Diffraction Pattern −50 −2 0 2 −100 −50 0 50 100 X, Position in Grating F, Position in Diffraction Pattern

Home , Fresnel diffraction

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 ∫ ∂B E · ds =6 0 H =6 0 ∇ × E = − y ∂t – Propagation alongz ˆ • Green Contour, Ey = Ez = 0 ∫ E · ds = 0 Hz = 0 0.5 0 x • Blue, Ex Constant, Ey = 0 −0.5 ∫ −1.5 −1 4 E · ds = 0 Hz = 0 −0.5 3 2 0 • 1 Blue–Dash Contour (Edge) 0.5 0 ∫ z y 1 −1 E · ds =6 0 Hz =6 0 E = Exeˆ jkz – Propagation Alongy ˆ 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 = |k| 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 |k| −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

∇ · (V ∇E) − ∇ · (E∇V ) =

∇ ∇ ∇2 − • Auxiliary Field ( V )( E) + V E ej(ωt+kr) (∇E)(∇V ) − E∇2V = 0 V (x, y, z, t) = r • E & V Satisfy Helmholz • Electric Field 2 2 ω jωt ∇ E = E E (x, y, z, t) = Es (x, y, z) e c2 • Green’s Theorem ω2 ∫ ∫ ∇2 = V 2 V (V ∇E − E∇V ) · ndˆ 2A = c A+A0 ∫ ∫ ∫ ∇ · (V ∇E) − ∇ · (E∇v) 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 ∫ ∫ (V ∇E − E∇V ) · ndˆ 2A = 0 A+A0 • Split Surface ∫ ∫ (V ∇E − E∇V ) · ndˆ 2A+ A ∫ ∫ 2 (V ∇E − E∇V ) · ndˆ A0 = 0 A0 • Radius of A0 → 0 ∫ ∫ (V ∇E − E∇V ) · ndˆ 2A = 4πE (x, y, z) A0 • Helmholtz–Kirchoﬀ Integral Theorem ∫ ∫ [ ( ) ] 1 ejkr ejkr E (x, y, z) = − E ∇ · nˆ − (∇E) · nˆ d2A 4π A r r

Sept. 2012 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides8–7 Fresnel–Kirchoﬀ Integral (3)

∫ ∫ [ ( ) ] 1 ejkr ejkr E (x, y, z) = − E ∇ · nˆ − (∇E) · nˆ d2A 4π A r r

• E0 Slowly Varying ( ) E (∇E) · nˆ = jkE − nˆ · rˆ0 r0 0 • Irregular Surface of A → ∞ • r >> λ, r >> λ 0 0 e−jkr ∇E → 0 1/r2 → 0 1/(r )2 → 0 E = E0 r0

∫ ∫ jkr ( ) jk e 0 E (x1, y1, z1) = E (x, y, z) nˆ · rˆ− nˆ · rˆ 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)

∫ ∫ jkr ( ) jk e 0 E (x1, y1, z1) = E (x, y, z) nˆ · rˆ− nˆ · rˆ dA 4π Aperture r

• Numerical Calculations Possible Here, or Simpify More

• Obliquity Factor (Removes Backward Wave)

• For Small Transverse Distances . . . ( ) 0 nˆ · rˆ− nˆ · rˆ = 2

Sept. 2012 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides8–9 Paraxial Approximation

∫ ∫ jk ejkr E (x1, y1, z1) = E (x, y, z) dA 2π Aperture r √ 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+ + √ 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

√ 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 √ 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

∫ ∫ − 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)

∫ ∫ 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 √ r j2π r √ 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 ∫ ∫ 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

√ √ 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 ∫ ∫ 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

∫ ∫ − 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

∫ ∫ 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. 2012 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides8–19 Fraunhofer Lens (1)

• Focusing

(x2+y2) −jk E (x, y, 0) = E0 (x, y, 0) e 2f

– Focused in Near Field of Aperture (f << D2/λ)

∫ ∫ x2+y2 2 2 jkz1 ( 1 1) (x +y ) (xx1+yy1) jke jk jk −jk E (x1, y1, z1) = e 2z1 E (x, y, 0) e 2Q e z1 dx1dy1 2πz1

– Modify Fresnel Radius Equation √ rf = 2λQ Q is the Defocus Parameter 1 1 1 = √ − √ rf 2λz1 2λf

Sept. 2012 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides8–20 Fraunhofer Lens (2)

