**4. Spatial multiplexing of dynamic holograms**

A multichannel adaptive interferometric demodulator proposed in (Kulchin, et al., 2000b) is built using the spatial multiplexing of dynamic holograms which are recorded in Bi12TiO20 crystal in its different parts (Fig.3). The holographic demodulation channels operate on the basis of the so called fanning effect (Voronov, et al., 1980) which appears as self-diffraction of light wave in photorefractive media. The nature of this effect consists in energy transfer between light beam entered the photorefractive crystal and beams scattered by crystal defects and inhomogeneities (Feinberg, 1982). Interference of initially weak scattered beams with the injected beam forms a set of dynamic holographic gratings. Diffraction of the input beam by these holograms leads to enhancement of those scattered waves which satisfy the phase-matching condition (Cronin-Golomb & Yariv, 1985). Due to holographic nature of fanning effect, any fast modulation of the phase of the injected light wave results in a variation of scattered light intensity; so the demodulation signal appears. Figure 4 illustrates light intensity distribution (fanning light) at the output of crystal when two holographic demodulation channels are formed. The channels optical beams are so close in a crystal that fanning waves are completely overlapped in the near field (Fig.4(a)) and can be separated only in the far field (Fig.4(b)), where they can be independently measured.

The authors show that demodulation signal in a channel decreases if the distance between light beams associated with multiplexed channels becomes below the lateral dimension of a channel (beam diameter) *d* (Fig.5). Such depletion is caused by reduction of interference pattern contrast *m* in a channel when light from another channel gets inside it. However this depletion is not critical – the output signal level reduces only by 2.5 dB even if optical fields from multiplexed channels completely overlap (*x* = 0).

Both the peculiarity and the advantage of holographic channel performance on the base of fanning effect are related to the fact that holograms are formed in a crystal without an external reference beam – the light waves scattered at the crystal defects act as a reference waves. Therefore radiation in a particular channel can be mutually non-coherent with other channels. Due to this, overlap of light waves from different channels does not lead to formation of cross-holograms and, as a sequence, does not produce cross-talk between channels. Figure 5 shows dependency of demodulation signal in 1st channel tuned to the

Fig. 3. Multi-channel adaptive demodulator based on spatial multiplexing of dynamic holograms in a photorefractive crystal: *d* is the characteristic dimension of a channel defined by light beam spot diameter, *x* is the distance between channel centers

A multichannel adaptive interferometric demodulator proposed in (Kulchin, et al., 2000b) is built using the spatial multiplexing of dynamic holograms which are recorded in Bi12TiO20 crystal in its different parts (Fig.3). The holographic demodulation channels operate on the basis of the so called fanning effect (Voronov, et al., 1980) which appears as self-diffraction of light wave in photorefractive media. The nature of this effect consists in energy transfer between light beam entered the photorefractive crystal and beams scattered by crystal defects and inhomogeneities (Feinberg, 1982). Interference of initially weak scattered beams with the injected beam forms a set of dynamic holographic gratings. Diffraction of the input beam by these holograms leads to enhancement of those scattered waves which satisfy the phase-matching condition (Cronin-Golomb & Yariv, 1985). Due to holographic nature of fanning effect, any fast modulation of the phase of the injected light wave results in a variation of scattered light intensity; so the demodulation signal appears. Figure 4 illustrates light intensity distribution (fanning light) at the output of crystal when two holographic demodulation channels are formed. The channels optical beams are so close in a crystal that fanning waves are completely overlapped in the near field (Fig.4(a)) and can be separated

The authors show that demodulation signal in a channel decreases if the distance between light beams associated with multiplexed channels becomes below the lateral dimension of a channel (beam diameter) *d* (Fig.5). Such depletion is caused by reduction of interference pattern contrast *m* in a channel when light from another channel gets inside it. However this depletion is not critical – the output signal level reduces only by 2.5 dB even if optical fields

Both the peculiarity and the advantage of holographic channel performance on the base of fanning effect are related to the fact that holograms are formed in a crystal without an external reference beam – the light waves scattered at the crystal defects act as a reference waves. Therefore radiation in a particular channel can be mutually non-coherent with other channels. Due to this, overlap of light waves from different channels does not lead to formation of cross-holograms and, as a sequence, does not produce cross-talk between channels. Figure 5 shows dependency of demodulation signal in 1st channel tuned to the

Fig. 3. Multi-channel adaptive demodulator based on spatial multiplexing of dynamic holograms in a photorefractive crystal: *d* is the characteristic dimension of a channel defined

demodulation channels

Fanning waves

*d*

PRC

*x* 

by light beam spot diameter, *x* is the distance between channel centers

Light beam 2 Holographic

only in the far field (Fig.4(b)), where they can be independently measured.

**4. Spatial multiplexing of dynamic holograms** 

from multiplexed channels completely overlap (*x* = 0).

PRC

Light beam 1

frequency of 2nd channel on relative distance between channels. As seen this signal does not depend on channels overlapping ratio, while its level does not exceed the inherent noise level of a channel (– 11 dB).

Fig. 4. Photographs of light field at the output of PR crystal in which two holographic channels are created: (a) near field image (T –beam transmitted through the crystal; F – fanning waves), (b) far field image of fanning waves (transmitted beam is blocked)

Fig. 5. Dependency of demodulation signals in two channels multiplexed in PR crystal on the relative distance between them. Channels provide processing of optical signals from two fiber-optical sensors detecting vibration at two different frequencies *f*1 and *f*2. The signal level in 1st channel tuned the frequency of 2nd channel does not exceed noise level of detection electronic equipment, which demonstrates the practical absence of cross-talk between channels

Multi-Channel Adaptive Interferometers Based on Dynamic Hologram Multiplexing 115

<sup>4</sup><sup>3</sup>

2

7 1 2

Fig. 7. Two-channel adaptive fiber-optical measurement system (a) and layout of sensing optical fibers in the object under the test – top view (b): 1, 2 – optical fibers; 3 – test object; 4 – polarizer; 5 – PR crystal Bi12TiO20; 6, 7 – channels photodetectors; 8 – recording system

As seen from the signal traces, crack frequency increases with growing of object deformation. It is more typical for the center part of the object where external pressure is applied and where the fiber #1 is placed (see Fig.7(b)). The object being stressed was broken at 325th second. The fiber placed in the center was broken together with the object. As a result the signal disappeared in the first channel. The second fiber remaining unbroken still detects weak signals indicating cracks which appear now in the object fragments. Start from 370th second, the press machine which creates mechanical stress in the object powders its

As seen from the oscilloscope traces, both channels operate independently without any

Fig. 8. Oscilloscope traces of photodetector signals in two channels of an adaptive fiber-optical measurement system which monitors crack the formation process in a solid state object

The authors of paper (Fomitchov, et al., 2002) realized a multi-channel measurement system on the basis of angular multiplexing of dynamic holograms in a photorefractive crystal. The

central part thus stopping cracks formation process.

He-Ne 1

5

**5. Angular multiplexing of dynamic holograms** 

detectable cross-talk.

8

6

(a) (b)

3

Pressure

Pressure

Fig. 6. Photographs of fanning waves at the output of two-channel holographic demodulator formed in PR crystal when a mechanical impact under measurement is applied to a sensor and is detected in the 1st channel (a) and in the 2nd channel (b)

Figure 6 shows photographs recorded in the far field of fanning waves intensity distribution for the case of two holographic channels multiplexed in BTO crystal. The distance between channels in a crystal is equal to a channel radius (*x/d* = ½), providing partial overlap of channel light fields. The radiation entering the crystal is coming from two independently operated fiber-optical sensors – single-fiber multimode interferometers which have speckled waves at their outputs (Kulchin, 2001). When one of sensors remained quiet, the other one was exposed to a sudden impact with amplitude sufficient for rearrangement of speckle pattern and, as consequence, to diminishing of fanning wave intensity in the corresponded channeling (Fig.6). At the same time the level of fanning wave intensity in the rest channel remained constant. This result demonstrates the independence of fanning-effect-based channels operation even in the case of significant overlap of channel light fields in the bulk photorefractive crystal.

The two-channel adaptive fiber-optical measurement system based on the considered multichannel holographic demodulator was tested for monitoring of cracks formation in an object under increasing mechanical stress (Kulchin, et al., 2000a). Two fiber-optical sensors based on multimode fibers were first embedded into the structure of the test object when it was created (see Fig.7). Mechanical stress caused by gradually increasing pressure applied to the object led to initiation of cracks whose appearance is accompanied by generation of sound pulses. In their turn sound pulses propagating in the object affect the optical fibers and modulate the phase of optical radiation guided by fibers. Phase demodulation is performed by means of dynamic holograms multiplexed in PRC. Figure 8 shows oscilloscope traces of demodulated signals in two channels.

It is worth noting that the adaptive property of a holographic demodulator formed in PRC allows not only protection of a measurement system from the influence of the environment (temperature drift, pressure variation, etc.) but also makes it insensitive to slow deformation of the object, providing a detection of only those moments in time when cracks appear.

(a) (b) Fig. 6. Photographs of fanning waves at the output of two-channel holographic demodulator formed in PR crystal when a mechanical impact under measurement is applied to a sensor

Figure 6 shows photographs recorded in the far field of fanning waves intensity distribution for the case of two holographic channels multiplexed in BTO crystal. The distance between channels in a crystal is equal to a channel radius (*x/d* = ½), providing partial overlap of channel light fields. The radiation entering the crystal is coming from two independently operated fiber-optical sensors – single-fiber multimode interferometers which have speckled waves at their outputs (Kulchin, 2001). When one of sensors remained quiet, the other one was exposed to a sudden impact with amplitude sufficient for rearrangement of speckle pattern and, as consequence, to diminishing of fanning wave intensity in the corresponded channeling (Fig.6). At the same time the level of fanning wave intensity in the rest channel remained constant. This result demonstrates the independence of fanning-effect-based channels operation even in the case of significant overlap of channel light fields in the bulk

The two-channel adaptive fiber-optical measurement system based on the considered multichannel holographic demodulator was tested for monitoring of cracks formation in an object under increasing mechanical stress (Kulchin, et al., 2000a). Two fiber-optical sensors based on multimode fibers were first embedded into the structure of the test object when it was created (see Fig.7). Mechanical stress caused by gradually increasing pressure applied to the object led to initiation of cracks whose appearance is accompanied by generation of sound pulses. In their turn sound pulses propagating in the object affect the optical fibers and modulate the phase of optical radiation guided by fibers. Phase demodulation is performed by means of dynamic holograms multiplexed in PRC. Figure 8 shows

It is worth noting that the adaptive property of a holographic demodulator formed in PRC allows not only protection of a measurement system from the influence of the environment (temperature drift, pressure variation, etc.) but also makes it insensitive to slow deformation of the object, providing a detection of only those moments in time when cracks appear.

and is detected in the 1st channel (a) and in the 2nd channel (b)

oscilloscope traces of demodulated signals in two channels.

photorefractive crystal.

