Comprehensive Study Notes on Fraunhofer and Fresnel Diffraction

Comparison Between Fresnel and Fraunhofer Diffraction

• Fresnel Diffraction:

  • Mathematical calculations involved in Fresnel diffraction are complicated.

  • Lenses are not required to focus light onto a screen or observation point.

• Fraunhofer Diffraction:

  • Mathematical calculations involved in Fraunhofer diffraction are comparatively easy.

  • To focus light onto the application plane or screen, a lens system (focusing lenses) must be used.

Fraunhofer Diffraction Due to Single Slit

• Experimental Setup and Geometry:

  • A monochromatic light source $S$ is positioned at the focal point of a convex lens $L_1$ to generate a parallel incident beam of light.

  • The parallel wavefront falls on a narrow single slit $AB$ of width $a$ (where $a = \text{slit width} = AB$).

  • A secondary convex lens $L_2$ is positioned after the slit to focus the diffracted light rays onto a screen located at its focal plane.

  • Undeviated light rays converge at the central point $P_0$ on the screen, while rays deviated by an angle $\theta$ converge at point $P_1$.

• Path Difference and Phase Difference Derivation:

  • Consider parallel rays diffracted at an angle $\theta$ with respect to the original direction of propagation.

  • To calculate the path difference between the diffracted rays originating from the extreme edges $A$ and $B$ of the slit, drop a perpendicular $AC$ from point $A$ onto the ray diffracted from point $B$.

  • In the right-angled triangle $\frac{}{}\triangle ABC$:

$\sin(\theta) = \frac{BC}{AB}$

$BC = AB \sin(\theta) = a \sin(\theta)$

*   Therefore, the path difference between the deviated rays originating from $A$ and $B$ is given by:

$\text{path difference} = BC = a \sin(\theta)$

*   The total phase difference $\delta$ between the diffracted rays from edge $A$ and edge $B$ is defined as:

$\delta = \frac{2\pi}{\lambda} \times \text{path difference}$

$\delta = \frac{2\pi}{\lambda} [a \sin(\theta)]$

Derivation of Resultant Amplitude and Intensity for Single Slit

• Division of Slit into Sub-sources:

  • Divide the primary slit $AB$ into $n$ equal sub-parts, where each part acts as an independent coherent wave source emitting light with amplitude $A'$.

  • The phase difference $d$ between light rays originating from two consecutive sub-slits is:

$d = \frac{\delta}{n} = \frac{2\pi}{\lambda} \frac{a \sin(\theta)}{n}$

• Vector Addition of $n$ Harmonic Waves:

  • Using the vector addition principle for $n$ vectors of equal amplitude $A'$ with a constant successive phase difference $d$, the resultant vector amplitude $R$ is:

$R = \frac{A' \sin\left(\frac{n d}{2}\right)}{\sin\left(\frac{d}{2}\right)}$

• Algebraic Simplification:

  • Substitute $d = \frac{2\pi a \sin(\theta)}{n \lambda}$ into the expression for $R$:

$R = \frac{A' \sin\left[\frac{n}{1} \times \frac{1}{2} \times \frac{2\pi a \sin(\theta)}{n \lambda}\right]}{\sin\left[\frac{1}{2} \times \frac{2\pi a \sin(\theta)}{n \lambda}\right]}$

$R = \frac{A' \sin\left(\frac{\pi a \sin(\theta)}{\lambda}\right)}{\sin\left(\frac{\pi a \sin(\theta)}{n \lambda}\right)}$

*   Define the parameter $\beta$ as:

$\beta = \frac{\pi a \sin(\theta)}{\lambda}$

*   Substituting $\beta$ yields:

$R = \frac{A' \sin(\beta)}{\sin\left(\frac{\beta}{n}\right)}$

*   For a continuous wavefront across the slit, $n$ is extremely large ($n \rightarrow \infty$), making $\frac{\beta}{n}$ an infinitely small angle.
*   Since $\theta$ is small, $\sin(\theta) \approx \theta$.
*   For very small angles, $\sin\left(\frac{\beta}{n}\right) \approx \frac{\beta}{n}$.
*   Substituting this approximation into the equation for $R$ gives:

$R = \frac{A' \sin(\beta)}{\frac{\beta}{n}} = n A' \frac{\sin(\beta)}{\beta}$

*   Let $n A' = A$, where $A$ represents the total combined amplitude of all wavelets across the entire slit width.
*   The final formula for resultant amplitude $R$ is:

$R = \frac{A \sin(\beta)}{\beta}$

• Resultant Intensity Formula:

