Topic 8: Holography Aim: To cover the basic of holographic recording - PDF document

I V N E U R S E I H T Modern Optics Y T O H F G R E U D B I N Topic 8: Holography Aim: To cover the basic of holographic recording and reconstruction and review holographic materials. Contents: � Photography � Holographic Recording � Hologram Formation � Reconstruction � Types of Holograms � Holographic Material � Mass Production of Holograms O P T I C D S E G I R L O P P U A P D S Holography -1- Autumn Term C E P I S A Y R H T P M f E o N T

I V N E U R S E I H T Modern Optics Y T O H F G R E U D B I N Photography Record optical distribution as Optical Density given by intensity only . Object Image = γ log 10 D ( x ; y ) ( E ( x ; y )) � D 0 where ( x ; y ) j 2 = τ ( x ; y ) j u E Do not record the Phase Information , so � No depth information � Two dimensional projection of three dimensional scene. � Similar for coherent and incoherent, (different transfer function) We have to do something different to retain phase information. O P T I C D S E G I R L O P P U A P D S Holography -2- Autumn Term C E P I S A Y R H T P M f E o N T

I V N E U R S E I H T Modern Optics Y T O H F G R E U D B I N Basic Holography To retain phase information we must encode complex distribution as intensity pattern. Encode by adding reference beam: θ Object Wave Holographic Plate Object P 0 Reference Wave At P 0 we have two optical distributions ( x ; y ) exp ( ı Φ ( x ; y )) ! o Scattered from object r exp ( ı κ x sin θ ) ! Reference Wave where r is a constant and θ is angle from plate normal Assume that the beams are coherent , then Amplitudes add to give, = r exp ( ı κ x sin θ ) ( x ; y ) exp ( ı Φ ( x ; y )) ( x ; y ) + o u Intensity in P 0 is given by ( x ; y ) j 2 ( x ; y ) = j u g which after some expansion is given by, j 2 ( x ; y ) j 2 + 2 ro ( x ; y ) cos ( κ x sin θ � Φ ( x ; y )) ( x ; y ) = j r + j o g O P T I C D S E G I R L O P P U A P D S Holography -3- Autumn Term C E P I S A Y R H T P M f E o N T

I V N E U R S E I H T Modern Optics Y T O H F G R E U D B I N There is no image , so for all practical cases: ( x ; y ) ! Varies slowly over ( x ; y ) o so we can assume that j 2 ( x ; y ) j 2 j r + j o � constant but we have that: θ ! NOT small the intensity can be written as: + 2 ro ( x ; y ) cos ( κ x sin θ � Φ ( x ; y )) ( x ; y ) = g 0 g which is high frequency cos () fringes in plane P 0 � Amplitude of fringes encodes o ( x ; y ) � Location of fringes encodes Φ ( x ; y ) We have encoded both the Amplitude and the Phase of the object ( x ; y ) as an intensity distribution. wave o O P T I C D S E G I R L O P P U A P D S Holography -4- Autumn Term C E P I S A Y R H T P M f E o N T

I V N E U R S E I H T Modern Optics Y T O H F G R E U D B I N Shape of Fringes Maxima of intensity when κ x sin θ = 2 n π � Φ ( x ; y ) so if θ large, then Φ ( x ; y ) displaces fringes from regular pattern φ d If Φ ( x ; y ) is a random variable, then mean separation λ = d sin θ = 30 = 633 nm (He-Ne) then Example: θ � , and λ = 2 λ � 1 : 3 µ m d or 700 lines/mm High Frequency Need a very high resolution photographic emulsion. Fine Grain, very slow photographic material needed. (special photo- graphic material) Need to record the fringe locations, so need a higher resolution than this, 1200 lines/mm is typical. O P T I C D S E G I R L O P P U A P D S Holography -5- Autumn Term C E P I S A Y R H T P M f E o N T

I V N E U R S E I H T Modern Optics Y T O H F G R E U D B I N Hologram Formation Expose emulsion in the linear region and develop to form negative. Amplitude Transmission is then: = 2 � γ = 2 � γ = 2 = 10 D 0 ( τ g ) = Kg ( x ; y ) T a we have that the intensity + 2 ro ( x ; y ) cos ( κ x sin θ � Φ ( x ; y )) ( x ; y ) = g 0 g ( x ; y ) j 2 is slow varying. This can be writ- j o where we have assumed ten as: + δ g ( x ; y ) = g 0 ( x ; y ) g where we have that: = 2 ro ( x ; y ) cos ( κ x sin θ δ g � Φ ( x ; y )) ( x ; y ) This gives the Amplitude Transmission as = 2 � γ + δ g = K ( g 0 ) T a which can then be written as � γ = 2 + δ g � � = 2 = 2 � γ � γ = 2 � γ 1 ( 1 + δ ˆ = Kg = Kg ) T a g 0 0 g 0 where = δ g δ ˆ g g 0 O P T I C D S E G I R L O P P U A P D S Holography -6- Autumn Term C E P I S A Y R H T P M f E o N T

