3/24/2017 1 PHYS 450 Spring Semester 2017 Lecture 11

3/24/2017

PHYS 450 Spring semester 2017

Lecture 11: Fresnel’s Theory for the Diffraction of Scalar Waves

Ron Reifenberger Birck Nanotechnology Center Purdue University

Lecture 11 1

Historical Context 1803 – Thomas Young reignites interest in wave theory of light. Increased interest in Huygens’ ideas from late 1600’s 1818 - French Academy launches a competition to explain the properties of light – Fresnel, a civil engineer, submits his wave theory Every unobstructed point on a wavefront becomes the source of a secondary spherical wavelet. The amplitude of the E-field at any point on a screen is given by the superposition of all secondary wavelets, taking into account their relative phases – Poisson, while evaluating the submissions to the French Academy competition, finds an apparent critical flaw in Fresnel’s theory. Concludes that Fresnel predicts the seemingly absurd result that a bright spot should appear directly behind a circular obstruction! – Arago experimentally verifies Fresnel’s theory and finds the Arago spot (aka Poisson spot) 1819 – enthusiasm for corpuscular theory of light begins to diminish 1882 – Kirchoff introduces more mathematical theory based on solutions to Maxwell’s wave equation 1905 – Einstein's explanation of photoeffect revives interest in the corpuscular theory of light 2

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The Generic Diffraction Problem

Viewing Screen ? Pattern in blocking screen blocks fraction of ? Fraunhofer incident EM wave ? Limit – far field b

Fresnel Limit – Illumination near field a Blocking Screen or Obstacle

After Systematic Experiments

intensity mirrors slit shape bright fringes immediately appear, just behind screen inside edge significant light broad pattern; penetration into complicated little resemblance shadow region profile to shape of original slit 4

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What we now understand

Viewing b>100A/ Screen Fraunhofer Limit

Fresnel Limit b Far Field b<100A/ b~100A/ (gray area) b

A=Aperture Area

Near Field

Pattern in blocking a monochromatic screen passes only a illumination,  fraction of the incident

EM wave 5

Definition Diffraction is a deviation from geometrical optics resulting from the obstruction of an optical wavefront by an obstacle or opening. We want the intensity I at an arbitrary point P located on an observation screen a distance b from the obstacle. Two Important Cases Various Openings Edges Viewing Viewing Screen Screen

P +

P + Blocking Slits or Screen Apertures The intensity at point P can be expressed in terms of two standard integrals: vvuu22  SduCv sinv cos du    6 v0022v 

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Fresnel’s Approach Simplest Case: What illumination will a diverging spherical wave produce at point P located a distance b ahead of wave? . L

b+3/2 s W 3 b+2/2 s2 b+/2 The E-field amplitude at P is the sum of all Monochromatic s1 the amplitudes of all light diverging b P the emitted wavelets from a point Vo s , s , … are arc lengths from the surface W. source 1 2

Fresnel’s approach: L’ divide wavefront into zones 7

Underlying Assumptions of Huygen-Fresnel Diffraction

2‐dimensional side view L What is E-field at observation point P at a W distance b from vertex Vo of wavefront? ds • Draw sphere of radius b Spherical V +2 etc. • Draw sphere of radius b + /2 Wavefront • Intersection of b + /2 sphere with wave- V+1 b   front defines 1st Fresnel zone s b   2 • Assume ALL Huygen wavelets in 1st zone P arrive at point P in phase Vo b • E-field amplitude due to ALL wavelets in a st S 1 zone is A1. b • Continue to divide incident wavefront into Viewing zones by drawing additional spheres with Screen radii b + , b + 3/2, etc.  • Assume wavelets from any given zone are approximately in phase when they arrive W L at point P and  radians out of phase from contribution from adjacent zones.