• Computational Considerations ∫ ∫ x2+y2 2 2 jkz1 ( 1 1) (x +y ) (xx1+yy1) jke jk jk −jk E (x1, y1, z1) = e 2z1 E (x, y, 0) e 2Q e z1 dx1dy1 2πz1

• Quadratic Phase Term Leads to Nyquist Issues

– For Large Apertures

– For Strong Defocusing

– Practical Limit

fz D2 8z2λ = 1 | − | 1 Q f z 2 |f − z1| 8λ D

Sept. 2012 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides8–21 Fraunhofer and Fresnel Zones

Sept. 2012 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides8–22 Spatial Frequency and Angle Deﬁnitions

• Spatial Frequency

k x1 k y1 fx = x1 = fy = y1 = 2πz1 λz1 2πz1 λz1

k dx1 k dy1 dfx = dfy = 2π z1 2π z1

−jkz • Angle or Wavefront Tilt, Ea = Ez1e 1, ∫ ∫ jk −jkxu+yv Ea (u, v) = E (x, y, 0) e w dxdy 2π cos θ

Sept. 2012 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides8–23 Fraunhofer Diﬀraction Equations

• Fraunhofer Diﬀraction Integral

x2+y2 ∫ ∫ jkz1 ( 1 1) − (xx1+yy1) jke jk 2z jk z E (x1, y1, z1) = e 1 E (x, y, 0) e 1 dxdy 2πz1

• Spatial Frequency Description

E (fx, fy, z1) = ( ) 2+ 2 ∫ ∫ 2 jkz (x1 y1) k jke 1 jk − 2 + e 2z1 E (x, y, 0) e j π(fxx fyy)dxdy 2πz1 2πz1

(x2+y2) ∫ ∫ 2 jkz1 1 1 j πz1e jk z −j2π(fxx+fyy) E (fx, fy, z1) = e 1 E (x, y, 0) e dxdy k

Sept. 2012 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides8–24 Camera Example

• Pixel Pitch: 7.4µm

1 − f = = 1.35 × 105m 1 (135 per mm.) sample 7.4 × 10−6m

• Nyquist Sampling: fmax = fsample/2 at umax = NA 2f λ NA = sample = f λ (Coherent Imaging) 2 sample

• Green light at 500nm,

NA = 0.068

(See Ch. 11 for Incoherent Imaging, which is More Complicated)

Sept. 2012 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides8–25 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. 2012 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides8–26 Rectangular Aperture (1)

• Irradiance Pattern P E2 (x, y, 0) = DxDy

• Fraunhofer Diﬀraction ∫ ∫ √ x2+y2 jkz1 ( ) Dx/2 Dy/2 (xx1+yy1) jke jk 1 1 P −jk E (x1, y1, z1) = e 2z1 e z1 dxdy 2 πz1 −Dx/2 −Dy/2 DxDy

• Separable in x and y √ ∫ ∫ x2+y2 jkz1 ( ) Dx/2 (xx1) Dy/2 (yy1) P jke jk 1 1 −jk −jk E (x1, y1, z1) = e 2z1 e z1 dx e z1 dy 2 DxDy πz1 −Dx/2 −Dy/2

Sept. 2012 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides8–27 Rectangular Aperture (2)

• Compare Rectangular Pulse in Signal Theory v ( ) ( ) u Dx Dy x2+y2 u sin kx1 sin ky1 ( 1 1) t DxDy 2z1 2z1 jk 2z E (x1, y1, z1) = P e 1 2 2 Dx Dy λ z1 kx1 ky 2z1 12z1 • Irradiance

 ( ) ( )2 sin kx Dx sin ky Dy DxDy  12z1 12z1  I (x1, y1, z1) = P   2 2 Dx Dy λ z1 kx1 ky 2z1 12z1

On Axis First Null First Sidelobe DxDy dx λ − I0 = I (0, 0, z1) = P 2 2 = I1 = 0.0174I0 ( 13dB) λ z1 2 Dx Check Math

Sept. 2012 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides8–28 Circular Aperture (1)

• Polar Coordinates

2 2 ∫ ∫ jkz1 (x +y ) D/2 2π jke jk 1 1 −j k(r cos ζx+r sin ζy) E (r1, z1) = e 2z1 E (r, 0) e z1 rdζdr 2πz1 0 0

• Hankel Transform ( ) r2 ∫ 2 jkz1 ( 1) D/2 − kr jke jk z j z krr1 E (r1, z1) = e 1 E (r, 0) e 1 J0 dr 2πz1 0 z1