Fig. 7. Two-channel adaptive fiber-optical measurement system (a) and layout of sensing optical fibers in the object under the test – top view (b): 1, 2 – optical fibers; 3 – test object; 4 – polarizer; 5 – PR crystal Bi12TiO20; 6, 7 – channels photodetectors; 8 – recording system

As seen from the signal traces, crack frequency increases with growing of object deformation. It is more typical for the center part of the object where external pressure is applied and where the fiber #1 is placed (see Fig.7(b)). The object being stressed was broken at 325th second. The fiber placed in the center was broken together with the object. As a result the signal disappeared in the first channel. The second fiber remaining unbroken still detects weak signals indicating cracks which appear now in the object fragments. Start from 370th second, the press machine which creates mechanical stress in the object powders its central part thus stopping cracks formation process.

As seen from the oscilloscope traces, both channels operate independently without any detectable cross-talk.

Fig. 8. Oscilloscope traces of photodetector signals in two channels of an adaptive fiber-optical measurement system which monitors crack the formation process in a solid state object

### **5. Angular multiplexing of dynamic holograms**

The authors of paper (Fomitchov, et al., 2002) realized a multi-channel measurement system on the basis of angular multiplexing of dynamic holograms in a photorefractive crystal. The

Multi-Channel Adaptive Interferometers Based on Dynamic Hologram Multiplexing 117

Fig. 10. Geometry of multi-wave mixing in PRC providing angular multiplexing of transmission dynamic holograms: all signal waves belong to horizontal plane (*xy*) and propagate in direction close to axis *y*; the reference wave belongs to plane (*yz*) and makes

The normalized ultrasonic signals are shown in Fig. 11(b). From the measured signals it can be seen that the bulk longitudinal waves are clearly detected by all three channels with good sensitivity. The longitudinal wave arrives at different times at different sensor locations, providing information about the wave speed (related to material properties of the composite). The measured longitudinal velocity of 2480 m/s agrees well with data obtained for the same composite material by a different technique (Fomitchov, et al., 2001). Furthermore, the absence of any spurious signals in sensors 2 and 3 at the time of arrival of the ultrasonic wave packet at sensor 1 is an indication of the absence of cross-talk in the multichannel adaptive system based on dynamic holograms multiplexed in a PR crystal.

(a) (b)

In spite of such a satisfactory performance the system proposed in (Fomitchov, et al., 2002) has a major drawback. This is the necessity of using strong external electric field which leads

Fig. 11. (a) Experimental arrangement used to detect bulk ultrasonic waves in a composite

sandwich structure, (b) detected ultrasonic signals for all three channels

the angle of 5º with plane (*xy*)

scheme of the multichannel holographic demodulator is shown in Fig.9. Several signal waves coming from fiber-optical sensors which transform a measurand (ultrasound waves) to light phase modulation are directed to the photorefractive crystal Bi12SiO20 where they are mixed with a common reference wave. Pair wise interference of all signal waves with the reference one forms a set of main dynamic holograms which provide phase demodulation of the corresponding signal waves.

Fig. 9. Optical schematic of laser source and typical arrangement of sensing fibers (a) and three-channel adaptive demodulator based on angular multiplexing of dynamic holograms in BSO crystal (b)

Use of a common reference beam for all multiplexed holograms implies that all signal waves are mutually coherent. In this case, overlap of signal beams could lead to formation of cross holograms and, as a sequence, to appearance of cross-talk between multiplexed holographic channels. In order to minimize the effect of cross holograms and to reduce a cross-talk the following approach is used. As seen from Fig.9 the holograms are recorded in so called transmission geometry – all mixed beams propagate in the crystal in same direction at small angles to each other. Having large spatial period Λ (or small wave number, 2 / *K* ) the holographic grating has low diffraction efficiency which is defined by space charge field *EK* proportional to *K* in accordance with Eqs.(2)-(3). In order to enhance the wave coupling efficiency in transmission holograms a strong external electric ac field with amplitude *E*0 = 6 kV/cm and frequency 3000 Hz was applied to the PRC. Meanwhile the geometry of beams propagating in the crystal has provided mutual orthogonality of grating vectors of main and cross-holograms (see Fig.10). Indeed, the incidence plane (*xy*) of signal beams forming crossholograms is orthogonal to the incidence plane (*yz*) of the reference beam which forms the main holograms. Thus, an external electric field applied to the crystal along grating vectors of main holograms has provided enhancement of their diffraction efficiency (see Eq.(2)), while cross-holograms remained unamplified. Moreover the angle between any pair of signal beams is much smaller than that between any signal and reference beams. This makes the cross-holograms weaker in comparison with the main holograms, thus reducing the cross-talk between channels

The three-channel adaptive measurement system based on the above considered approach of dynamic holograms angular multiplexing was applied to the detection of ultra-sound pulse propagation in a composite structure material (CSM). Three sensing fibers were embedded between CSM layers as shown in Fig.11(a). A 5-MHz contact transducer was used to launch bulk waves into the composite. The ultrasonic wave packet was detected by the sensing fibers, which were located at different depths within the composite.

scheme of the multichannel holographic demodulator is shown in Fig.9. Several signal waves coming from fiber-optical sensors which transform a measurand (ultrasound waves) to light phase modulation are directed to the photorefractive crystal Bi12SiO20 where they are mixed with a common reference wave. Pair wise interference of all signal waves with the reference one forms a set of main dynamic holograms which provide phase demodulation of

(a) (b) Fig. 9. Optical schematic of laser source and typical arrangement of sensing fibers (a) and three-channel adaptive demodulator based on angular multiplexing of dynamic holograms

Use of a common reference beam for all multiplexed holograms implies that all signal waves are mutually coherent. In this case, overlap of signal beams could lead to formation of cross holograms and, as a sequence, to appearance of cross-talk between multiplexed holographic channels. In order to minimize the effect of cross holograms and to reduce a cross-talk the following approach is used. As seen from Fig.9 the holograms are recorded in so called transmission geometry – all mixed beams propagate in the crystal in same direction at small angles to each other. Having large spatial period Λ (or small wave number, 2 / *K*

holographic grating has low diffraction efficiency which is defined by space charge field *EK* proportional to *K* in accordance with Eqs.(2)-(3). In order to enhance the wave coupling efficiency in transmission holograms a strong external electric ac field with amplitude *E*0 = 6 kV/cm and frequency 3000 Hz was applied to the PRC. Meanwhile the geometry of beams propagating in the crystal has provided mutual orthogonality of grating vectors of main and cross-holograms (see Fig.10). Indeed, the incidence plane (*xy*) of signal beams forming crossholograms is orthogonal to the incidence plane (*yz*) of the reference beam which forms the main holograms. Thus, an external electric field applied to the crystal along grating vectors of main holograms has provided enhancement of their diffraction efficiency (see Eq.(2)), while cross-holograms remained unamplified. Moreover the angle between any pair of signal beams is much smaller than that between any signal and reference beams. This makes the cross-holograms weaker in comparison with the main holograms, thus reducing the

The three-channel adaptive measurement system based on the above considered approach of dynamic holograms angular multiplexing was applied to the detection of ultra-sound pulse propagation in a composite structure material (CSM). Three sensing fibers were embedded between CSM layers as shown in Fig.11(a). A 5-MHz contact transducer was used to launch bulk waves into the composite. The ultrasonic wave packet was detected by

the sensing fibers, which were located at different depths within the composite.

) the

the corresponding signal waves.

in BSO crystal (b)

cross-talk between channels

Fig. 10. Geometry of multi-wave mixing in PRC providing angular multiplexing of transmission dynamic holograms: all signal waves belong to horizontal plane (*xy*) and propagate in direction close to axis *y*; the reference wave belongs to plane (*yz*) and makes the angle of 5º with plane (*xy*)

The normalized ultrasonic signals are shown in Fig. 11(b). From the measured signals it can be seen that the bulk longitudinal waves are clearly detected by all three channels with good sensitivity. The longitudinal wave arrives at different times at different sensor locations, providing information about the wave speed (related to material properties of the composite). The measured longitudinal velocity of 2480 m/s agrees well with data obtained for the same composite material by a different technique (Fomitchov, et al., 2001). Furthermore, the absence of any spurious signals in sensors 2 and 3 at the time of arrival of the ultrasonic wave packet at sensor 1 is an indication of the absence of cross-talk in the multichannel adaptive system based on dynamic holograms multiplexed in a PR crystal.