Arc length (and angles) measured from point Vo • Each zone has nearly the same area can be either positive (CCW) or negative (CW). • Denote the E-field from each zone as A2, A , etc. 3 8

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Using Zones in the Huygen-Fresnel Theory of Diffraction – A Few Simple Examples

I. Light through a circular aperture II. Zones & Barriers

EXAMPLE I: What is the E-field and Intensity at point P? Adding up the contributions from the various zones

o • A2 is 180 out of phase with A1, so it tends to cancel

• A2 is slightly smaller than A1 (area is essentially equal, but distance has increased) • The amplitude from the nth zone is to a good approximation the average value of that from the n-1 and n+1 zones

• Resultant E-field amplitude Anet(at P) given to good approximation by

A1 A3 AatPAAAAnet 1234..... A5 at point P A7 11 On average,,,etc A A A A  A A 232213 24 . . . . ~0 ~0 11 1 1 1 Anet=A1/2 AatPAnet 1123 AA  A AA 345  A ... A6 A4 22 2 2 2 A2 1 resultant  AatPAnet  1 2 dominated by A1 A2 I at P A2  at P 1 unblocked net 4

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Insert Blocking Screen AatPAnet    1 I at P A22 at P A L blocked net 1

2 . I at P A b+3/2  blocked 1 4 2 s Iunblocked at P A1 W 3 b+2/2  4 s2

s 1 s1 b+/2 at point P b P Vo

Fresnel number: N = s 2/(b) = 1 Blocking screen allows F 1 light from only 1st Note bright spot in L’ zone to pass. center of pattern 11

EXAMPLE II: How the Zones Interact with Barriers (no obstruction)

L A . W . .

a b +s +φ P Vo a S -s

No Barrier

. . . W A L

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The Effect of a Knife Edge

L A W

a s φ P Vo S a b

W A L

Intensity on Screen Above the Knife Edge

L A W

Integrate x I(y) over s a s Vo φ y

S a P

W A L

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Intensity on Screen Below the Knife Edge

L A W

Integrate a b over s φ s a P S -y Vo

x I(-y)

W A L

Contribution from an Arbitrary Point Q on the Wavefront

=extra distance traveled L A

W The wavelet from Vo travels ds d distance b to P, Q ∆ The wavelet from an arbitrary b a s point Q travels a distance φ P d=b+Δ(s) S ab Vo before arriving at the same point P.

Need to know (s).

W L A

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Details: Generalizing the formalism

1. Define amplitude (E-field) contribution from each zone 2. Isolate terms with (s) - extra path length  3. Calculate (s) 1 ab s s2 2 ab 4. Change variables: convert s to a dimensionless parameter 

2 ab vs  ab 5. After considerable analytical analysis, the amplitude (E-field) contribution can be define in terms of C(v) and S(v) – Fresnel Integrals vvuu22  SduCv sinv cos du vv0022  6. From E-field amplitude, calculate time averaged irradiance at point P; additional factor of ½ 7. Introduce the Cornu Spiral: E(v)  C(v) + iS(v) 8. Normalization to remove arbitrary constants 9. Relate v to y – problem specific

The Fresnel Integrals 2. Using the phasor form for E(v ) vvuu22  SduCduvvsin cos  v 2 N 1  u2 22ii u j vv00  Ev ee22 du u You can show that : v0 j0 222 E v C  v i S  v rectangular form uuoN u1 1 ii22 i 2 Then you can write  euu e  ....  e  u 2 2 vv22 ii u 4 u uu  uu e2  u e2  .... vis contour Evcos dui sin  du vv0022  3. Graphical representation: C(v ) length along v 2 i  u Cornu spiral   e 2 du phasor form v0 Imag 2

Intensity  Ev v u E(v) . . . S(v ) What’s it mean? u 4 u2 1. Break the integral limits from 0 to v into N u  2  intervals u  2 Real

u u u  2  u ...... 4. Physical meaning u u u u u v o 1 2 3 4 E vCviSvCviSv      v 22 uujujN j ,0,1,2...1   Cv Sv  N 2 Intensity E v C22 v S  v 18 uuuuuetco 0;12 ; 2 ; .

