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全像攝影術(Holography)

Chen-Ching Ting(丁振卿)

Mechanical Engineering, National Taipei University of Technology(國立台北科技大學機械系)

Homepage: http://cct.me.ntut.edu.tw/E-mail: [email protected]

December 11, 2007

Chen-Ching Ting (丁振卿) Mechanical Engineering, National Taipei University of Technology (國立台北科技大學機械系) Homepage: http://cct.me.ntut.edu.tw/ E-mail: [email protected] ()全像攝影術(Holography) December 11, 2007 1 / 42

課程大綱

1 HolographyPhotography (Recording)Image Reconstruction

2 Holographical InterferometryReal TimeDouble Exposure


參考文獻 I

E. Hecht, Optics, Addison Wesley Co., New York 2001.

W. Lauterborn, T. Kurz, and M. Wiesenfeldt, Coherent Optics, Fundamentals andApplications. Springer Verlag, Berlin 2003

R. J. Collier, C. B. Burckhardt, and L. H. Lin, Optical Holography, Academic Press, 1971.

http://www.nsc.gov.tw/ newfiles/popular science.asp?add year=2005&popsc aid=17

http://en.wikipedia.org/wiki/Holography


Holography

Holography (1948)Dennis Gabor (1900-1979)


Holography

Holography

全像攝影術(Holography)簡單的說乃一項三度空間的攝影技術，不同於一般的攝影僅儲存了亮度(Intensity)的資料，全像攝影儲存了亮度(Intensity)及相(Phase)的資料。

全像攝影術乃在1948年，由匈牙利人(Hungarian)物理學家Dennis Gabor (1900–1979)所發明，他也因此在1971年獲得諾貝爾物理獎(Nobel Prize in physics)。但是，全像攝影術一直到1960年Laser光出現後才真正突飛猛進。


Holography

Laser

The first working laser was demonstrated in May 1960 by TheodoreMaiman at Hughes Research Laboratories.


Holography

Coherent Light、Interference、Holography相互之關係

Coherent Light為產生Interference現象的唯一光源，而產生Interference為同時儲存入射光之振幅與相的唯一方法，Holography則是同時儲存入射光之振幅與相的照像技術。


Holography

Holographical Photos


Holography

Holography

1 Photography (Recording)

2 Hologram(底片)

3 Image Reconstruction

4 Phase Conjugation

5 Holographic Arrangements


Holography Photography (Recording)

Photography (Recording)



Setup of Holography



Photography (Recording)



Superposition

The monofrequency light (Coherent Light):Object Wave:

S(x , y , t) = S(x , y)exp(−iωt) (1)Reference Wave:

R(x , y , t) = R(x , y)exp(−iωt) (2)At a point (x,y) of the hologram:Superposition:

E (x , y , t) = S(x , y , t) + R(x , y , t) (3)

Intensity:

I (x , y) = |E (x , y)|2 (4)= |R(x , y) + S(x , y)|2 (5)= (R + S)(R + S)∗ (6)

= RR∗ + SS∗ + R∗S + RS∗ (7)Chen-Ching Ting (丁振卿) Mechanical Engineering, National Taipei University of Technology (國立台北科技大學機械系) Homepage: http://cct.me.ntut.edu.tw/ E-mail: [email protected] ()全像攝影術(Holography) December 11, 2007 13 / 42


Hologram


Exposure

*

0 0

( , ) ( , , ) ( , , ) ( , , )B Bt t

B x y I x y t dt E x y t E x y t dt= =∫ ∫

B (Energy/Area), Bt :Exposure Time

The complex amplitude transmittance τ is given by

exp( ) | | exp( )ae

E T i iE

τ ϕ τ ϕ= = = (From Fabry-Perot Interferometer)

Amplitude Phase




Laser共共共振振振腔腔腔(Etalon)


共振腔(Etalon)Holography Photography (Recording)


Fabry-Perot Interferometer

0 exp( )etE ikL

0 exp( 2 )etrE ik L2

0 exp( 3 )etr E ik L

0eE2

0 exp( )et E ikL0erE

2 20 exp( 3 )et r E ik L

20 exp( 2 )et rE ik L





2 2( 1)0 exp[ (2 1) ]

net r E ik n L

− −

1E

2E

3E

nE

Etalon



Fabry-Perot Interferometer2 2( 1)

02 2( 1)

02( 1)

1

exp[ (2 1) ]

exp( ) exp( 2 )

exp( 2 )

nn e

ne

n

E t r E ik n L

t E ikL r ik nL

E r ik nL

−

−

−

= −

=

=

2 2( 1)0

2 2 3 20

21

exp[ (2 1) ]

exp[ (2 3) ] exp( 2 )

exp( 2 )

nn e

ne

n

E t r E ik n L

t r E ik n L r ik L

E r ik L

−

−

−

= −

= −

=



Hologram

Amplitude Hologram

Phase Hologram.constϕ =

.T const=



Hologram

The complex amplitude transmittance is usually a function of the position on the hologram.