In spite of such a satisfactory performance the system proposed in (Fomitchov, et al., 2002) has a major drawback. This is the necessity of using strong external electric field which leads

Multi-Channel Adaptive Interferometers Based on Dynamic Hologram Multiplexing 119

crystallographic axis (see Eq.(5)). One can show that for the geometry depicted in Fig.12 the

0 1 <sup>ˆ</sup> 1 0 

coupled at hologram by the grating vector parallel to the principal crystallographic axis [001]. As seen in Fig.12 this requirement is satisfied for any pair of signal and reference

On the contrary, the grating vector **K12** of the cross-hologram formed by any pair of two signal waves is orthogonal to the axis [001]. By using Eq.(5) one can show that all

that, in such geometry, there is no coupling between signal waves even if they completely overlap, and hence there is no cross-talk between channels associated with these signal

However the situation changes if the orthogonality of a cross-hologram grating vector **K12** to direction [001] is violated (this becomes true if signal beams enter the crystal at different angles). Thus the efficiency of two signal waves coupling in this case will depends on the [001]-axis component of the vector **K12**. Taking into account that the efficiency of waves mixing at the main hologram (and hence the level of demodulation signal in a channel) is determined by wave number *K* (i.e. *K*1 or *K*2) in accordance with Eq.(3), one can show that

> sin 20log <sup>2</sup> *C h*

where is the angle between vector **K12** and normal to axis [001], 1 *h* is the coefficient

Figure 13 shows the dependency *C* from which it is seen that violation of orthogonality [001] **K12** leads to a rise in cross-talk. However, its level remains below the inherent noise of a channel (–30 dB) over a wide range of angles . The last provides wide angular aperture for entry of signal beams to a crystal (Fig.14). Thus, for the CdTe crystal having refractive index ~2,8 the aperture within which a cross-talk is practically absent can reach 60 degrees. Moreover the signal beams can belong to different planes (e.g., spaced in fantail way) in contrast to the previous multiplexing technique where they lay in same plane, see Fig.9(b). The wide angular range of signal beams input makes the formation of holographic channels

As it was shown in the paper (Di Girolamo, et al., 2008) another possible reason for crosstalk in the geometry of reflection holograms multiplexing is related to residual mechanical stresses or defects in PR crystal which locally disturb its symmetry and partially enable interaction of signal beams even in forbidden geometry (when ]001[ **K12** ). In this case matrix **H**ˆ can have non-zero components. However the cross-talk level still does not exceed channel inherent noise. A low level of cross-talk is obtained by low spatial frequency of cross hologram grating *K*12 which is much smaller than that of main holograms, *K*1 and *K*2. It worth noting that cross-talk related to crystal defects/stresses can be excluded (or at least

components of the coupling matrix **H**ˆ in this case will be zero,

the cross-talk level can be estimated with good accuracy by following

which takes into account degree of signal beams overlap in a crystal.

easy and increases their potential number.

minimized) by using crystals of better quality.

**H** ) only for counter propagating waves

0 0 <sup>ˆ</sup> 0 0 

, (14)

**H** . This means

matrix **H**ˆ has non-zero components (

waves.

waves.

to the technical problems common to all systems based on drift recorded dynamic holograms mentioned in the Introduction. Moreover, in order to realize a linear regime of phase demodulation the retarder plate installed after the crystal is used together with polarizer which introduces optical losses and produces polarization noise (Di Girolamo, et al., 2007b). In its turn, it was shown in the paper (Di Girolamo, et al., 2007a) that high efficiency of dynamic hologram (and, as sequence, high sensitivity of adaptive interferometer) can be achieved even without any external electric field if the hologram is recorded in reflection geometry at high spatial frequencies, *K*. Moreover, the linear mode of phase demodulation can be realized in this case also without any polarization filtering – due to VWM of waves with different polarization states (e.g., linear and elliptical) in a geometry which supports anisotropic diffraction of light (see Section 2). As was shown further in our paper (Di Girolamo, et al., 2008) a multiplexing of diffusion dynamic holograms of the reflection type can became a basis for an efficient multi-channel adaptive measurement system. Consider this approach in more details.

The geometry of reflection holograms multiplexing in a PR crystal of cubic symmetry is shown in Fig.12. Two signal waves S1 and S2 propagate in the crystal along its principal crystallographic axis [001] and mix with the reference wave R propagating in the opposite direction toward to them. As in the previous case, all mixed waves are mutually coherent. Thus their pair wise interference can lead in general to formation of dynamic holograms of two kinds – (i) the main DHs recorded by a reference and a signal beam and (ii) cross holograms recorded by pair of signal beams. The main holograms form demodulation channels while the presence of cross-holograms can become a source of cross-talk between channels. Moreover, the multiple recording of DHs in a single crystal can lead in general to reduction of diffraction efficiency of a particular dynamic hologram and, as sequence, to depletion of the sensitivity in particular channel. Both the rate of sensitivity decrease and the cross-talk level determine the performance of a multi-channel holographic demodulator. Consider this in more details.

Fig. 12. The geometry of multiplexing of two reflection dynamic holograms in a PR crystal: 1 2 , , **kk k** *RS S* are wave vectors of reference (R) and signal (S1, S2) light waves, respectively; 1 2 **K K**, are grating vectors of main holograms; **K**12 is the grating vector of cross-hologram Recall that two-wave mixing in PR crystal is described by system of coupled-wave equations (1) or (6) where **H**ˆ is the coupling matrix whose components are determined by the orientations of the grating vector and wave propagation directions with respect to the

to the technical problems common to all systems based on drift recorded dynamic holograms mentioned in the Introduction. Moreover, in order to realize a linear regime of phase demodulation the retarder plate installed after the crystal is used together with polarizer which introduces optical losses and produces polarization noise (Di Girolamo, et al., 2007b). In its turn, it was shown in the paper (Di Girolamo, et al., 2007a) that high efficiency of dynamic hologram (and, as sequence, high sensitivity of adaptive interferometer) can be achieved even without any external electric field if the hologram is recorded in reflection geometry at high spatial frequencies, *K*. Moreover, the linear mode of phase demodulation can be realized in this case also without any polarization filtering – due to VWM of waves with different polarization states (e.g., linear and elliptical) in a geometry which supports anisotropic diffraction of light (see Section 2). As was shown further in our paper (Di Girolamo, et al., 2008) a multiplexing of diffusion dynamic holograms of the reflection type can became a basis for an efficient multi-channel adaptive measurement

The geometry of reflection holograms multiplexing in a PR crystal of cubic symmetry is shown in Fig.12. Two signal waves S1 and S2 propagate in the crystal along its principal crystallographic axis [001] and mix with the reference wave R propagating in the opposite direction toward to them. As in the previous case, all mixed waves are mutually coherent. Thus their pair wise interference can lead in general to formation of dynamic holograms of two kinds – (i) the main DHs recorded by a reference and a signal beam and (ii) cross holograms recorded by pair of signal beams. The main holograms form demodulation channels while the presence of cross-holograms can become a source of cross-talk between channels. Moreover, the multiple recording of DHs in a single crystal can lead in general to reduction of diffraction efficiency of a particular dynamic hologram and, as sequence, to depletion of the sensitivity in particular channel. Both the rate of sensitivity decrease and the cross-talk level determine the performance of a multi-channel holographic demodulator.

Fig. 12. The geometry of multiplexing of two reflection dynamic holograms in a PR crystal: 1 2 , , **kk k** *RS S* are wave vectors of reference (R) and signal (S1, S2) light waves, respectively; 1 2 **K K**, are grating vectors of main holograms; **K**12 is the grating vector of cross-hologram Recall that two-wave mixing in PR crystal is described by system of coupled-wave equations (1) or (6) where **H**ˆ is the coupling matrix whose components are determined by the orientations of the grating vector and wave propagation directions with respect to the

[100]

S1

S2

**k***R*

[010] [001]

**K1 k***<sup>S</sup>***<sup>1</sup>**

**K2**

**k***<sup>S</sup>***<sup>2</sup>**

**K12** R

system. Consider this approach in more details.

Consider this in more details.

crystallographic axis (see Eq.(5)). One can show that for the geometry depicted in Fig.12 the matrix **H**ˆ has non-zero components ( 0 1 <sup>ˆ</sup> 1 0 **H** ) only for counter propagating waves

coupled at hologram by the grating vector parallel to the principal crystallographic axis [001]. As seen in Fig.12 this requirement is satisfied for any pair of signal and reference waves.

On the contrary, the grating vector **K12** of the cross-hologram formed by any pair of two signal waves is orthogonal to the axis [001]. By using Eq.(5) one can show that all components of the coupling matrix **H**ˆ in this case will be zero, 0 0 <sup>ˆ</sup> 0 0 **H** . This means

that, in such geometry, there is no coupling between signal waves even if they completely overlap, and hence there is no cross-talk between channels associated with these signal waves.

However the situation changes if the orthogonality of a cross-hologram grating vector **K12** to direction [001] is violated (this becomes true if signal beams enter the crystal at different angles). Thus the efficiency of two signal waves coupling in this case will depends on the [001]-axis component of the vector **K12**. Taking into account that the efficiency of waves mixing at the main hologram (and hence the level of demodulation signal in a channel) is determined by wave number *K* (i.e. *K*1 or *K*2) in accordance with Eq.(3), one can show that the cross-talk level can be estimated with good accuracy by following

$$C \approx 20\log\left(h\frac{|\sin\Theta|}{2}\right),\tag{14}$$

where is the angle between vector **K12** and normal to axis [001], 1 *h* is the coefficient which takes into account degree of signal beams overlap in a crystal.

Figure 13 shows the dependency *C* from which it is seen that violation of orthogonality [001] **K12** leads to a rise in cross-talk. However, its level remains below the inherent noise of a channel (–30 dB) over a wide range of angles . The last provides wide angular aperture for entry of signal beams to a crystal (Fig.14). Thus, for the CdTe crystal having refractive index ~2,8 the aperture within which a cross-talk is practically absent can reach 60 degrees. Moreover the signal beams can belong to different planes (e.g., spaced in fantail way) in contrast to the previous multiplexing technique where they lay in same plane, see Fig.9(b). The wide angular range of signal beams input makes the formation of holographic channels easy and increases their potential number.

As it was shown in the paper (Di Girolamo, et al., 2008) another possible reason for crosstalk in the geometry of reflection holograms multiplexing is related to residual mechanical stresses or defects in PR crystal which locally disturb its symmetry and partially enable interaction of signal beams even in forbidden geometry (when ]001[ **K12** ). In this case matrix **H**ˆ can have non-zero components. However the cross-talk level still does not exceed channel inherent noise. A low level of cross-talk is obtained by low spatial frequency of cross hologram grating *K*12 which is much smaller than that of main holograms, *K*1 and *K*2. It worth noting that cross-talk related to crystal defects/stresses can be excluded (or at least minimized) by using crystals of better quality.

