Download presentation

Presentation is loading. Please wait.

Published byIsaiah Hickey Modified over 4 years ago

1
**Today • Diffraction from periodic transparencies: gratings**

• Grating dispersion • Wave optics description of a lens: quadratic phase delay • Lens as Fourier transform engine

2
**Diffraction from periodic array of holes**

Spatial frequency: 1/Λ A spherical wave is generated at each hole; we need to figure out how the periodically-spaced spherical waves interfere incident plane wave

3
**Diffraction from periodic array of holes**

Spatial frequency: 1/Λ Interference is constructive in the direction pointed by the parallel rays if the optical path difference between successive rays equals an integral multiple of λ (equivalently, the phase delay equals an integral multiple of 2π) incident plane wave Optical path differences

4
**Diffraction from periodic array of holes**

Spatial frequency: 1/Λ From the geometry we find Therefore, interference is constructive iff

5
**Diffraction from periodic array of holes**

incident plane wave Grating spatial frequency: 1/Λ Angular separation between diffracted orders: Δθ ≈λ/Λ 2nd diffracted order 1st diffracted “straight-through” order (aka DC term) –1st diffracted several diffracted plane waves “diffraction orders”

6
**Fraunhofer diffraction from periodic array of holes**

7
**Sinusoidal amplitude grating**

8
**Sinusoidal amplitude grating**

incident plane wave Only the 0th and ±1st orders are visible

9
**Sinusoidal amplitude grating**

three plane waves far field one plane wave three converging spherical waves +1st order 0th order –1st order diffraction efficiencies

10
Dispersion

11
**Dispersion from a grating**

12
**Dispersion from a grating**

13
**Prism dispersion vs grating dispersion**

Blue light is refracted at larger angle than red: normal dispersion Blue light is diffracted at smaller angle than red: anomalous dispersion

14
**The ideal thin lens as a Fourier transform engine**

15
**Q: how can we “undo” the convolution optically?**

Fresnel diffraction Reminder coherent plane-wave illumination The diffracted field is the convolution of the transparency with a spherical wave Q: how can we “undo” the convolution optically?

16
**Fraunhofer diffraction**

Reminder The “far-field” (i.e. the diffraction pattern at a large longitudinal distance l equals the Fourier transform of the original transparency calculated at spatial frequencies Q: is there another optical element who can perform a Fourier transformation without having to go too far (to ∞ ) ?

17
**The thin lens (geometrical optics)**

f (focal length) object at ∞ (plane wave) Ray bending is proportional to the distance from the axis point object at finite distance (spherical wave)

18
**The thin lens (wave optics)**

incoming wavefront a(x,y) outgoing wavefront a(x,y) t(x,y)eiφ(x,y) (thin transparency approximation)

19
**The thin lens transmission function**

20
**The thin lens transmission function**

this constant-phase term can be omitted where is the focal length

21
**Example: plane wave through lens**

plane wave: exp{i2πu0x} angle θ0, sp. freq. u0≈ θ0 /λ

22
**Example: plane wave through lens**

back focal plane spherical wave, converging off–axis wavefront after lens : ignore

23
**Example: spherical wave through lens**

front focal plane spherical wave, diverging off–axis spherical wave (has propagated distance ) : lens transmission function :

24
**Example: spherical wave through lens**

front focal plane spherical wave, diverging off–axis plane wave at angle wavefront after lens ignore

25
**Diffraction at the back focal plane**

diffraction pattern gf(x”,y”) thin transparency g(x,y) thin lens

26
**Diffraction at the back focal plane**

1D calculation Field before lens Field after lens Field at back f.p.

27
**Diffraction at the back focal plane**

1D calculation 2D version

28
**Diffraction at the back focal plane**

spherical wave-front Fourier transform of g(x,y)

29
**Fraunhofer diffraction vis-á-vis a lens**

30
**Spherical – plane wave duality**

plane wave oriented point source at (x,y) amplitude gin(x,y) towards ... of plane waves corresponding to point sources in the object each output coordinate (x’,y’) receives ... ... a superposition ...

31
**Spherical – plane wave duality**

produces a spherical wave converging a plane wave departing from the transparency at angle (θx, θy) has amplitude equal to the Fourier coefficient at frequency (θx/λ, θy /λ) of gin(x,y) produces a spherical wave converging towards each output coordinate (x’,y’) receives amplitude equal to that of the corresponding Fourier component

32
Conclusions When a thin transparency is illuminated coherently by a monochromatic plane wave and the light passes through a lens, the field at the focal plane is the Fourier transform of the transparency times a spherical wavefront The lens produces at its focal plane the Fraunhofer diffraction pattern of the transparency When the transparency is placed exactly one focal distance behind the lens (i.e., z=f ), the Fourier transform relationship is exact.

Similar presentations

Presentation is loading. Please wait....

OK

Essential Cell Biology

Essential Cell Biology

© 2018 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on depth first search algorithm Ppt on travel agency management Ppt on eco friendly agriculture factors Download ppt on sectors of economy Ppt on respiration in plants and animals for class 7 Ppt on amplitude modulation and demodulation Ppt on relations and functions for class 11th result Ppt on information security policy Ppt on cross-sectional study examples Ppt on big data and hadoop