• Fraunhofer Diﬀraction for Source, E2 (r, 0) = P π(D/2)2

2 v ( ) jkz (r ) ∫ u jke 1 jk 1 D/2 u P krr1 z1 t E (r1, z1) = e 2J0 dr 2πz1 0 π (D/2) z1

Sept. 2012 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides8–29 Circular Aperture (2)

v ( ) u u 2 J kr D r2 t πD 1 12z1 jk 1 E (r1, z1) = P e 2z 4λ2z2 kr D 1 12z1

 ( )2 2 J kr D πD  1 12z1  I (r1, z1) = P   4λ2z2 kr D 1 12z1

On Axis First Null First Sidelobe 2 2 2 − I0 = P πD /4λ z1 2.44z1λD I = 0.02I0 ( 17dB)

Sept. 2012 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides8–30 Gaussian

• Plane–Wave Gaussian Source (Assume Inﬁnite Aperture) √ 2P − 2 2 0 = r /w0 E (x, y, ) 2e π(w0)

• Polar Coordinates (Arbitrary Choice) ∫ ∫ √ r2 ∞ 2 jkz1 ( 1) π (r1 cos ζ+r1 sin ζ) jke jk 2P −r2/w2 −jkr E (r , z ) = e 2z1 e 0 e z1 rdζdr 1 1 2 2πz1 0 0 π(w0)

• Result: Gaussians are Forever. (More in Chapter 9)

√ 2P − 2 2 z1λ = r1/w = E (x1, y1, z1) 2e w π(w) πw0

Sept. 2012 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides8–31 Summary of Common Fraunhofer Patterns

Pupil Pattern Fraunhofer Pattern Slice

−4 1 80 SQUARE: −15 −3 3000 0.8 −10 70 λ −2 d = 2 (1st Nulls) −5 2000 D −1 0.6 60 2 PD 0 0 1000 0.4 50 I = 1 0 2 2 5 λ z 0 1 2 0.2 40 10 −1000 I /I = −13dB 3 1 0 15 0 30 −4 −2 0 2 −15 −10 −5 0 5 10 15 0 2 4 6 8 10

−4 1 100 −15 3000 CIRCULAR: −3 2500 0.8 −10 90 λ −2 2000 = 2 44 (1st Nulls) −5 1500 d . −1 0.6 80 D 1000 2 0 0 500 πP D 0.4 70 1 = 5 I0 4 2 2 0 λ z1 2 −500 0.2 10 60 = −17 3 −1000 I1/I0 dB 15 0 −1500 50 −4 −2 0 2 −15 −10 −5 0 5 10 15 0 2 4 6 8 10

7 −4 x 10 70 3 GAUSSIAN: −3 1.4 −10 2.5 4 λ −2 −2 1.2 = ( Width) 2 d e −1 1 60 π d0 0 0.8 0 πP d2 1.5 0 1 0.6 50 I0 = 2 2 1 2λ z1 2 0.4 10 0.5 3 0.2 No sidelobes 0 −4 −2 0 2 −10 0 10 −20 −15 −10 −5 0 5 10 15 20

Sept. 2012 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides8–32 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 ∫ ∫ √ x2+y2 jkz1 ( ) D/2 2π (x1 cos ζ+y1 sin ζ) jke jk 1 1 2P − 4r2 −jkr E (x , y , z ) = e 2z1 e h2D2 e z1 rdζdr 1 1 1 2 2πz1 0 0 π(hD/2)

Sept. 2012 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides8–33 Truncated Gaussian Approximations

• Small Beam, h → 0 • Large Beam, h → ∞ – Ignore Aperture – Near Uniform Illumination – Integrate in Closed Form – Power Through Aperture – Gaussian Pattern 2P D2 2P = 4 λ 2 π 2 d = z1 π(hD) /4 4 h π hD – Use Circular Aperture ( ) ( )2 2 πhD P πD I0 = P I0 = λz1 2π hλz1 – Low Axial Irradiance • Low Axial Irradiance 2 ∝ −2 → I0 ∝ h → 0 I0 h 0

• Between These Limits, Use Numerical Calculation

Sept. 2012 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides8–34 Numerical Integration Results: Gaussian in an Aperture

