Файл: Ersoy O.K. Diffraction, Fourier optics, and imaging (Wiley, 2006)(ISBN 0471238163)(427s) PEo .pdf

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ANALYSIS OF HOLOGRAPHIC IMAGING

207

Figure 13.8. Holographic reconstruction of the twin images.

The recording and reconstruction wavelengths are l1 and l2, respectively. l1 may be different from l2.

The final hologram may be of a different size from the initial hologram generated by recording. A point on the initial hologram at z ¼ 0 will be denoted by ½x; y& whereas the corresponding point on the final hologram will be denoted by ½x0; y0&.

The phase due to the object wave at a hologram point ½x; y& can be written relative to the origin as

foðx; yÞ ¼ 2lp ðx x0Þ2 þ ðy y0Þ2 þ ðz z0Þ2&1=2 ½x2o þ y2o þ z2o&1=2g

1

ð13:5-1Þ

Keeping the first two terms of the Taylor series expansion, the last equation can be written for nonconstant terms as

2p

1

2

foðx; yÞ ¼

x2 þ y2 2xx0 2yy0 þ zo

þ bðx; yÞ

ð13:5-2Þ

l1

2z0

Where the third-order term bðx; yÞ is given by

0

x4 þ y4 þ 2x2y2 4x3x0

1

1

4y3y0 4x2yy0 4xy2x0

B

2

2

3

C

b

x; y

3

B

þ

6x2xo2

þ

6y2yo2

þ

2x2yo2

C

ð

13:5-3

Þ

ð

Þ ¼ 8z

B

C

B

þ

þ

C

B

3

2

2

C

@

4xxo

A

B

2y xo

8xyx0y0

C

B

4yyo

4xx0yo

4xxoy0

C

208

HOLOGRAPHY

bðx; yÞ is related to the aberrations discussed in Section 13.5. Similar equations can be written at the hologram point ðx; yÞ relative to the origin for fcðx; yÞ, the phase due to the reconstruction wave at wavelength l2, and frðx; yÞ, the phase due to the recording reference wave at wavelength l1.

With respect to Eqs. (13.2-8) and (13.2-9), the important phase terms for U3 and U4 can be written as

fV ¼ fc þ fo fr

ð13:5-4Þ

fR ¼ fc fo þ fr

ð13:5-5Þ

At this point, terms of order higher than 1 in 1=z or 1=zc or 1=z0 are neglected. Writing fI for f3or f4, we get

2p 1

2p 1

fI ðx; yÞ ¼

ðx02 þ y02

2x0xr 2y0yrÞ þ

ðx2 þ y2 2xxo 2yyoÞ

l2

2zc

l1

2zo

2p 1

2

2

ðx

þ y

2xxr 2yyrÞ

ð13:5-6Þ

l1

2zr

where þ sign is for f3 and – sign is for f4.

The hologram magnification Mh is defined by

x0

y0

Mh ¼

¼

x

y

and the wavelength ratio is given by

m ¼

l2

l1

Then, Eq. (13.5-6) can be written as

1

m

m

2

ðx02 þ y02Þ

þ

3

zc

Mh2zo

Mh2zr

p

6

mxo

mxr

7

fI

ð

x; y

Þ ¼ l2

6

2x

zc

þ Mhzo

Mhzr

7

6

2y

7

6

m

m

7

6

yo

yr

7

4

5

6

yc

7

6

zc

þ

Mhzo

Mhzr

7

ð13:5-7Þ

ð13:5-8Þ

ð13:5-9Þ

Equation (13.5-9) can be interpreted as the phase corresponding to another spherical wave originating from the point ðxI ; yI ; zI Þ. The relevant phase for this wave within the Fresnel approximation is given by

fI0

p

ðx02 þ y02 2x0xI 2y0yI Þ

¼ l2zI

ð13:5-10Þ

ANALYSIS OF HOLOGRAPHIC IMAGING

209

Setting f0I ¼ fI yields

zI ¼

Mh2zczozr

ð13:5-11Þ

Mh2zozr þ mzczr mzczo

x

Mh2xczozr þ mMhxozczr mMhxrzczo

ð

13:5-12

Þ

Mh2xczozr þ mzczr mzczo

I ¼

y

Mh2yczozr þ mMhyozczr mMhyrzczo

ð

13:5-13

Þ

Mh2yczozr þ mzczr mzczo

I ¼

Which image is virtual and which image is real is determined by the signs of zr and zc, respectively. The transverse magnification is given by

