Holography.pdf

8/18/2019 Holography.pdf

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Wave front sensing or 3D imaging

• Both amplitude and phase of the wavefront is

detected/recorded

• Phase contain information about the state of thewavefront.

• So can be used to image various characteristics of

the object with which the wavefront interacts.• Has immense applications ranging from microscopy

to stress analysis and photoelastic studies.


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Whole field (3D) imaging

digital techniques

• Semiconductor arrays as detectors (CCD/CMOS

arrays)• Optical recording and numerical reconstruction.

• Numerical Reconstruction (by simulation ofdiffraction using diffraction integrals) of bothamplitude and phase

• Since reconstructions are done numerically phasemanipulation is possible

• Methods include digital holography, phase retrieval,Optical coherence tomography etc.


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Phase• Complex amplitude distribution at any plane for

monochromatic waves can be written as

U(P)= A(P) exp[i (P)] (A- amplitude, -phase)

• Phase contains spatial and temporal information onchanges in surface shape (height), optical path

length etc.


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• Detectors are quadratic – measures the absolute

square of the light complex amplitude.

• Need fast detectors even to measure the phase

distribution (~10-15s response time).

• Fortunately one can convert this phase information

into an intensity pattern.

• This is done by combining (interfering) the object

beam with a known reference.


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Holography

• Inventor – Dennis Gabor in 1948

• Holo → entire graphien → write• Means for recording amplitude and phase of a wave

field.

• Hologram is the recorded micro-interference patternbetween an object beam and a coherent known

background.

• Object is reconstructed by illuminating the hologram

with reference beam

• Main difference with photography is that phase

information is recorded.


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• Hologram is an interference pattern between the

wavefront of interest and a known background.

• Usually recorded on a flat surface but containsinformation about the entire 3D wavefield.

• It can then be envisioned as a randomly oriented

phase or amplitude grating.

• Hologram diffracts any light incident upon it.

• So reconstructed by illuminating it by the reference

beam.

• It diffracts the reference beam in the direction of the

object, 3D object can be observed.

Hologram


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Interference pattern due to two plane waves

incident at an angle


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Light incident on such a grating gets

diffracted in the direction of the two beams


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Interference pattern due to two plane and an

off axis point source


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Light incident on such a grating gets diffracted in the

direction of the two beams one a point source another

undiffracted plane wave.


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Generation of interference pattern due to two point sources


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Light incident on such a set of grating gets diffracted in the

direction of the two beams two point sources and

undiffracted plane wave.


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Viewing virtual image using a lens


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Hologram Formation


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Reference wave

y xi y xr y x R R ,exp,,

Object wave

y xi y xo y xO O ,exp,, H ol o gr a ph

i cf r i n g e s

Recording

Medium

Object and reference wave interfere at the hologram plane.

This pattern is recorded as a spatially varying intensitypattern in the recording medium h(x, y)

Hologram Formation


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Intensity at the hologram plane

2, , ,

, , , , *

, * , , * , , * , , * ,

I x y O x y R x y

O x y R x y O x y R x y

R x y R x y O x y O x y O x y R x y R x y O x y

• Holograms are reconstructed by illuminating them withthe reference beam.

• This is equivalent to multiplying the recorded

holographic interference pattern (intensity pattern) withthe complex amplitude of the reference wave.

(1)


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Hologram Reconstruction


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2 20

2 2

, , ,

, , * ,

R x y h x y h r o R x y

r O x y R x y O x y

Reference wave

complex amplitude Reconstructed object

wave (virtual image)Distorted real image


Hologram

(intensity pattern)

Undiffracted reference beam

(zero diffraction order)

• Amplitude transmittance of the developed photographic

plate is

(3)

(2)


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Reference wave

y xi y xr y x R R ,exp,,


Object wave

(virtual image)

Object wave

(real image)

Undiffracted

reference beam


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Recording setups

In-line hologram

recording setup

(Gabor)

• Object as well as reference beams lie in the same line.

• During reconstruction, undiffracted reference, virtual as

well as real images are superposed.