3500 0.4 5 4.5 0.6 4 3000 0.8 6 3.5 1 3 4 1.2 2.5 2500 Field

h, Fill Factor 2 1.4 2 Axial Irradiance 1.5 1.6 0 2000 1 0 1.8 1 1 0.5 0.5 0 max = 3257.483 at h = 0.89 2 −0.5 1500 −0.5 0 0.5 −1 2 h, Fill Factor 0 0.5 1 1.5 2 x, Location x, Location h, Fill Factor A. Source C. Source E. Axial I

70 4 0.4 65 1400 0.6 1200 3.5 0.8 60 4000 1000 55 3 1 3000 800 600 1.2 50 2000 2.5 h, Fill Factor

Field 400 1.4 45 1000 200 0 2 1.6 40 0 0 −1000 1.8 35 4 1 1.5 2 −200 0 2 −2 −3 −2 −1 0 1 2 3 −4 2 h, Fill Factor f , Spatial Frequency f , Spatial Frequency 1 x x 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 B. Diﬀraction D. Diﬀraction F. Null Locations

Sept. 2012 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides8–35 Depth of Focus

• Fraunhofer Diﬀraction ∫ ∫ x2+y2 2 2 jkz1 ( 1 1) (x +y ) (xx1+yy1) jke jk jk −jk E (x1, y1, z1) = e 2z1 E (x, y, 0) e 2Q e z1 dx1dy1 2πz1 • Defocus (Diopters) 1 1 1 1 f − z = − ≈ 1 2 Q z1 f Q f • Numerical Result 12000 • Equation −5 10000 8000 0 6000 λ 4000 DOF ≈ NA2 5 2000

x, Transverse Distance −20 −10 0 10 20 z−f, Defocus Distance

Sept. 2012 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides8–36 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, ex- pressed as the smallest separation so distinguishable, . . . ” • Resolution is Not Number of Pixels or Width of Point–Spread Function

Sept. 2012 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides8–37 Resolution Analysis

−4 • Diﬀraction Patterns of Two −2 “Point Objects” – Point–Spread Functions 0 – From Fraunhofer Diﬀraction 2 at Pupil γ = 2 −4 −2 0 2

– Fourier Transforms (See Ch. 11) 1 • Add 0.8 • Set Criterion for Valley 0.6 – Noise Analysis? 0.4

– Contrast? Relative Irradiance 0.2 – Arbitrary Decision? 0 −8 −6 −4 −2 0 2 4 6 8 x D/λ, Position

A complete and consistent deﬁnition would require knowledge we are not likely to have.

Sept. 2012 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides8–38 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 – Circular Aperture

δ = z1λ/D λ δθ = 1.22 D δθ = λ/D – Valley Depth [ ] λ 2 = 0 61 sin (π/2) δ . 2 = NA π/2 – Valley Depth 8 = 0.81 π2 0.73 Sept. 2012 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides8–39 Resolution at the Rayleigh Limit

1 • Issues −4

0.8 – Noise −2 0.6 – Contrast 0 0.4 – Statistics Relative Irradiance – Sampling* 2 0.2 • Other 0 −8 −6 −4 −2 0 2 4 6 8 −4 −2 0 2 x D/λ, Position Deﬁnitions Square Aperture, λ/D, 81% Valley 1 – MTF −4 (Ch. 11) 0.8 −2 – Any Valley 0.6 0 (Sparrow) 0.4

Relative Irradiance – 81% Valley 2 0.2 (Wadsworth) 0 −8 −6 −4 −2 0 2 4 6 8 −4 −2 0 2 x D/λ, Position – PSF FWHM Circular Aperture, 1.22λ/D, 73% Valley (Houston)* * Not really resolution

Sept. 2012 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides8–40 The Diﬀraction Grating (1)

• The Grating Equation

Nλ = d (sin θi + sin θd)

λ sin θ = − sin θ + N d i d Sept. 2012 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides8–41 The Diﬀraction Grating (2)

• Grating Equation • Anti–Aliasing Filter

Nλ = d (sin θi + sin θd) – e.g. Colored Glass – e.g. “Filter Wheel” • Grating Dispersion • Monochromator d δλ = δ (sin θ ) (More Later) N d • Applications – Monochromator – Spectrometer • Aliasing Issues – N to N + 1 N λ +1 = λ × N N N + 1 – Maximum Width: Factor of 2

Sept. 2012 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides8–42 Fourier Analysis of Grating

