kb/data/en.wikipedia.org/wiki/Diffraction-2.md

---
title: "Diffraction"
chunk: 3/5
source: "https://en.wikipedia.org/wiki/Diffraction"
category: "reference"
tags: "science, encyclopedia"
date_saved: "2026-05-05T10:54:57.453682+00:00"
instance: "kb-cron"
---

An illuminated slit that is wider than a wavelength produces interference effects in the space downstream of the slit. Assuming that the slit behaves as though it has a large number of point sources spaced evenly across the width of the slit, the interference effects can be calculated. If the incident light is a single wavelength and coherent, these sources all have the same phase. Light in the space downstream of the slit is made up of contributions from each of these point sources. If the relative phases of contributions from each point source vary due to differences in the path lengths, the resulting intensity will vary. Imagining the slit as the y axis with the z across the slit and x pointing downstream, intensity minima and maxima can be seen along z for large values of x.
For points very close to 
  
    
      
        x
      
    
    {\displaystyle x}
  
 all the point sources are in phase. This is the undiffracted beam, forming a maximum. Moving away fom the 
  
    
      
        x
      
    
    {\displaystyle x}
  
 axis, the path length from the point sources in the center and those on the edges of the slit differ. When the path difference equals 
  
    
      
        λ
        
          /
        
        2
      
    
    {\displaystyle \lambda /2}
  
, the central sources cancel the edge sources in destructive interference. For angles off the 
  
    
      
        x
      
    
    {\displaystyle x}
  
 axis of 
  
    
      
        θ
      
    
    {\displaystyle \theta }
  
, the path difference is approximately 
  
    
      
        
          
            
              d
              sin
              ⁡
              (
              θ
              )
            
            2
          
        
      
    
    {\displaystyle {\frac {d\sin(\theta )}{2}}}
  
 so that the minimum intensity occurs at an angle 
  
    
      
        
          θ
          
            min
          
        
      
    
    {\displaystyle \theta _{\text{min}}}
  
 given by

  
    
      
        d
        
        sin
        ⁡
        
          θ
          
            min
          
        
        =
        λ
        ,
      
    
    {\displaystyle d\,\sin \theta _{\text{min}}=\lambda ,}
  

where 
  
    
      
        d
      
    
    {\displaystyle d}
  
 is the width of the slit and 
  
    
      
        λ
      
    
    {\displaystyle \lambda }
  
 is the wavelength of the light.
The entire intensity profile can be calculated using the Fraunhofer diffraction equation as

  
    
      
        I
        (
        θ
        )
        =
        
          I
          
            0
          
        
        
        
          sinc
          
            2
          
        
        ⁡
        
          (
          
            
              
                
                  d
                  π
                
                λ
              
            
            sin
            ⁡
            θ
          
          )
        
        ,
      
    
    {\displaystyle I(\theta )=I_{0}\,\operatorname {sinc} ^{2}\left({\frac {d\pi }{\lambda }}\sin \theta \right),}
  

where 
  
    
      
        I
        (
        θ
        )
      
    
    {\displaystyle I(\theta )}
  
 is the intensity at a given angle, 
  
    
      
        
          I
          
            0
          
        
      
    
    {\displaystyle I_{0}}
  
 is the intensity at the central maximum (
  
    
      
        θ
        =
        0
      
    
    {\displaystyle \theta =0}
  
), which is also a normalization factor of the intensity profile that can be determined by an integration from 
  
    
      
        θ
        =
        −
        
          
            π
            2
          
        
      
    
    {\textstyle \theta =-{\frac {\pi }{2}}}
  
 to 
  
    
      
        θ
        =
        
          
            π
            2
          
        
      
    
    {\textstyle \theta ={\frac {\pi }{2}}}
  
 and conservation of energy, and 
  
    
      
        sinc
        ⁡
        x
        =
        
          
            
              sin
              ⁡
              x
            
            x
          
        
      
    
    {\displaystyle \operatorname {sinc} x={\frac {\sin x}{x}}}
  
, which is the unnormalized sinc function.
This analysis applies only to the far field (Fraunhofer diffraction), that is, at a distance much larger than the width of the slit.