• This geometry is not prone to external noise like

mechanical vibrations.


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Recording setups

off-axis hologramrecording setup

• Object as well as reference beams interfere at an angle.

• During reconstruction, undiffracted reference, virtual as

well as real images are spatially separated.

• This setup is prone to mechanical vibrations.


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Holographic interferometry

• One of the advantages of holography is that it cancompare wavefronts existing at two time instances.

• This is holographic interferometry.

• Finds application in varying fields.

• Coherent wavefield from the medium of interest attwo time instances interfere.

• In conventional holography this can be realized intwo ways 1) double exposure and 2) Real time.

• The interference phase between these two states willmanifest as intensity fringes and can be analyzed to

gain information about the object.


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Complex amplitude of the object in the initial state

Deformation of the object produces optical path lengthchanges equivalent to phase change from to +

The intensity of holographic interference pattern is

given by

1 , , exp ,O x y o x y i x y

2 , , exp , ,O x y o x y i x y x y

2 21 2 1 2 1 2, * 2 1 cos I x y O O O O O O o

(4)

(5)

(6)


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Hologram recording mediums

• Amplitude hologram: Holographic fringes are recorded as amplitudechanges of the recording medium

• Phase holograms: Holographic fringes are recorded as phase changes(refractive index) of the recording medium

• Any photosenstive medium can record holograms.

Photographic plates Amplitude/Phase hologram

Dichromated Gelatin Phase

Photoresist PhasePhotothermoplastics Phase

Photopolymers Phase

Photochromics Amplitude

Photorefractives PhaseSemiconductors Amplitude


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Digital Holography

• Holography is a two step process 1) formation byinterference and 2) reconstruction by diffraction.

• Hologram recording using semiconductors photo-detector arrays and their numerical reconstructionusing diffraction theory is Digital Holography.

• The recording process eliminates the need of wet

process as in conventional holography.• Numerical reconstruction provides both phase andamplitude of the object wavefront directly.

• Conventional interferometry including holography

require phase shifting techniques to obtain thephase information.

Theoretical aspects of holographic


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Theoretical aspects of holographic

Reconstruction

• In conventional holography reconstruction isachieved by illuminating the hologram by thereference wave. This is Equivalent to multiplying the

amplitude transmission function by the Referencebeam.

• In digital holography the reconstruction is achievedby simulating the diffraction process using scalar diffraction theory.

• Digital holograms can be considered as an aperturekept perpendicular to the reconstructing reference

beam.• Either Fresnel-Kirchoff diffraction integral or Angular Spectrum approximation to scalar diffraction theorycould be utilized to simulate the diffraction process.

• Both methods have their own advantages anddisadvantages.

Theoretical look on Reconstruction using Fresnel-Kirchoff


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Theoretical look on Reconstruction using Fresnel-Kirchoff

Integral

dxdyr

r i

y x R y xhi

cos

21

21

2exp

),(),(,

Complex amplitude of the diffracted wave at the image planeis given by

Reference wavecomplex amplitude

Hologram(transmittance)

(7)


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Numerical Reconstruction using Fresnel-Kirchoff Integral

• If the image plane is far from the CCD (d>>x or y),

Fresnel approximation gives

• The Fresnel-Kirchoff integral can be written as

d

y

d

xd r

22

22

dxdy y xd

i y x

d

i y xh y x R

d id i

d i

2expexp,,

exp2exp,

22

22

22exp,, y x

d

i y xh y x R

(8)

(9)


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Intensity and phase

• Numerical reconstruction using diffraction integral

provides the complex amplitude

• The intensity at the reconstruction plane is

• Phase at the reconstruction plane is

,

2,, I

,Re

,Imarctan,

(9)

• Numerical reconstruction using diffraction integral

provides the complex amplitude

• The intensity at the reconstruction plane is

• Phase at the reconstruction plane is

(10)

(11)

(12)

H l h f diff bj t


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Holography of diffuse objects

Hologram recording setup

N i l f i (di it l h l )


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Undiffracted

reference

Reconstructed

Object

Hologram of a coin placed1.18m from a CCD camera

Numerical reconstruction using

Fresnel approximation (intensity

and phase)

Phase

corresponding to

reconstructed

object

Numerical focusing (digital hologram)

By varying d in Eq. (9)


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Holographic interferometry (HI)


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Holographic interferometry (HI)

• HI compares wavefronts existing at two time instances.

