November, 2000

**David Andrews**

University College

**Third Year Science** **Third Year Science**

**Imaging** **Imaging**

**Imaging**

**Imaging**

**Holography**

**Holography**

**Introduction** **Introduction**

**Introduction**

**Introduction**

**Photograph ** **Photograph **

**ÐImage intensity distribution of light .**

++ **Does not record direction.****Does not record direction.**

++ **Two****Two****-****-****dimensional image.****dimensional image.**

**Hologram ** **Hologram **

**ÐRecords intensity & direction of light.**

++ **Information in interference pattern.****Information in interference pattern.**

++ **Reconstruct image by passing original light ****Reconstruct image by passing original light **
**through hologram.**

**through hologram.**

++ **Need laser so that light interferes.****Need laser so that light interferes.**

**Diffraction Grating** **Diffraction Grating**

**Diffraction Grating**

**Diffraction Grating**

**Series of very closely spaced slits** **Series of very closely spaced slits**

*f*

*d*

*dsinθ*

θ

**Diffraction Grating**

**Diffraction Grating**

**Diffraction Grating**

**Diffraction Grating**

**Diffraction Grating** **Diffraction Grating**

**Diffraction Grating**

**Diffraction Grating**

**N** **N** **narrow slits** **narrow slits**

**N**

**N**

**ÐWidth = a**

**ÐSeparation = d**

**Intensity diffraction pattern is** **Intensity diffraction pattern is**

**ÐA**^{2}**is diffraction pattern from each slit **

### ( )

### ( )

2 2

2

sin sin

sin sin

∝ θ

θ
*I* *A* *Nkd*

*kd*

### ( )

### ( )

2 1 2 2

1 2 2

sin sin θ

∝ *ka*

*A*

*ka*

**Diffraction Grating** **Diffraction Grating**

**Diffraction Grating**

**Diffraction Grating**

**Intensity pattern** **Intensity pattern**

**Ðset of very sharp peaks**

++ **at angles given by****at angles given by**

**ÐAmplitude of peaks**

++ **modulated by single slit pattern****modulated by single slit pattern**

### sin θ = 0, , , , ± λ ± 2λ ± 3λ L

**Nd** **Nd** **Nd**

**Nd**

**Nd**

**Nd**

### ( )

### ( )

2 1 2 1 2

2

### sin *ka* sin θ

*ka*

**Diffraction Grating** **Diffraction Grating**

**Diffraction Grating**

**Diffraction Grating**

*Intensity, I*

Single slit envelope

0 sinθ

λ * Nd*
λ

**Nd*** Nd* −

− 2λ

2λ **Nd**

Diffraction peaks

**Diffraction grating** **Diffraction grating**

**Diffraction grating**

**Diffraction grating**

**Can be used to display spectrum** **Can be used to display spectrum**

**ÐUsually use first order peak**

**ÐSince angle depends on wavelength**

++ **spectrum spread out.****spectrum spread out.**

sin θ = ± λ

**Nd**

**Simple Hologram** **Simple Hologram**

**Simple Hologram**

**Simple Hologram**

**Two beams cross at an angle ** **Two beams cross at an angle ** θ θ

**Photographic plate**
**Photographic plate**

θθ

*zz*
* xx*
laser

laser **beam****beam****a****a**

laser

laser **be****am**** b****be****am**** b**

**Simple Hologram** **Simple Hologram**

**Simple Hologram**

**Simple Hologram**

**Extra path is** **Extra path is**

**Displacements of two beams are** **Displacements of two beams are**

*z**z*
θθ
θθ

ll

= sin θ
*l z*

### ( )

### ( )

### ( )

cos

cos sin

*a* *o*

*b* *o*

*E* *E* *kx* *t*

*E* *E* *k x* *z* *t*

= − ω

= + θ − ω

**Thus displacement at film is:** **Thus displacement at film is:**

**ÐUsing the trig identity**

**ÐAmplitude varies as**
**ÐIntensity varies as**

### ( ) ( [ ] )

### { }

### [ ]

### ( ) ^{(} ^{)}

1 2

1 1

2 2

cos cos

2 cos cos

= + = − ω + + − ω

= + − ω

l

l l

*o*

*o*

*E* *E* *E* *E* *kx* *t* *k x* *t*

*E* *k x* *t* *k*

**Simple Hologram** **Simple Hologram**

**Simple Hologram**

**Simple Hologram**

### ( ) ( )

1 1

2 2

cos *A* + cos *B* = 2 cos *A* + *B* cos *A B*−

### (

^{1}

_{2}

### )

cos *k*l

### ( ) ( )

2 1 2 1

2 2