Multi-Channel Adaptive Interferometers Based on Dynamic Hologram Multiplexing 121

It is worth noting that maximum sensitivity of an adaptive interferometer based on a dynamic hologram is achieved at a certain ratio of mixed beam intensities (Di Girolamo, et al., 2008). Thus, taking into account that a common reference beam is used in the considered geometry of hologram multiplexing the optimized multichannel system will have signal beams of equal intensity, ,( 1, ) *j S I I j N* . Then, Eq.(15) for contrast of interference pattern in one of *N* multiplexed channels can be reduced with good accuracy to the following

<sup>2</sup> ( ) *S R*

By solving the system of coupled wave equations (1) talking into account Eq.(16) one can analyze one channel performance under conditions of decreasing number of channels.

number *N*. As seen, increase of *N* is accompanied by increase of detection limit (equivalent to reduction of sensitivity, see Eq.(11)) in each channel. However this increase is not critical and depends on the crystal sample. Thus, formation of 10 channels in one sample of CdTe

amounts to just 23%. Taking into account the extremely high sensitivity of an adaptive interferometer which approaches that of classical interferometer (Kamshilin, et al., 2009), one can conclude that even a reduction of sensitivity by a factor of two can be considered as not significant and remains acceptable in most practical applications. Additionally mutual influence of multiplexed channels can be diminished by reduction of the degree of signal

Fig. 15. Dependence of relative detection limit in one channel of holographic multi-channel

The scheme of a two-channel adaptive interferometer based on angular multiplexing of reflection dynamic holograms in a photorefractive crystal CdTe in the above-considered geometry is shown in Fig. 16. Two signal beams coming from two sensing optical fibers are focused by lenses into the crystal where they are mixed with a reference beam propagating in the opposite direction along the crystallographic axis [001]. Calibrated piezoelectric transducers attached to the fibers simulate the action of measurand and introduce phase modulation with amplitudes 0.30 and 0.15 rad in the first and the second channels, respectively. This is equivalent to a detection of mechanical vibration with amplitude of 25

*m N*

Figure 15 shows dependency of the relative detection limit *rel*

beams overlap *h* (see Eq.(15)) using their tighter focusing.

system on number of multiplexed channels

crystal leads to doubling of *rel*

*R S I I*

*I hNI* . (16)

on multiplexed channels

while for another sample of same crystal this increase

Fig. 13. Cross-talk between channels as function of angle between cross-hologram grating vector and normal to the crystal axis [001]: line represents theoretical dependency calculated in accordance with Eq.(14); markers are experimental data

As seen from Eqs.(2), (6), the efficiency of wave coupling at a dynamic hologram (and hence demodulation signal amplitude) is defined by the interference pattern contrast \* 12 0 *m I* 2 / **A A** . However this expression is valid only for the two-wave mixing case. Introduction of additional signal waves (related with other channels) changes the contrast.

Fig. 14. Spatial arrangement of beams providing multiple recording of reflection dynamic holograms in a photorefractive crystal without noticeable crosstalk

In spite of mutual coherence of all light beams crossed in the crystal, the change of contrast is related mostly to increase of total light intensity. High angular selectivity of volume holograms allows one to exclude contributions to the variable part of the contrast from additional interference patterns having spatial frequencies and orientation different from those of main hologram. Thus the contrast of the interference pattern in the one of *N* channels can be estimated by

$$m(N) = \frac{2\sqrt{I\_S I\_R}}{I\_R + I\_S + h\sum\_{j=1}^{N-1} I\_j} \tag{15}$$

where *IR* , *IS* and *Ij* are the intensities of reference beam, signal beam of a particular channel and signal beam of the *j-*th additional channel, respectively.

Fig. 13. Cross-talk between channels as function of angle between cross-hologram grating vector and normal to the crystal axis [001]: line represents theoretical dependency calculated

As seen from Eqs.(2), (6), the efficiency of wave coupling at a dynamic hologram (and hence demodulation signal amplitude) is defined by the interference pattern contrast \* 12 0 *m I* 2 / **A A** . However this expression is valid only for the two-wave mixing case. Introduction of additional signal waves (related with other channels) changes the contrast.

Fig. 14. Spatial arrangement of beams providing multiple recording of reflection dynamic

In spite of mutual coherence of all light beams crossed in the crystal, the change of contrast is related mostly to increase of total light intensity. High angular selectivity of volume holograms allows one to exclude contributions to the variable part of the contrast from additional interference patterns having spatial frequencies and orientation different from those of main hologram. Thus the contrast of the interference pattern in the one of *N*

> 2 ( ) *S R*

where *IR* , *IS* and *Ij* are the intensities of reference beam, signal beam of a particular channel

1

, (15)

*N R S j j*

*I I*

*IIhI*

1

holograms in a photorefractive crystal without noticeable crosstalk

*m N*

and signal beam of the *j-*th additional channel, respectively.

channels can be estimated by

in accordance with Eq.(14); markers are experimental data

It is worth noting that maximum sensitivity of an adaptive interferometer based on a dynamic hologram is achieved at a certain ratio of mixed beam intensities (Di Girolamo, et al., 2008). Thus, taking into account that a common reference beam is used in the considered geometry of hologram multiplexing the optimized multichannel system will have signal beams of equal intensity, ,( 1, ) *j S I I j N* . Then, Eq.(15) for contrast of interference pattern in one of *N* multiplexed channels can be reduced with good accuracy to the following

$$m(\text{N}) \approx \frac{2\sqrt{I\_S I\_R}}{I\_R + hN I\_S} \,. \tag{16}$$

By solving the system of coupled wave equations (1) talking into account Eq.(16) one can analyze one channel performance under conditions of decreasing number of channels. Figure 15 shows dependency of the relative detection limit *rel* on multiplexed channels number *N*. As seen, increase of *N* is accompanied by increase of detection limit (equivalent to reduction of sensitivity, see Eq.(11)) in each channel. However this increase is not critical and depends on the crystal sample. Thus, formation of 10 channels in one sample of CdTe crystal leads to doubling of *rel* while for another sample of same crystal this increase amounts to just 23%. Taking into account the extremely high sensitivity of an adaptive interferometer which approaches that of classical interferometer (Kamshilin, et al., 2009), one can conclude that even a reduction of sensitivity by a factor of two can be considered as not significant and remains acceptable in most practical applications. Additionally mutual influence of multiplexed channels can be diminished by reduction of the degree of signal beams overlap *h* (see Eq.(15)) using their tighter focusing.

Fig. 15. Dependence of relative detection limit in one channel of holographic multi-channel system on number of multiplexed channels

The scheme of a two-channel adaptive interferometer based on angular multiplexing of reflection dynamic holograms in a photorefractive crystal CdTe in the above-considered geometry is shown in Fig. 16. Two signal beams coming from two sensing optical fibers are focused by lenses into the crystal where they are mixed with a reference beam propagating in the opposite direction along the crystallographic axis [001]. Calibrated piezoelectric transducers attached to the fibers simulate the action of measurand and introduce phase modulation with amplitudes 0.30 and 0.15 rad in the first and the second channels, respectively. This is equivalent to a detection of mechanical vibration with amplitude of 25

Multi-Channel Adaptive Interferometers Based on Dynamic Hologram Multiplexing 123

transmittivity (equivalently reflectivity). The spectral shifts can be monitored in several ways with an appropriate demodulator. For spectrally encoded FBG sensors, extant demodulation schemes can be classified into three categories: scanning type, spectrometry based, and interferometry based. Scanning-type techniques include Fabry–Perot scanning filters (Kersey, et al., 1993), acoustooptic tunable filters (Xu, et al., 1993), and tunable laser sources (Fomitchov & Krishnaswamy, 2003). All these scanning-type techniques suffer from the fact that at any instant only one FBG sensor can be interrogated. Such approaches are not applicable if all the sensors have to be interrogated *simultaneously* for the purpose of monitoring impact signals and acoustic emissions (Perez, et al., 2001). Spectrometric methods (Davis & Kersey, 1995) suffer from low sensitivity and are not suitable for dynamic measurements if several sensors have to be active at all times. Interferometric methods such as the Mach–Zehnder interferometer (Kersey, et al., 1992) are ideally suited to monitor dynamic strains; however, they require electronic feedback to actively compensate for any quasi-static drift to maintain the interferometer at quadrature. This makes the cost of multiplexing high since each sensor requires its own feedback system. Therefore, a cost-effective multichannel demodulator capable of providing not only simultaneous interrogation of FBG-sensor arrays but also stable

In the paper (Qiao, et al., 2006) authors show that such a demodulator can be developed on the basis of dynamic holograms formed in a photorefractive crystal. Consider this in more

Due to its adaptive properties a PRC-based demodulator can compensate all slow (including quasi-static) impacts on sensors and provide stable detection of high-frequency or pulse measurands (e.g., vibration, ultra-sonic waves, acoustic emission, etc.) without using any feedback loops or external stabilizing electronic circuits. As will be considered below, the demodulator based on two-wave mixing in a PR crystal can provide both spectral decoding

The spectral shift induced by strain, *εF*, and temperature change, *T*, in a Bragg-grating

where, *λ<sup>B</sup>* is the center wavelength of the FBG sensor, *λB,* is the wavelength shift caused by strain or temperature, *n*eff is the effective refractive index of the fiber, *pij* (*i*, *j* = 1, 2) are the

1550 nm, it has been estimated that one microstrain (1×10–6) will lead to a 1.2 pm change in wavelength, and a 1 °C change in temperature will lead to about 13 pm change in wavelength (Othonos & Kalli, 1999). This fact emphasizes again the necessity of

The adaptive interferometer based on dynamic holograms provides transformation of transient phase shift to intensity modulation of signal beam, while the signal of FBG-sensor is encrypted in a shift of central wavelengths in the reflection spectra. At the same time the adaptive interferometer also can be configured as a *spectral* demodulator. Indeed, a spectral shift

the signal *and* reference beams can be effectively converted into a relative phase shift

the beams due to travel through unbalanced optical paths (Othonos & Kalli, 1999):

 

*<sup>N</sup>* is the thermo-optic coefficient of the fiber. For typical Bragg sensors at

*T*

, (17)

*<sup>B</sup>* of

*between* 

*<sup>T</sup>* is the thermal expansion

12 11 12 1 ( )( ) <sup>2</sup> *B B F TN*

performance in unstable environment is required.

sensor can be written as (Murray, et al., 2000):

 

of FBG-sensor signals and their effective spectral demultiplexing.