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The Cornu Spiral – an elegant graphical representation

v =+∞ Contributions from different +3 +1 zones S(v) +2 |E(v)| v v specifies the arc length along C(v) the Cornu spiral

-2 v =0

-1 -3 v =-∞

Standard Geometry

Cylindrical wave encountering some arbitrary obstacle

Obstacle Viewing Plane Screen L  A

y W x z slit Assume cylindrical wavefront: equivalent W to a line source L (turns a 2d problem A into 1d) a b

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Obtaining a coherent wave, diverging from a point, using a laser

objective viewing screen laser HeNe laser beam

Spatial Make sure you position filter pinhole, typically the central part of the 20 m diam. beam over the photodiode spatial filter assembly entrance slit (requires careful alignment)

Laser stability is an issue

Laser Head No. 1373707 =633 nm 30 +10% shift 75 mins after turn on 25 +5% shift 45 mins after turn on 20 15 mins after turn on 15 Intensity (% max) (% Intensity 0 10203040506070 Time (s)

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Example I: No Obstacle

222 Ixyz,,  A2  C C S S      A is a constant to be determiend 2 22 2 11 11  A   22 22  2 AA221122 4   Rquire the inensity for the unblocked wave

to equal Io , the intensity of incident wave 2  Io  4A I A2  o 4

Example II: The Knife Edge

L A W

Integrate I(y) over s x a s φ y Vo S a P

W A L

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Result II: Knife Edge

I 22 2 2 Ix0, z  0, y o  C  Cv   S  Sv   C  C     S  S    4  2222 Io 11 1111 Cv  Cv  42 2 2 2 2 2 22 Io 11 Cv  Sv 22 2  b yabv  2 a where Io is intensity of incident wave

Use the scanning photodiode to collect data Scan photodiode . Use Data Studio software to collect data across pattern . Apply +3 V or -3 V to motor to scan Photodiode, photodiode across diffraction pattern. b entrance slit 1; . Need to measure scanning velocity. width ~0.1 mm . Position single slit (mask 1) in front of photodiode, width ~0.1 mm. Razor . Photodiode gain is typically set to 100, or a blade perhaps 10 if laser is bright.

HeNe laser w. spatial filter. How to measure a? Photo: Diffraction from knife edge in Fresnel configuration. Alignment is critical. 26

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Scanning photodiode - details motion

scanning photodiode with entrance slit No. 1 Center laser beam over slit

linear displacement dc linear translator (mm)

Calibration: apply dc voltage to linear translator linear translator slope=velocity

displacement time 27

Using the Cornu Spiral to Predict Location of Max and Mins

v =+∞ Typical results (a=0.2 m, b=0.1 m, =650 nm) Label various max (k=0.1.2.3…) and mins (j=0,1,2,3….) Diffraction near straight edge k=0 Expected +3 +1 1.50 d 1 2 … +2 a 1.25 1.00 0.75 c e b 1 2 … c 0.50 j=0 e 0.25 a b v =0 0.00

-2 Irradiance Normalized -1500 -1000 -500 0 500 1000 1500 -1 -3 v =-∞ y (m) d

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Fitting Data Positive y (above edge) Above the edge 22 D 2 1 11 I yCvSv time yy v = -0.6 d 2 2 2 y

typically10 vy 0

Below the edge 22 D '12 1  1 I yCvSvyy time d 2 22 Negative y

typically010 vy (geometrical shadow)

Scaling parameter  b yabv   y  2 a  Cv Cv  vy = +0.6 Sv   Sv 

Origin of the max and mins in the diffraction pattern

1st minimum defines y=0

1st maximum

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Matching the max and mins in the diffraction pattern near an edge Typical results (a=0.2 m, b=0.1 m, =650 nm) Diffraction near straight edge 1.50 1.25

Converting dimensionless vy to y (in m) 1.00 0.75 22ab ab 0.50 -1.225 vsayy ab ab 0.25 -1.871 -2.345

1 Normalized Irradiance 0.00  tan yab yab -6.0 -4.0 -2.0 0.0 2.0 4.0 6.0 v (dimensionless)  b y yabv  y 2 a Diffraction near straight edge 3 1.50 vkkmax 40,1,2,3...  . . . .k y 2 1.25 1.00 min 7 0.75 . . . .j vjjy 40,1,2,3...  2 0.50 0.25 0.00 NormalizedIrradiance -1500 -1000 -500 0 500 1000 1500 y (m) 31