• Convolution in Grating is Multiplication in Diﬀraction Pattern

• Multiplication in Grating is Convolution in Diﬀraction Pattern

Grating Diﬀraction Pattern ∫ 0 0 0 g (x, 0) = f (x − x , 0) h (x , 0) dx g (x1, z1) =f (x1, z1) h (x1, z1) , ∫ ( ) ( ) − 0 0 0 g (x, 0) =f (x, 0) h (x, 0) g (x1, z1) = f x1 x1, z1 h x1, z1 dx1,

• Approach

– Convolve Slit with Comb to Produce Grating

– Multiply Grating by Illumination or Apodization Pattern

Sept. 2012 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides8–43 Fourier Analsysis Equations (1)

Grating Pattern Diﬀraction Pattern 150

Slit Pattern 1 100

50 Slit 0.5 0

0 Slit Diffraction Pattern −50 −2 0 2 −100 −50 0 50 100 x, Position in Grating f, Position in Diffraction Pattern

40 Repetition 1 Pattern 30 20

Comb 0.5 Comb 10

0 0 −2 0 2 −100 −50 0 50 100 x, Position in Grating f, Position in Diffraction Pattern

4000 1 Convolve above 3000 Multiply above two to obtain 0.8 two to obtain 0.6 2000

grating pattern. Grating diﬀraction 0.4 1000 0.2 pattern. 0 0 −2 −1 0 1 2 Grating Diffraction Pattern −100 −50 0 50 100 x, Position in Grating f, Position in Diffraction Pattern Zeros Depend on Slit Width and Repetition Interval

Sept. 2012 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides8–44 Fourier Analsysis Equations (2)

Grating Pattern Diﬀraction Pattern 4000 1 3000 0.8 0.6 2000

Grating 0.4 1000 0.2 0 0 −2 −1 0 1 2 Grating Diffraction Pattern −100 −50 0 50 100 x, Position in Grating f, Position in Diffraction Pattern 1000 1 Apodization or 800 0.8 illumination 600 0.6 400 0.4 Apodization Apodization 0.2 200 0 0 −2 −1 0 1 2 −100 −50 0 50 100 x, Position in Grating f, Position in Diffraction Pattern

500 1 Multiply above 400 Convolve above 0.8 two to obtain 300 two to obtain 0.6 grating output. 200 diﬀraction 0.4 100 pattern. Apodized Grating 0.2 0 0 −2 −1 0 1 2 −100 −50 0 50 100 x, Position in Grating f, Position in Diffraction Pattern Wider Apodization Provides Better ResolutionApodized Grating Diffraction Pattern

Sept. 2012 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides8–45 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 Apodization Gaussian d Gaussian 4 λ 0 π d0 * ﬁrst nulls

• Finesse: (Order Spacing)/(Width of a Single Order) 4 λ 1 4 d = π d0 = F λ π d d 0

• Rule of Thumb

– Roughly F = Number of Grooves Illuminated

Sept. 2012 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides8–46 Grating Spectrometeter Example (1)

• Laser Source with Two Wavelengths

λ1 = λ − δλ/2 λ2 = λ + δλ/2

• Beam Diameters Equal to (Or Expanded to) d0

• Spacing in First–Order Spectrum (N = 1)

λ2 − λ1 δλ δθ = θ 2 − θ 1 ≈ sin θ 2 − sin θ 1 = = d d d d d d

• Width 4 λ α = π d0

Sept. 2012 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides8–47 Grating Spectrometeter Example (2)

• Sum of Irradiances − − 2 2 − − 2 2 I (θ) = e 2(θ θd1) /α + e 2(θ θd2) /α

• Resolution Criterion

– Not Rayleigh (No Nulls)

– Try Sparrow: Check First Derivative dI (θ) θ + θ = 0 1 or 3 Solutions including θ = d2 d1 dθ 2 – Check Second Derivative (Positive for Valley) ( ) 2 [ ] −2 δθ α 2 2 d2I 8e α − (δθ) = dθ2 α2 θ=(θd1+θd2)/2

Sept. 2012 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides8–48 Sparrow Criterion for Grating

• Second Derivative ( ) 2 [ ] −2 δθ α 2 2 d2I 8e α − (δθ) = dθ2 α2 θ=(θd1+θd2)/2

Negative (Peak) for α < δθ Positive (Valley) for α > δθ

• Boundary

4 λ δλ0 = Subscript on δλ0 for Onset of Valley π d0 d

δλ 4 d 1 4 10−6m 1 0 = = e.g. ≈ −3 λ π d0 F π 10 m 1300 For λ = 830nm, δλ0 = 830nm/1300 ≈ 0.61nm

