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Module 5.1 Fraunhofer Diffraction in Double Slit

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Simulation 1

Fraunhofer Diffraction at a Double Slit

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 Mathematical Problem Formulation

                                            
Fraunhofer Diffraction at a Double Slit: Intensity Distribution and Simulated Fringe Pattern

                                        

                                    
                                                                            

                                            
 Theoretical Background & Explanation

                                            

    

        

             Fraunhofer Double-Slit Diffraction & Interference
        

        

            Fraunhofer diffraction at a double slit illustrates the fundamental interplay between single-slit diffraction and two-beam Young's interference. The resulting pattern features rapid interference oscillations enveloped by a broad diffraction profile.
        

    


    

        1. Total Intensity Distribution
    

    

        Consider a monochromatic plane wave of wavelength $\lambda$ incident normally upon two long parallel slits, each of width $a$, with center-to-center separation $d$ ($d > a$). At a diffraction angle $\theta$ on a distant screen at distance $D$, the resultant intensity is given by:
    

    

        $$I(\theta) = I_0 \left(\frac{\sin\beta}{\beta}\right)^2 \cos^2\alpha$$
    

    

        where the dimensionless phase variables $\beta$ and $\alpha$ are defined by:
    

    

        $$\beta = \frac{\pi a}{\lambda}\sin\theta \approx \frac{\pi a x}{\lambda D}, \qquad \alpha = \frac{\pi d}{\lambda}\sin\theta \approx \frac{\pi d x}{\lambda D}$$
    

    

        • Diffraction Factor $\left(\frac{\sin\beta}{\beta}\right)^2$: Arises from phase differences between secondary wavelets originating across the width $a$ of each individual slit.

        • Interference Factor $\cos^2\alpha$: Arises from phase differences between the two slits separated by distance $d$.
    


    

        2. Interference Maxima & Fringe Width
    

    

        Bright interference maxima occur when $\cos^2\alpha = 1$, which requires:
    

    

        $$\alpha = m\pi \implies d\sin\theta = m\lambda \implies x_m \approx \frac{m\lambda D}{d}, \quad m = 0, \pm 1, \pm 2, \dots$$
    

    

        The linear fringe spacing $\Delta x$ between consecutive interference maxima on the screen is:
    

    

        $$\Delta x = \frac{\lambda D}{d}$$
    


    

        3. Diffraction Minima & Envelope Zeros
    

    

        The overall diffraction envelope drops to zero whenever $\sin\beta = 0$ with $\beta \neq 0$:
    

    

        $$\beta = p\pi \implies a\sin\theta = p\lambda \implies x_p \approx \frac{p\lambda D}{a}, \quad p = \pm 1, \pm 2, \dots$$
    

    

        The central diffraction peak spans between the first minima at $p = \pm 1$, with total angular width:
    

    

        $$2\theta_0 = \frac{2\lambda}{a}$$
    


    

        4. Missing Orders (Absent Spectra)
    

    

        A crucial feature studied in optics laboratories is missing orders. If an interference maximum occurs at the exact same angle as a diffraction minimum, no light reaches that point, and the expected interference fringe vanishes completely:
    

    

        $$\frac{d\sin\theta}{a\sin\theta} = \frac{m\lambda}{p\lambda} \implies \frac{d}{a} = \frac{m}{p}$$
    

    

        For instance, when $d = 5a$ (as set initially in this simulation), the missing interference orders are $m = \pm 5, \pm 10, \pm 15, \dots$. Within the central diffraction peak, exactly $2(d/a) - 1 = 9$ bright interference fringes are visible!
    


    

        5. Interactive Parameter Controls
    

    

        Use the live sliders below to explore how physical parameters alter the optical pattern:
        
• Wavelength $\lambda$ (400–750 nm): Alters the laser color in real time and scales fringe spacing proportionally ($\Delta x \propto \lambda$).
        
• Slit Width $a$ (0.01–0.10 mm): Narrower slits widen the envelope, revealing more interference fringes inside the central peak.
        
• Slit Separation $d$ (0.08–0.60 mm): Greater separation packs the interference fringes closer together.
        
• Screen Distance $D$ (0.5–2.5 m): Increases linear magnification on the detection screen.