( , ) ( , ) exp[ ( , )]x y T x y i x yτ τ ϕ= =

Amplitude Hologram:.constϕ =

Phase Hologram:.T const=

T

B

B

ϕ



Amplitude HologramThe usual processing of a photographic plate yields a realamplitude transmittance T which depends on the exposureB.

BT a bIt= −T a bB= −BB It=

* * * *( , ) ( )( , )BT x y a bt RR SS R S RS x y= − + + +



Amplitude TransmittanceHolography Photography (Recording)


Phase Hologram

( ) ' ' BI a b t Iϕ = −exp[ ( )]i Iτ ϕ=

( )2

I πϕ exp[ ( )] cos[ ( )] sin[ ( )]1 ( )

i I I i Ii Iϕ ϕ ϕϕ

= +≈ +

* * * *1 ( ) (1 ') ' ( )Bi I ia ib t RR SS R S RSϕ≈ + = + + + + +



HologramHolography Photography (Recording)



Hologram

Holography 為利用干涉現象的攝影技術，而Hologram乃所使用的底片，因此，未使用CoherentLight進行影像重建的Hologram，在Incoherent 的白色光下呈現干涉現象的明暗相間條紋。



Hologram


Holography Image Reconstruction

Image Reconstruction



Image Reconstruction


ReconstructionRecording

Reconstruction

虛像



Reconstruction of Virtual ImageHolography Image Reconstruction


Reconstruction of Real ImageHolography Image Reconstruction



成像

1 Real Image(實像)：成像光來自真實的像所發出。

2 Virtual Image(虛像)：成像光來自虛擬的像所發出。



Amplitude Hologramm

以參考光束(R)進行重建：

R(x , y , t) = R(x , y)exp(−iωt)Ea(x , y) = T (x , y)Ee = T (x , y)R

= R [a − btB(RR∗ + SS∗ + R∗S + RS∗)]= R(a − btBRR∗) − btBRSS∗ − btBRR∗S − btBR2S∗

= R(a − btB |R|2) − btBR|S |2 − btB |R|2S − btBR2S∗

以參考光之共軛光束(R∗)進行重建：

R∗(x , y , t) = R(x , y)exp(iωt)

Ea(x , y) = T (x , y)Ee = T (x , y)R∗

= R∗ [a − btB(RR∗ + SS∗ + R∗S + RS∗)]= R∗(a − btBRR∗) − btBR∗SS∗ − btB(R∗)2S − btBR∗RS∗

= R∗(a − btB |R|2) − btBR∗|S |2 − btB(R∗)2S − btB |R|2S∗


Amplitude HologramThe reference wave for reconstruction:

( , , ) ( , ) exp( )R x y t R x y i tω= −

* * * *

* * * *

2 2 2 2 *

( , ) ( , ) ( , )

( )

( )

( | | ) | | | |

a e

B

B B B B

B B B B

E x y T x y E T x y R

Ra bt R RR SS R S RS

R a bt RR bt RSS bt RR S bt RRS

R a bt R bt R S bt R S bt R S

= =

= − + + +

= − − + +

= − − + +

.constϕ =

直接穿透光

原物光

共軛光

雜訊



Location of the ImagesHolography Image Reconstruction


Plane Waves

0( ) exp( )E r E ik r= ⋅

0( ) exp( sin )E r E ikx γ=

γ

Sx

cos 0 cos( )2

sin

k r kr kx

kx

π γ

γ

⋅ = = −

=

Plane Waves:

2π γ−



Location of the Images

The source light is a plane wave with incident angle :γ

The wave S* can be interpreted as a wave leaving the hologram at the angle .γ−

γπ γ+

S

*S

sin( ) sin( ) sinπ γ γ γ+ = − = −

The reference and reconstruction light are also a plane wave with incident angle :α



Conjugate image

sin 2 2sin cosα α α=1α ﹐

Conjugate image:



Conjugate image

With increasing α γ , the conjugate image becomesmore and more distorted until, for

or

Disappear at

an angle does no longer exists to satisfy the relation. Then, the corresponding parts of the conjugate image disappear.

