Fresnel diffraction occurs when light waves are observed close to their source. This near-field effect creates complex patterns that depend on distance and aperture shape. Understanding Fresnel diffraction is crucial for analyzing light behavior in optical systems.
Calculating Fresnel patterns involves complex integrals and special functions. As the observation distance increases, Fresnel diffraction gradually transitions to Fraunhofer diffraction. This concept is vital for applications like near-field microscopy and zone plate design.
Fresnel Diffraction and Near-Field Effects
Characteristics of Fresnel diffraction
Top images from around the web for Characteristics of Fresnel diffraction Single Slit Diffraction | Physics View original
Is this image relevant?
Near and far field - Wikipedia View original
Is this image relevant?
Single Slit Diffraction | Physics View original
Is this image relevant?
Near and far field - Wikipedia View original
Is this image relevant?
1 of 2
Top images from around the web for Characteristics of Fresnel diffraction Single Slit Diffraction | Physics View original
Is this image relevant?
Near and far field - Wikipedia View original
Is this image relevant?
Single Slit Diffraction | Physics View original
Is this image relevant?
Near and far field - Wikipedia View original
Is this image relevant?
1 of 2
Occurs when observation distance is comparable to aperture size and light wavelength
Near-field region defined as z ≪ a 2 λ z \ll \frac{a^2}{\lambda} z ≪ λ a 2 , z z z is observation distance, a a a is aperture size, λ \lambda λ is wavelength
Exhibits complex structure with bright and dark fringes
Pattern not a simple Fourier transform of aperture shape
Depends on distance between aperture and observation plane
Wavefront curvature cannot be neglected in near-field region
Calculation of Fresnel diffraction patterns
Fresnel diffraction integral U ( x , y ) = e i k z i λ z ∬ Σ U 0 ( ξ , η ) e i k 2 z [ ( x − ξ ) 2 + ( y − η ) 2 ] d ξ d η U(x,y) = \frac{e^{ikz}}{i\lambda z} \iint_\Sigma U_0(\xi, \eta) e^{\frac{ik}{2z}[(x-\xi)^2 + (y-\eta)^2]} d\xi d\eta U ( x , y ) = iλ z e ik z ∬ Σ U 0 ( ξ , η ) e 2 z ik [( x − ξ ) 2 + ( y − η ) 2 ] d ξ d η
U ( x , y ) U(x,y) U ( x , y ) complex amplitude at observation point ( x , y ) (x,y) ( x , y )
U 0 ( ξ , η ) U_0(\xi, \eta) U 0 ( ξ , η ) complex amplitude at aperture
k = 2 π λ k = \frac{2\pi}{\lambda} k = λ 2 π wavenumber
Fresnel integrals C ( v ) = ∫ 0 v cos ( π 2 t 2 ) d t C(v) = \int_0^v \cos(\frac{\pi}{2}t^2) dt C ( v ) = ∫ 0 v cos ( 2 π t 2 ) d t and S ( v ) = ∫ 0 v sin ( π 2 t 2 ) d t S(v) = \int_0^v \sin(\frac{\pi}{2}t^2) dt S ( v ) = ∫ 0 v sin ( 2 π t 2 ) d t
Calculate diffraction pattern for simple apertures (rectangular, circular)
Rectangular aperture U ( x , y ) = U 0 2 [ ( C ( v 2 ) − C ( v 1 ) ) + i ( S ( v 2 ) − S ( v 1 ) ) ] U(x,y) = \frac{U_0}{2} \left[ \left(C(v_2) - C(v_1)\right) + i\left(S(v_2) - S(v_1)\right) \right] U ( x , y ) = 2 U 0 [ ( C ( v 2 ) − C ( v 1 ) ) + i ( S ( v 2 ) − S ( v 1 ) ) ]
v 1 = 2 λ z ( x − a 2 ) v_1 = \sqrt{\frac{2}{\lambda z}}(x - \frac{a}{2}) v 1 = λ z 2 ( x − 2 a ) and v 2 = 2 λ z ( x + a 2 ) v_2 = \sqrt{\frac{2}{\lambda z}}(x + \frac{a}{2}) v 2 = λ z 2 ( x + 2 a ) , a a a is aperture width
Circular aperture U ( r ) = U 0 [ 1 2 − C ( v ) + i ( 1 2 − S ( v ) ) ] U(r) = U_0 \left[ \frac{1}{2} - C(v) + i\left(\frac{1}{2} - S(v)\right) \right] U ( r ) = U 0 [ 2 1 − C ( v ) + i ( 2 1 − S ( v ) ) ]
v = 2 λ z r v = \sqrt{\frac{2}{\lambda z}}r v = λ z 2 r , r r r is radial distance from diffraction pattern center
Transition from Fresnel to Fraunhofer
Fraunhofer diffraction occurs when observation distance much larger than aperture size and wavelength (z ≫ a 2 λ z \gg \frac{a^2}{\lambda} z ≫ λ a 2 )
As observation distance increases, Fresnel pattern gradually transitions to Fraunhofer pattern
Wavefront curvature becomes less significant
Diffraction pattern becomes Fourier transform of aperture shape
Transition characterized by Fresnel number N F = a 2 λ z N_F = \frac{a^2}{\lambda z} N F = λ z a 2
Fresnel diffraction dominates when N F ≫ 1 N_F \gg 1 N F ≫ 1
Fraunhofer diffraction dominates when N F ≪ 1 N_F \ll 1 N F ≪ 1
Transition occurs when N F ≈ 1 N_F \approx 1 N F ≈ 1
Applications of Fresnel diffraction theory
Analyze near-field diffraction phenomena in optical systems
Diffraction by apertures and obstacles in near-field region
Light propagation through optical components (lenses, mirrors) in near-field region
Near-field imaging and lithography techniques
Fresnel zone plate
Alternating transparent and opaque concentric rings
Focuses light by diffraction in near-field region
Focal length depends on wavelength and zone plate geometry
Near-field scanning optical microscopy (NSOM)
Uses subwavelength aperture or tip to probe sample's near-field region
Achieves resolution beyond diffraction limit by exploiting evanescent waves in near-field region
Crucial for designing and optimizing near-field optical systems