Optoelectronics EE 430.423.001 2016. 2nd Semester
Chapter 5. Diffraction Part 1
2016. 10. 18.
Changhee Lee School of Electrical and Computer Engineering Seoul National Univ. [email protected]
1/27 Changhee Lee, SNU, Korea Optoelectronics 5.1 General description of diffraction EE 430.423.001 2016. 2nd Semester • Diffraction is defined as the bending of light around the corners of an obstacle or aperture into the region of geometrical shadow of the obstacle. • The essential features of diffraction can be explained qualitatively by Huygens’ principle. The Huygens’ principle states that every point on a wavefront actd as the source of a secondary wave that spreads out in all directions. The envelope of all the secondary waves is the new wave front. Augustin Jean Fresnel (1788-1827) in 1818 explained the diffraction phenomena using the Huygens’ principle and Young’s principle of interference Huygens-Fresnel principle • We use a more quantitative approach, the Fresnel-Kirchhoff formula to various cases of diffraction of light by obstacles and apertures.
Diffraction of a plane wave at a slit whose Diffraction of a plane wave when the slit width width is several times the wavelength. equals the wavelength https://en.wikipedia.org/wiki/Huygens%E2%80%93Fresnel_principle 2/27 Changhee Lee, SNU, Korea Optoelectronics 5.2 Fundamental theory EE 430.423.001 2016. 2nd Semester Green’s theorem ∇ − ∇ = ∇2 − ∇2 ∫∫(V nU U nV )dA ∫∫∫(V U V U )dV ∇ = ∇ ⋅ Divergence theorem ∫∫ nFdA ∫∫∫ FdV , F = U∇V −V∇U, ∇ ⋅(U∇V ) = U∇2V + (∇U )⋅(∇V )
If both U and V are wave functions and have a harmonic time dependence of the form eiωt. 1 ∂2U ∇2U = u 2 ∂t 2 1 ∂2V ∇2V = u 2 ∂t 2
∇ − ∇ = ∫∫(V nU U nV )dA 0
3/27 Changhee Lee, SNU, Korea Optoelectronics 5.2 Fundamental theory EE 430.423.001 2016. 2nd Semester ei(kr+ωt) Suppose that we take V to be the wave function V = V 0 r Since V becomes infinite at P, we must exclude that point from the integration. Subtract an integral over a small sphere of radius r=ρ centered at P and then let ρ shrink to zero. Ω = π ∫∫U Pd 4 U P ikr ikr ikr ikr e e e ∂U ∂ e 2 ( ∇ U −U∇ )dA − ( −U ) =ρ ρ dΩ = 0 ∫∫ r n n r ∫∫ r ∂r ∂r r r
Kirchhoff integral theorem 1 eikr eikr U = − (U∇ − ∇ U )dA P 4π ∫∫ n r r n
U = optical disturbance
4/27 Changhee Lee, SNU, Korea Optoelectronics Fresnel-Kirchhoff formula EE 430.423.001 2016. 2nd Semester Determine optical disturbance reaching the receiving point P from the source S. V
Two basic simplifying assumptions: (1) The wave function U and its gradient contribute negligible amounts to the integral except at the aperture opening itself. (2) The values of U and grad U at the aperture are the same as they would be in the absence of the partition. ei(kr'−ωt) The wave function U at the aperture U = U 0 r' U e−iωt eikr eikr' eikr' eikr U = 0 ( ∇ − ∇ )dA P 4π ∫∫ r n r' r' n r Smaller than the 1st term if r, r’>> λ. eikr ∂ eikr ikeikr eikr ∇ = = − n cos(n,r) cos(n,r) 2 r ∂r r r r eikr' ∂ eikr' ikeikr' eikr' ∇ = = − n cos(n,r') cos(n,r') 2 r' ∂r' r' r' r' Smaller than the 1st term if r, r’>> λ.
