(a) In a double-slit experiment, what largest ratio of d to...

(a) In a double-slit experiment, what largest ratio of $d$ to $a$ causes diffraction to eliminate the fifth bright side fringe? (b) What other bright fringes are also eliminated? (c) How many other ratios of $d$ to $a$ cause the diffraction to (exactly) eliminate that bright fringe?

Key Concepts

Double-Slit Interference

In a double?slit experiment, light waves from the two narrow slits combine to form an interference pattern on a distant screen. The condition for bright fringes (constructive interference) is given by d·sin? = m?, where d is the slit separation, ? is the wavelength of light, m is an integer (the order of the fringe), and ? is the angle relative to the central maximum. This equation determines the positions of the bright fringes on the screen.

Single-Slit Diffraction

Each slit also produces its own diffraction pattern, characterised by a broad envelope. The minima (dark fringes) in the diffraction pattern occur when a·sin? = n?, where a is the width of the slit, n is a nonzero integer, and ? is the wavelength of light. This diffraction envelope modulates the overall intensity of the interference fringes, sometimes completely cancelling a fringe if the conditions for interference and diffraction overlap.

Missing Orders (Interplay of Interference and Diffraction)

Missing orders occur when an interference maximum coincides with a diffraction minimum. This happens when the condition for the mth bright fringe from interference, m?/d, equals an angle corresponding to a diffraction minimum, n?/a. Algebraically, this overlap gives d/a = m/n, meaning that for certain ratios of the slit separation to the slit width, specific interference maxima do not appear because they fall exactly at a point of zero intensity in the diffraction pattern.

Parameter Ratios and Order Coincidence

The ratio d/a plays a crucial role in determining which interference fringes are ‘missing.’ By setting d/a equal to m/n, one finds the specific conditions under which a particular bright fringe (such as the fifth order, m = 5) is eliminated by the diffraction envelope. The largest possible ratio for eliminating a given fringe corresponds to using the smallest possible integer n, and other valid ratios (subject to physical constraints like d > a and sin? ? 1) occur for larger values of n. This rational relationship explains how geometrical parameters control the overall pattern.

Physical Constraints on Angular Positions

Not every solution of the form d/a = m/n is physically meaningful, as the resulting value of sin? (which determines the fringe position on the screen) must lie between -1 and 1. This operational limit ensures that only certain orders, and thus certain ratios of d to a, lead to observable missing fringes. It also means that when choosing parameters to eliminate a specific interference maximum, one must take into account the allowed angular range of the light’s propagation.

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