  • Since optical intensity $I$ is proportional to the square of the resultant amplitude ($I = R^2$):

$I = \left[\frac{A \sin(\beta)}{\beta}\right]^2 = A^2 \left[\frac{\sin(\beta)}{\beta}\right]^2$

Analysis of Diffraction Conditions

• Condition for Central Maxima:

  • The central maximum is formed by undeviated light rays where $\theta \rightarrow 0$, which implies $\beta \rightarrow 0$

  • When $\beta$ approaches zero, $\lim_{\beta \to 0} \frac{\sin(\beta)}{\beta} = 1$, or $\sin(\beta) \approx \beta$

  • The resultant intensity at the center $P_0$ is:

$I = A^2 \left(\frac{\beta}{\beta}\right)^2 = A^2 = I_0$

*   Thus, maximum peak intensity $I_0 = A^2$ occurs at the central position.

• Condition for Minima (Zero Intensity):

  • Minimum intensity ($I = 0$) is obtained when $\sin(\beta) = 0$

  • This condition is satisfied when:

$\beta = \pm m \pi$

*   Where $m = 1, 2, 3, \dots$
*   The order $m \neq 0$ because $m = 0$ yields $\beta = 0$, which corresponds to the central maximum.
*   Substituting $\beta = \frac{\pi a \sin(\theta)}{\lambda}$:

$\frac{\pi a \sin(\theta)}{\lambda} = \pm m \pi$

$a \sin(\theta) = \pm m \lambda$

• Condition for Secondary Maxima:

  • To find the positions of secondary maxima, differentiate the intensity $I$ with respect to $\beta$ and set it equal to zero ($\frac{dI}{d\beta} = 0$):

$\frac{d}{d\beta} \left[ A \frac{\sin(\beta)}{\beta} \right]^2 = 0$

$A^2 \frac{d}{d\beta} \left[ \frac{\sin(\beta)}{\beta} \right]^2 = 0$

$A^2 \left[ 2 \frac{\sin(\beta)}{\beta} \right] \frac{d}{d\beta} \left[ \frac{\sin(\beta)}{\beta} \right] = 0$

$2 A^2 \frac{\sin(\beta)}{\beta} \left[ \frac{\beta \cos(\beta) - \sin(\beta)}{\beta^2} \right] = 0$

$2 A^2 \frac{\sin(\beta)}{\beta^3} [\beta \cos(\beta) - \sin(\beta)] = 0$

*   Since $2 A^2 \frac{\sin(\beta)}{\beta^3} = 0$ gives the positions of intensity minima (where $\sin(\beta) = 0$), the non-trivial condition for secondary maxima is:

$\beta \cos(\beta) - \sin(\beta) = 0$

$\beta \cos(\beta) = \sin(\beta)$

$\beta = \tan(\beta)$

*   The roots of $\beta = \tan(\beta)$ are solved graphically by finding the points of intersection between the line $y = \beta$ and the tangent curves $y = \tan(\beta)$.
*   The intersection points occur approximately at:

$\beta = \pm \frac{3\pi}{2}, \pm \frac{5\pi}{2}, \pm \frac{7\pi}{2}, \dots$

$\beta = \pm 1.5\pi, \pm 2.5\pi, \pm 3.5\pi, \dots$

Intensity Distribution and Double Slit Fraunhofer Diffraction

• Intensity Profile Characteristics:

  • The central maximum occurs at $\beta = 0$ with maximum intensity $I_0$.

  • Minima ($I = 0$) occur at integer multiples of $\pi$: $\beta = \pm \pi, \pm 2\pi, \pm 3\pi, \dots$

  • Secondary maxima occur near odd half-integer multiples of $\pi$: $\beta = \pm 1.5\pi, \pm 2.5\pi, \pm 3.5\pi, \dots$

• Fraunhofer Diffraction Due to Double Slit:

  • Let $a$ equal the width of each individual slit.

  • Let $d$ equal the width of the opaque space separating the two slits.

  • The distance of separation between consecutive corresponding positions of the two slits is $a + d$

  • Each slit individually produces a diffracted beam of amplitude $A \frac{\sin(\beta)}{\beta}$.

  • The path difference between two diffracted rays originating from corresponding points of the double slit is:

$\sin(\theta) = \frac{BC}{AB}$

$\text{path difference} = (a + d) \sin(\theta)$

*   The phase difference $\delta$ between diffracted rays originating from corresponding points of the two slits is given by:

$\delta = \frac{2\pi}{\lambda} \times (\text{path difference})$

$\delta = \frac{2\pi}{\lambda} (a + d) \sin(\theta)$