M/# of pure index holograms

Chromophore 0 Chromophore 0

4.3 M/# of absorption and index holograms in SHB media

4.3.3 M/# of pure index holograms

An interesting property of wavelength selective materials is that light can be diffracted by both an absorption modulation and the induced index modulation. The advantage of index gratings is that the diffraction efficiency can reach 100% if the absorption at the read-out frequency is equal to zero. Here, we analyze the AI

1#

when absorption modulation is transformed to index modulation. A pure index grating can be created by using two holographic exposures at two optical frequencies with a 7r

phase shift between the two exposures. The hologram generated is the so-called

II-hologram. We assume that the absorption modulation 0:1 has a Lorentzian shape:

(4.20)

where, is the homogeneous linewidth and Wb the writing (burning) optical frequency.

The index grating strength /31 induced by the absorption grating is equal to:

W - W

/31 = 0:1 P

,

. ^(4.21)

where wp is the read (probe) frequency. A II-hologram is recorded with a first exposure at frequency Wb_j and a second exposure (same exposure energy) at frequency Wb2' The diffracted field Ed of the II-hologram is equal to:

where

cPl

and

cP2

are the phases of the grating in exposures 1 and 2 respectively. If we choose

cP2 - cPl

^{= 7r,} the diffraction efficiency computed from eqn. 4.22 becomes:

If h t e o II h 1 ogram . IS rea -out at requency d f wp = Wbj +Wb.) 2 , t e a sorptlOn gratmg h b ' . cancels and only the index grating prevails:

( 20:od) d² 4

2 [(.6.w) 2 ,2 ]2

T}=exp - - - , 0 : ] - - - - - " - , - -

cosf) 4cos²f) 2, (Wb- W )2+,2 (4.24)

where .6.w

=

^{Wb2 - Wb2'} The function in brackets has a maximum for

.6.w ⁼

^2,.

vVhen read-out for a long time at frequency Wb) ;Wb2

, the index grating is not erased and the remaining absorption is equal to the non burnable absorption parameter B.

Therefore, in the exponential we replace 0:₀ with B. Thus, the diffraction efficiency

of a II hologram becomes:

2.0 non burnable absorption B = 0

1.5

!

^1.0

0.5

0.0

o

Ghologram

Aread =(Aw1 +~)I2 __

2

~ Pure absorption hologram A=A _{w read}

4 6 8

Optical density: a" d

( 4.25)

10

Figure 4.11: .M# comparison between pure absorption holograms and II-holograms when the non-burnable background absorption is equal to zero (B=O).

The assumption that the index grating is not erased during read-out at frequency

Wbl ;Wb2 is not an accurate description. The tail of the Lorentzian absorption curve

affects the absorption holograms recorded at frequencies Wbl and Wb2. A decay of the absorption hologram's strength induces a proportional decay of the induced index hologram's strength. When the two absorption holograms are recorded many ho- mogeneous linewidths away, the decay of the index grating is much slower than the decay of the remaining absorption at the read-out frequency Wbl +Wb2 /2; therefore, the assumption is valid. However, the maximum induced index modulation is achieved when ~w = 2

r

and the assumption ceases to be valid. Nevertheless, this assumption provides an insight for the upper bound of the

M/#

achievable using II-holograms for multiplexing.

Eqn. 4.25 is similar to the expression found for a pure absorption hologram, except for the absorption term in the exponential. \Vhen the non-burnable background B is

equal to 0, the medium is transparent at the read-out wavelength and the diffraction efficiency is proportional to the optical density. Fig. 4.11 shows a comparison between the 111# obtained with IT-holograms and pure absorption holograms. The result of fig. 4.11 suggests that IT-holograms yields larger 111/# than for pure absorption holograms. When the optical density is larger than approximately 3, the holograms become non-uniform inside the medium. In fact, the dynamic range is peeled off at the front of the material and the hologram is built up towards the back. These non-uniformities decrease the value of the 111/# plotted in fig. 4.11 when a_o d

>

3.

Fig. 4.12 plots the

M#

for different values of the non-burnable background.

1.8 , - - - , 1.6

1.4 1.2 1.0

I

0.8 0.6 0.4 0.2 0.0

_. _. - non-bumable absorption B= 50% a

o d , , , , , B=10% ad _o " "

" ^.'

~'., .

/ . '

".

~---

a

^{2 4 6 8 10}

Optical density: a_o d

Figure 4.12: !vI

#

versus optical density when multiplexing IT-holograms. The pa- rameter B is the non-burnable background absorption.

The M

#

obtainable in SHB polymer material can be estimated by fitting the absorption kinetics with a bi-exponential function and computing the resulting grating strength a1 and DC absorption a_{o .} The absorption is measured at 1.9 Kelvin in a 400fLm thick sample of H2TBN in polyvinyl butyral (PVB) with a bleaching intensity of 10 fL~l//cm2 (inset of fig. 4.13) and concentration of 5.10-.⁵ mol/I. Fig 4.13 illustrates the !vI

#

that can be obtained using the above material as a function of the sample thickness. An lvI

#

of 0.04 can be reached with that particular sample.

0.05 0.04 0.03

'*I:

::E 0.02 0.01 0.00

M number computed from experimental absorption curve

0.39

o

"' \

\

\

\

, , , , ,

400/-!m

2

,

- - - - absorption holograms - - index holograms

1 . o , - - - ,

09

~ 08

i ⁰⁷

! i 0.6 05

H₂-TBN:PVB dye concentration:

310.⁵ mol/l

0.4"--::-0 ---=20~40:--60=---=80~,00::-'

bl8aching lime (s]

---

4 6 8 10

Sample thickness d [mm]

Figure 4.13:

111#

versus optical density derived from the experimental absorption kinetics shown in the inset of the plot.

The remaining absorption B is the dominant factor in the M /

#

equations for absorption and index holograms. The computed values for the

111/#

of index and ab- sorption holograms are close. However, it is advantageous to use II-holograms instead of pure absorption holograms because the frequency dependence of II-holograms fall off as the square of the frequency while absorption holograms fall off proportionally to the frequency. We have shown that multiplexing holograms in a wavelength se- lective material yields an M /

#

of 0.05 using the polymer materials presented above.

Potentially, an

111/#

of 0.5 can be achieved by recording II-holograms and using a material with low non-burnable background. Experimentally, the number of multi- plexed holograms at a single frequency and location is limited to about 10. With an

M/#

of 0.05 and M=10 holograms, the maximum diffraction efficiency is equal to 2.5· 10-^5, which is enough to obtain good SNR values.