2 eff

*p up p*

*n*

components of the strainoptic tensor, *u* is Poisson's ratio,

compensation for environment effects in the measurement system.

details.

coefficient, and

and 13 nm (at wavelength 1.06 m), respectively. Oscilloscope traces of detected signals in two channels of the adaptive interferometer are shown in Fig.17. As seen the measurement system provides good SNR in both channels in spite of the weak level of measured vibration and wide-band detection mode (Δ*f* = 50 MHz). Moreover, the two channels operate independently without any detectable cross-talk.

Fig. 16. Layout (a) and partial photo (b) of two-channel adaptive interferometer based on dynamic reflection holograms angularly multiplexed in PR crystal: PD1 and PD2 are phototdetectors which register signal beam intensities in 1st and 2nd channels

Fig. 17. Snapshots of oscilloscope traces demonstrating the operation of a two-channel adaptive measurement system: the first channel (left) and the second channel (right); (A) modulation signal which excites vibration in sensing fibers, (B) signal detected in a channel

### **6. Spectral multiplexing of dynamic holograms**

Spectral (or wavelength division) multiplexing is one of effective techniques for building multichannel optical and fiber-optical measurement systems. In particular, this approach is highly demanded if fiber Bragg gratings (FBG) are used as sensors (Grattan & Meggitt, 1998). FBG sensors are an increasingly important emerging technology in the area of intelligent structural health monitoring (SHM) of civil, mechanical, naval, and aerospace structures (Claus, 1992; Culshaw & Dakin, 1996). A large number of FBG sensors can be easily written in a single fiber. A measurand (for SHM applications, FBG sensors are typically used to monitor static or dynamic strains) causes spectral shifts in the FBG sensor

and 13 nm (at wavelength 1.06 m), respectively. Oscilloscope traces of detected signals in two channels of the adaptive interferometer are shown in Fig.17. As seen the measurement system provides good SNR in both channels in spite of the weak level of measured vibration and wide-band detection mode (Δ*f* = 50 MHz). Moreover, the two channels operate

(a) (b)

Fig. 16. Layout (a) and partial photo (b) of two-channel adaptive interferometer based on dynamic reflection holograms angularly multiplexed in PR crystal: PD1 and PD2 are

Fig. 17. Snapshots of oscilloscope traces demonstrating the operation of a two-channel adaptive measurement system: the first channel (left) and the second channel (right); (A) modulation signal which excites vibration in sensing fibers, (B) signal detected in a channel

Spectral (or wavelength division) multiplexing is one of effective techniques for building multichannel optical and fiber-optical measurement systems. In particular, this approach is highly demanded if fiber Bragg gratings (FBG) are used as sensors (Grattan & Meggitt, 1998). FBG sensors are an increasingly important emerging technology in the area of intelligent structural health monitoring (SHM) of civil, mechanical, naval, and aerospace structures (Claus, 1992; Culshaw & Dakin, 1996). A large number of FBG sensors can be easily written in a single fiber. A measurand (for SHM applications, FBG sensors are typically used to monitor static or dynamic strains) causes spectral shifts in the FBG sensor

**6. Spectral multiplexing of dynamic holograms** 

phototdetectors which register signal beam intensities in 1st and 2nd channels

independently without any detectable cross-talk.

transmittivity (equivalently reflectivity). The spectral shifts can be monitored in several ways with an appropriate demodulator. For spectrally encoded FBG sensors, extant demodulation schemes can be classified into three categories: scanning type, spectrometry based, and interferometry based. Scanning-type techniques include Fabry–Perot scanning filters (Kersey, et al., 1993), acoustooptic tunable filters (Xu, et al., 1993), and tunable laser sources (Fomitchov & Krishnaswamy, 2003). All these scanning-type techniques suffer from the fact that at any instant only one FBG sensor can be interrogated. Such approaches are not applicable if all the sensors have to be interrogated *simultaneously* for the purpose of monitoring impact signals and acoustic emissions (Perez, et al., 2001). Spectrometric methods (Davis & Kersey, 1995) suffer from low sensitivity and are not suitable for dynamic measurements if several sensors have to be active at all times. Interferometric methods such as the Mach–Zehnder interferometer (Kersey, et al., 1992) are ideally suited to monitor dynamic strains; however, they require electronic feedback to actively compensate for any quasi-static drift to maintain the interferometer at quadrature. This makes the cost of multiplexing high since each sensor requires its own feedback system. Therefore, a cost-effective multichannel demodulator capable of providing not only simultaneous interrogation of FBG-sensor arrays but also stable performance in unstable environment is required.

In the paper (Qiao, et al., 2006) authors show that such a demodulator can be developed on the basis of dynamic holograms formed in a photorefractive crystal. Consider this in more details.

Due to its adaptive properties a PRC-based demodulator can compensate all slow (including quasi-static) impacts on sensors and provide stable detection of high-frequency or pulse measurands (e.g., vibration, ultra-sonic waves, acoustic emission, etc.) without using any feedback loops or external stabilizing electronic circuits. As will be considered below, the demodulator based on two-wave mixing in a PR crystal can provide both spectral decoding of FBG-sensor signals and their effective spectral demultiplexing.

The spectral shift induced by strain, *εF*, and temperature change, *T*, in a Bragg-grating sensor can be written as (Murray, et al., 2000):

$$
\Delta\lambda\_{\rm B} = \lambda\_{\rm B} \left\{ 1 - \frac{n\_{\rm eff}^2}{2} \left[ p\_{12} - \mu (p\_{11} + p\_{12}) \right] \varepsilon\_{\rm F} + (a\_{\rm T} + a\_{\rm N}) \Delta T \right\}, \tag{17}
$$

where, *λ<sup>B</sup>* is the center wavelength of the FBG sensor, *λB,* is the wavelength shift caused by strain or temperature, *n*eff is the effective refractive index of the fiber, *pij* (*i*, *j* = 1, 2) are the components of the strainoptic tensor, *u* is Poisson's ratio, *<sup>T</sup>* is the thermal expansion coefficient, and*<sup>N</sup>* is the thermo-optic coefficient of the fiber. For typical Bragg sensors at 1550 nm, it has been estimated that one microstrain (1×10–6) will lead to a 1.2 pm change in wavelength, and a 1 °C change in temperature will lead to about 13 pm change in wavelength (Othonos & Kalli, 1999). This fact emphasizes again the necessity of compensation for environment effects in the measurement system.

The adaptive interferometer based on dynamic holograms provides transformation of transient phase shift to intensity modulation of signal beam, while the signal of FBG-sensor is encrypted in a shift of central wavelengths in the reflection spectra. At the same time the adaptive interferometer also can be configured as a *spectral* demodulator. Indeed, a spectral shift *<sup>B</sup>* of the signal *and* reference beams can be effectively converted into a relative phase shift *between*  the beams due to travel through unbalanced optical paths (Othonos & Kalli, 1999):

Multi-Channel Adaptive Interferometers Based on Dynamic Hologram Multiplexing 125

and as the OPD increases the signal amplitude increases to a maximum beyond which the signal starts to drop due to decreasing fringe visibility. For each given line width of the FBG sensor, there exists an optimum value of the OPD that maximizes the wavelengthdemodulated signal. It is also clear from Fig. 18 that the narrower the linewidth, the larger the optimum OPD, and therefore the larger the demodulated signal. However, it is not always better to use a narrower linewidth FBG sensor. First, a narrower linewidth FBG sensor is usually longer in length (Othonos & Kalli, 1999), which decreases the highest frequency to which the FBG can respond (Coppola, et al., 2001). Second, a larger OPD

Fig. 18. Wavelength demodulated signal amplitude as a function of the OPD for different

The scheme of the measurement system based on FBG sensor and photorefractive dynamic hologram proposed in (Qiao, et al., 2006) is shown in Fig. 19. The FBG sensor is illuminated by a broadband amplified spontaneous emission (ASE) source in the C band (1530 to 1570 nm), and the reflected light is coupled by a circulator into an erbium-doped fiber amplifier (EDFA) that works in saturation mode (output 500 mW). The amplified light is split using a 1 × 2 coupler (splitting ratio 95/5) into reference (95*%*) and signal (5*%*) beams that travel unbalanced optical paths to the InP:Fe photorefractive crystal. The InP:Fe crystal is oriented for two-wave mixing in the direct detection configuration (Delaye, et al., 1997), in which both beams enter the crystal by the (110) face. An external dc field (6 kV/cm) was applied along [001] direction to provide the quadrature condition of phase detection. The authors note that to apply a continuous dc field across the InP:Fe crystal, a Peltier cooler was used to

In the work (Qiao, et al., 2006) a FBG sensor centered at 1552 nm with a linewidth of 0.1 nm, length of 10 mm, and reflectivity of 50% was glued onto a piezoelectric transducer (PZT) stretcher that was used to induce known dynamic strains (with amplitude 10 strain at frequency 10 kHz). The light reflected from the FBG sensor undergoes spectral shift due to strain-induced changes in the Bragg reflectivity. The sensor by itself is sensitive to both quasi-static and dynamic strains, and is also subject to temperature drift. However the TWM

decreases the dynamic range that the FBG sensor can measure.

prevent electrical breakdown due to crystal overheating.

linewidth FBG sensors

$$
\Delta\phi(t) = \frac{2\pi b}{\mathcal{J}\_{\text{B}}^2} \Delta\mathcal{J}\_{\text{B'}} \tag{18}
$$

where *b* is the optical path difference (OPD). It is noteworthy that a similar principle of spectra change transformation to phase modulation is used as well in Mach-Zehnder and other unbalanced interferometers (Kersey, et al., 1992).

