Chapter 8 Optics for Engineers

Transcription

1 Chapter 8 Optics for Engineers Charles A. DiMarzio Northeastern University Sept. 22

2 Diffraction Collimated Beam: Divergence in Far Field Focused Beam: Minimum Spot Size and Location α = C λ D Sept. 22 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides8

3 Slit Experiments A. Two slits B. Aperture Angular Divergence of Light Waves Alternating Bright and Dark Regions Near and Far Field Behavior λ = 8nm. Axis Units are µm Sept. 22 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides8 2

4 Diffraction and Maxwell s Equations x Faraday s Equation E = B t y.5 z Red Contour E ds H y Propagation along ẑ Green Contour, E y = E z = E ds = H z = Blue, E x Constant, E y = E ds = H z = Blue Dash Contour (Edge) E ds H z E = Eˆxe jkz Propagation Along ŷ too Sept. 22 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides8 3

5 Diffraction and Quantum Mechanics p = hk/(2π) x, Transverse Distance, µ m 5 5 x, Transverse Distance, µ m 5 5 k x = k sin α δx δp x h x, Transverse Distance, µ m Gaussian Beam, d = 2 µ m, λ = µ m z, Axial Distance, µ m 5 5 Gaussian Beam, d = 3 µ m, λ = µ m z, Axial Distance, µ m x, Transverse Distance, µ m Gaussian Beam, d = 6 µ m, λ = µ m z, Axial Distance, µ m 5 5 Gaussian Beam, d =.5 µ m, λ = µ m z, Axial Distance, µ m δx δk x 2π δx k sin δα 2π sin δα 2π δx k sin δα λ δx Sept. 22 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides8 4

6 A Peek at Gaussian Beams The Correct Quantum Approach Position Probability Density Function, P (x) Momentum Probability Density Function, P (k) = F T (P (x)) Uncertainty Measurement (δx) 2 = x 2 x 2 (δp x ) 2 = p 2 x px 2, Gaussian Beam (Minimum Uncertainty Wave) Otherwise sin δα = λ δx sin δα > λ δx Sept. 22 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides8 5

7 Fresnel Kirchoff Integral () Expand RHS (V E) (E V ) = Auxiliary Field ( V ) ( E) + V 2 E V (x, y, z, t) = ej(ωt+kr) r Electric Field E (x, y, z, t) = E s (x, y, z) e jωt Green s Theorem A+A (V E E V ) ˆnd 2 A = A+A (V E) (E v) d 3 V ( E) ( V ) E 2 V = E & V Satisfy Helmholz 2 E = ω2 c 2 E 2 V = ω2 c 2 V RHS = = LHS Sept. 22 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides8 6

8 Fresnel Kirchoff Integral (2) From Prev. Page Split Surface Radius of A A+A (V E E V ) ˆnd 2 A = A (V E E V ) ˆnd2 A+ A (V E E V ) ˆnd 2 A = A (V E E V ) ˆnd 2 A = 4πE (x, y, z) Helmholtz Kirchoff Integral Theorem E (x, y, z) = 4π A [ E ( ejkr r ) ˆn ejkr r ( E) ˆn ] d 2 A Sept. 22 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides8 7

9 Fresnel Kirchoff Integral (3) E (x, y, z) = 4π A [ E ( ejkr r ) ˆn ejkr r ( E) ˆn ] d 2 A E Slowly Varying ( E) ˆn = (jke Er ) ˆn ˆr Irregular Surface of A E = E e jkr r r >> λ, r >> λ E /r 2 /(r ) 2 E (x, y, z ) = jk 4π Aperture E (x, y, z) ejkr r (ˆn ˆr ˆn ˆr ) da Sept. 22 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides8 8

10 Obliquity Factor Fresnel Kirchoff Integral Theorem (Previous Page) E (x, y, z ) = jk 4π Aperture E (x, y, z) ejkr r (ˆn ˆr ˆn ˆr ) da Numerical Calculations Possible Here, or Simpify More Obliquity Factor (Removes Backward Wave) For Small Transverse Distances... (ˆn ˆr ˆn ˆr ) = 2 Sept. 22 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides8 9

