Engineering Physics Notes on Interference of Light and Coherence

Fundamental Principles of Interference

• Interference of light is defined as the superimposition of two or more light waves traveling either in the same phase or maintaining a constant phase relationship between them.

• This process results in a non-uniform distribution of intensity within the medium.

• Constructive Interference: Occurs in regions where the light intensity is maximum.

• Destructive Interference: Occurs in regions where the light intensity is minimum.

• Fringes: The alternate bright and dark bands formed when two light waves are made to interfere.

Coherence and Coherent Sources

• Coherence Definition: Two light waves of the same or nearly the same frequency and wavelength are considered coherent when the phase difference between them is constant or they travel in the same phase.

• Incoherence Definition: Two waves are incoherent if there is a random or changing phase relationship between them.

• Practical Limitation: Two independent sources cannot produce coherent light.

• Methods to Produce Coherent Sources:

  1. Lloyd’s Single mirror method.

  2. Fresnel’s Double mirror method.

  3. Michelson interferometer.

  4. Young’s double slit experiment.

  5. Fresnel Biprism experiment.

Mathematical Expression of the Resultant Wave

• To find the expression of interference, consider two waves $Y_1$ and $Y_2$ of the same frequency ($\omega$) with slight differences in their amplitudes, designated as $a_1$ and $a_2$.

• According to the superposition theorem:   $Y = Y_1 + Y_2$

• Substituting the wave equations where $\phi$ is the phase difference between the interfering waves:   $Y = a_1 \sin(\omega t) + a_2 \sin(\omega t + \phi)$

• Expanding the second term using the trigonometric identity $\sin(A+B) = \sin(A)\cos(B) + \cos(A)\sin(B)$:   $Y = a_1 \sin(\omega t) + a_2 (\sin(\omega t)\cos(\phi) + \sin(\phi)\cos(\omega t))$   $Y = \sin(\omega t) (a_1 + a_2 \cos(\phi)) + a_2 \sin(\phi) \cos(\omega t)$

• By substitution, let:   $A \cos(\theta) = a_1 + a_2 \cos(\phi)$ … [Equation 1]   $A \sin(\theta) = a_2 \sin(\phi)$ … [Equation 2]

• The resultant wave equation becomes:   $Y = A \sin(\omega t)\cos(\theta) + A \cos(\omega t)\sin(\theta)$   $Y = A \sin(\omega t + \theta)$

• This confirms that the resultant is a sinusoidal wave with amplitude $A$.

Determination of Resultant Amplitude and Intensity

• To calculate the Resultant Amplitude ($A$), square and add Equation 1 and Equation 2:   $(A \sin(\theta))^2 + (A \cos(\theta))^2 = (a_2 \sin(\phi))^2 + (a_1 + a_2 \cos(\phi))^2$   $A^2 (\sin^2(\theta) + \cos^2(\theta)) = a_2^2 \sin^2(\phi) + a_1^2 + a_2^2 \cos^2(\phi) + 2a_1 a_2 \cos(\phi)$   $A^2 = a_1^2 + a_2^2 (\sin^2(\phi) + \cos^2(\phi)) + 2a_1 a_2 \cos(\phi)$   $A^2 = a_1^2 + a_2^2 + 2a_1 a_2 \cos(\phi)$

• Intensity Relationship: The intensity of a sinusoidal wave is given by $2\pi^2 n^2 a^2 \rho v$. Since the frequency ($n$) is fixed, and for electromagnetic waves, the density of the medium ($\rho$) and velocity ($v$) are fixed, the only variable is the amplitude.

• Therefore, Intensity ($I$) is proportional to the square of the amplitude ($A^2$):   $\text{Intensity} \propto \text{Amplitude}^2$

• Resultant Intensity ($I$):   $I = A^2 = a_1^2 + a_2^2 + 2a_1 a_2 \cos(\phi)$

• This expression shows that intensity depends on the individual intensities of the two coherent sources ($I_1 = a_1^2$ and $I_2 = a_2^2$) and the phase difference ($\phi$) between them.

Specific Conditions for Constructive and Destructive Interference

• Constructive Interference ($I_{max}$):

  • Occurs when $\phi = 0, 2\pi, 4\pi \dots 2n\pi$.

  • Maximum Intensity: $I_{max} = a_1^2 + a_2^2 + 2a_1 a_2 = (a_1 + a_2)^2$.

  • Path Difference ($\Delta x$): Determined by $\Delta x = \frac{\lambda}{2\pi} \times \phi$.

  • Condition: $\Delta x = \frac{\lambda}{2\pi} \times 2n\pi = 2n \frac{\lambda}{2}$.

  • Summary: Phase difference must be an even multiple of $\pi$, and path difference must be an even multiple of $\frac{\lambda}{2}$.