Sept. 2012 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides8–49 Blaze Angle

• Linear Phase Shift in Slit • Littrow Grating: θi = θd – Prism for Transmission λ = 2d sin θ (N = 1) – Tilt for Reflection i • Translation of Diffraction • Backward Reflection Pattern • e.g. Tunable Laser 150

Phase Ramp 1 100 Translation Here Here Align Peak with 50 Slit 0.5 First Order for 0 Chosen λ

0 Slit Diffraction Pattern −50 −2 0 2 −100 −50 0 50 100 x, Position in Grating f, Position in Diffraction Pattern

Q: Add a phase ramp to the Matlab code

Sept. 2012 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides8–50 Monochromator

• Entrance Slit Limits θi • Exit Slit Selects Limits θd • Choose One Order 500

400

300

200

100

0 −100 −50 0 50 100 f, Position in Diffraction Pattern Apodized Grating Diffraction Pattern • Symmetry Minimizes Aberrations • Blazed Grating Enhances Signal • Colored–Glass Filter Eliminates Aliasing

Sept. 2012 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides8–51 Monochromator Resolution

• Entrance Slit d + i = 2wi

• Input Angles: −wi/(2f) ≤ sin θi ≤ wi/(2f) • Transmission Through Entrance Slit

d T (δλ) = 1 δλ − δ sin θ < w /2, i N i i • Transmission Through Exit Slit

d T (δλ) = 1 δλ − δ sin θ < w /2 d N d d • Overall Transmission Spectrum

Ti (δλ) ⊗ I (δλ) ⊗ Td (δλ) • Issues of Source Spatial Coherence (Ch. 10 and 11) in in Apodization for I (δλ)

Sept. 2012 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides8–52 Fresnel Diﬀraction

• Focused Light in Geometric Optics 2 |z1 − f| P f DI = 2 2 f π (D/2) (z1 − f)

• 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

∫ − 2 ∫ − 2 jkz1 (x x1) (y y1) jke jk 2z jk 2z E (x1, y1, z1) = Ex (x, 0) e 1 dx Ey (y, 0) e 1 dy 2πz1

Sept. 2012 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides8–53 Uniformly Illuminated Rectangular Aperture

• Aperture Size 2a by 2b √ jkejkz1 P E (x1, y1, z1) = I (x1, a, z1) I (y1, b, z1) 2πz1 4ab ∫ 2 a (x−x1) jk 2z I (x1, a, z1) = e 1 dx1 −a ( ) ( ) ∫ 2 ∫ 2 a (x − x1) a (x − x1) I (x, a, z1) = cos k dx+j sin k dx −a 2z1 −a 2z1

1 • 0.8 Fresnel Cosine and Sine Integral 0.6 • Cosine Interand Contributes 0.4 0.2 Near x = x 0 1 −0.2 Curvature Term • Sine is Zero at x = x1 −0.4 −0.6 • Away from Edges, I(x, y, z1) = I(x, y, 0) −0.8 −1 0 0.5 1 1.5 2 2.5 3 3.5 4 Radius in Fresnel Zones

Sept. 2012 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides8–54 Fresnel Sine and Cosine Integrals (1)

• Dimensionless Coordinates √ √ k k u = (x − x1) du = dx1 2z1 2z1

• Limits √ √ k k u1 = (x + a) u2 = (x − a) 2z1 2z1

• Integrals √ √ ∫ ∫ u u 2z1 2 2 2z1 2 2 I (x1, a, z1) = cos u du + j sin u du k u1 k u1 √ 2z1 I (x1, a, z1) = [C (u2) − C (u1) + S (u2) − S (u1)] k

Sept. 2012 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides8–55 Fresnel Sine and Cosine Integrals (2)

√ 2z1 I (x1, a, z1) = [C (u2) − C (u1) + S (u2) − S (u1)] k ∫ ∫ u u 1 2 1 2 C (u1) = cos u du S (u1) = sin u du 0 0

F (u1) = C (u1) + jS (u1)

• Above 3 Are Odd Functions C (u) = −C (−u) S (u) = −S (−u) F (u) = −F (−u)

• Approximations 2 F (u) ≈ uejπu /2 u ≈ 0 ≈ 1+j − j jπu2/2 → ∞ F (u) 2 πue u (1) Sept. 2012 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides8–56 Fresnel Diﬀraction Results

jkejkz1 P 2z E (x , y , z ) = 1 × 1 1 1 2 4 { πz[ 1 ab k√ ] [ √ ]} k k F (x1 − a) − F (x1 + a) × 2z 2z { [ √ 1] [ √ 1]} k k F (y1 − b) − F (y1 + b) 2z1 2z1

• Large x1 or y1: E → 0: Q: How Close? Use Approximations on Previous Page.