11 Paraxial Approximation E (x, y, z ) = jk 2π r = Aperture ejkr E (x, y, z) r da (x x ) 2 + (y y ) 2 + z 2 Paraxial Approximation for r in Denominator r in Exponent r z (x ) 2 x r = z + z ( ) 2 y y + (x x ) << z (y y ) << z z Taylor s Series (First Order) r z [ + (x x ) 2 2z 2 + (y y ) 2 2z 2 r z + (x x ) 2 2z + (y y ) 2 Easier Closed Form Computation Amenable to Further Approximations 2z ] Sept. 22 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides8

12 Error in Paraxial Approximation r = (x x ) 2 + (y y ) 2 + z 2 r z + (x x ) 2 2z + (y y ) 2 2z dr/dx 2 3 Paraxial True Difference Angle, Degrees Sept. 22 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides8

13 Pupil Coordinates Pupil Position Coordinates (Length Units) x, y Angular Coordinates (Dimensionless) max u = sin θ cos ζ v = sin θ sin ζ u 2 + v 2 = NA Spatial Frequency (Inverse Length) 2πf x = d(kr) dx = 2π λ dr dx Spatial Frequency Approximation f x x rλ = sin θ cos ζ λ f y y sin θ sin ζ = rλ λ For Fourier Optics (Ch. ) Sept. 22 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides8 2

14 Coordinate Relationships Spatial Pupil Direction Frequency Location Cosines Spatial Frequency x = λzf x u = f x λ y = λzf y v = f y λ Pupil Location f x = x λz u = x z f y = y λz v = y z Direction cosines f x = λ u x = uz Angle f y = v λ y = vz u = sin θ cos ζ v = sin θ sin ζ Sept. 22 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides8 3

15 Two Approaches Fresnel Diffraction E (x, y, z ) = jkejkz 2πz E (x, y, ) e jk( x x ) 2 2z + ( y y ) 2 2z dxdy Fraunhofer Diffraction (x, x both small) (x x ) 2 + (y y ) 2 = ( x 2 + y 2) + ( x 2 + y2 ) 2 (xx + yy ) E (x, y, z ) = jkejkz 2πz e jk(x2 +y2 ) 2z E (x, y, ) e jk ( x2+y2 ) 2z e jk ( xx +yy ) z dxdy Sept. 22 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides8 4

16 Fresnel Radius and the Far Field Curvature Terms e jk( x2 +y2 ) 2z e jk( x 2 +y2 ) 2z Fresnel Radius in Integrand (See Plot on Next Page) e jk( x2 +y2 ) 2z = e j2π Simplified Integral ( ) 2 ( r 2λz = e j2π r E (x, y, z ) = jkejkz e jk(x2 2πz +y2 ) 2z r f ) 2 r f = 2λz j2π x2 +y 2 E (x, y, ) e r 2 f e jk( xx +yy ) z dxdy Sept. 22 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides8 5

17 Integrand in Fresnel Zones e j2π ( r r f ) 2 Phase Increases With r 2 Real: Blue Solid Line, Imag: Red Broken Line Curvature Term Radius in Fresnel Zones Fresnel Number N f = r max r f Near Field N f Far Field N f Almost Far Field N f Different Approximations Sept. 22 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides8 6

18 Far Field: Fraunhofer Zone z D 2 /λ (Far Field Condition) r f = 2z λ 2D N f = r/r f (Fraunhofer Zone) e j2π( x2 +y2 ) r f Fraunhofer Diffraction Fresnel Kirchoff Integral is a Fourier Transform E (x, y, z ) = jkejkz e jk(x2 2πz +y2 ) 2z E (x, y, ) e jk( xx +yy ) z dxdy Sept. 22 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides8 7

19 Near Field: Fresnel Zone z D 2 /λ (Near Field Condition) N f = r/r f Fresnel Diffraction E (x, y, z ) = jkejkz E (x, y, ) e jk( x x ) 2 2z + ( y y ) 2 2z dxdy 2πz Integrand Averages to Zero Except near x = x, y = y Sept. 22 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides8 8