=== Circular aperture ===

The far-field diffraction pattern of a plane wave incident on a circular aperture is known as the Airy disk. The Airy disk has the following intensity distribution as a function of angle θ:

  
    
      
        I
        (
        θ
        )
        =
        
          I
          
            0
          
        
        
          
            (
            
              
                
                  2
                  
                    J
                    
                      1
                    
                  
                  (
                  k
                  a
                  sin
                  ⁡
                  θ
                  )
                
                
                  k
                  a
                  sin
                  ⁡
                  θ
                
              
            
            )
          
          
            2
          
        
        ,
      
    
    {\displaystyle I(\theta )=I_{0}\left({\frac {2J_{1}(ka\sin \theta )}{ka\sin \theta }}\right)^{2},}
  

where 
  
    
      
        a
      
    
    {\displaystyle a}
  
 is the radius of the circular aperture, 
  
    
      
        k
      
    
    {\displaystyle k}
  
 is equal to 
  
    
      
        2
        π
        
          /
        
        λ
      
    
    {\displaystyle 2\pi /\lambda }
  
 and 
  
    
      
        
          J
          
            1
          
        
      
    
    {\displaystyle J_{1}}
  
 is a Bessel function. The smaller the aperture, the larger the spot size at a given distance, and the greater the divergence of the diffracted beams. For aperture diameters close to the wavelength of the light, the Airy disk begins to act like a point source with very large divergence of diffracted beams.

=== Babinet's principle ===

An opaque body and a hole of the same size and shape as the opaque body are called complementary apertures: as diffraction apertures they sum to a completely open space. For example, a screen with a circular hole is complementary to a circular disk the same size as the hole. The optical effect 
  
    
      
        E
      
    
    {\displaystyle E}
  
 of these apertures add, 

  
    
      
        
          E
          
            
              opaque
            
          
        
        +
        
          E
          
            
              hole
            
          
        
        =
        
          E
          
            0
          
        
      
    
    {\displaystyle E_{\textrm {opaque}}+E_{\textrm {hole}}=E_{0}}
  
 
giving the optical effect, 
  
    
      
        
          E
          
            0
          
        
      
    
    {\displaystyle E_{0}}
  
 of no obstacle. This is Babinet's principle; it works well in the Fraunhofer diffraction limit. When 
  
    
      
        
          E
          
            0
          
        
        ≈
        0
        ,
      
    
    {\displaystyle E_{0}\approx 0,}
  
 
  
    
      
        
          E
          
            
              opaque
            
          
        
        =
        −
        
          E
          
            
              hole
            
          
        
      
    
    {\displaystyle E_{\textrm {opaque}}=-E_{\textrm {hole}}}
  
 so the intensity in the two patterns are the same everywhere. By imaging a point source through either aperture, the same pattern results every except right at the focal point.

=== Knife edge ===
The knife-edge effect or knife-edge diffraction is a truncation of a portion of the incident radiation that strikes a sharp well-defined obstacle, such as a mountain range or the wall of a building.
Knife-edge diffraction is an outgrowth of the "half-plane problem", originally solved by Arnold Sommerfeld using a plane wave spectrum formulation. A generalization of the half-plane problem is the "wedge problem", solvable as a boundary value problem in cylindrical coordinates. The solution in cylindrical coordinates was then extended to the optical regime by Joseph B. Keller, who introduced the notion of diffraction coefficients through his geometrical theory of diffraction (GTD). In 1974, Prabhakar Pathak and Robert Kouyoumjian extended the (singular) Keller coefficients via the uniform theory of diffraction (UTD).

=== Gratings ===

A diffraction grating is an optical component with a regular pattern. The form of the light diffracted by a grating depends on the structure of the elements and the number of elements present, but all gratings have intensity maxima at angles θm which are given by the grating equation
  
    
      
        d
        
          (
          
            sin
            ⁡
            
              
                θ
                
                  m
                
              
            
            ±
            sin
            ⁡
            
              
                θ
                
                  i
                
              
            
          
          )
        
        =
        m
        λ
        ,
      
    
    {\displaystyle d\left(\sin {\theta _{m}}\pm \sin {\theta _{i}}\right)=m\lambda ,}