Chapter 8 Optics for Engineers Charles A. DiMarzio Northeastern University Sept. 2012 Diffraction Collimated Beam: Divergence in Far Field Focused Beam: Minimum Spot Size and Location λ α = C D Sept. 2012 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides8{1 Slit Experiments −5 1 −5 1 0.5 0.5 0 0 0 0 −0.5 −0.5 5 −1 5 −1 −5 0 5 −5 0 5 A. Two slits B. Aperture • Angular Divergence of Light Waves • Alternating Bright and Dark Regions • Near{ and Far{Field Behavior λ = 800nm. Axis Units are µm Sept. 2012 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides8{2 Diffraction and Maxwell's Equations • • Red Contour Faraday's Equation Z @B E · ds =6 0 H =6 0 r × E = − y @t { Propagation alongz ^ • Green Contour, Ey = Ez = 0 Z E · ds = 0 Hz = 0 0.5 0 x • Blue, Ex Constant, Ey = 0 −0.5 Z −1.5 −1 4 E · ds = 0 Hz = 0 −0.5 3 2 0 • 1 Blue{Dash Contour (Edge) 0.5 0 Z z y 1 −1 E · ds =6 0 Hz =6 0 E = Exe^ jkz { Propagation Alongy ^ too Sept. 2012 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides8{3 Diffraction and Quantum Mechanics 10 10 p = hk=(2π) m 5 m 5 µ µ kx = jkj sin α 0 0 ≥ −5 −5 δx δpx h x, Transverse Distance, x, Transverse Distance, Gaussian Beam, d = 12 µ m, λ = 1 µ m. Gaussian Beam, d = 6 µ m, λ = 1 µ m. 0 0 −10 −10 0 10 20 30 40 50 0 10 20 30 40 50 ≥ z, Axial Distance, µ m z, Axial Distance, µ m δx δkx 2π 10 10 δx k sin δα ≥ 2π m 5 m 5 µ µ 2π 0 0 sin δα ≥ δx jkj −5 −5 x, Transverse Distance, x, Transverse Distance, Gaussian Beam, d = 3 µ m, λ = 1 µ m. Gaussian Beam, d = 1.5 µ m, λ = 1 µ m. 0 0 λ −10 −10 0 10 20 30 40 50 0 10 20 30 40 50 sin δα ≥ z, Axial Distance, µ m z, Axial Distance, µ m δx Sept. 2012 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides8{4 A Peek at Gaussian Beams • The Correct Quantum Approach { Position Probability Density Function, P (x) { Momentum Probability Density Function, P (k) = FT (P (x)) { Uncertainty Measurement D E D E 2 2 2 2 2 2 (δx) = x − hxi (δpx) = px − hpxi ; • Gaussian Beam (Minimum Uncertainty Wave) λ sin δα = δx • Otherwise λ sin δα > δx Sept. 2012 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides8{5 Fresnel–Kirchoff Integral (1) • Expand RHS r · (V rE) − r · (ErV ) = r r r2 − • Auxiliary Field ( V )( E) + V E ej(!t+kr) (rE)(rV ) − Er2V = 0 V (x; y; z; t) = r • E & V Satisfy Helmholz • Electric Field 2 2 ! j!t r E = E E (x; y; z; t) = Es (x; y; z) e c2 • Green's Theorem !2 ZZ r2 = V 2 V (V rE − ErV ) · nd^ 2A = c A+A0 ZZZ r · (V rE) − r · (Erv) d3V RHS = 0 = LHS A+A0 Sept. 2012 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides8{6 Fresnel–Kirchoff Integral (2) • From Prev. Page ZZ (V rE − ErV ) · nd^ 2A = 0 A+A0 • Split Surface ZZ (V rE − ErV ) · nd^ 2A+ A ZZ 2 (V rE − ErV ) · nd^ A0 = 0 A0 • Radius of A0 ! 0 ZZ (V rE − ErV ) · nd^ 2A = 4πE (x; y; z) A0 • Helmholtz–Kirchoff Integral Theorem ZZ " ! # 1 ejkr ejkr E (x; y; z) = − E r · n^ − (rE) · n^ d2A 4π A r r Sept. 2012 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides8{7 Fresnel–Kirchoff Integral (3) ZZ " ! # 1 ejkr ejkr E (x; y; z) = − E r · n^ − (rE) · n^ d2A 4π A r r • E0 Slowly Varying ( ) E (rE) · n^ = jkE − n^ · r^0 r0 0 • Irregular Surface of A ! 1 • r >> λ, r >> λ 0 0 e−jkr rE ! 0 1=r2 ! 0 1=(r )2 ! 0 E = E0 r0 ZZ jkr ( ) jk e 0 E (x1; y1; z1) = E (x; y; z) n^ · r^− n^ · r^ dA 4π Aperture r Sept. 2012 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides8{8 Obliquity Factor • Fresnel–Kirchoff Integral Theorem (Previous Page) ZZ jkr ( ) jk e 0 E (x1; y1; z1) = E (x; y; z) n^ · r^− n^ · r^ dA 4π Aperture r • Numerical Calculations Possible Here, or Simpify More • Obliquity Factor (Removes Backward Wave) • For Small Transverse Distances . ( ) 0 n^ · r^− n^ · r^ = 2 Sept. 2012 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides8{9 Paraxial Approximation ZZ jk ejkr E (x1; y1; z1) = E (x; y; z) dA 