5/27 Changhee Lee, SNU, Korea Optoelectronics Fresnel-Kirchhoff formula EE 430.423.001 2016. 2nd Semester Fresnel-Kirchhoff integral formula
ikU e−iωt eik (r+r') U = − 0 [cos(n,r ) − cos(n,r')]dA [cos(n,r) − cos(n,r')]=obliquity factor P 4π ∫∫ rr' ik U ei(kr−iωt) Circular aperture, r'= constant, cos(n,r') = −1 U = − A [cos(n,r ) +1]dA, P 4π ∫∫ r U eikr' U = 0 A r'
6/27 Changhee Lee, SNU, Korea Optoelectronics Complementary apertures. Babinet’s principle EE 430.423.001 2016. 2nd Semester
If the aperture is divided into two portions A1 and A2 such that A= A1 + A2. The two apertures A1 and A2 are said to be complementary.
From the Fresnel-Kirchhoff integral formula,
UP= U1P + U2P (Babinet’s principle)
If UP=0, U1P = - U2P The complementary apertures yield identical optical disturbances, except that they differ in phase by 180o. The intensity at P is
therefore the same for the two apertures.
7/27 Changhee Lee, SNU, Korea Optoelectronics Babinet’s principle EE 430.423.001 2016. 2nd Semester
http://userdisk.webry.biglobe.ne.jp/006/095/15/N000/000/004/136844068624713202721.JPG 8/27 Changhee Lee, SNU, Korea Optoelectronics Babinet’s principle EE 430.423.001 2016. 2nd Semester
http://userdisk.webry.biglobe.ne.jp/006/095/15/N000/000/004/136844073251013202889_Corona.JPG 9/27 Changhee Lee, SNU, Korea Optoelectronics 5.3 Fraunhofer and Fresnel Diffraction EE 430.423.001 2016. 2nd Semester Fraunhofer diffraction occurs when both the incident and diffracted waves are effectively plane. This will be the case when the distances from the source to the diffracting aperture and from the aperture to the receiving point are both large enough
for the curvatures of the incident and diffracted waves to be neglected.
If either the source or the receiving point is close enough to the diffracting aperture so that the curvature of the wave front is significant, then one has Fresnel Diffraction.
10/27 Changhee Lee, SNU, Korea Optoelectronics 5.3 Fraunhofer and Fresnel Diffraction EE 430.423.001 2016. 2nd Semester The variation of the quantity r+r’ from one edge of the aperture to the other is given by ∆ = d'2 +(h'+δ )2 + d 2 + (h +δ )2 − d'2 +h'2 − d 2 + h2 h' h 1 1 1 = ( + )δ + ( + )δ 2 +... d' d 2 d' d
The quadratic term is a measure of the curvature of the wave front. The wave is effectively plane over the aperture if 1 1 1 ( + )δ 2 << λ 2 d' d Criterion for Fraunhofer diffraction
11/27 Changhee Lee, SNU, Korea Optoelectronics 5.4 Fraunhofer Diffraction Patterns EE 430.423.001 2016. 2nd Semester Simplifying assumptions: (1) The angular spread of the diffracted light is small enough for the obliquity factor not to vary appreciably over the aperture and to be taken outside the integral. (2) eikr’/r ’ is nearly constant and can be taken outside the integral. (3) The variation of eikr/r over the aperture comes principally from the exponential part, so that the factor 1/r can be replaced by its mean value and taken outside the integral. ikU e−iωt eik (r+r') U = − 0 [cos(n,r) − cos(n,r')]dA P 4π ∫∫ rr'
= ikr U P C∫∫e dA
12/27 Changhee Lee, SNU, Korea Optoelectronics 5.4 Fraunhofer diffraction patterns for the single slit EE 430.423.001 2016. 2nd Semester
F. A. Jenkins and H. E. White, Fundamentals of Optics, 3rd ed. (McGraw-Hill, 1957) 13/27 Changhee Lee, SNU, Korea Optoelectronics 5.4 Fraunhofer diffraction patterns for the single slit EE 430.423.001 2016. 2nd Semester For a single slit of length L and width b, dA=Ldy.