20 Almost Far field Condition N f small but not that small Fraunhofer Diffraction E (x, y, z ) = jkejkz 2πz e jk(x2 +y2 ) 2z E (x, y, ) e jk ( x2+y2 ) 2z e jk ( xx +yy ) z dxdy Looks Like Fourier Transform with Curvature Computation Feasible if N f is Not Too Large (Watch for Nyquist Criterion in Sampling E(x, y, )) Sept. 22 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides8 9

21 Fraunhofer Lens () Focusing E (x, y, ) = E (x, y, ) e jk( x 2 +y 2 ) 2f Focused in Near Field of Aperture (f << D 2 /λ) E (x, y, z ) = jkejkz 2πz e jk(x2 +y2 ) 2z E (x, y, ) e jk ( x2+y2 ) 2Q e jk ( xx +yy ) z dx dy Modify Fresnel Radius Equation r f = 2λQ Q is the Defocus Parameter = r f 2λz 2λf Sept. 22 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides8 2

22 Fraunhofer Lens (2) Computational Considerations E (x, y, z ) = jkejkz 2πz e jk(x2 +y2 ) 2z E (x, y, ) e jk ( x2+y2 ) 2Q e jk ( xx +yy ) z dx dy Quadratic Phase Term Leads to Nyquist Issues For Large Apertures For Strong Defocusing Practical Limit Q = fz f z D2 8λ f z 8z2 λ D 2 Sept. 22 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides8 2

23 Fraunhofer and Fresnel Zones Sept. 22 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides8 22

24 Spatial Frequency and Angle Definitions Spatial Frequency f x = k 2πz x = x λz f y = k 2πz y = y λz df x = k dx 2π z df y = k dy 2π z Angle or Wavefront Tilt, E a = Ez e jkz, E a (u, v) = jk 2π cos θ E (x, y, ) e jkxu+yv w dxdy Sept. 22 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides8 23

25 Fraunhofer Diffraction Equations Fraunhofer Diffraction Integral E (x, y, z ) = jkejkz e jk(x2 2πz +y2 ) 2z E (x, y, ) e jk( xx +yy ) z dxdy Spatial Frequency Description E (f x, f y, z ) = ( k 2πz ) 2 jke jkz 2πz e jk(x2 +y2 ) 2z E (x, y, ) e j2π(f xx+f y y) dxdy E (f x, f y, z ) = j2πz e jkz k e jk( x2+y2 ) z E (x, y, ) e j2π(f xx+f y y) dxdy Sept. 22 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides8 24

26 Camera Example Pixel Pitch: 7.4µm f sample = m =.35 5 m (35 per mm.) Nyquist Sampling: f max = f sample /2 at u max = NA NA = 2f sampleλ 2 = f sample λ (Coherent Imaging) Green light at 5nm, NA =.68 (See Ch. for Incoherent Imaging, which is More Complicated) Sept. 22 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides8 25

27 Some Useful Fraunhofer Patterns Frequently Used Patterns Uniform Illumination through Rectangular Aperture (Separable Integrals in x and y) Uniform Illumination through Circular Aperture (Frequently Encountered in Imaging) Gaussian Illumination with no Aperture (Many Laser Beams, Minimum Uncertainty, Closed Form) Parameters of Interest Some Measure of Width Axial Irradiance Nulls and Sidelobes if Any Sept. 22 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides8 26

28 Rectangular Aperture () Irradiance Pattern E 2 (x, y, ) = P D x D y Fraunhofer Diffraction E (x, y, z ) = jkejkz e jk(x2 +y2 ) Dx /2 2z 2πz D x /2 Dy /2 D y /2 P e jk D x D y (xx +yy ) z dxdy Separable in x and y E (x, y, z ) = P jke jkz e jk(x2 +y2 ) Dx /2 2z D x D y 2πz D x /2 e jk ( xx) Dy /2 z dx D y /2 e jk ( yy) z dy Sept. 22 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides8 27