Talbert/Leeman data November 2016

1.50 Data 1.25 Max Loc. Min. Loc. 1.00 Fresnel Theory Io 0.75

0.50 Relative Intensity Relative 0.25

0.00 -500 0 500 1000 1500 2000 2500 Photodiode Displacement (microns)

Scaled photodiode voltage by 16 to match b=0.3 m heights of maxima and minima “a” adjusted to optimize fit =543.5 nm Shift data by 870 m to make origin match velocity=134.8 m/s 0.25Io 32

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Centering spatially filtered laser beam =633 nm 25 off-center 20 good on center 15 better 10 Misalignment of peaks in data means velocity 5 in forward and backward directions 0 are different. Intensity (% max) (% Intensity 0 102030405060 Time (s)

Example III: Long, narrow single slit, width w

L Top view, looking down A W

Integrate over s I(y) Vo x that passes +s through slit a y φ -s w S a P

W A L 34

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Result III: Single Slit, width w

I 22 2 2 Ix0, z 0, y o  Cu  Cv   Su  Sv   C  C    S S   4  22 I 2211 11  o Cu Cv Su Sv        42222 

Io 22 2Cu Cv   Su Sv  4  I 22 o Cu Cv   Su Sv  2  b 2  2111 w y v a   b 2 1 b w a b 2  2111 w y u  a   bw2 1 b a

where Io is intensity of incident wave

Excel Utility to Fit Data

Enter four numbers

Copy/Paste these two columns

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Use single slits on Cornell Slit Film

Use these two slits to investigate Fresnel diffraction. Must center laser beam over slit.

to calculate width of slits, see Appendix S in PHYS 450 Lab Manual 37

Results - diffraction through a wide and narrow slit vs. y

Slit width=1406 μm; a=b=0.4 m; λ=632.8 nm Slit width=703 μm; a=b=0.4 m; λ=632.8 nm 16 12 Data Data 12 Theory 9 Theory

8 6

Intensity (a.u.) Intensity 3

Intensity (a.u.) 0 -2500.0 -1250.0 0.0 1250.0 2500.0 0 -2500.0 -1250.0 0.0 1250.0 2500.0 Displacement along Viewing Screen (μm) Displacement along Viewing Screen (μm)

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Graphical Interpretation (Situation drawn for y=0)

b 2  -y 2111  w y u a    bw2 1 b a b 2  2111 w y v a   bw2 1 b a +y b 2   211111 w y y vua      bww2211bb  aa b 2 2 1 w  a constant  b

Example III: Fraunhofer diffraction through single slit

Fresnel Number

2⁄ 4

If NF <<1, Fraunhofer analysis is warranted.

If NF 1, Fresnel theory must be used.

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(a) Diffraction throughEvolution long slit of Single Slit Diffraction 1.50 Fraunhofer-like* NF=0.17 1.00

0.50 Calculated diffraction through a w=200 μm

Relative Irradiance Relative 0.00 wide slit positioned a=30 cm from a source -400 -300 -200 -100 0 100 200 300 400 with =580 nm. y- position along screen (in m) * Line shape derived in Appendix of Lecture 9 (b) Diffraction through long slit In (a), the pattern when b=10 cm (screen to 2.00 between slit distance). NF=0.57 1.50 Fresnel and 1.00 Fraunhofer In (b), the pattern when b=3 cm (screen to

0.50 slit distance).

Relative Irradiance Relative 0.00 -400 -300 -200 -100 0 100 200 300 400 In (c), the pattern when b=1 cm. y- position along screen (in m)

(c) Diffraction through long slit The square box indicates the location of the 2.00 geometrical shadow of the slit. N =1.72 1.50 Fresnel-like F

1.00

0.50

Relative Irradiance Relative 0.00 -400 -300 -200 -100 0 100 200 300 400 41 y- position along screen (in m)

Observing Fraunhofer Diffraction: Parallel light in, parallel but converging light out

Problem: Impractical to walk down a long hallway to view a far-field diffraction pattern. Tough to make adjustments.