• Near the Center: E (x1, y1, z1) = E (x1, y1, 0) (Using the Odd Property of F )

Sept. 2012 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides8–57 Cornu Spiral

0.8

0.6

0.4

0.2

−0.2

−0.4

−0.6

−0.8 −0.5 0 0.5

Sept. 2012 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides8–58 Example of Fresnel Diﬀraction: Near Field Diﬀraction

13 0.8 x 10 2.5

0.6

2 0.4 2 0.2 1.5

0 1 −0.2 I, Irradiance, W/m

−0.4 0.5

−0.6 0 −0.8 −20 −15 −10 −5 0 5 10 15 20 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 x, mm A. Cornu Spiral B. Irradiance

13 x 10 135

−15 2.2 130 2 −10 125 1.8 −5 1.6 120

1.4 115 0 1.2

y, mm 110 5 1 0.8 105 10 Relative Irradiance, dB 0.6 100 15 0.4 95 0.2 20 90 −20 −15 −10 −5 0 5 10 15 20 −20 −15 −10 −5 0 5 10 15 20 x, mm x, mm C. Image D. Irradiance, dB

Sept. 2012 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides8–59 Fresnel Diﬀraction Includes Fraunhofer Diﬀraction

0.56 150

0.54 100 0.52 50 0.5 0 0.48 Slit Diffraction Pattern −50 0.46 −100 −50 0 50 100 0.44 0.46 0.48 0.5 0.52 0.54 f, Position in Diffraction Pattern A. Cornu Spiral B. Field 13 x 10 2 3

I, Irradiance, W/m 0 −40 −20 0 20 40 x, mm C. Irradiance

Sept. 2012 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides8–60 Fresnel Lens

10 −1 8 −1 60 −0.8 0.8

6 −0.6 0.6

4 −0.5 −0.4 0.4 40 2 −0.2 0.2

0 0 0 0 y, cm y, cm height, mm −2 20 0.2 −0.2

−4 0.5 0.4 −0.4 −6 0.6 −0.6 −8 1 0 −1 0 1 0.8 −0.8 −10 −2 0 2 1 x, cm −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 OPD, µ m x, cm Lens Unwrapped Phase Phase for for Regular Lens Fresnel Lens

Sept. 2012 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides8–61 Fresnel Zone Plate

−1

−0.5 Fresnel zone plate with Diameter 0

y, cm 2cm at distance of 10m. The zone plate acts as a lens. 0.5

1 −1 −0.5 0 0.5 1 x, cm 1

Integrand of Cosine Integral pass- 0.5 ing through zone plate. All negative

Zone Plate contributions are blocked. 0 0 5 10 u

Sept. 2012 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides8–62 Phase in Fresnel Zone Plate

−1 −1 2 −0.5 −0.5

0 0 0 y, cm y, cm 0.5 −2 0.5 1 1 −1 0 1 −1 0 1 x, cm x, cm Raw Phase in Radians Zone Plate Transmission

−1 1 −1

−0.5 −0.5 0.5

0 0.5 0 0 y, cm y, cm 0.5 0.5 −0.5

1 0 1 −1 0 1 −1 0 1 x, cm x, cm Real Transmitted Field Imag. Transmitted Field Sept. 2012 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides8–63 Focusing With Fresnel Zone Plate

−50 40 −50 40 −50 40

−40 38 −40 38 −40 38

−30 36 −30 36 −30 36

−20 34 −20 34 −20 34

−10 32 −10 32 −10 32

0 30 0 30 0 30 , /cm , /cm , /cm y y y f f f 10 28 10 28 10 28

20 26 20 26 20 26

30 24 30 24 30 24

40 22 40 22 40 22

50 20 50 20 50 20 −50 0 50 −50 0 50 −50 0 50 f , /cm f , /cm f , /cm x x x Unfocused Focused Focused with Pattern With Lens Zone Plate

Sept. 2012 c C. DiMarzio (Based on Optics for Engineers, CRC Press) slides8–64