r = r0 + y sinθ
r0 = the value of r for y = 0
b + ikr0 2 ikysinθ U = Ce b e Ldy ∫− 2 1 sin( kbsinθ ) = 2Ceikr0 L 2 k sinθ sin β = C'( ) β 1 β = kbsinθ, C'= eikr0 CbL 2
14/27 Changhee Lee, SNU, Korea Optoelectronics 5.4 Fraunhofer diffraction patterns for the single slit EE 430.423.001 2016. 2nd Semester The irradiance distribution in the focal plane is
2 sin β I = U = I ( )2 0 β 2 irradiance for θ = 0, I0 = CbL ∝ area of the slit
The maximum value occurs at θ=0, and minimum values occur for β=mπ=±π, ±2π, ±3π, … The 1st minimum, β=π, sinθ=2π/kb=λ/b.
15/27 Changhee Lee, SNU, Korea Optoelectronics 5.4 Fraunhofer diffraction patterns for the single slit EE 430.423.001 2016. 2nd Semester (Prob. 5.5) The secondary maxima occur at θ for which β=tanβ. β=1.43π, 2.46π, 3.47π, ...
F. A. Jenkins and H. E. White, Fundamentals of Optics, 3rd ed. (McGraw-Hill, 1957) 16/27 Changhee Lee, SNU, Korea Optoelectronics 5.4 Fraunhofer diffraction for the rectangular aperture EE 430.423.001 2016. 2nd Semester For a rectangular aperture of width a and height b, dA=dxdy.
sinα sin β I = I ( )2 ( )2 0 α β 1 1 α = kasinφ, β = kbsinθ, 2 2 The minimum values occur for α=±π, ±2π, … and β=±π, ±2π, … 17/27 Changhee Lee, SNU, Korea Optoelectronics 5.4 Fraunhofer diffraction for the circular aperture EE 430.423.001 2016. 2nd Semester For a circular aperture of radius R, dA = 2 R2 − y 2 dy
+ R θ U = Ceikr0 eikysin 2 R2 − y 2 dy ∫−R
y u = , ρ = kRsinθ R +1 iρu 2 e 1− u du = πJ1(ρ) / ρ ∫−1
J1(ρ) = Bessel function of the 1st kind
J1(ρ) / ρ →1/ 2, as ρ → 0
2 2J1(ρ) 2 2 I = I0 , where I0 = (CπR ) ρ
18/27 Changhee Lee, SNU, Korea Optoelectronics 5.4 Fraunhofer diffraction for the circular aperture EE 430.423.001 2016. 2nd Semester
The bright central area is known as the Airy disk. 1st zero of the Bessel function ρ=3.832. The angular radius of the 1st dark ring is
3.832 1.22λ sinθ = = ≈ θ kR D D = 2R = diameter of the aperture
19/27 Changhee Lee, SNU, Korea Optoelectronics Optical Resolution EE 430.423.001 2016. 2nd Semester The image of a distant point source formed at the focal plane of a camera lens is a Fraunfoffer diffraction pattern for which the aperture is the lens opening D.
Thus the image of a composite source is a superposition of many Airy disks. The resolution in the image depends on the size of the individual Airy disks.
Rayleigh criterion: minimum angular separation between two equal point sources such that they can be just barely resolved. At this angular separation the central maximum of the image of one source falls on the 1st minimum of the other.