• Destructive Interference ($I_{min}$):

  • Occurs when $\phi = \pi, 3\pi \dots (2n+1)\pi$.

  • Minimum Intensity: $I_{min} = a_1^2 + a_2^2 - 2a_1 a_2 = (a_1 - a_2)^2$.

  • Condition: $\Delta x = \frac{\lambda}{2\pi} \times (2n+1)\pi = (2n+1) \frac{\lambda}{2}$.

  • Summary: Phase difference must be an odd multiple of $\pi$, and path difference must be an odd multiple of $\frac{\lambda}{2}$.

Historical Context of Light Research

• 1670: Huygens explained the laws of reflection and refraction.

• 1801: Young’s Experiment demonstrated the interference property.

• 1873: Maxwell proved that light is an electromagnetic (EM) wave.

• 1905: Albert Einstein used the concept of energy packets (photons) to explain the photoelectric effect.

• 1923: Arthur H. Compton demonstrated the corpuscular (particle) nature of X-RAYS.

Average Intensity and Superposition Details

• Average Intensity ($I_{avrg}$) calculation over a cycle from $0$ to $2\pi$:   $I_{avrg} = \frac{\int_0^{2\pi} I \,d\delta}{\int_0^{2\pi} d\delta} = \frac{\int_0^{2\pi} (a_1^2 + a_2^2 + 2a_1 a_2 \cos(\delta)) \,d\delta}{\int_0^{2\pi} d\delta}$   $I_{avrg} = \frac{(a_1^2 [\delta]_0^{2\pi}) + (a_2^2 [\delta]_0^{2\pi}) + (2a_1 a_2 [\sin(\delta)]_0^{2\pi})}{[\delta]_0^{2\pi}}$   $I_{avrg} = \frac{2\pi (a_1^2 + a_2^2)}{2\pi} = a_1^2 + a_2^2 = I_1 + I_2$

• In the case where $a_1 = a_2 = a$:

  • $I_{avrg} = a^2 + a^2 = 2a^2$

  • $I_{max} = (a+a)^2 = (2a)^2 = 4a^2$

• Note: The phenomenon of interference is in accordance with the Law of Conservation of Energy; it represents a redistribution of intensity rather than a loss or gain of net energy.

Young’s Double Slit Experiment (YDSE) and Fringe Properties

• In YDSE, interference occurs due to the division of wave fronts from slits $S_1$ and $S_2$.

• Superimposition leads to energy redistribution:

  • Constructive Interference: Where waves superimpose in the same phase, resulting in maximum intensity.

  • Destructive Interference: Where waves superimpose in opposite phase, resulting in zero or minimum intensity.

• Shape of Fringes: Theoretically, the fringes are hyperbolas. However, due to the large eccentricity of these hyperbolas, they appear as straight lines on the screen.

• Spacing: All bright and dark fringes formed are equally spaced.

Conditions for Sustained and Distinct Interference

• Conditions for Sustained Interference:

  • The two superimposing waves must be coherent.

  • The two waves must have the same or nearly the same frequency.

  • If waves are polarized, the plane of polarization must be identical for both.

• Conditions for Distinct Interference:

  • The amplitudes of the waves should be same ($a_1 = a_2 = a$). This ensures $I_{max} = 4a^2$ and $I_{min} = 0$, creating extremely bright maxima and completely dark minima.

  • The distance between the source and the screen ($D$) must be large ($\beta \propto D$).

  • The distance between the two slits ($d$) must be small ($\beta \propto \frac{1}{d}$).

Coherency Types: Temporal and Spatial

• Temporal Coherency: Two points ($C$ and $D$) are taken within the coherence length in the direction of wave propagation. If the phase difference ($\Delta \phi$) at these points remains the same at different instances (e.g., time $t_1$ and $t_2$), the source has temporal coherency.

• Spatial Coherency: Two points ($C$ and $D$) are taken on a plane introduced perpendicular to the direction of wave propagation. If the phase difference ($\Delta \phi$) at these points remains the same at different instances ($t_1$ and $t_2$), the source possesses spatial coherency.

Holographic Image Reconstruction

• A hologram is reconstructed by illuminating it with the original reference beam.

• Individual zone plates in the hologram reconstruct the object wave that initially produced them.

• These individual wavefronts combine to reconstruct the entire object beam.

• The viewer perceives a wavefront identical to the one scattered from the original object, making the object appear as if it is still in place even if removed.

Questions and Discussion

• What is light (define)?

• Why are the fringes hyperbolic? (Refer to textbook for the eccentricity explanation).

• What will happen if radio waves of the same frequency with slight amplitude differences are cast in a medium simultaneously by two broadcasters? (This would result in interference, assuming coherence criteria are met).

• What are the conditions to obtain a sustained interference pattern?