29 Rectangular Aperture (2) Compare Rectangular Pulse in Signal Theory E (x, y, z ) = Irradiance P D xd y λ 2 z 2 I (x, y, z ) = P D xd y λ 2 z 2 sin ( kx D x kx D x 2z 2z ) sin ( kx D x kx D x 2z sin ( ) D ky y 2z 2z ) ky D y 2z sin ( ky D y jk ( x 2 +y2 ) e 2z ) 2 2z D ky y 2z On Axis First Null First Sidelobe I = I (,, z ) = P D xd y λ 2 z 2 d x 2 = λ D x I =.74I ( 3dB) Check Math Sept. 22 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides8 28

30 Circular Aperture () Polar Coordinates E (r, z ) = jkejkz e jk(x2 +y2 ) D/2 2z 2πz 2π E (r, ) e j k(r cos ζx+r sin ζy) z rdζdr Hankel Transform E (r, z ) = jkejkz ) e jk(r2 2πz z D/2 ( j kr2 E (r, ) e z krr J z ) dr Fraunhofer Diffraction for Source, E 2 (r, ) = E (r, z ) = jkejkz ) e jk(r2 2πz D/2 z P π(d/2) 2 ( P krr π (D/2) 2J z ) dr Sept. 22 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides8 29

31 Circular Aperture (2) E (r, z ) = P πd2 4λ 2 z 2 J ( kr D 2z ) kr D 2z e jk r 2 2z I (r, z ) = P πd2 4λ 2 z 2 ) 2 2z kr D 2z J ( kr D On Axis First Null First Sidelobe I = P πd 2 /4λ 2 z z λd I =.2I ( 7dB) Sept. 22 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides8 3

32 Gaussian Plane Wave Gaussian Source (Assume Infinite Aperture) E (x, y, ) = 2P π(w ) 2e r2 /w 2 Polar Coordinates (Arbitrary Choice) E (r, z ) = jkejkz ) 2z 2πz e jk(r2 2π 2P π(w ) 2e r2 /w 2 e jkr ( r cos ζ+r sin ζ ) z rdζdr Result: Gaussians are Forever. (More in Chapter 9) E (x, y, z ) = 2P π(w) 2e r2 /w2 w = z λ πw Sept. 22 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides8 3

33 Summary of Common Fraunhofer Patterns Pupil Pattern Fraunhofer Pattern Slice SQUARE: d = 2 λ (st Nulls) D I = P D2 λ 2 z 2 I /I = 3dB CIRCULAR: d = 2.44 λ (st Nulls) D πp D2 I = 4λ 2 z 2 I /I = 7dB GAUSSIAN: d = 4 (e 2 Width) π λ d I = πp d2 2λ 2 z 2 No sidelobes x Sept. 22 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides8 32

34 Numerical Integration Example: Gaussian in an Aperture Given Aperture, D Choose Gaussian Size, hd Fill Factor, h Optimize Pattern Somehow (e.g. Axial Irradiance) Integrate in Polar Coordinates E (x, y, z ) = jkejkz e jk(x2 +y2 ) D/2 2z 2πz 2π 2P π(hd/2) 2e 4r 2 h 2 D 2 e jkr ( x cos ζ+y sin ζ ) z rdζdr Sept. 22 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides8 33

35 Truncated Gaussian Approximations Small Beam, h Ignore Aperture Integrate in Closed Form Gaussian Pattern d = 4 π λ hd z ) 2 ( πhd I = P λz Low Axial Irradiance I h 2 Large Beam, h Near Uniform Illumination Power Through Aperture 2P π(hd) 2 /4 πd2 4 = 2P h 2 Use Circular Aperture I = P ( ) 2 πd 2π hλz Low Axial Irradiance I h 2 Between These Limits, Use Numerical Calculation Sept. 22 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides8 34

