Optics Conditions

phy-1109 PHY-1109 Physics

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CONDITIONS FOR MAXIMA AND MINIMA IN OPTICS

I. Interference Phenomena (Based on Path and Phase Difference)

Interference occurs due to the superposition of coherent waves. The intensity ($I$) is proportional to the square of the amplitude, given generally by $I\propto {\cos }^2(\delta /2)$.

A. Young’s Double Slit Experiment (YDSE) / General Interference

This applies to experiments like Young’s Double Slit or Fresnel’s Biprism, where interference results from the superposition of two waves traveling slightly different paths ($\Delta x$) from coherent sources.

Result	Maxima (Bright Fringe / Constructive Interference)	Minima (Dark Fringe / Destructive Interference)
Path Difference ($\Delta x$)	Integral multiple of the wavelength: $\mathbf{\Delta x}=\mathbf{n}\lambda$	Odd half multiple of the wavelength: $\mathbf{\Delta x}=(\mathbf{n}+\frac{1}{2})\lambda$ or $(2n+1)\frac{\lambda }{2}$
Phase Difference ($\delta$)	Integral multiple of $2\pi$: $\delta =\mathbf{n}(2\pi )$	Odd multiple of $\pi$: $\delta =(2n+1)\pi$
Fringe Position ($x_n$)	${\mathbf{x}}_{\mathbf{n}}=\frac{\mathbf{n}\lambda \mathbf{D}}{\mathbf{d}}$	${\mathbf{x}}_{\mathbf{n}}=\frac{(2n+1)\lambda \mathbf{D}}{2d}$
Intensity (I)	$\mathbf{I}=4{\mathbf{a}}^2$ (Maximum)	$\mathbf{I}=0$ (Minimum)
Integer $n=0,1,2,3,\dots$

B. Thin Films (Reflected Light)

The conditions for thin film interference (e.g., soap bubbles, oil slicks) involve the geometric path difference $2\mu t\cos r$ AND an additional phase shift ($\lambda /2$) if reflection occurs at the surface of an optically denser medium.

Result	Maxima (Bright Appearance)	Minima (Dark Appearance)
Correct Path Difference ($x$)	$x=n\lambda$	$x=(2n+1)\frac{\lambda }{2}$
Condition (Using $2\mu t\cos r$ geometric path)	$2\mu \mathbf{t}\cos \mathbf{r}=(2n+1)\frac{\lambda }{2}$	$2\mu \mathbf{t}\cos \mathbf{r}=\mathbf{n}\lambda$
Key Principle: The presence of the $\lambda /2$ phase shift causes the conditions to be reversed compared to YDSE (i.e., odd half multiples yield brightness).

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II. Diffraction Phenomena (Based on Angular Conditions)

Diffraction involves the superposition of continuous wavelets originating from different points on a single wavefront.

A. Diffraction Grating (Principal Maxima)

A grating consists of a large number of slits separated by a grating element $d$.

Result	Maxima (Principal Maxima)	Minima (Condition for Missing Orders/Absent Spectra)
Condition (Angle $\theta$)	$\mathbf{d}\sin \theta =\mathbf{m}\lambda$ (Where $d$ is the grating element, $d=a+b$)	Occurs when an interference maximum coincides with a diffraction minimum.
Missing Order $n$	N/A	$\frac{\mathbf{a}+\mathbf{b}}{\mathbf{a}}={\mathbf{n}}_{\text{missing}}$ (Where $a$ is slit width and $b$ is opaque width).

B. Single Slit and Double Slit Diffraction

The sources discuss Fraunhofer diffraction primarily in terms of complexity, drawing attention to the intensity pattern. The following derivation focus is stated in the course material for the double slit pattern:

Phenomenon	Maxima (General Secondary Maxima)	Minima (Destructive/Zero Intensity)
Single Slit	(Derivations for intensity of first, second, and third secondary maxima were asked in PYQs, but the general angular formula for minima is not provided in the sources).	(The condition $\frac{a\sin \theta }{\lambda }=\pm n$ is typically used, but not provided in your source materials.)
Double Slit	The intensity distribution formula for Fraunhofer diffraction at double slits is discussed.	The missing order condition above ($\frac{a+b}{a}=n$) relates directly to the occurrence of minima in the double slit pattern.