Interference of Light

Coherent And Incoherent Addition Of Waves

When two or more waves overlap in a region of space, their effects add up according to the principle of superposition. The nature of the resulting wave depends significantly on the relationship between the phases of the individual waves. This leads to the concepts of coherent and incoherent addition of waves.

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Coherent Sources

Two sources of waves are said to be coherent if they produce waves of the same frequency and have a constant phase difference between them over time. If the phase difference is always zero, they are in phase. If the phase difference is constant but non-zero, they maintain that relative phase over time.

Coherent sources are required to observe sustained interference patterns. In a coherent wave, the phase relationship between different points in space or different points in time is predictable and constant.

Examples of coherent sources (or how to obtain them):

• Light from a laser is highly coherent.
• To obtain coherent light from a single source (like a bulb), the light is often split into two parts that then travel different paths and are finally recombined. The two resulting beams are coherent because they originate from the same wave segments of the original source, and any random phase changes in the original source affect both paths equally, maintaining a constant relative phase difference. This is the technique used in Young's double-slit experiment.

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Incoherent Sources

Two sources of waves are said to be incoherent if they produce waves whose phase difference varies randomly and rapidly with time. For example, light from a conventional bulb consists of waves emitted by individual atoms, which radiate light pulses randomly and independently. The phase relationship between the waves from different atoms (or even different pulses from the same atom) changes rapidly and unpredictably.

If two incoherent sources are used, while interference occurs instantaneously according to superposition, the pattern averages out over time due to the rapid random phase changes. No stable interference pattern is observed.

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Addition of Waves

• Coherent Addition: When waves from coherent sources interfere, the superposition leads to sustained regions of constructive interference (where waves add up to produce a larger amplitude) and destructive interference (where waves cancel out to produce a smaller or zero amplitude). The amplitude of the resultant wave varies in space (or time) depending on the relative phase of the interfering waves. The intensity of the resultant wave is related to the square of the resultant amplitude and is not simply the sum of the intensities of the individual waves. $I_{resultant} \ne I_1 + I_2$. In constructive interference, $I_{resultant}$ is greater than $I_1 + I_2$; in destructive interference, it is less than $I_1 + I_2$.
• Incoherent Addition: When waves from incoherent sources interfere, the instantaneous interference pattern changes rapidly. Over any measurable time interval, the average effect is simply the sum of the intensities of the individual waves, as if they passed through each other without interfering. The total intensity is the sum of the individual intensities. $I_{resultant} = I_1 + I_2$.

Interference phenomena are observed only with coherent sources because a stable phase relationship is needed for the points of constructive and destructive interference to remain fixed over time, creating an observable pattern of varying intensity.

Interference Of Light Waves And Young’S Experiment ($ \beta = \frac{\lambda D}{d} $)

Interference of light is the phenomenon that occurs when two or more light waves superpose, resulting in a redistribution of light intensity in the region of overlap. This redistribution creates an interference pattern, which typically consists of alternating bright and dark fringes.

Interference is a strong evidence for the wave nature of light. It can only occur if the interfering waves are coherent.

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Young's Double-Slit Experiment (YDSE)

Thomas Young performed his famous double-slit experiment in 1801, which provided compelling evidence for the wave theory of light. In this experiment, monochromatic light (light of a single wavelength or frequency) from a single source (S) is passed through two narrow, closely spaced parallel slits ($S_1$ and $S_2$) in an opaque screen. These two slits act as two coherent secondary sources of light waves because they are illuminated by the same wavefront from the original source.

(Image Placeholder: A diagram showing a single slit illuminated by a source. A screen with two narrow parallel slits S1 and S2 is placed behind the first slit. A screen is placed some distance away from the double slit. Show waves from S1 and S2 spreading out and overlapping on the screen. Show points of constructive (bright fringes) and destructive (dark fringes) interference on the screen, forming alternating bands.)

The waves from $S_1$ and $S_2$ are coherent and travel to a screen placed at a distance $D$ from the double slit. At any point P on the screen, the waves from $S_1$ and $S_2$ interfere. The outcome of the interference at point P (constructive or destructive) depends on the path difference between the waves travelling from $S_1$ to P and from $S_2$ to P, i.e., $\Delta x = S_2P - S_1P$.

Assuming the distance $D$ to the screen is much larger than the distance between the slits $d$ ($D \gg d$), and the point P is at a distance $y$ from the center of the screen (O), the path difference $\Delta x$ can be approximated as:

$ \Delta x = S_2P - S_1P \approx \frac{yd}{D} $ (for small angles, assuming P is close to the central axis)

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Conditions for Constructive and Destructive Interference

• Constructive Interference (Bright Fringes): Occurs when the path difference is an integer multiple of the wavelength ($\lambda$). The waves arrive in phase, and their amplitudes add up, resulting in maximum intensity.

  $ \Delta x = n\lambda $

  where $n$ is an integer (0, $\pm 1, \pm 2, ...$). $n=0$ corresponds to the central bright fringe.
• Destructive Interference (Dark Fringes): Occurs when the path difference is an odd multiple of half the wavelength. The waves arrive out of phase (180° phase difference), and their amplitudes cancel out (if amplitudes are equal), resulting in minimum (usually zero) intensity.

  $ \Delta x = (n + \frac{1}{2})\lambda = (2n+1)\frac{\lambda}{2} $

  where $n$ is an integer (0, $\pm 1, \pm 2, ...$).

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Position of Bright and Dark Fringes

Using the approximation $\Delta x \approx yd/D$, we can find the positions of the fringes on the screen:

• Position of Bright Fringes ($y_n^{bright}$):

  $ \frac{y_n^{bright} d}{D} = n\lambda \implies y_n^{bright} = \frac{n\lambda D}{d} $

  For $n=0$, $y_0^{bright} = 0$ (central bright fringe). For $n=\pm 1, \pm 2, ...$ the positions of the higher order bright fringes.
• Position of Dark Fringes ($y_n^{dark}$):

  $ \frac{y_n^{dark} d}{D} = (n + \frac{1}{2})\lambda \implies y_n^{dark} = (n + \frac{1}{2})\frac{\lambda D}{d} $

  For $n=0$, $y_0^{dark} = \frac{1}{2}\frac{\lambda D}{d}$ (first dark fringe). For $n=1$, $y_1^{dark} = \frac{3}{2}\frac{\lambda D}{d}$, etc.

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Fringe Width ($\beta$)

The distance between two consecutive bright fringes or two consecutive dark fringes is called the fringe width ($\beta$) or bandwidth. For bright fringes: $\beta = y_{n+1}^{bright} - y_n^{bright} = \frac{(n+1)\lambda D}{d} - \frac{n\lambda D}{d} = \frac{\lambda D}{d}$. For dark fringes: $\beta = y_{n+1}^{dark} - y_n^{dark} = (n+1+\frac{1}{2})\frac{\lambda D}{d} - (n+\frac{1}{2})\frac{\lambda D}{d} = \frac{\lambda D}{d}$.

The formula for fringe width is:

$ \beta = \frac{\lambda D}{d} $

where:

• $\lambda$ is the wavelength of light.
• $D$ is the distance between the double slit and the screen.
• $d$ is the distance between the two slits.

This formula shows that the fringe width is directly proportional to the wavelength and the screen distance, and inversely proportional to the slit separation. By measuring the fringe width, the wavelength of light can be determined.

Young's experiment provided compelling evidence for the wave nature of light and allowed for the first determination of the wavelengths of visible light.

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