36 Numerical Integration Results: Gaussian in an Aperture Axial Irradiance Field h, Fill Factor x, Location max = at h = fx, Spatial Frequency 3 B. Diffraction Field h, Fill Factor Sept. 22 h, Fill Factor E. Axial I C. Source h, Fill Factor x, Location A. Source D. Diffraction 2 2 h, Fill Factor fx, Spatial Frequency F. Null Locations c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides8 35 2

37 Depth of Focus Fraunhofer Diffraction E (x, y, z ) = jkejkz Defocus (Diopters) x, Transverse Distance 2πz e jk(x2 Numerical Result +y2 ) 2z Q = z f E (x, y, ) e jk ( x2+y2 ) 2Q e jk ( xx +yy ) z dx dy Q f z f z f, Defocus Distance Equation DOF λ NA 2 Sept. 22 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides8 36

38 Resolution of an Imaging System To Resolve or Resolution Defined... to distinguish parts or components of (something) that are close together in space or time; the process or capability of rendering distinguishable the component parts of an object or image; a measure of this, expressed as the smallest separation so distinguishable,... Resolution is Not Number of Pixels or Width of Point Spread Function Sept. 22 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides8 37

39 Resolution Analysis Diffraction Patterns of Two Point Objects Point Spread Functions From Fraunhofer Diffraction at Pupil Fourier Transforms (See Ch. ) Add Set Criterion for Valley Noise Analysis? Contrast? Arbitrary Decision? Relative Irradiance γ = x D/λ, Position A complete and consistent definition would require knowledge we are not likely to have. Sept. 22 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides8 38

40 Rayleigh Criterion Frequently Used, but Arbitrary Defined by Nulls of Point Spread Function Peak of One over Valley of Other Produces Inconsistent Valley Square Aperture δ = z λ/d δθ = λ/d Valley Depth [ sin (π/2) 2 π/2 8 π 2 =.8 ] 2 = Circular Aperture Valley Depth δθ =.22 λ D δ =.6 λ NA.73 Sept. 22 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides8 39

41 Resolution at the Rayleigh Limit Relative Irradiance x D/λ, Position Square Aperture, λ/d, 8% Valley Relative Irradiance x D/λ, Position Circular Aperture,.22λ/D, 73% Valley Issues Noise Contrast Statistics Sampling* Other Definitions MTF (Ch. ) Any Valley (Sparrow) 8% Valley (Wadsworth) PSF FWHM (Houston)* * Not really resolution Sept. 22 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides8 4

42 The Diffraction Grating () The Grating Equation Nλ = d (sin θ i + sin θ d ) sin θ d = sin θ i + N λ d Sept. 22 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides8 4

43 The Diffraction Grating (2) Grating Equation Nλ = d (sin θ i + sin θ d ) Grating Dispersion δλ = d N δ (sin θ d) Anti Aliasing Filter e.g. Colored Glass e.g. Filter Wheel Monochromator (More Later) Applications Monochromator Spectrometer Aliasing Issues N to N + λ N+ = λ N N N + Maximum Width: Factor of 2 Sept. 22 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides8 42

44 Fourier Analysis of Grating Convolution in Grating is Multiplication in Diffraction Pattern Multiplication in Grating is Convolution in Diffraction Pattern Grating Diffraction Pattern g (x, ) = f (x x, ) h (x, ) dx g (x, z ) =f (x, z ) h (x, z ), g (x, ) =f (x, ) h (x, ) g (x, z ) = f ( x x, z ) h ( x, z ) dx, Approach Convolve Slit with Comb to Produce Grating Multiply Grating by Illumination or Apodization Pattern Sept. 22 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides8 43

45 Fourier Analsysis Equations () Slit Pattern Repetition Pattern Slit Comb Grating Pattern x, Position in Grating x, Position in Grating Diffraction Pattern Slit Diffraction Pattern Comb f, Position in Diffraction Pattern f, Position in Diffraction Pattern Convolve above two to obtain grating pattern. Grating x, Position in Grating Grating Diffraction Pattern f, Position in Diffraction Pattern Zeros Depend on Slit Width and Repetition Interval Multiply above two to obtain diffraction pattern. Sept. 22 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides8 44