From Eq. (18), it appears that the greater the OPD, the larger the equivalent phase shift, and therefore the stronger the interference signal should be. However, typically broadband light sources are used to illuminate the FBG sensors, and the FBG reflection spectrum typically has a finite line width of the order of 0.1–0.4 nm. This implies that coherence of the two interfering beams needs to be taken into account both in the photorefractive grating creating process, and in the subsequent interference between the transmitted signal and the diffracted reference beams. The fringe visibility due to the interference of two beams of finite spectral width *k* is given by

$$m(b) = \frac{2\sqrt{\beta}}{\beta + 1} \exp\left\{-\frac{\Delta k^2 b^2}{16\ln 2}\right\} \,\, \, \, \tag{19}$$

where *β* is the intensity ratio of reference and signal beams. The factor before *exp*-function in Eq.(19) is nothing other than interference fringe contrast *m* defining space charge field in PR crystal (see Eq.(2)). Incorporating the degradation in fringe visibility due to low coherence in the TWM analysis (by solving the coupled-wave equations) leads to the following expression for spectral-shift demodulated interference signal:

$$
\Delta I\_{\text{-S}} \propto \left[ \exp(\gamma' L) \sin(\gamma'' L) \right] \exp\left\{ -\frac{\Delta k^2 b^2}{16 \ln 2} \right\} \frac{b}{\lambda\_{\text{B}}^2} \Delta \lambda\_{\text{B}} \,\,\,\tag{20}
$$

where *i* is the TWM complex gain and *L* is the crystal length in the beam propagation direction.

The effect of decreased fringe visibility on the photorefractive grating formation process is not a significant factor, since the preferred mode of two-wave mixing in PRCs is in the low intensity modulation regime (this avoids the creation of higher order index gratings that could lead to cross talk when the system is multiplexed). Furthermore, the system is operated at near quadrature, when *L* / 2 (Solymar, et al., 1996), and since the TWM energy gain, , is typically very small, the variations in the first two terms with the OPD are not significant. The dependence of the interference signal on the OPD is therefore predominantly given by

$$
\Delta I\_{\text{\tiny \text{S}}} \propto \exp\left\{-\frac{\Delta k^2 b^2}{16\ln 2}\right\} \frac{b}{\lambda\_{\text{\tiny \text{B}}}^2} \Delta \lambda\_{\text{\tiny \text{B}}} \,\text{.}\tag{21}
$$

As seen from Eq.(21), the wavelength demodulation signal is a strong function of the OPD *d*  with the exponential decay arising from loss of coherence, and the linear increase arising from the increase with OPD in the phase shift due to spectral change. Figure 18 is a plot of the signal amplitude versus the OPD for different linewidth FBG sensors. As indicated in Fig. 18, when the OPD equals zero, there is no wavelength-demodulated signal detected,

other unbalanced interferometers (Kersey, et al., 1992).

finite spectral width *k* is given by

where

 

energy gain,

propagation direction.

operated at near quadrature, when

predominantly given by

2 <sup>2</sup> ( ) *<sup>B</sup> B <sup>b</sup> <sup>t</sup>* 

where *b* is the optical path difference (OPD). It is noteworthy that a similar principle of spectra change transformation to phase modulation is used as well in Mach-Zehnder and

From Eq. (18), it appears that the greater the OPD, the larger the equivalent phase shift, and therefore the stronger the interference signal should be. However, typically broadband light sources are used to illuminate the FBG sensors, and the FBG reflection spectrum typically has a finite line width of the order of 0.1–0.4 nm. This implies that coherence of the two interfering beams needs to be taken into account both in the photorefractive grating creating process, and in the subsequent interference between the transmitted signal and the diffracted reference beams. The fringe visibility due to the interference of two beams of

> 2 2 <sup>2</sup> ( ) exp 1 16ln 2 *k b m b*

where *β* is the intensity ratio of reference and signal beams. The factor before *exp*-function in Eq.(19) is nothing other than interference fringe contrast *m* defining space charge field in PR crystal (see Eq.(2)). Incorporating the degradation in fringe visibility due to low coherence in the TWM analysis (by solving the coupled-wave equations) leads to the following

<sup>2</sup> exp( )sin( ) exp 16ln 2 *S B*

The effect of decreased fringe visibility on the photorefractive grating formation process is not a significant factor, since the preferred mode of two-wave mixing in PRCs is in the low intensity modulation regime (this avoids the creation of higher order index gratings that could lead to cross talk when the system is multiplexed). Furthermore, the system is

are not significant. The dependence of the interference signal on the OPD is therefore

*kb b <sup>I</sup>*

 

As seen from Eq.(21), the wavelength demodulation signal is a strong function of the OPD *d*  with the exponential decay arising from loss of coherence, and the linear increase arising from the increase with OPD in the phase shift due to spectral change. Figure 18 is a plot of the signal amplitude versus the OPD for different linewidth FBG sensors. As indicated in Fig. 18, when the OPD equals zero, there is no wavelength-demodulated signal detected,

2 2 <sup>2</sup> exp 16ln 2 *S B*

*kb b I LL*

 

2 2

*i* is the TWM complex gain and *L* is the crystal length in the beam

, is typically very small, the variations in the first two terms with the OPD

*B*

*B*

 

/ 2 (Solymar, et al., 1996), and since the TWM

. (21)

 *L* 

expression for spectral-shift demodulated interference signal:

, (18)

, (19)

, (20)

and as the OPD increases the signal amplitude increases to a maximum beyond which the signal starts to drop due to decreasing fringe visibility. For each given line width of the FBG sensor, there exists an optimum value of the OPD that maximizes the wavelengthdemodulated signal. It is also clear from Fig. 18 that the narrower the linewidth, the larger the optimum OPD, and therefore the larger the demodulated signal. However, it is not always better to use a narrower linewidth FBG sensor. First, a narrower linewidth FBG sensor is usually longer in length (Othonos & Kalli, 1999), which decreases the highest frequency to which the FBG can respond (Coppola, et al., 2001). Second, a larger OPD decreases the dynamic range that the FBG sensor can measure.

Fig. 18. Wavelength demodulated signal amplitude as a function of the OPD for different linewidth FBG sensors

The scheme of the measurement system based on FBG sensor and photorefractive dynamic hologram proposed in (Qiao, et al., 2006) is shown in Fig. 19. The FBG sensor is illuminated by a broadband amplified spontaneous emission (ASE) source in the C band (1530 to 1570 nm), and the reflected light is coupled by a circulator into an erbium-doped fiber amplifier (EDFA) that works in saturation mode (output 500 mW). The amplified light is split using a 1 × 2 coupler (splitting ratio 95/5) into reference (95*%*) and signal (5*%*) beams that travel unbalanced optical paths to the InP:Fe photorefractive crystal. The InP:Fe crystal is oriented for two-wave mixing in the direct detection configuration (Delaye, et al., 1997), in which both beams enter the crystal by the (110) face. An external dc field (6 kV/cm) was applied along [001] direction to provide the quadrature condition of phase detection. The authors note that to apply a continuous dc field across the InP:Fe crystal, a Peltier cooler was used to prevent electrical breakdown due to crystal overheating.

In the work (Qiao, et al., 2006) a FBG sensor centered at 1552 nm with a linewidth of 0.1 nm, length of 10 mm, and reflectivity of 50% was glued onto a piezoelectric transducer (PZT) stretcher that was used to induce known dynamic strains (with amplitude 10 strain at frequency 10 kHz). The light reflected from the FBG sensor undergoes spectral shift due to strain-induced changes in the Bragg reflectivity. The sensor by itself is sensitive to both quasi-static and dynamic strains, and is also subject to temperature drift. However the TWM

Multi-Channel Adaptive Interferometers Based on Dynamic Hologram Multiplexing 127

Fig. 21. Plot of wavelength demodulated signal amplitude and TWM gain versus OPD

Also, as was mentioned above, the TWM energy gain

the intensity noise.

communication systems.

reference beam angle *θ*:

Figure 21 is a plot of the wavelength demodulated signal amplitude and TWM energy gain versus the OPD. The optimum OPD that maximizes the wavelength demodulated signal for the 0.1 nm linewidth FBG sensor is found to be 8 mm. For an OPD of 8 mm, Eq. (18) indicates a wavelength-to-phase shift conversion sensitivity of about 21 radians per nanometer wavelength shift at 1550 nm wavelength. This translates to 0.0252 radian/με. Such phase shifts are readily detectable by the TWM interferometer. Also note that the trend of the signal amplitude curve is similar to that of the theoretical curve shown in Fig. 18.

experimentally is found to vary from 0.47 to 0.1 cm-1 as the OPD changes from 0 to 12 mm. This causes the first exponential term in Eq. (20) to vary from 1.6 to 1.1, which is much

The minimum detectable strain with the setup of adaptive interferometer proposed and realized in (Qiao, et al., 2006) was measured to be 0.25 με, corresponding to a spectral shift of 0.3 pm. The minimum detectable spectral shift, which is limited by the ASE source and EDFA intensity noise, can be further improved by using balanced photodetection to cancel

Let us consider now how the multi-channel measurement system can be created on the basis of decoding of spectral signals from FBG sensors in PR crystal. The general principle is following (Qiao, et al., 2006). Optical signals from FBG sensors with distinct spectral reflectivities and center wavelength separation, *λC*, propagate in common optical path and enter a single PR crystal where they record a set of dynamic holograms which in its turn provides demodulation of spectrally encoded signals. Then the signals coming from different FBGs are separated by means of band-drop filters widely used in optical

The distance between FBG central wavelengths *λC* is chosen to be sufficiently large so that stationary optical interference between the multiple channels cannot occur. In this case, inside the PRC each channel creates its own index grating of different grating pitch. The change in the index grating pitch, , is related to the channel separation and the signal and

> 2sin( /2) *C*

. (22)

smaller than the change in the second exponential term that decreases from 1 to 0.03.

is indeed a small number and

demodulator system will automatically compensate for quasi-static drifts and track only the dynamic strains.

Oscilloscope traces of demodulated signal obtained at different values of the optical path difference are shown in Fig. 20. As seen, an intermittent dc field is applied from 1 to 6 ms with respect to a reference trigger, and the photorefractive grating initially builds up. The dynamic strain is applied as a tone burst starting from 2 to 6 ms. When the OPD equals to zero, although the TWM energy gain is at its maximum, there is no detected wavelength demodulated signal because there is no OPD to convert the wavelength shift into phase shift. As the OPD increases, the wavelength demodulated signal starts to appear.