I V N E U R S E I H T Modern Optics Y T O H F G R E U D B I N j δ g � ( x ; y ) j . (Assume low contrast fringes on a Assume: that g 0 large background). Expand the term, to second order to get; + 2 � γ + γ ( γ ) = 2 � γ ) 2 ( 1 + δ ˆ = 1 2 δ ˆ ( δ ˆ ) g g g 8 Substituting this back into the expression the T a we get + 2 � γ + γ ( γ � ) � = 2 � γ ) 2 1 2 δ ˆ ( δ ˆ = Kg T a g g 0 8 which we will write as: ) 2 � a δ ˆ ( δ ˆ = T 0 + b T a g g where T 0 , a and b are constants given by: = 2 � γ = T 0 K g 0 K γ � γ = 2 = a 2 g 0 + 2 K γ ( γ ) = 2 � γ = b g 0 8 � 1 so T 0 For most emulsions γ � a � b , but δ ˆ � 1 g so that g ) 2 j a δ ˆ ( δ ˆ � g j � j b j T 0 O P T I C D S E G I R L O P P U A P D S Holography -7- Autumn Term C E P I S A Y R H T P M f E o N T

I V N E U R S E I H T Modern Optics Y T O H F G R E U D B I N Reconstruction Reconstruct with the original reference beam, T a Exposed and Developed θ Holographic film/plate Reconstruction Beam which is = r exp ( ı κ x sin θ ) ( x ; y ) u The Complex Amplitude transmitted by the hologram is then v ( x ; y ) = T a ( x ; y ) u ( x ; y ) = 0 ), Look at First Two Terms: (assume b ( x ; y ) a δ ˆ v ( x ; y ) = u ( x ; y ) T 0 � u ( x ; y ) g ( x ; y ) and δ ˆ which with substitution for u g , gives T 0 r exp ( ı κ x sin θ ) v ( x ; y ) = + ar exp ( ı κ x sin θ ) 2 ro ( x ; y ) cos ( κ x sin θ � Φ ( x ; y )) g 0 If we new expand the cos () term and cancel term, be get three terms = T 0 r exp ( ı κ x sin θ ) v ( x ; y ) � (1) ar 2 ( x ; y ) exp ( ı Φ ( x ; y )) � o (2) g 0 ar 2 ( x ; y ) exp ( � ı Φ ( x ; y )) exp ( ı 2 κ x sin θ ) o (3) g 0 O P T I C D S E G I R L O P P U A P D S Holography -8- Autumn Term C E P I S A Y R H T P M f E o N T

I V N E U R S E I H T Modern Optics Y T O H F G R E U D B I N Look at the three terms. 1. Partially transmitted reconstruction beam in direction θ . 2. Reconstruction of original complex object wave. Both ampli- � sign, which gives phase tude and phase reconstructed. Note shift or π . (discussed later). 3. Conjugate Reconstruction Similar to Reconstruction, but com- plex conjugate. In direction φ where sin φ = 2sin θ So provided that θ is NOT small, three terms will be separated. (3) Conjugate Reconstruction (1) DC Term Hologram Reconstruction Beam (2) Reconstruction T a Three terms separated. Only want (2) which is full three dimensional reconstruction of object wave. O P T I C D S E G I R L O P P U A P D S Holography -9- Autumn Term C E P I S A Y R H T P M f E o N T

I V N E U R S E I H T Modern Optics Y T O H F G R E U D B I N We “see in 3-D” We live in a 3-D world, and we see in “3-D”. 3-D Object Right Image Left Image We have two eyes separated by about 65 mm. We see two images of the same object from different directions, Left Eye Right Eye Brain “matches up” the vertical disparities and interperates the differ- ence as “depth”. Because of our two eyes we can see in 3-D. O P T I C D S E G I R L O P P U A P D S Holography -10- Autumn Term C E P I S A Y R H T P M f E o N T

I V N E U R S E I H T Modern Optics Y T O H F G R E U D B I N Scattered Light from Object If we consider our wave model, then we have: Object Wave 3-D Object Right Image Left Image o(x,y) Plane P 0 See two different images, and again the brain makes the 3-D scene. Green Record Amplitude distribution in plane P 0 , and play-it-back . Reconstructed Object Wave 3-D Object (Vitrual) Right Image Left Image Plane P 0 Reconstruction Wave Reconstruct Amplitude Distribution in plane P 0 we will still see the two images, and hence a 3-D virtual image of the original object. O P T I C D S E G I R L O P P U A P D S Holography -11- Autumn Term C E P I S A Y R H T P M f E o N T