46 Fourier Analsysis Equations (2) Apodization or illumination Grating Apodization Grating Pattern x, Position in Grating x, Position in Grating Diffraction Pattern Grating Diffraction Pattern Apodization f, Position in Diffraction Pattern f, Position in Diffraction Pattern Multiply above two to obtain grating output. Apodized Grating x, Position in Grating podized Grating Diffraction Patt f, Position in Diffraction Pattern Wider Apodization Provides Better Resolution Convolve above two to obtain diffraction pattern. Sept. 22 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides8 45

47 Example of Grating Analysis Equations Item Grating Size Diffraction Angle Pattern Pattern Basic Shape Uniform D sinc λ/d* Repetition Comb d Comb λ/d * first nulls Apodization Gaussian d Gaussian 4 π λ d Finesse: (Order Spacing)/(Width of a Single Order) F = 4 π λ d λ d = 4 π d d Rule of Thumb Roughly F = Number of Grooves Illuminated Sept. 22 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides8 46

48 Grating Spectrometeter Example () Laser Source with Two Wavelengths λ = λ δλ/2 λ 2 = λ + δλ/2 Beam Diameters Equal to (Or Expanded to) d Spacing in First Order Spectrum (N = ) δθ = θ d2 θ d sin θ d2 sin θ d = λ 2 λ d = δλ d Width α = 4 π λ d Sept. 22 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides8 47

49 Grating Spectrometeter Example (2) Sum of Irradiances I (θ) = e 2(θ θ d) 2 /α 2 + e 2(θ θ d2) 2 /α 2 Resolution Criterion Not Rayleigh (No Nulls) Try Sparrow: Check First Derivative di (θ) dθ = or 3 Solutions including θ = θ d2 + θ d 2 Check Second Derivative (Positive for Valley) d 2 I dθ 2 θ=(θd +θ d2 )/2 = 8e 2 ( δθα ) 2 [α 2 (δθ) 2] α 2 Sept. 22 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides8 48

50 Sparrow Criterion for Grating Second Derivative d 2 I dθ 2 θ=(θd +θ d2 )/2 = 8e 2 ( δθα ) 2 [α 2 (δθ) 2] α 2 Negative (Peak) for α < δθ Positive (Valley) for α > δθ Boundary 4 π λ = δλ d d Subscript on δλ for Onset of Valley δλ λ = 4 d = π d F e.g. 4 π 6 m 3 m 3 For λ = 83nm, δλ = 83nm/3.6nm Sept. 22 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides8 49

51 Blaze Angle Linear Phase Shift in Slit Prism for Transmission Tilt for Reflection Translation of Diffraction Pattern Phase Ramp Here Slit x, Position in Grating Slit Diffraction Pattern Littrow Grating: θ i = θ d λ = 2d sin θ i (N = ) Backward Reflection e.g. Tunable Laser f, Position in Diffraction Pattern Translation Here Align Peak with First Order for Chosen λ Q: Add a phase ramp to the Matlab code Sept. 22 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides8 5

52 Monochromator Entrance Slit Limits θ i Exit Slit Selects Limits θ d Choose One Order podized Grating Diffraction Patt f, Position in Diffraction Pattern Symmetry Minimizes Aberrations Blazed Grating Enhances Signal Colored Glass Filter Eliminates Aliasing Sept. 22 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides8 5

53 Monochromator Resolution Entrance Slit d + i = 2w i Input Angles: w i /(2f) sin θ i w i /(2f) Transmission Through Entrance Slit T i (δλ) = Transmission Through Exit Slit T d (δλ) = Overall Transmission Spectrum δλ d N δ sin θ i < w i/2, δλ d N δ sin θ d < w d/2 T i (δλ) I (δλ) T d (δλ) Issues of Source Spatial Coherence (Ch. and ) in in Apodization for I (δλ) Sept. 22 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides8 52

