13. Fresnel diffraction
Remind! Diffraction regimes
Fresnel-Kirchhoff diffraction formula E E ( P0 ) = 0 iλ
exp ( ikr ) ∫∫∑ r F (θ )dA r
z Obliquity factor : F (θ ) = cos θ = r
zE 0 E ( x, y ) = iλ r=
∫∫ ∑
exp ( ikr ) d ξ dη 2 r
z2 + (x − ξ ) + ( y −η ) 2
Aperture (ξ,η)
2
2 2 ⎡ 1 ⎛ x − ξ ⎞2 1 ⎛ y − η ⎞2 ⎤ x − ξ ) ( y −η ) ( ≈ z ⎢1 + ⎜ + ⎟ + ⎜ ⎟ ⎥= z+ 2 2 2 2z z z z ⎝ ⎠ ⎝ ⎠ ⎢⎣ ⎥⎦
⎛ x 2 y 2 ⎞ ⎛ ξ 2 η 2 ⎞ ⎛ xξ yη ⎞ = z +⎜ + ⎟+⎜ + ⎟−⎜ + ⎟ 2 2 2 2 z z z z z z ⎠ ⎝ ⎠ ⎝ ⎠ ⎝
E ( x, y ) =
E0 ⎡ k ⎤ exp ( ikz ) exp ⎢ i x 2 + y 2 )⎥ ( iλ z ⎣ 2z ⎦ ⎡ k ⎤ ⎡ k ⎤ × ∫∫ exp ⎢ i ξ 2 + η 2 ) ⎥ exp ⎢ − i ( xξ + yη ) ⎥ d ξ d η ( ⎣ 2z ⎦ ⎣ z ⎦ ∑
Screen (x,y)
E ( x, y ) =
E0 ⎡ k ⎤ x 2 + y 2 )⎥ exp ( ikz ) exp ⎢ i ( iλ z ⎣ 2z ⎦ ⎡ k ⎤ ⎡ k ⎤ × ∫∫ exp ⎢ i ξ 2 + η 2 ) ⎥ exp ⎢ − i ( xξ + yη ) ⎥ d ξ d η ( ⎣ 2z ⎦ ⎣ z ⎦ ∑
r
Aperture (ξ,η)
⎡ k ⎤ ⎡ k ⎤ = C ∫∫ exp ⎢ i ξ 2 + η 2 ) ⎥ exp ⎢ − i ( xξ + yη ) ⎥ d ξ d η ( ⎣ 2z ⎦ ⎣ z ⎦ ∑
Fresnel diffraction ⎡ k ⎤ ⎡ k ⎤ E ( x , y ) = C ∫∫ U (ξ , η ) exp ⎢ i ξ 2 + η 2 ) ⎥ exp ⎢ − i ( xξ + yη ) ⎥ d ξ d η ( ⎣ 2z ⎦ ⎣ z ⎦ k j (ξ 2 +η 2 ) ⎫ ⎧ E ( x, y ) ∝ F ⎨U (ξ ,η ) e 2 z ⎬ ⎩ ⎭
Fraunhofer diffraction ⎡ k ⎤ E ( x , y ) = C ∫∫ U (ξ , η ) exp ⎢ − i ( xξ + yη ) ⎥ d ξ d η ⎣ z ⎦ = C ∫∫ U (ξ , η ) exp ⎡⎣ − ik (ξ sin θ ξ + η sin θ η ) ⎤⎦ d ξ d η
E ( x, y ) ∝ F {U (ξ ,η )}
Screen (x,y)
Fresnel (near-field) diffraction This is most general form of diffraction – No restrictions on optical layout • near-field diffraction • curved wavefront – Analysis somewhat difficult
k j (ξ 2 +η 2 ) ⎫ ⎧ U ( x, y ) ≈ F ⎨U (ξ ,η ) e 2 z ⎬ ⎩ ⎭
z Fresnel Diffraction
Screen
Curved wavefront (parabolic wavelets)
Accuracy of the Fresnel Approximation
[
π 2 2 (x − ξ ) + ( y − η ) z 〉〉 4λ 3
]
2 max
• Accuracy can be expected for much shorter distances for U (ξ ,η ) smooth & slow varing function; 2 x − ξ = D ≤ 4 λ z
D2 z≥ 16λ
Fresnel approximation
In summary, Fresnel diffraction is …
13-7. Fresnel Diffraction by Square Aperture
2w 2a
Fresnel Diffraction from a slit of width D = 2a. (a) Shaded area is the geometrical shadow of the aperture. The dashed line is the width of the Fraunhofer diffracted beam. (b) Diffraction pattern at four axial positions marked by the arrows in (a) and corresponding to the Fresnel numbers NF=10, 1, 0.5, and 0.1. The shaded area represents the geometrical shadow of the slit. The dashed lines at x = (Îť / D )d represent the width of the Fraunhofer pattern in the far field. Where the dashed lines coincide with the edges of the geometrical shadow, the Fresnel number NF=0.5.
