Observing screen is quite close to aperture Lecture 21 Geometrical Optics: Sharp image of aperture expected (as light waves treated as rays propagating along straight lines) Observing screen is quite close to aperture Fraunhofer (Far-Field) Diffraction: Fringed image bearing little resemblance to the aperture Observing screen is far from the aperture Wavefront at diffracting aperture is considered planar FRESNEL (Near-Field) Diffraction: Image is essentially an image of aperture, but with fringed edges Diffraction patterns form continuity between patterns characterizing geometrical optics and Fraunhofer diffraction at two extremes Wavefront curvature is considered

Fresnel Diffraction (Near-field) Fresnel Diffraction occurs when: Spherical waves (point source is near) impinge on aperture - OR - Parallel waves (point source is distant) impinge on aperture, but observation screen is parallel with, and close to aperture, image takes on limiting shape of aperture, with fringed edges  of incoming radiation is sufficiently reduced, fringes as in Fraunhofer patterns would disappear, and image takes on the limiting shape of aperture Nearby: Fresnel diffraction Faraway: Fraunhofer diffraction Image of single slit as observation screen is moved from far to near the slit Geometrical optics considered if observation screen is still nearer to slit. Note - Fraunhofer diffraction Criterion :

 Thus r and r’ variation must be considered Fresnel Diffraction (Near-field) O da r P r’ S Area of aperture = AP Both r and r’ vary with different aperture points O and different observation points P  Thus r and r’ variation must be considered Schematic defining parameters for typical Fresnel diffraction Spherical wavefronts The direction from various aperture points O to given field point P is different  thus, must consider the dependence of the amplitude on direction of Huygens’ wavelets originating at the aperture This is dealt with through the inclination or obliquity factor: (21-1) where  = angle made with the normal to primary wavefront

Contribution to resultant field at P due to elemental area da is: Fresnel Diffraction (Near-field) Contribution to resultant field at P due to elemental area da is:  Secondary wavelet Primary wave P (spherical wave representation) But wave amplitude dE0 at aperture  elemental area: where amplitude EL at point O = amplitude of spherical wave that originates from source given by: Angle  defined in obliquity factor F() Combining (and ignoring constant of proportionality): (21-2)

This equation is incomplete because it does not take into account: Fresnel Diffraction (Near-field) Therefore, the field at P due to secondary wavelets from entire aperture is: (21-3) This equation is incomplete because it does not take into account: 1. The Obliquity factor K() which describes attenuation of diffracted waves on their direction 2. The phase shift of 90 of diffracted waves relative to primary incident wave The corrected version of Eq. (21-3) is the Fresnel-Kirchhoff diffraction formula: (21-4) Factor i= e  i/2 represents the required phase shift; and K() limits amplitude ES

Fresnel Diffraction (Near-field) Fresnel-Kirchhoff diffraction formula is simplified for the limiting case of Fraunhofer diffraction by assuming: (1) The Obliquity factor is approximately constant over the aperture due to the small spread in the diffracted light (2) Variation of distances r and r’ is small relative to that of the exponential function Taking all constant (or almost constant) terms out of the integral in Fresnel-Kirchhoff diffraction formula to be represented by an overall constant C, Eq. (21-4) becomes Corresponds to integral used to calculate Fraunhofer diffraction patterns (take example for circular aperture): Integration in Fresnel-Kirchhoff diffraction formula is difficult, and Fresnel introduced a simplified method to deal with it … to be shown in later slides.

Criterion for Fresnel Diffraction Consider simple case whereby both S and P are on the central axis through the aperture. S r’ h p  P r q Curvature of incident wavefronts Curvature of diffracted wavefronts  = 0 for planar wavefront. Binomial expansion was used: Since p  r’, condition for near-field (where there is significant curvature of incident wave) is: Similarly for the diffracted wave curvature: (21-5) (21-6)

d represents either p or q ; A  area of aperture Criterion for Fresnel Diffraction Combining (21-5) & (21-6), the criterion for Fresnel (near-field) diffraction may be expressed as: near field: (21-7) (in 1-D) This condition also applies to the perpendicular dimension of aperture (transverse to h). Assuming h  maximum extent in either direction of aperture (or radius of circular aperture), Eq. (21-5) or (21-6) can be approximated as: near field: (21-8) (in 2-D) d represents either p or q ; A  area of aperture