cos cos sin

∝ l = θ

*I* *k* *kz*

**Simple Hologram** **Simple Hologram**

**Simple Hologram**

**Simple Hologram**

**Plate is developed, and horizontal lines ** **Plate is developed, and horizontal lines ** **appear.**

**appear.**

**Beam a is shone through plate** **Beam a is shone through plate**

**Ð amplitude varies as** ^{cos}^{2}

### (

^{1}

_{2}

^{kz}^{sin}

^{θ}

### )

Intensity

*z*

**Sinusoidal Diffraction Grating**

**Sinusoidal Diffraction Grating**

**Sinusoidal Diffraction Grating**

**Sinusoidal Diffraction Grating**

**Amplitude of beam is:** **Amplitude of beam is:**

**ÐSince**
**ÐAnd**

**Ðsince**

**Simple Hologram** **Simple Hologram**

**Simple Hologram**

**Simple Hologram**

### ( ) ( )

### ( )

### { } ^{(} ^{)}

2 1 2 1

2

cos sin cos

1 cos sin cos

= θ − ω

= + θ − ω

*o*

*o*

*E* *E* *kz* *kx* *t*

*E* *kz* *kx* *t*

( )

2 1

cos *C* = 2 1 cos 2+ *C*

### ( ) ( ) ( )

### ( ) ( ( ) )

### ( )

### ( )

1 1

2 0 2

1 1

0 0

2 4

1 4 0

cos cos sin cos

cos cos sin

cos sin

= − ω + θ − ω

= − ω + + θ − ω

+ − θ − ω

*E* *E* *kx* *t* *E**o* *kz* *kx* *t*

*E* *kx* *t* *E* *k x* *z* *t*

*E* *k x* *z* *t*

( ) ( )

1 1

2 2

cos *A*cos *B* = cos *A*+ *B* + cos *A B*−

**Simple Hologram** **Simple Hologram**

**Simple Hologram**

**Simple Hologram**

**The three parts are:** **The three parts are:**

**ÐBeam continuing in direction of **beam a
**ÐBeam in direction +θ.**

**ÐBeam in direction -θ.**

**Three beams emerge, one in direction ** **Three beams emerge, one in direction ** **0, one at +**

**0, one at +** θ θ **and one at ** **and one at ** **-** **-** θ θ **.** **.**

### ( )

### ( )

1

4 *E*0 cos *k x* − *z* sin θ − ω*t*

### ( )

1

2 *E*0 cos *kx* − ω*t*

### ( )

### ( )

1

4 *E*0 cos *k x* + *z* sin θ − ω*t*

**Simple Hologram** **Simple Hologram**

**Simple Hologram**

**Simple Hologram**

**You will get:** **You will get:**

**Ð+θ recreation of beam b (virtual image)**
**Ð-θ beam is real image**

θθ

**Developed **
**Developed **

**Photographic plate**

**Photographic plate**

### θ θ **laser beam a**

**laser beam a**

**+****+**θθ

**-θ****-**θ

**Recording a Hologram** **Recording a Hologram**

**Recording a Hologram**

**Recording a Hologram**

**laser**
**laser**

**beam****beam**
**splitter**
**splitter**

**Mirror**

**reference**
**beam**

**Mirror**

**object**

**beam** **Mirror**

**object**

**holographic plate**

**Reconstructing a hologram** **Reconstructing a hologram**

**Reconstructing a hologram**

**Reconstructing a hologram**

**Observer**
**Image**

**In line Hologram** **In line Hologram**

**In line Hologram**

**In line Hologram**

**laser**

**holographic film**

**object**

**Hologram is interference pattern ** **Hologram is interference pattern ** **between direct beam and light **

**between direct beam and light ** **scattered form object**

**scattered form object**

**In line Hologram** **In line Hologram**

**In line Hologram**

**In line Hologram**

**Hologram produced by developing film** **Hologram produced by developing film**

**ÐImage seen when original beam shone on **
**hologram**

**laser**

**hologram**

**image**

**Holography** **Holography**

**Holography**

**Holography**

**Many different optical arrangements.** **Many different optical arrangements.**

**Recording requirements:** **Recording requirements:**

**ÐLaser light source(coherent light)**

**ÐHolographic film needs small grains.**

**ÐGood stability (no movements during **
**exposure)**

**Reconstruction requirements:** **Reconstruction requirements:**

**ÐMuch less strict.**

**ÐSome do not need laser.**

**‘** **‘** **Conventional’ light** **Conventional’ light**

**‘**

**‘**

**Conventional’ light**

**Conventional’ light**

Short, random wave Short, random wave

trains (Incoherent) trains (Incoherent)