N F = a 2 / Îť z : Fresnel number
U ( x, y ) =
jkz
∞
[
e ⎧ k 2 2 ( ) ( ) ( ) U j ξ , η exp x ξ y η − + − ⎨ ∫ ∫ jλz −∞ ⎩ 2z
]⎫⎬⎭ dξdη
Δv = N F = a 2 / Ν z : Fresnel number
Fresnel diffraction from a wire
Fresnel diffraction from a straight edge
From Huygens’ principle to Fresnel-Kirchhoff diffraction
Huygens’ principle Every point on a wave front is a source of secondary wavelets. i.e. particles in a medium excited by electric field (E) re-radiate in all directions i.e. in vacuum, E, B fields associated with wave act as sources of additional fields
New wavefront Construct the wave front tangent to the wavelets
Secondary wavelet
r = c ∆t ≈ λ
secondary wavelets
Given wave-front at t
Allow wavelets to evolve for time ∆t
What about –r direction? (π-phase delay when the secondary wavelets, Hecht, 3.5.2, 3nd Ed)
Huygens’ wave front construction
Incompleteness of Huygens’ principle
Fresnel’s modification Î Huygens-Fresnel principle
Huygens-Fresnel principle
P
O
E p = Es ∫∫ Ap
1 ik ( r + r ') ⎡1 ⎤ 1 e F (θ )da = Es ⎢ e ikr ' ⎥ ∫∫ e ikr F (θ )da rr ' ⎣ r' ⎦ Ap r
Spherical wave from the point source S
Obliquity factor: unity where θ=0 zero where θ = π/2
Huygens’ Secondary wavelets on the wavefront surface O
Kirchhoff modification Fresnel’s shortcomings : He did not mention the existence of backward secondary wavelets, however, there also would be a reverse wave traveling back toward the source. He introduce a quantity of the obliquity factor, but he did little more than conjecture about this kind.
E p = Es
1 ikr ' 1 ikr π⎞ ⎛ π e ∫∫ e F (θ )da , ⎜ - < θ < ⎟ 2⎠ r' r ⎝ 2 Ap
Gustav Kirchhoff : Fresnel-Kirchhoff diffraction theory A more rigorous theory based directly on the solution of the differential wave equation. He, although a contemporary of Maxwell, employed the older elastic-solid theory of light. He found F(θ) = (1 + cosθ )/2. F(0) = 1 in the forward direction, F(π) = 0 with the back wave.
Fresnel-Kirchhoff diffraction formula
Fresnel-Kirchhoff diffraction integral − ikEs Ep = 2π
⎧1 + cos θ ⎫ 1 ik ( r + r ') ∫∫A ⎨⎩ 2 ⎬⎭ rr 'e da , ( -π < θ < π ) p
Arnold Johannes Wilhelm Sommerfeld : Rayleigh-Sommerfeld diffraction theory A very rigorous solution of partial differential wave equation. The first solution utilizing the electromagnetic theory of light.
