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Question #43590

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The Question

Consider a modified single-slit Fraunhofer diffraction experiment. A monochromatic plane wave of wavelength $\lambda$ (wave vector $k = 2\pi/\lambda$ ) is incident normally on a slit of width $a$ located from $x = -a/2$ to $x = a/2$ . Due to a special absorbing filter placed across the slit, the amplitude of the transmitted electric field is not uniform. Instead, it varies spatially as $E(x) = E_0 e^{-\alpha x}$ for $-a/2 \le x \le a/2$ , where $x$ is the distance from the center of the slit and $\alpha$ is a positive constant ( $\alpha > 0$ ). Let $I_0$ be the intensity of the diffracted light at the central maximum ( $\theta = 0$ ). Which of the following expressions correctly represents the intensity $I(\theta)$ of the diffracted light at an angle $\theta$ with respect to the central axis?

Options

A.
$I_0 \left( \frac{\alpha^2}{\alpha^2 + k^2 \sin^2\theta} \right) \left[ 1 + \frac{\sin^2\left(\frac{k a \sin\theta}{2}\right)}{\sinh^2\left(\frac{\alpha a}{2}\right)} \right]$
B.
$I_0 \left( \frac{\alpha^2}{\alpha^2 + k^2 \sin^2\theta} \right) \left[ 1 - \frac{\sin^2\left(\frac{k a \sin\theta}{2}\right)}{\cosh^2\left(\frac{\alpha a}{2}\right)} \right]$
C.
$I_0 \left( \frac{\alpha^2}{\alpha^2 + k^2 \sin^2\theta} \right) \frac{\sin^2\left(\frac{k a \sin\theta}{2}\right)}{\left(\frac{k a \sin\theta}{2}\right)^2}$
D.
$I_0 \left( \frac{\alpha^2}{\alpha^2 + k^2 \sin^2\theta} \right) \left[ \frac{\sinh^2\left(\frac{\alpha a}{2}\right) + \cos^2\left(\frac{k a \sin\theta}{2}\right)}{\sinh^2\left(\frac{\alpha a}{2}\right) + 1} \right]$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Intuition_Breaker

Train your brain to resist "obvious" answers that are designed to deceive your intuition.

Concepts Behind This Question

Diffraction Width|Damped Oscillation Equation|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the fundamental origin of diffraction patterns, moving beyond the rote memorization of the standard $I = I_0 \text{sinc}^2(\beta a/2)$ formula.

Students often incorrectly assume that any single-slit diffraction will simply have its intensity modulated by an envelope function (like Option C), failing to realize that the diffraction pattern is the exact squared magnitude of the Fourier transform of the aperture's spatial field distribution.

By introducing a spatially damped amplitude $E(x) = E_0 e^{-\alpha x}$ (apodization), the student is forced to perform the complex integration and correctly evaluate the squared magnitude of a complex-argument hyperbolic function.

The trap lies in either guessing a phenomenological combination of a Lorentzian and a sinc function or making algebraic errors when separating the real and imaginary parts of the integrated field.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Wave Optics - Class 12
Concept	Diffraction Width
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free