• In conventional holography there are two methods for HI 1)Double exposure technique 2) Real-time technique

• DOUBLE EXPOSURE HI – Holograms recorded on the same

photo-sensitive medium for initial and final state of the object.The developed hologram is illuminated with reference beam.One can see a fringe pattern corresponding to amount of deformation. Interference between two virtual images of theobject at two time instances. The change of wavefront between

only two time instances can be studied.• REAL-TIME HI – Hologram in the initial state of the object is

recorded. This is developed and fixed and kept back exactly atthe position where the initial hologram was recorded. Theoriginal object is still there. The interference is between thewave scattered from the original object and the wavefrontscattered towards the virtual object. Fringes will form in real-time over the object. Advantage over double exposuretechnique is that wavefronts at several time instances can be

studied.



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Reconstructed

intensity before tilt

Reconstructedintensity after tilt

Holographic

interference

pattern

Digital holographic interferometry


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Digital holographic interferometry

x (cm)

y (cm)

-1 0 1

-1

1

x (cm)

y (cm)

-1 0 1

-1

1

x

y

x (cm)

y (cm)

-1 0 1

-1

1

Phase before tilt 1

2 -1

Unwrapped

Phase after tilt 2

Measured tilt


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Measured tilt

Along x-direction Along y-direction

Deformation measurement


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Deformation measurement

Deformation of a cantilever


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Deformation of a cantilever

Phase change with loading


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Phase change with loading

y (cm)

z0 5

1

2

3

0

x (cm)

Cantilever deformation


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Cantilever deformation

UnwrappedChange in deformation

along x-direction

Complex deformation


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Complex deformation

Phase map for the complex deformation


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Phase map for the complex deformation

y (cm)

x (cm)

5

2

z0

0

Removal of tilt component


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Removal of tilt component

y

x

z

y

x

z

x

y

x

z

c (deformation + tilt)

t (tilt)

deform = c- t

Deformation

Types of holograms based on diffraction


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yp g

• Fresnel – If recording plane lies with in the region of Fresnel diffraction

• Fraunhoffer – If the transformation from object tohologram plane is best described by Fraunhoffer diffraction equation

• Image – Hologram is image plane of the object. So alens is used.

• Fourier – Recording plane resides in the Fourier Transform plane of the object. So definitely a lens isrequired.



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yp g

Fourier Hologram



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yp g

Lens Less Fourier Transform Hologram

Sampling criteria


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p g

• Path length at difference at any point = x sin ()

• Change in path length between points x1 and x2 is

• (x2-x1) also corresponds to one bright and one dark fringe,i.e. 2 fringes with width d.

• So (x2-x1)=2d• Also x2 sin () – x1 sin ()=

• This leads to (x2 – x1) sin ()= and d= / [2*sin ()]

• Fringe density is 1/d = [2*sin ()] /

Path difference= x2 sin() = (at

this point) and x1 sin() =0d/2 d/2d

Sampling criteria


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p g

• This is also the maximum spatial frequency to beimaged/resolved.

• Here max=2 or max /2. =max /2 is the angle at whichmaximum resolvable frequency results.

Sampling criteria


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p g

• Nyquist sampling criteria states that each fringe shouldoccupy at least 2 pixels in the sensor.

• If the sensor pixel size is x, then the spatial frequency of the sensor is 1/x

• The sampling frequency (1/x) should be at least greater than or equal to twice the signal frequency (1/d) for thefringes to be resolved.

• This leads to 1/(2 x)=1/d

• Means

• This leads to

• When the angles involved are small

• And the maximum possible angle between the object andreference beam is max= /(2 x)

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Holography.pdf