Mt ¼

qxI

¼

Mh

qxo

1 þ

M2z

o

zo

h

mzc

zr

Ma

qzI

Mh2

d

2

zo

3

¼ qzo ¼ m dzo

4

2

Mh2

1

5

61

zo

mzc

þ zr

7

6

7

1

M

h

¼

m

1 þ

Mh2zo

zo

2

mzc

zr

¼ m1 Mt2

ð13:5-14Þ

ð13:5-15Þ

EXAMPLE 13.2 Determine m so that Ma ¼ Mt in magnitude.

Solution: We set

jMaj ¼ m1 Mt2 ¼ Mt

Hence,

m ¼ Mt

ð13:5-16Þ

This means the changes in hologram size and the reference wave origin can be compensated by choosing a new wavelength satisfying Eq. (13.5-16). By the same token, if the wavelength is changed, the hologram size and/or the reference wave origin can be changed to make the two types of magnification equal to each other as much as possible.

210

HOLOGRAPHY

13.6ABERRATIONS

In discussing aberrations, it is more convenient to replace the rectangular coordinates x and y by the polar coordinates r and y. Wave front aberrations are defined by the phase difference fAðr; yÞ between the ideal spherical wave front and the actual wave front with source at ðxI ; yI ; zI Þ.

We will consider the aberrations of the image U3 due to the third-order terms as in Eq. (13.5-3). The third-order terms due to fc; fo; fr are combined to give the thirdorder term in the actual wave front. The aberration wave function becomes

2

1

r4S

þ

1

r3ðCx cos y þ Cy sin yÞ

3

8

2

2p

6

7

6

7

fA

r; y

Þ ¼ l2

6

1

r2

ð

Ax cos2 y

þ

Ay sin2 y

þ

2AxAy cos y sin y

Þ

7

ð

6

2

2

1

7

6

1

7

4

4

þ

2

ð

þ

Þ

5

6

7

6

r F

r

Dx cos y

Dy sin y

7

where the parameters belong to the following aberrations: S: spherical aberration

Cx, Cy: coma

Ax, Ay: astigmatism F: field curvature Dx, Dy: distortion

They are given by the following equations:

1

m

m

1

S ¼

þ

zc3

Mh4zo3

Mh4zr3

zI3

xc

mxo

mxr

xI

Cx

¼

þ

zc3

Mh3zo3

Mh3zr3

zI3

yc

myo

myr

yI

Cy

¼

þ

zc3

Mh3zo3

Mh3zr3

zI3

xc2

mxo2

mxr2

xI2

Ax

¼

þ

zc3

Mh2zo3

Mh2zr3

zI3

yc2

myo2

myr2

yI2

Ay

¼

þ

zc3

Mh2zo3

Mh2zr3

zI3

F

¼

xc2

þ yc2

mðxo2 þ yo2Þ

mðxr2 þ yr2Þ

xI2 þ yI2

zc3

Mh2zo3

þ

Mh2zr3

zI3

D

¼

xc3

þ xcyc2

mðxo3 þ xoyo2Þ

mðxr3 þ xryr2Þ

xI3

þ xI yI2

zI3

x

zc3

Mhzo3

þ

Mhzr3

D

¼

yc3

þ ycxc2

mðyo3 þ yoxo2Þ

mðyr3 þ yrxr2Þ

yI3

þ yI xI2

zI3

y

zc3

Mhzo3

þ

Mhzr3

ð13:6-1Þ

ð13:6-2Þ

ð13:6-3Þ

ð13:6-4Þ

ð13:6-5Þ

ð13:6-6Þ

ð13:6-7Þ

ð13:6-8Þ

ð13:6-9Þ

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