Fig. 19. Experimental configuration of the FBG sensor and the TWM wavelength demodulator

Fig. 20. Wavelength demodulated signal at different values of OPD

demodulator system will automatically compensate for quasi-static drifts and track only the

Oscilloscope traces of demodulated signal obtained at different values of the optical path difference are shown in Fig. 20. As seen, an intermittent dc field is applied from 1 to 6 ms with respect to a reference trigger, and the photorefractive grating initially builds up. The dynamic strain is applied as a tone burst starting from 2 to 6 ms. When the OPD equals to zero, although the TWM energy gain is at its maximum, there is no detected wavelength demodulated signal because there is no OPD to convert the wavelength shift into phase

shift. As the OPD increases, the wavelength demodulated signal starts to appear.

Fig. 19. Experimental configuration of the FBG sensor and the TWM wavelength

Fig. 20. Wavelength demodulated signal at different values of OPD

dynamic strains.

demodulator

Fig. 21. Plot of wavelength demodulated signal amplitude and TWM gain versus OPD

Figure 21 is a plot of the wavelength demodulated signal amplitude and TWM energy gain versus the OPD. The optimum OPD that maximizes the wavelength demodulated signal for the 0.1 nm linewidth FBG sensor is found to be 8 mm. For an OPD of 8 mm, Eq. (18) indicates a wavelength-to-phase shift conversion sensitivity of about 21 radians per nanometer wavelength shift at 1550 nm wavelength. This translates to 0.0252 radian/με. Such phase shifts are readily detectable by the TWM interferometer. Also note that the trend of the signal amplitude curve is similar to that of the theoretical curve shown in Fig. 18. Also, as was mentioned above, the TWM energy gain is indeed a small number and experimentally is found to vary from 0.47 to 0.1 cm-1 as the OPD changes from 0 to 12 mm. This causes the first exponential term in Eq. (20) to vary from 1.6 to 1.1, which is much smaller than the change in the second exponential term that decreases from 1 to 0.03.

The minimum detectable strain with the setup of adaptive interferometer proposed and realized in (Qiao, et al., 2006) was measured to be 0.25 με, corresponding to a spectral shift of 0.3 pm. The minimum detectable spectral shift, which is limited by the ASE source and EDFA intensity noise, can be further improved by using balanced photodetection to cancel the intensity noise.

Let us consider now how the multi-channel measurement system can be created on the basis of decoding of spectral signals from FBG sensors in PR crystal. The general principle is following (Qiao, et al., 2006). Optical signals from FBG sensors with distinct spectral reflectivities and center wavelength separation, *λC*, propagate in common optical path and enter a single PR crystal where they record a set of dynamic holograms which in its turn provides demodulation of spectrally encoded signals. Then the signals coming from different FBGs are separated by means of band-drop filters widely used in optical communication systems.

The distance between FBG central wavelengths *λC* is chosen to be sufficiently large so that stationary optical interference between the multiple channels cannot occur. In this case, inside the PRC each channel creates its own index grating of different grating pitch. The change in the index grating pitch, , is related to the channel separation and the signal and reference beam angle *θ*:

$$
\Delta\Lambda = \frac{\Delta\mathcal{J}\_{\mathbb{C}}}{2\sin(\theta/2)}.\tag{22}
$$

Multi-Channel Adaptive Interferometers Based on Dynamic Hologram Multiplexing 129

Dynamic strains with equal amplitude (5 με) and different frequencies were simultaneously applied to all FBG sensors: 10 kHz on FBG sensor 1 (1548 nm), 5 kHz on FBG sensor 2 (1552 nm), 2 kHz on FBG sensor 3 (1556 nm), and 20 kHz on FBG sensor 4 (1560 nm). Figure 23(a) shows that the four channels can be demodulated simultaneously. Note that the low frequency fluctuations seen in the signals are due to environmental noise-induced *intensity*  fluctuations as explained earlier, and these can be removed using a balanced detection scheme if necessary. Also note that although the applied signal amplitudes for all four channels are the same, the demodulated signal amplitude for each channel is slightly different. This is mainly due to each channel having different optical intensities (due to nonuniform EDFA gain). In practice, this can either be precalibrated, or a gain-flattened

(a) (b)

Fig. 23. Output demodulated signals (a) and their Fourier spectra (b) in four channels of adaptive measurement system based on FBG sensors and dynamic holograms spectrally

The cross talk between these four channels can be inferred from the Fourier spectra of the signals shown in Fig.23(b). If there were cross talk, we would expect all the frequency components, namely 2, 5, 10, and 20 kHz, to show up in the signal spectrum of each channel. However, in the spectrum of channel 1, there is only the expected 10 kHz component, and no other frequency components are observed. This is true also for the other three channels. It is safe to conclude that the cross talk between these four channels is at most comparable to the noise level in the Fourier spectrum, which is at least 30 dB below the signal level in this

To conclude this Section we note that, apart from band-drop filters with fixed central wavelengths used in the paper (Qiao, et al., 2006), other WDM-demultiplexers including

EDFA can be used.

multiplexed in PR crystal

particular case.

It is worth noting that the FBG channel spacing *λ<sup>C</sup>* should be neither too small to avoid cross talk due to closely packed index gratings, nor too large to maximize the number of sensors that can be used within the limited bandwidth of the light source and that of spectral sensitivity of the PR crystal. Thus, for the C band (1530–1570 nm), the channel separation *λC* was 4 nm, which, according to Eq.(22), will give rise to an index grating pitch shift of 76 nm at a beam angle of 3°. In the context of holographic storage elements using PRCs, it has been shown that multiple gratings can be written in a single PRC with negligible cross talk if the grating pitches were to differ by as little as 0.03 nm (Kume, et al., 1998). Thus, a 4 nm channel spacing allows 10 channels with very small cross talk in the C band. In practice, this can be increased by using narrower channel spacing, at the expense of decreased dynamic range.

The experimental configuration of a four-channel TWM wavelength demodulator demonstrated in (Qiao, et al., 2006) is shown in Fig. 22. Four 0.1 nm linewidth FBG sensors are connected in series and are centered at 1548, 1552, 1556, and 1560 nm, respectively. The experimental configuration is similar to that of the single channel configuration shown in Fig. 19 except that after the PRC, there are four band-drop filters which transmit a certain band (i.e., from 1546.8 to 1549.4 nm) and reflect all the other wavelengths. The band-drop filters therefore decouple the TWM-demodulator signals from the various FBG sensors prior to photodetection. The width of the band-drop filter should be chosen to be wider than the expected quasi-static phase shifts of the FBG sensors, and to be slightly narrower than the FBG channel spacing *λC*.

Fig. 22. Experimental configuration of the four-channel adaptive measurement system based on spectral multiplexing of FBG sensors and TWM demodulation of their signals at dynamic holograms in PRC

Performance of the four-channel adaptive measurement system is illustrated by Fig.23 which shows oscilloscope traces of demodulation signal in each channel of the system.

It is worth noting that the FBG channel spacing *λ<sup>C</sup>* should be neither too small to avoid cross talk due to closely packed index gratings, nor too large to maximize the number of sensors that can be used within the limited bandwidth of the light source and that of spectral sensitivity of the PR crystal. Thus, for the C band (1530–1570 nm), the channel separation *λC* was 4 nm, which, according to Eq.(22), will give rise to an index grating pitch shift of 76 nm at a beam angle of 3°. In the context of holographic storage elements using PRCs, it has been shown that multiple gratings can be written in a single PRC with negligible cross talk if the grating pitches were to differ by as little as 0.03 nm (Kume, et al., 1998). Thus, a 4 nm channel spacing allows 10 channels with very small cross talk in the C band. In practice, this can be increased by using narrower channel spacing, at the expense of

The experimental configuration of a four-channel TWM wavelength demodulator demonstrated in (Qiao, et al., 2006) is shown in Fig. 22. Four 0.1 nm linewidth FBG sensors are connected in series and are centered at 1548, 1552, 1556, and 1560 nm, respectively. The experimental configuration is similar to that of the single channel configuration shown in Fig. 19 except that after the PRC, there are four band-drop filters which transmit a certain band (i.e., from 1546.8 to 1549.4 nm) and reflect all the other wavelengths. The band-drop filters therefore decouple the TWM-demodulator signals from the various FBG sensors prior to photodetection. The width of the band-drop filter should be chosen to be wider than the expected quasi-static phase shifts of the FBG sensors, and to be slightly narrower than the

Fig. 22. Experimental configuration of the four-channel adaptive measurement system based on spectral multiplexing of FBG sensors and TWM demodulation of their signals at dynamic

Performance of the four-channel adaptive measurement system is illustrated by Fig.23 which shows oscilloscope traces of demodulation signal in each channel of the system.

decreased dynamic range.

FBG channel spacing *λC*.

holograms in PRC

Dynamic strains with equal amplitude (5 με) and different frequencies were simultaneously applied to all FBG sensors: 10 kHz on FBG sensor 1 (1548 nm), 5 kHz on FBG sensor 2 (1552 nm), 2 kHz on FBG sensor 3 (1556 nm), and 20 kHz on FBG sensor 4 (1560 nm). Figure 23(a) shows that the four channels can be demodulated simultaneously. Note that the low frequency fluctuations seen in the signals are due to environmental noise-induced *intensity*  fluctuations as explained earlier, and these can be removed using a balanced detection scheme if necessary. Also note that although the applied signal amplitudes for all four channels are the same, the demodulated signal amplitude for each channel is slightly different. This is mainly due to each channel having different optical intensities (due to nonuniform EDFA gain). In practice, this can either be precalibrated, or a gain-flattened EDFA can be used.

Fig. 23. Output demodulated signals (a) and their Fourier spectra (b) in four channels of adaptive measurement system based on FBG sensors and dynamic holograms spectrally multiplexed in PR crystal

The cross talk between these four channels can be inferred from the Fourier spectra of the signals shown in Fig.23(b). If there were cross talk, we would expect all the frequency components, namely 2, 5, 10, and 20 kHz, to show up in the signal spectrum of each channel. However, in the spectrum of channel 1, there is only the expected 10 kHz component, and no other frequency components are observed. This is true also for the other three channels. It is safe to conclude that the cross talk between these four channels is at most comparable to the noise level in the Fourier spectrum, which is at least 30 dB below the signal level in this particular case.