54 Fresnel Diffraction Focused Light in Geometric Optics z f D I = f P f 2 π (D/2) 2 (z f) 2 Fails at Edges of Shadow and at Focus Focal Region Done in Fraunhofer Diffraction Fresnel Diffraction Used at Edges Assume Integrals Separable in x and y E (x, y, z ) = jkejkz 2πz E x (x, ) e jk( x x ) 2 2z dx E y (y, ) e jk( y y ) 2 2z dy Sept. 22 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides8 53

55 Uniformly Illuminated Rectangular Aperture Aperture Size 2a by 2b E (x, y, z ) = jkejkz 2πz P 4ab I (x, a, z ) I (y, b, z ) I (x, a, z ) = a I (x, a, z ) = a cos ( a k (x x ) 2 2z a ejk ) (x x ) 2 2z dx dx+j a a sin ( k (x x ) 2 2z ) dx Fresnel Cosine and Sine Integral Cosine Interand Contributes Near x = x Sine is Zero at x = x Away from Edges, I(x, y, z ) = I(x, y, ) Curvature Term Radius in Fresnel Zones Sept. 22 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides8 54

56 Fresnel Sine and Cosine Integrals () Dimensionless Coordinates k u = (x x ) du = dx 2z k 2z Limits u = (x + a) k k u 2 = (x a) 2z 2z Integrals I (x, a, z ) = 2z k u2 u cos 2 u du + j 2z k u2 u sin 2 u du I (x, a, z ) = 2z k [C (u 2) C (u ) + S (u 2 ) S (u )] Sept. 22 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides8 55

57 Fresnel Sine and Cosine Integrals (2) I (x, a, z ) = 2z k [C (u 2) C (u ) + S (u 2 ) S (u )] C (u ) = u cos2 u du S (u ) = u sin2 u du F (u ) = C (u ) + js (u ) Above 3 Are Odd Functions C (u) = C ( u) S (u) = S ( u) F (u) = F ( u) Approximations F (u) ue jπu2 /2 u F (u) +j 2 j πu ejπu2 /2 u () Sept. 22 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides8 56

58 Fresnel Diffraction Results E (x, y, z ) = jkejkz P 2z 2πz 4ab k { [ F (x a) { [ F (y b) k 2z k 2z ] [ ] F (x + a) [ F (y + b) ]} k k 2z 2z ]} Large x or y : E : Q: How Close? Use Approximations on Previous Page. Near the Center: E (x, y, z ) = E (x, y, ) (Using the Odd Property of F ) Sept. 22 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides8 57

59 Cornu Spiral Sept. 22 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides8 58

60 Example of Fresnel Diffraction: Near Field Diffraction x I, Irradiance, W/m y, mm x, mm A. Cornu Spiral B. Irradiance x, mm x Relative Irradiance, db x, mm C. Image D. Irradiance, db Sept. 22 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides8 59

61 Fresnel Diffraction Includes Fraunhofer Diffraction I, Irradiance, W/m Slit Diffraction Pattern f, Position in Diffraction Pattern A. Cornu Spiral B. Field 3 x x, mm C. Irradiance Sept. 22 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides8 6

62 Fresnel Lens height, mm OPD, µ m y, cm x, cm y, cm x, cm Lens Unwrapped Phase Phase for for Regular Lens Fresnel Lens Sept. 22 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides8 6

63 Fresnel Zone Plate.5 y, cm.5 Fresnel zone plate with Diameter 2cm at distance of m. The zone plate acts as a lens..5.5 x, cm Zone Plate.5 5 u Integrand of Cosine Integral passing through zone plate. All negative contributions are blocked. Sept. 22 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides8 62

64 Phase in Fresnel Zone Plate y, cm x, cm Raw Phase in Radians y, cm.5.5 x, cm Zone Plate Transmission y, cm.5 y, cm x, cm x, cm Real Transmitted Field Imag. Transmitted Field Sept. 22 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides8 63

65 Focusing With Fresnel Zone Plate f y, /cm 3 f y, /cm 3 f y, /cm f, /cm x f, /cm x f, /cm x Unfocused Focused Focused with Pattern With Lens Zone Plate Sept. 22 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides8 64