The Obliquity Factor According to Huygens, wavelets from a secondary point source radiate in all directions equally - this implies new wavefronts propagate both forward and backward, but reverse wave does not exist The obliquity factor takes care of this problem; K() is maximum in forward direction K(0) = 1; and dispenses with back wave as K() = 0 Thus, the amplitude of the secondary wavelets arriving at P is: (21-9)

Fresnel Diffraction from Circular Apertures Z1 Z2 ZN Spherical wavefront (centred at S) emerging from circular aperture is divided into annular zones symmetrical about axis SOP Each zone is a circle on the wavefront, spaced such that the each successive circle is larger by a distance /2 to point P If OP = r0, then Z1P = r0 + /2, Z2P = r0 + 2(/2), … ZNP = r0 + N(/2) Each successive zone’s contribution is then exactly out of phase with that of the preceding one These half-period zones  Fresnel zones

Resultant phase difference from beginning to end of zone =  or 180 Fresnel Diffraction from Circular Apertures Each Fresnel zone can be sub-divided further into smallerM sub-zones (e.g. 15 sub-zones) Resultant phase difference from beginning to end of zone =  or 180 Individual phasors indicate the average phase angle of the subzones and are progressively shorter by 5% (due to obliquity factor), thus it spirals inward Amplitude a1  1st half-period zone; A  resultant of 5½ half-periods Each small phasor  one subzone /15 Effective phasor from 1st zone has phase of /2

Summation of waves at P from each half-period zone is: Fresnel Diffraction from Circular Apertures When M  , the polygon of vectors blends into a segment of a smooth spiral called a vibration curve. Circle does not close as obliquity factor ensures each succeeding phasor is slightly shorter than the preceding one. Summation of waves at P from each half-period zone is: or (21-10)

Successive zonal amplitude are affected by 3 different considerations: Fresnel Diffraction from Circular Apertures Successive zonal amplitude are affected by 3 different considerations: (1) gradual increase with N due to slightly increasing zonal areas (2) gradual decrease with N due to inverse square law effect as distances from P increase (3) gradual decrease with N due to obliquity factor What is the quantitative contribution from these 3 considerations? With regard to (1), the surface area SN of Nth Fresnel zone is expressed as: (/r0) is usually very small, 2nd term in brackets can be neglected compared to the first, and we have zones of equal areas independent of N: (21-11) (21-12)

Fresnel Diffraction from Circular Apertures The neglected 2nd term in (21-11), though small, indicates increases in zonal areas with N and correspondingly, each successive terms of Eq. (21-10) increases a little However, these increases are canceled by the decreases arising from the consideration (2) due to inverse square law. Since the increases in consideration (1) and the decreases in consideration (2) cancel out, that leaves only the 3rd consideration that will result in the systematic decreases in the amplitudes as N increases (due to obliquity factor)

Phasor diagram for amplitude of Fresnel half-period zones (Eq. 21-10): Fresnel Diffraction from Circular Apertures Phasor diagram for amplitude of Fresnel half-period zones (Eq. 21-10): Resultant phasors at each step of half-period zones For small N, large changes in resultant phasor AN observed. Individual phasors For large N, resultant amplitude AN  AR = A1/2, i.e. ½ of 1st contributing zone Each phasor out of phase with its predecessor by 180, and is also shorter (due to obliquity factor)

Resultant amplitude for N zones are approximately: N even: Fresnel Diffraction from Circular Apertures Resultant amplitude for N zones are approximately: N even: N odd: (21-13a) (21-13b) Conclusions: (1) If N is small, a1  aN, resultant amplitude for odd N = a1 = amplitude of 1st zone alone; resultant amplitude for even N  0 (2) If N is large (unlimited aperture), aN  0 regardless of whether N is odd or even, resultant amplitude = a1/2 = half that of 1st contributing zone

What are the physical meanings of the conclusions? Fresnel Diffraction from Circular Apertures What are the physical meanings of the conclusions? When a circular aperture coincides with the 1st Fresnel zone, an amplitude AP = a1 may be measured at P Opening the aperture wider to include the 2nd Fresnel zone as well will result in almost zero amplitude at P If the aperture is totally opened, amplitude at P becomes a1/2 (With irradiance  amplitude2, unobstructed irradiance at P = I1/4; I1 = irradiance due to 1st Fresnel zone alone)