To conclude this Section we note that, apart from band-drop filters with fixed central wavelengths used in the paper (Qiao, et al., 2006), other WDM-demultiplexers including

Multi-Channel Adaptive Interferometers Based on Dynamic Hologram Multiplexing 131

Fig. 26. The wavelength position of the filter transfer function versus the angle *θ* between the

Fig. 27. The wavelength position of the filter transfer function versus the external electric field: A, *Eext* = 0; B, *Eext* = −370 V cm<sup>−</sup>1; C, *Eext* = +389 V cm<sup>−</sup>1; D, *Eext* = −614 V cm<sup>−</sup>1; E, *Eext* = +653 V cm-1

Fig. 25. The transfer function of the simple dynamic Bragg grating

writing beams

those of scanning type can be used for spectral division of channels. Note as well that such a demultiplexer can be also built on the basis of dynamic holographic grating recorded in a PR crystal (Hukriede, et al., 2003; Petrov, et al., 2001; Petrov, et al., 2003; Runde, et al., 2005). The scheme of such photorefractive band-drop filter proposed in (Petrov, et al., 2003) is shown in Fig. 24. Two coherent light beams A and B (*λw* = 532 nm) making the angle 2*θ* enter the photorefractive crystal BaTiO3:Co where they form a dynamic holographic grating with the index pitch (2sin ) *F w* . Radiation to be spectrally filtered enters the crystal along the wave vector of holographic grating **KF** (see Fig. 24) which acts as a Bragg mirror with central wavelength

$$\mathcal{A}\_{\mathcal{B}} = \frac{n\_{\mathcal{C}} \mathcal{A}\_{w}}{\sin \theta} \,' \,. \tag{23}$$

where *nC* is the refractive index of crystal.

Figure 25 shows typical transmission spectrum of band-drop filter based on a holographic grating formed in PRC. As seen, the filter has high Q-factor due to narrow line width ~0.1 nm.

Fig. 24. Geometry of dynamic band-drop Bragg filter recording in PR crystal (a) and orientation of external electric field applied to a crystal (b)

The adaptive property of a dynamic grating formed in PRC allows one to perform fine tuning of the filter parameter thus providing on-line switching between different FBG sensors. Thus, change of the angle *θ* between recording beams leads to erasure of the previous grating and recording of a new one with new pitch corresponded to new central wavelength *<sup>B</sup>* . As seen from Fig. 26 which shows experimental dependence of central wavelength on the angle *θ*, the change of the latter leads to shift of *<sup>B</sup>* by 80 nm (from 1480 to 1560 nm). Authors (Petrov, et al., 2003) note that shift of the Bragg wavelength has no observable effect on the form of the transfer function and the diffraction efficiency. The speed of tuning was determined by the recording time of the dynamic hologram and was about 0.5 s.

those of scanning type can be used for spectral division of channels. Note as well that such a demultiplexer can be also built on the basis of dynamic holographic grating recorded in a PR crystal (Hukriede, et al., 2003; Petrov, et al., 2001; Petrov, et al., 2003; Runde, et al., 2005). The scheme of such photorefractive band-drop filter proposed in (Petrov, et al., 2003) is shown in Fig. 24. Two coherent light beams A and B (*λw* = 532 nm) making the angle 2*θ* enter the photorefractive crystal BaTiO3:Co where they form a dynamic holographic grating with

the wave vector of holographic grating **KF** (see Fig. 24) which acts as a Bragg mirror with

sin *C w*

Figure 25 shows typical transmission spectrum of band-drop filter based on a holographic grating formed in PRC. As seen, the filter has high Q-factor due to narrow line width

*B n* 

(a) (b)

wavelength on the angle *θ*, the change of the latter leads to shift of

orientation of external electric field applied to a crystal (b)

Fig. 24. Geometry of dynamic band-drop Bragg filter recording in PR crystal (a) and

The adaptive property of a dynamic grating formed in PRC allows one to perform fine tuning of the filter parameter thus providing on-line switching between different FBG sensors. Thus, change of the angle *θ* between recording beams leads to erasure of the previous grating and recording of a new one with new pitch corresponded to new central

to 1560 nm). Authors (Petrov, et al., 2003) note that shift of the Bragg wavelength has no observable effect on the form of the transfer function and the diffraction efficiency. The speed of tuning was determined by the recording time of the dynamic hologram and was

*<sup>B</sup>* . As seen from Fig. 26 which shows experimental dependence of central

*<sup>B</sup>* by 80 nm (from 1480

. Radiation to be spectrally filtered enters the crystal along

, (23)

the index pitch (2sin ) *F w*

central wavelength

~0.1 nm.

wavelength

about 0.5 s.

where *nC* is the refractive index of crystal.

Fig. 25. The transfer function of the simple dynamic Bragg grating

Fig. 26. The wavelength position of the filter transfer function versus the angle *θ* between the writing beams

Fig. 27. The wavelength position of the filter transfer function versus the external electric field: A, *Eext* = 0; B, *Eext* = −370 V cm<sup>−</sup>1; C, *Eext* = +389 V cm<sup>−</sup>1; D, *Eext* = −614 V cm<sup>−</sup>1; E, *Eext* = +653 V cm-1

Multi-Channel Adaptive Interferometers Based on Dynamic Hologram Multiplexing 133

Davis, M. A. and Kersey, A. D. (1995). Application of a fiber Fourier transform spectrometer

Delaye, P.; Blouin, A.; Drolet, D.; de Montmorillon, L.-A.; Roosen, G. and Monchalin, J.-P.

Delaye, P.; Roosen, G.; Ramaz, F.; Forget, B. C.; Atlan, M.; Boccara, A. C. and Gross, M.

*Materials, and Devices PR-05*, Vol.99, Sanya, Hainan, China, 19-23 July, 2005 de Montmorillon, L.-A.; Delaye, P.; Launay, J.-C. and Roosen, G. (1997). Novel theoretical

Dewhurst, R. J. and Shan, Q. (1999). Optical remote measurement of ultrasound. *Meas. Sci.* 

Di Girolamo, S.; Kamshilin, A. A.; Romashko, R. V.; Kulchin, Y. N. and Launay, J.-C. (2007a).

Di Girolamo, S.; Kamshilin, A. A.; Romashko, R. V.; Kulchin, Y. N. and Launay, J.-C. (2007b).

Di Girolamo, S.; Romashko, R. V.; Kulchin, Y. N.; Launay, J.-C. and Kamshilin, A. A. (2008).

Di Girolamo, S.; Romashko, R. V.; Kulchin, Y. N. and Kamshilin, A. A. (2010). Orthogonal

Feinberg, J. (1982). Asymmetric self-defocusing of an optical beam from the photorefractive

Feng, W.; Yan, Y.; Jin, G.; Wu, M. and He, Q. (2000). Multiplexing of volume holographic wavelet correlation processor. *Opt. Comm.*, Vol.176, pp.49-59, ISSN 0030-4018 Fomitchov, P. A.; Kim, Y. K.; Kromin, A.; Krishnaswamy, S.; Achenbach, J. D. and Daniel, I.

Fomitchov, P. A.; Murray, T. W. and Krishnaswamy, S. (2002). Intrinsic fiber-optic ultrasonic

Fomitchov, P. A. and Krishnaswamy, S. (2003). Response of a fiber Bragg-grating ultrasound

Forward, R. L. (1978). Wideband laser-interferometer graviational-radiation experiment.

Grattan, K. T. V. and Meggitt, B. T. (Eds). (1998). *Optical Fiber Sensor Technology: Devices and* 

*Lightwave Technol.*, Vol.13, pp.1289-1295, ISSN 0733-8724

*Technol.*, Vol.10, pp.R139-R168 ISSN 0957-0233

*Express*, Vol.15, No.2, pp.545-555, ISSN 1094-4087

*Lett.*, Vol.32, No.13, pp.1821-1823, ISSN 0146-9592

effect. *J. Opt. Soc. Am.*, Vol.72, No.1, pp.46-51

*Health Monitoring"*, Vol.4335, July, 2001

No.7, pp.1262-1266, ISSN 1559-128X

*Phys. Rev.*, Vol.17, No.2, pp.379-390

sensor. *Opt. Eng.*, Vol.42, pp.956-963, ISSN 0091-3286

*Technology*. Chapman and Hall, ISBN 0-412-78290-1, London

*Opt. Express*, Vol.16, No.22, pp.18041-18049, ISSN 1094-4087

demodulation. *Opt. Comm.*, Vol.283, pp.128-131, ISSN 0030-4018

ISSN 1520-8540

7550

to the detection of wavelength encoded signals from Bragg-grating sensors. *J.* 

(1997). Detection of ultrasonic motion of a scattering surface using photorefractive InP:Fe under an applied dc field. *J. Opt. Soc. Am. B*, Vol.14, No.7, pp.1723-1734,

(2005). Photorefractive two wave mixing detection for acousto-optical imaging of biological thick tissues, *Proceedings of 10th Int. Conf. on Photorefractive Effects,* 

aspects on photorefractive ultrasonic detection and implementation of a sensor with an optimum sensitivity. *J. Appl. Phys.*, Vol.82, No.12, pp.5913-5922, ISSN 1089-

Fast adaptive interferometer on dynamic reflection hologram in CdTe:V. *Opt.* 

Sensing of multimode-fiber strain by a dynamic photorefractive hologram. *Opt.* 

Fiber sensors multiplexing using vectorial wave mixing in a photorefractive crystal.

geometry of wave interaction in a photorefractive crystal for linear phase

M. (2001). Distributed photoacoustic system for cure monitoring of composites, *Proceedings of SPIE "Advanced Nondestructive Evaluation for Structural and Biological* 

sensor array using multiplexed two-wave mixing interferometry. *Appl. Opt.*, Vol.41,

The central wavelength of the filter based on a photorefractive grating can be additionally tuned electrically by application of external electric field to the photorefractive crystal. Figure 27 demonstrates such electrical tuning of the filter. This is an example of fast tuning via the electro-optically induced variations in the average refractive index. One can see that the application of voltages ranging from −614 to +655 V cm−1 provides tuning in the 0.55 nm range. The speed of tuning reached 2.2 nm *μ*s−1 and was limited only by the available equipment (e.g. switching time of the power source used).
