens.

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For two lenses in contact,

⇒ \(\frac{1}{F}=\frac{1}{f_1}+\frac{1}{f_2} \text { and } P=P_1+P_2\)

Path of a ray through a prism: The emergent ray bends towards the base of a prism.

A = r + r’ and A + δ = i + i’

Condition for minimum deviation:

i = i’ and r = r’.

⇒ \(\mu=\frac{\sin \frac{A+\delta_m}{2}}{\sin \frac{A}{2}}\)

Deviation through a thin prism (with A small) = δ = (μ-1)A.

Dispersion (Cauchy’s relation): \(\mu=A+\frac{B}{\lambda^2}\)

Mean deviation = δ = (μ_y-1)A.

Angular dispersion = θ = δ_V– δ_R =(μ_V – μ_R.)A.

⇒ Dispersive power = \(\omega=\frac{\text { angular dispersion }}{\text { mean deviation }}=\frac{\theta}{\delta_{\mathrm{y}}}=\frac{\mu_{\mathrm{V}}-\mu_{\mathrm{R}}}{\mu_{\mathrm{y}}-1}\)

Dispersion without deviation (direct-vision spectroscope):

1. Condition: \(\frac{A_1}{A_2}=-\frac{\mu_{2 y}-1}{\mu_{1 y}-1} \)
2. Net angular dispersion = θ = A₁(μ_1y – 1)(ω₁ – ω₂).

Deviation without dispersion (achromatic combination of prisms):

Condition: \(\frac{A_1}{A_2}=-\frac{\omega_2\left(\mu_{2 y}-1\right)}{\omega_1\left(\mu_{1 y}-1\right)}\)

Achromatic combination of lenses: \(\frac{\omega_1}{f_1}+\frac{\omega_2}{2}=0\)

Optical instruments: These are devices to increase the visual angle.

Magnification (m) of a simple microscope:

1. \(m=1+\frac{D}{f}\) when the final image is at a distance D.
2. \(m=\frac{D}{f}\) for normal adjustment, i.e., when the eye is most relaxed (v = ∞).

Magnification (m) of a compound microscope:

1. \(m=\frac{-v_{\mathrm{o}}}{u_{\mathrm{o}}}\left(1+\frac{D}{f_{\mathrm{e}}}\right) \approx-\frac{L}{f_{\mathrm{o}}}\left(1+\frac{D}{f_{\mathrm{e}}}\right)\)
2. For normal adjustment, m = \(-\frac{L}{f_0} \frac{D}{f_e}\)

Astronomical telescope:

1. Magnification = m = \(-\frac{f_0}{f_{\mathrm{e}}}\)
2. Tube length = L = f₀ +f_e.

Wave Optics

Light (or any other electromagnetic wave) is a transverse wave motion.

A wavefront is an imaginary surface containing all the points vibrating in the same phase.

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Coherent sources: Sources are said to be coherent if the phase difference between them remains invariant with time.

Path difference is

Δx = S₂P-S₁P ≈ S₂N = d sin θ,

where d = separation between the coherent sources S₁ and S₂.

Phase difference = \(\phi=\frac{2 \pi}{\lambda}(\Delta x)\)

⇒ General relation: \(\frac{\Delta x}{\lambda}=\frac{\phi}{2 \pi}=\frac{\Delta t}{T}\), where Δt = time delay.

Optical path = p(path length in the medium).

Intensity ∝ (amplitude)².

Resultant intensity (I):

A₁ = A1₂ + A₂² + 2A₁A₂ Cos Φ

or I = I₁ +1₂ + 2√I₁I₂cos Φ.

Maxima and minima:

For maxima, cos Φ = +1 ⇒ Φ = 2nπ for n = 0, 1, 2, … .

For minima, cos Φ = -1 ⇒ Φ = (2n +1)π for n = 0, 1, 2, … .

The following table shows the conditions for maxima and minima.

Fringe width = \(\beta=\frac{\lambda D}{d}\), where λ = wavelength, D = distance between the screen and the plane of the slits, and d = slit-separation.

Angular width of a fringe = θ = \(\frac{\beta}{D}=\frac{\lambda}{d}\)

⇒ Maximum number of observable fringes = \(N_{\max }=\frac{d}{\lambda}\)

Shift of a fringe system: When a thin film of thickness t and refractive index μ is pasted over one slit, the optical path introduced is Δx = (μ-1)t.

number of fringes shifted = N = \(=\frac{(\mu-1) t}{\lambda}\)

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The angular position of minima in diffraction:

For a single slit of width a:

⇒ a sin θ_n = nλ ⇒ \(\theta_n \approx n\left(\frac{\lambda}{a}\right)\)

For a circular aperture of diameter b:

⇒ b sin θ =1.22λ ⇒ \(\theta \approx 1.22\left(\frac{\lambda}{b}\right)\)

Airy disc: It is the central bright circular region of the diffraction pattern formed due to a circular aperture (objective lens, pupil, etc.).

1. Angular width of the principal maximum due to a single slit = \(2 \theta_1=2\left(\frac{\lambda}{a}\right)\)
2. Angular radius of the Airy disc = θ = \(1.22\left(\frac{\lambda}{b}\right)\)

Rayleigh criterion: Two-point sources cannot be resolved if their separation is less than the radius of the Airy disc.

For resolution, the angular separation is \(\theta \geq \ 1.22\left(\frac{\lambda}{b}\right)\).

Resolving power (RP):

1. For a telescope, RP = \(\frac{1}{\Delta \theta}=\frac{b}{1.22 \lambda}\) where b = diameter of the objective.
2. For a microscope,r RP = \(\frac{1}{d_{\min }}=\frac{2 \mu \sin \beta}{1.22 \lambda}=\frac{2 \mathrm{NA}}{1.22 \lambda}\), where NA = numerical aperture = μsinβ

Polarization:

Brewster’s law: μ = tanθ_p.

Intensity of the polarized light = I = \(\frac{1}{2} I_0\), where I₀ = intensity of the unpolarized light.

Malus’s law: I = I₀cos²θ, where I₀ = initial intensity of the incident polarized light, I = intensity after its emergence and θ = angle between the pass axes.