β



Location of the ImagesHolography Image Reconstruction


Phase Hologram

* * * *

( , ) ( , ) exp[ ( , )] [1 ( ( , ))]

(1 ') ' ( )a

B

E x y x y R i x y R i I x y R

ia R ib t R RR SS R S RS

τ ϕ ϕ= = ≈ +

≈ + + + + +

exp( ) cos sini iϕ ϕ ϕ= + exp( ) 1i iϕ ϕ≈ +1ϕ

As a result of the additional approximation, the nonlinear effects are more pronounced with phase holograms than amplitude holograms.

exp( ) 1i iϕ ϕ≈ +




Phase Conjugation


Phase Conjugation

Phase conjugating mirrors can be built from nonlinear crystals such as

3 3 12 20, ,LiNbO BaTiO Bi SiO



Phase Conjugation

Plane wave:

To obtain the phase conjugate wave, the wave vector k has to be replaced by –k.(反方向傳播)

Phase conjugate wave:



Phase ConjugationHolography Image Reconstruction



Holographic Arrangements



Holographic Arrangements

1 In-line Holograms

2 Reflection Holograms

3 Transmission Holograms

4 White light Holograms



In-line Holograms



Reflection Holograms


Reflection HologramsHolography Image Reconstruction



Transmission Holograms


Transmission HologramsHolography Image Reconstruction



White Light Holograms


White Light Holograms

Spectrum of white light of halogen metal vapor lamp



White light Holograms

2



White light Holograms

Source Light

Reference Light

多層膜底片

Bragg Reflection: 2dλ ≤



Arrangement of White Light Holograms

利用Bragg’s Law進行像重建




Rainbow Holography


Rainbow Holograms

彩虹全像術的拍攝與重建過程為使用物波(OB) 與參考波 (RB) 產生干涉圖案，由全像底 H1 記錄成離軸式全像片。把拍攝好的全像片 H1 以一水平狹縫遮蓋（或把它剪出約三或四毫米高的長條來）後，再以參考波RB 的共軛波照射，重建出原物的實像到記錄底片 H2 附近（底片前、後，或底片上）。這時再加入另一道同調參考波來產生干涉，形成第二張離軸式全像片 H2，這張全像片就是所謂的彩虹全像片。



Rainbow HolographyHolography Image Reconstruction


Rainbow Holograms

彩虹全像片 (Rainbow Holograms)乃全像片以全光譜白光來照射，因不同波長光的繞射角度不同，產生一系列互相錯開的影像，而使得所觀察到的影像呈現一片模糊，若要看到清楚的影像，就必須限制眼睛所能看到的波長頻帶，彩虹全像術正是為了這個目的而被提出。



Rainbow Holograms

以第二次參考波的共軛波來照射這全像片，會重建出原狹縫的影像，觀察者把雙眼擺在狹縫影像的位置朝全像片觀看，可以看到原物的立體影像。由於不同波長的光的繞射角度不同，白光照射所重建的狹縫實際上是一系列互相平行的彩虹狹縫組，而限於眼睛的尺寸，所看到的波長頻帶會因此變小很多。



Rainbow Holograms

如果參考波與物波的夾角很大，那麼彩虹狹縫組分布的範圍就很廣，眼睛在某一種顏色的狹縫後觀看，就會看到該顏色的影像。否則，影像從上到下就會呈現不同波長的分布，最嚴重的情況則是彩虹狀的分布，當然，這是我們最不想要的情況。



Rainbow Holograms

簡單來說，彩虹全像的影像來自方向非常專一的影像光，此一影像光在與參考光進行干涉而將影像儲存在全像底片後，影像的重建過程將隨參考光入射後的折射角度變化而有不同結果，全光譜白色光因具備可見光的不同波長光線，在波長與折射率成反比的條件下﹐眼睛以不同角度觀看此重建的全像片時便會顯現不同顏色的影像。



Arrangement of Rainbow Holography



Reconstruction of Rainbow Holograms




Recording of Rainbow Holography



Reconstruction of Rainbow Holography



Recording of Denisyuk Holography



Reconstruction of Denisyuk Holography (White Light)


Holographical Interferometry

Holographic Interferometry


Holographical Interferometry

Holographic Interferometry

1 Real Time

2 Double Exposure


Holographical Interferometry Real Time

Real Time


Real Time

Holographical Interferometry Real Time


Holographical Interferometry Double Exposure

Double Exposure


Double Exposure

Holographic Interferometry Setup




結結結論論論



結論

全像攝影為一真正三度空間的攝影技術﹐所謂三度空間指的是空間的距離﹐即全像攝影儲存的資訊包含物體縱深﹐不同一般攝影技術只儲存光的亮度資訊﹐全像攝影乃儲存了光的亮度與相﹐儲存光之相的唯一方法乃透過干涉結果﹐而Coherent光為進行干涉的唯一光源。全像攝影的技術發展不僅可實現真正三度空間的影像顯示﹐更可開創虛凝真實影像的新世界。


HolographyPhotography (Recording)Image Reconstruction

Holographical InterferometryReal TimeDouble Exposure

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