Intensity

Wavelength

Large spread in Large spread in

wavelengths wavelengths

Light emerges in Light emerges in manymany directionsdirections

**Laser Light** **Laser Light**

**Laser Light**

**Laser Light**

Long wave trains Long wave trains

(Coherent) (Coherent)

Small spread in Small spread in

wavelengths wavelengths

Narrow, intense Narrow, intense

light beam light beam

LASER LASER

Often plane Often plane

polarized polarized

Intensity

Wavelength

**Coherence** **Coherence**

**Coherence**

**Coherence**

**Interference needs fixed relationship ** **Interference needs fixed relationship ** **between phases.**

**between phases.**

**ÐConventional light sources produce short **
**wavetrains at random intervals.**

++ **Random variation of phase****Random variation of phase**

++ **Little interference****Little interference**

**ÐLasers produce long wavetrains**

++ **Phase relation between light beams fairly ****Phase relation between light beams fairly **
**constant.**

**constant.**

++ **Good interference****Good interference**

November 2000 David Andrews

**Applications of Holography** **Applications of Holography**

**Applications of Holography**

**Applications of Holography**

**Artistic creations.** **Artistic creations.**

**Storing & transporting delicate images** **Storing & transporting delicate images**

**ÐRussian icons are shown as holograms.**

**Holographic Interferometry.** **Holographic Interferometry.**

**ÐStrain analysis of objects under stress**
**ÐUsed for measuring shape of objects.**

**Data storage.** **Data storage.**

**ÐContain large amount of visual **
**information**

**ÐSimilar technique for storing digital data**

**Holographic Interferometery** **Holographic Interferometery**

**Holographic Interferometery**

**Holographic Interferometery**

**Measures strain on an object.** **Measures strain on an object.**

**Two methods:** **Two methods:**

**ÐDouble exposure holographic **
**interferometry.**

**ÐReal time holographic interferometry.**

**Holographic Interferometery** **Holographic Interferometery**

**Holographic Interferometery**

**Holographic Interferometery**

**Double exposure holographic ** **Double exposure holographic ** **interferometry.**

**interferometry.**

**ÐTwo holograms on photographic plate.**

**ÐObject is stressed between exposures.**

**ÐMovement of object appears as **
**interference fringes.**

++ **Light & dark bands across image.****Light & dark bands across image.**

**Holographic Interferometery** **Holographic Interferometery**

**Holographic Interferometery**

**Holographic Interferometery**

**Real time holographic interferometry.** **Real time holographic interferometry.**

**ÐStandard hologram of image made.**

**ÐReconstruct image on top of object.**

**ÐStress object & interference fringes **
**appear.**

**Double Exposure Interference ** **Double Exposure Interference **

**Double Exposure Interference**

**Double Exposure Interference**

**Holography** **Holography**

**Holography**

**Holography**

**Method is as follows** **Method is as follows**

**ÐSet up as for standard holography.**

**ÐTake one exposure of object.**

**ÐStress object.**

**ÐTake second exposure.**

**ÐDevelop hologram.**

**ÐReconstruct images.**

**Two images are on top of each other.** **Two images are on top of each other.**

**Analysis** **Analysis**

**Analysis**

**Analysis**

**From each point on two images, light ** **From each point on two images, light ** **will have the displacements.**

**will have the displacements.**

Ð∆x is movement of that point when object
**was stressed.**

**ÐResultant displacement is sum of the two:**

### ( )

### ( )

### ( )

1

2

cos cos

*o*

*o*

*E* *E* *kx* *t*

*E* *E* *k x* *x* *t*

= − ω

= + ∆ − ω

### ( ) ( ( ) )

### { }

1 2

2 2

cos cos

cos cos

*o*

*o*

*E* *E* *E* *E* *kx* *t* *k x* *x* *t*

*x* *x* *k x*

*E* *k* *t*

= + = − ω + + ∆ − ω

+ ∆ ∆

= − ω

**Analysis** **Analysis**

**Analysis**

**Analysis**

**Intensity varies as** **Intensity varies as**

**ÐIntensity shows how much (∆x) object **
**has moved.**

2

cos *k x*∆2