Light Waves

• Light waves do not require a material medium to propagate; they can travel in a vacuum.
• The electric field is the quantity that changes with space and time in a light wave. It is a vector quantity and transverse to the direction of propagation.
• Because light waves are transverse, they can be polarized.
• The speed of light in a vacuum (c) is a universal constant, approximately equal to 3 x 10^8 m/s.
• When a light wave travels in a transparent material, its speed decreases by a factor µ, the refractive index of the material.
• Equation of a light wave travelling along the x-direction: E = E₀ sin ω(t − x/v)
• For a spherical wave originating from a point source: E = (a/r) sin ω(t − r/v) where a is a constant. The amplitude is inversely proportional to the distance, and intensity is inversely proportional to the square of the distance.
• Visible light has frequencies from approximately 3800 x 10^11 Hz to 7800 x 10^11 Hz, corresponding to wavelengths of 380 nm to 780 nm.
• Different wavelengths of light correspond to different colors (e.g., 780 nm is red, 380 nm is violet).
• Monochromatic light consists of a single wavelength.
• White light is a mixture of all wavelengths from about 380 nm to 780 nm in appropriate proportions.

• The optical disturbance reaches the particles on a surface σ at time t=0 and lasts for a short interval.
• These particles on σ then send spherical wavelets which spread beyond σ.
• At time t, each of these wavelets has a radius vt.
• The sphere Σ is the geometrical envelope of all the secondary wavelets which were emitted at time t=0 from the primary wavefront σ.
• The secondary wavelets from σ superpose in such a way that they produce a new wavefront at the geometrical envelope of the secondary wavelets.
• Secondary wavelets are emitted only in the forward direction unless there is a change in medium where reflection occurs, in which case backward wavelets are also considered.

• A parallel light beam is incident upon a reflecting plane surface σ.
• The wavefronts of the incident wave are planes perpendicular to the direction of incidence.
• After reflection, the light returns in the same medium.
• As the various points of σ are reached by the wavefront AB, they become sources of secondary wavelets emitted in both forward and backward directions.
• For reflection, the wavelets emitted in the backward directions are considered.
• At time t, the wavelet from point A has a radius vt.

• Fraunhofer Diffraction: Occurs when the source and screen are effectively at infinite distance from the diffracting element. This condition is often achieved using lenses to collimate the light from the source and focus the diffracted light onto the screen.
• Huygens' Principle: Each point on the wavefront within the slit acts as a source of secondary wavelets.
• Path Difference: The optical path difference between waves from the top and center of the slit, arriving at an angle θ, is (b/2)sinθ, where b is the slit width.
• Condition for First Minimum: Occurs when (b/2)sinθ = λ/2, leading to b sinθ = λ. This is because the waves from the top half of the slit cancel the waves from the bottom half due to destructive interference.
• General Condition for Minima (Dark Fringes): b sinθ = nλ, where n is an integer (n = 1, 2, 3,...).
• Amplitude of Electric Field: The amplitude E' of the electric field at a point P on the screen is given by E' = E₀ (sinβ)/β, where E₀ is the amplitude at θ = 0, and β = (π/λ)b sinθ.
• Intensity: The intensity I at a point P is proportional to the square of the amplitude.

• A converging lens can never form a perfect point image of a distant point source; it produces a bright disc surrounded by dark and bright rings due to diffraction.
• The radius of this diffraction disc is given by R = 1.22λD/b, where λ is the wavelength of light, D is the distance from the lens to the focal point, and b is the diameter of the lens.
• Fresnel Diffraction at a Straight Edge: When light passes by an opaque obstacle with a sharp edge, the light is diffracted. The intensity gradually decreases inside the geometrical shadow. Above the shadow, the intensity alternates between maximum and minimum.
• Limit of Resolution: The diffraction disc formed by a lens limits the ability to resolve two neighboring points. If the image discs of two points are too close and overlap significantly, the points are not resolved.
• Two points are considered just resolved when their image discs are sufficiently separated to be distinguished.
• The angular radius (θ) of the diffraction disc is given by sinθ ≈ 1.22λ/b, where b is the radius of the lens. A larger lens radius improves resolution.

Linearly Polarized Light:

• When the phase difference (δ) between E_y and E_z is 0 or π, the light is linearly polarized.
• The electric field is always parallel to a fixed direction.
• The plane containing the electric field and the direction of propagation is called the plane of polarization.

Circularly Polarized Light:

• When δ = π/2 and E_1 = E_2, the light is circularly polarized.
• The electric field rotates at a uniform angular speed ω.
• The magnitude of the electric field remains constant.
• The tip of the electric field traces out a circle.

Elliptically Polarized Light:

• When δ = π/2 but E_1 ≠ E_2, the light is elliptically polarized.
• The tip of the electric field traces out an ellipse.

Unpolarized Light:

• Light emitted by ordinary sources (electric lamp, mercury tube, candle, sun) is generally unpolarized.
• The electric field at a point keeps changing its direction randomly and rapidly.
• E_y and E_z have a phase difference δ that changes randomly with time: E_y = E_1 sin(ωt − kx + δ) and E_z = E_2 sin(ωt − kx).

Polaroids:

• Polarizers are used to produce polarized light from unpolarized light.
• Polaroid sheets transmit light with the E-vector parallel to a special direction (transmission axis).
• Hydrocarbon chains in polaroids absorb the electric field parallel to the chains and transmit the field perpendicular to the chains.
• If linearly polarized light is incident on a polaroid with the E-vector parallel to the transmission axis, the light is completely transmitted.
• If the E-vector is perpendicular to the transmission axis, the light is completely stopped.
• If the E-vector is at an angle θ with the transmission axis, light is partially transmitted.