2π Aperture r q 2 2 2 r = (x − x1) + (y − y1) + z1 • Paraxial Approximation for • Taylor's Series (First Order) [ ] 2 2 (x − x1) (y − y1) r ≈ z 1 + + • in Denominator 1 2 2 r 2z1 2z1 ≈ r z 2 2 (x − x1) (y − y1) • r in Exponent r ≈ z1+ + s 2z1 2z1 ( )2 ( )2 x − x1 y − y1 • Easier Closed{Form r = z1 + + 1 z1 z1 Computation • Amenable to Further (x − x1) << z1 (y − y1) << z1 Approximations Sept. 2012 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides8{10 Error in Paraxial Approximation q 2 2 2 2 2 (x − x1) (y − y1) r = (x − x1) + (y − y1) + z1 r ≈ z1+ + 2z1 2z1 1 10 0 10 −1 10 dr/dx −2 10 Paraxial True −3 Difference 10 0 10 20 30 40 50 60 70 80 90 Angle, Degrees Sept. 2012 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides8{11 Pupil Coordinates • Pupil Position Coordinates (Length Units) x1; y1 • Angular Coordinates (Di- mensionless) u = sin θ cos ζ v = sin θ sin ζ • Spatial Frequency q Approximation 2 2 max u + v = NA x1 sin θ cos ζ fx ≈ = • Spatial Frequency (Inverse rλ λ Length) y1 sin θ sin ζ fy ≈ = d(kr) 2π dr rλ λ 2πfx = = dx λ dx • For Fourier Optics (Ch. 11) Sept. 2012 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides8{12 Coordinate Relationships Spatial Pupil Direction Frequency Location Cosines Spatial Frequency x1 = λzfx u = fxλ y1 = λzfy v = fyλ x1 x1 Pupil Location fx = λz u = z y1 y1 fy = λz v = z u Direction cosines fx = λ x1 = uz v fy = λ y1 = vz Angle u = sin θ cos ζ v = sin θ sin ζ Sept. 2012 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides8{13 Two Approaches • Fresnel Diffraction ZZ − 2 − 2 jkz1 (x x1) (y y1) jke jk 2z + 2z E (x1; y1; z1) = E (x; y; 0) e 1 1 dxdy 2πz1 • Fraunhofer Diffraction (x, x1 both small) ( ) ( ) 2 2 2 2 2 2 (x − x1) + (y − y1) = x + y + x1 + y1 − 2 (xx1 + yy1) ZZ x2+y2 2 2 jkz1 ( ) (x +y ) (xx1+yy1) jke jk 1 1 jk −jk E (x1; y1; z1) = e 2z1 E (x; y; 0) e 2z1 e z1 dxdy 2πz1 Sept. 2012 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides8{14 Fresnel Radius and the Far Field • Curvature Terms (x2+y2) (x2+y2) jk jk 1 1 e 2z1 e 2z1 • Fresnel Radius in Integrand (See Plot on Next Page) ( )2 ( )2 2 2 (x +y ) j2π p r j2π r q jk 2z 2λz1 rf e 1 = e = e rf = 2λz1 • Simplified Integral x2+y2 jkz1 ( 1 1) jke jk 2z E (x1; y1; z1) = e 1 × 2πz1 x2+y2 ZZ j2π (xx +yy ) r2 −jk 1 1 E (x; y; 0) e f e z1 dxdy Sept. 2012 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides8{15 Integrand in Fresnel Zones ( )2 r j2π r • Fresnel e f Phase Increases With r2 Number Real: Blue Solid Line, Imag: Red Broken Line rmax 1 Nf = rf 0.8 { Near Field 0.6 ≈ 0 0.4 Nf 0.2 { Far Field 0 Nf 0 −0.2 Curvature Term { Almost{ −0.4 Far Field −0.6 N 0 −0.8 f • −1 Different 0 0.5 1 1.5 2 2.5 3 3.5 4 Radius in Fresnel Zones Approximations Sept. 2012 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides8{16 Far Field: Fraunhofer Zone q p 2 z1 D /λ (Far{Field Condition) rf = 2z1λ 2D (x2+y2) j2π rf Nf = r=rf 1 (Fraunhofer Zone) e ≈ 1 Fraunhofer Diffraction Fresnel–Kirchoff Integral is a Fourier Transform x2+y2 ZZ jkz1 ( 1 1) − (xx1+yy1) jke jk 2z jk z E (x1; y1; z1) = e 1 E (x; y; 0) e 1 dxdy 2πz1 Sept. 2012 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides8{17 Near Field: Fresnel Zone 2 z1 D /λ (Near{Field Condition) Nf = r=rf 1 • Fresnel Diffraction ZZ − 2 − 2 jkz1 (x x1) (y y1) jke jk 2z + 2z E (x1; y1; z1) = E (x; y; 0) e 1 1 dxdy 2πz1 • Integrand Averages to Zero Except near x1 = x, y1 = y Sept. 2012 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides8{18 Almost{Far field • Condition Nf small but not that small • Fraunhofer Diffraction ZZ x2+y2 2 2 jkz1 ( ) (x +y ) (xx1+yy1) jke jk 1 1 jk −jk E (x1; y1; z1) = e 2z1 E (x; y; 0) e 2z1 e z1 dxdy 2πz1 • Looks Like Fourier Transform with Curvature • Computation Feasible if Nf is Not Too Large (Watch for Nyquist Criterion in Sampling E(x; y; 0)) Sept.