Solution:

. Use a first lens (f1) to produce a collimated plane wavefront when located at focal distance from point source . Insert grating to produce diffracted beam

. A second lens (f2) brings plane waves to a focus one focal length behind the lens. . Same pattern will be observed that would appear in far field of grating except the pattern will be scaled by the ratio of the focal

length (f2) to the distance of the lens from the grating.

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Fraunhofer Diffraction: Optical Setup

Collimating Focusing lens, f lens, f 1 2 . Spatial Filter Viewing . Screen (point source) .

Laser

. .

f1 f2 . Slit Pattern

Set-up for Fraunhofer Diffraction

Cylindrical Viewing screen on Slit Pattern Lens, fc photodiode

y y  fc yy'   W

fc W

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The Effect of Lens Relating y’ to y . Imagine a beam of rays (dotted lines) incident on lens, all parallel to the ray of interest . Lens will focus all rays in parallel beam to same location in back focal plane of lens . Use chief ray (dashed line, the ray through center of lens) to locate focal spot; the chief ray is not deviated by the lens . All other rays parallel to chief ray must be focused to same point

Top view, looking down yy' Slit tan   pattern Wf Lens c

fc yy'   W

y  y ray of interest

Screen W fc 45

Set-up for Fraunhofer Diffraction

Fraunhofer diffraction implies plane Cylindrical lens, Copy lens, produces waves i.e. NO wavefront curvature: focuses pattern parallel incident beam; parallel beam in, parallel beam out onto photodiode slit set to smallest aperture

Photodiode, entrance slit 1; width ~0.1 mm

PASCO OS-9165 slit set: • Electroformed Ni foil ~29 cm • Dimensions given in mm • Tolerances: 0.005 mm • Slit Space dimension is Photo: Diffraction from Pasco Slits HeNe laser w. center-to-center of each slit slits in Fraunhofer (vertical slits) spatial filter 46 configuration

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Use PASCO Slits for Fraunhofer Diffraction

Table 1: Dimensions of Pasco Electroformed slits Slit Set Pattern # slits Width (mm) Spacing (mm) Spacing A B 1 0.04 A D 1 0.16

B A 2 0.04 0.250 B C 2 0.08 0.250 B D 2 0.08 0.500

C B 3 0.04 0.125 C D 5 0.04 0.125 qualitatively explore systematics between

appropriate slits 47

A more quantitative comparison between theory and experiment can be made using the results on the next slide

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Single Slit Two Slits Three Slits

Fraunhofer Diffraction - One Slit Fraunhofer Diffraction - Two Slits Fraunhofer Diffraction - Three Slits 1.00 1.00 1.00

Fraunhofer Fraunhofer Fraunhofer 0.75 Theory 0.75 Theory 0.75 Theory

0.50 0.50 0.50

Intensity (a.u.) Intensity 0.25 (a.u.) Intensity 0.25 Intensity (a.u.) Intensity 0.25

0.00 0.00 0.00 -5.0 -3.0 -1.0 1.0 3.0 5.0 -8.0 -6.0 -4.0 -2.0 0.0 2.0 4.0 6.0 8.0 -8.0 -6.0 -4.0 -2.0 0.0 2.0 4.0 6.0 8.0 Displacement along Screen (mm) Displacement along Screen (mm) Displacement along Screen (mm)

bsin bdsin 3 sin 2  22bdsin 2 sin 22 sin  sin sin sin sin I(y)  I(y)  I(y)      I 2 2 2  d sin o bsin Io bsin 2  d sin Io bsin 2   sin  sin      

Fraunhofer Diffraction - Five Slits Five Slits 1.00

Fraunhofer 22bdsin 5 sin 0.75 Theory sin sin Results plotted using: I(y)  d=300 m  0.50 2 b=50 m Io bsin 2  d sin sin  =632.8 nm

 (a.u.) Intensity   0.25

0.00 -8.0 -6.0 -4.0 -2.0 0.0 2.0 4.0 6.0 8.0 Displacement along Screen (mm) 49

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