1 Ep = iλ
e ikr ∫∫A EO r cosθ da p
This final formula looks similar to the Fresnel-Kirchhoff formula, therefore, now we call this the revised Fresnel-Kirchhoff formula, or, just call the Fresnel-Kirchhoff diffraction integral.
: Fresnel Zones
Spherical wave from source Po
Obliquity factor: unity where χ=0 at C zero where χ=π/2 at high enough zone index Huygens’ Secondary wavelets on the wavefront surface S
Z2
Z1
λ/2
Z3
: Fresnel Zones
Z2 The average distance of successive zones from P differs by λ/2 -> half-period zones. Thus, the contributions of the zones to the disturbance at P alternate in sign,
Z3
Z1
For an unobstructed wave, the last term ψn=0.
(1/2 means averaging of the possible values, more details are in 10-3, Optics, Hecht, 2nd Ed)
Therefore, one can assume that the complex amplitude of Whereas, a freely propagating spherical wave from the source Po to P is
=
1 ⎛ exp(iks ) ⎞ ⎜ ⎟ iλ ⎝ s ⎠
: Diffraction of light from circular apertures and disks (a) The first two zones are uncovered, 1
(consider the point P at the on-axis P)
2
(b) The first zone is uncovered if point P is placed father away, 1
: Babinet principle (c) Only the first zone is covered by an opaque disk, 1
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P R R Variation of on-axis irradiance
Diffraction patterns from circular apertures
Fresnel diffraction from a circular aperture
Poisson spot
Babinet principle At screen
ψS
At complementary screen
without screen
ψ CS
Amplitude of {ψ S }
Amplitude of {ψ CS }
Phase of {ψ S }
Phase of {ψ CS }
ψ S +ψ CS =ψUN
: Straight edge
Damped oscillating At the edge
Monotonically decreasing
13-6. The Fresnel zone plate The average distance of successive zones from P differs by λ/2 -> half-period zones. Thus, the contributions of the zones to the disturbance at P alternate in sign,
RN
Assume plane wavefronts
R4 R3 R2 R1
r0 + N
O
λ 2
r0
⎡ λ n ⎛λ⎞ λ⎞ ⎛ Rn2 = ⎜ r0 + n ⎟ − r02 = r02 ⎢ n + ⎜ ⎟ 2⎠ ⎝ ⎢⎣ r0 4 ⎝ r0 ⎠ 2
Z2 If the even zones (n=even) are blocked
Z1
2
2
⎤ ⎥ ⎥⎦
P
Rn ≈ nr0 λ
( r0 >> λ )
Z3
ψ ( P) = ψ 1 +ψ 3 +ψ 5 +
Bright spot at P It acts as a lens!
Fresnel zone-plate lens
RN
Rn ≈ nr0 λ R4 R3 R2 R1
r0 + N
O
( r0 >> λ )
Rn2 r0 = nλ
λ 2
r0
f1 = r0 (n = 1) =
R12
λ
P
Fresnel zone-plate lens has multiple foci. Rn
n 1 = R1 2 n 2 n R mλ ( Rn ) sin θ m = mλ ⇒ sin θ m ∼ tan θ m = n = fm Rn 1 ⎛ R ⎞ 1 = nR1 ⎜ 1 ⎟ f m = ( Rn )( Rn ) mλ ⎝ 2 n ⎠ mλ Rn = r0 λ
f3
f2
f1
(
R12 fm = mλ
)
Fresnel zone-plate lens
Binary zone plate: The areas of each ring, both light and dark, are equal. It has multiple focal points. For soft X-ray focusing
Sinusoidal zone plate: This type has a single focal point.
Fresnel lens: This type has a single focal point. Focusing efficiency approaches 100%.