Rayleigh criterion for the resolution 1.22λ θ ≈ D
20/27 Changhee Lee, SNU, Korea Optoelectronics 5.4 Fraunhofer diffraction for the double slit EE 430.423.001 2016. 2nd Semester For a circular aperture of radius R, dA = 2 R2 − y 2 dy
b h+b eikysinθ dy = eikysinθ dy + eikysinθ dy ∫Aperture ∫0 ∫h 1 = (eikbsinθ −1+ eik (h+b)sinθ − eikbsinθ ) ik sinθ ikbsinθ e −1 ikhsinθ = (1+ e ) ik sinθ
iβ iγ sin β = 2be e cosγ β 1 1 β = kbsinθ, γ = khsinθ 2 2
2 sin β 2 I = I0 cos γ β
21/27 Changhee Lee, SNU, Korea Optoelectronics 5.4 Fraunhofer diffraction for the double slit EE 430.423.001 2016. 2nd Semester The single-slit factor (sinβ/β)2 appears as the envelope for the interference firnges given by the term cos2γ. Bright fringes occur for γ=0, ±π, ±2π, … The angular separation between fringes is given by ∆γ=π. 1 2π λ ∆γ = khcosθ ∆θ = π ∆θ ≈ = 2 kh h
22/27 Changhee Lee, SNU, Korea Optoelectronics 5.4 Fraunhofer diffraction for the double slit EE 430.423.001 2016. 2nd Semester
F. A. Jenkins and H. E. White, Fundamentals of Optics, 3rd ed. (McGraw-Hill, 1957) 23/27 Changhee Lee, SNU, Korea Optoelectronics 5.4 Fraunhofer diffraction for the double slit EE 430.423.001 2016. 2nd Semester
F. A. Jenkins and H. E. White, Fundamentals of Optics, 3rd ed. (McGraw-Hill, 1957) 24/27 Changhee Lee, SNU, Korea Optoelectronics Multiple slits, Diffraction gratings EE 430.423.001 2016. 2nd Semester
b h+b 2hb (N −1)h+b eikysinθ dy = + + +....+ eikysinθ dy ∫Aperture ∫0 ∫h ∫2h ∫(N −1)h eikbsinθ −1 = [1+ eikhsinθ +....+ eik (N −1)hsinθ ] ik sinθ eikbsinθ −1 1− eikNhsinθ = ⋅ ik sinθ 1− eikhsinθ
iβ i(N −1)γ sin β sin Nγ = be e β sin γ 1 1 β = kbsinθ, γ = khsinθ 2 2
2 2 sin β sin Nγ I = I0 β N sin γ The factor N has been inserted in order to
normalize the expression, so that I=I0 when θ=0. A diffraction pattern of a 633 nm laser through a grid of 150 slits https://en.wikipedia.org/wiki/Diffraction 25/27 Changhee Lee, SNU, Korea Optoelectronics Multiple slits, Diffraction gratings EE 430.423.001 2016. 2nd Semester
2 2 sin β sin Nγ I = I0 β N sin γ
The single-slit factor (sinβ/β)2 appears as the envelope of the diffraction pattern.
Principal maxima occur within the envelope for γ=nπ , n=0, π, 2π, …
nλ = hsinθ n=order of diffraction
Secondary maxima occur for γ=3π/2Ν, 5π/2Ν, …
Zeros occur for γ=π/Ν, 2π/Ν, …
26/27 Changhee Lee, SNU, Korea Optoelectronics Multiple slits, Diffraction gratings EE 430.423.001 2016. 2nd Semester The angular width of a principal fringe is found by setting the change of Nγ equal to π. π 1 γλ ∆γ = = khcosθ ∆θ, ∴∆θ = N 2 Nhcosθ If N is made very large, then ∆θ is very small, and the diffraction pattern consists of a series of sharp fringes corresponding to the different orders n=0, ±π, ±2π, …
For a given order the dependence of θ on the wavelength gives by differentiation n∆λ nλ = hsinθ ∆θ = hcosθ
Resolving power of a grating spectroscope according to the Rayleigh criterion λ RP = = Nn ∆λ For a typical grating with 600 lines/mm ruled over a total width of 10 cm, N=60,000 and the theoretical resolving power can be 60,000n.
27/27 Changhee Lee, SNU, Korea