**3. Adaptive ineterferometer sensitivity**

108 Advanced Holography – Metrology and Imaging

provisionally that the applied field is parallel to the grating vector the space charge-field can be found, within the linear approximation in the contrast of the interference pattern, as

*K*

(0) *G sE H i ij* <sup>0</sup> *ij*

is the wavelength; *ρ* is the rotatory power; *ijz*

(0) 0 41 0 *ijl l*

*r E*

*r E* ,

*ij*

*z*

 

*z*

 

*H*

where \*

here <sup>3</sup> *s nr* 0 41 /

are respectively:

following

matrix **H**ˆ being non-zero.

coefficient;

characteristic fields,

*D Kk T <sup>E</sup>*

*B*

*<sup>e</sup>* (diffusion electric field),

dielectric constant. The matrices **G**ˆ and **V**ˆ are given by

indices *i* and *j* assume independently the values *x* and *y*.

*E im*

0

( ) *<sup>D</sup> <sup>q</sup>*

*E E iE* 

12 0 *m I* 2 / **A A** is the contrast of the interference pattern, *I*0 is the total light intensity, *E*0 is the amplitude of external electric field applied to the crystal, while *ED* and *Eq* are the

*E iE E*

*D q*

where *e* is the elementary charge, *kB* is the Boltzmann constant, *T* is the absolute temperature, *Nt* is the effective concentration of photorefractive centers (traps), *εε*0 is the

> *ijz*

rank tensor; *E*<sup>0</sup> **E0** ; *EK* **EK** . The Latin subscripts denote Cartesian components, and

The tensors ˆ (0) **H** and **H**ˆ responsible for electro-optic contribution to variation of crystal dielectric permittivity caused by external electric field **E0** and internal space-charge field **EK**,

where *rijl* is the electro-optic tensor. Taking into account that the space-charge field appears due to interference of mixed waves as well as a form of coupled-wave equations (1), one can conclude that tensor **H**ˆ has the character of the wave-coupling matrix. The last becomes more clear in the case of waves mixing in non-gyrotropic crystal ( 0

without application of external electric field ( <sup>0</sup> *E* 0 ) when the system (1) is reduced to the

1 2

*K*

<sup>ˆ</sup> . <sup>2</sup>

**A HA**

<sup>ˆ</sup> , <sup>2</sup>

*g smE*

As seen from Eqs.(6) in order for the waves to couple they must interfere ( 0 *m* ) with

By solving Eqs.(1), one can find amplitudes of mixed waves at the output of the PR crystal. Note that a particular solution depends on wave mixing geometry which, in its turn, is

*smE*

**A HA**

2 1

*K*

0

0 *<sup>t</sup> <sup>q</sup> eN <sup>E</sup>* 

is a material parameter; *n*0 is the refractive index; *r*41 is the electro-optic

41 *ijl Kl*

*r E*

*K*

*ij*

*H*

, (2)

*<sup>K</sup>* (traps saturation field), (3)

, *V sH ij ij* , (4)

is the unit antisymmetric third

*r E* , (5)

(6)

) Since in any interferometer a physical parameter, which has to be measured, is finally encoded in the phase modulation of the light wave, its sensitivity to small phase excursions is a parameter serving as a primary criterion for comparison of different systems. It is not only the configuration of the optical scheme but also its particular realisation (the output laser power, the generated wavelength, the type of the crystal, etc.) which determines the sensitivity of an adaptive interferometer. Comparison of different adaptive interferometers is usually done by estimation the extent to which their sensitivity is worse than that of the classical lossless interferometer (Delaye, et al., 1997; de Montmorillon, et al., 1997). It was shown in early papers devoted to analysis of a classical interferometer that its sensitivity can be extremely high if the available light power is not limited and measurements of the phase modulation is carried out within a very narrow frequency band (Bershtein, 1954; Forward, 1978). Evidently, the minimum detectable phase difference *min* is defined by the noise level of the measuring system. There are several sources of the noise in an optical interferometer: laser noise, thermal and shot noise of the photodetector, and noise of amplifying electronics. When the light power arriving at the photo-detector is high enough, shot noise (which is proportional to the square root of the received light power) of the photo-excited charge carriers prevails over all the other noise levels (Bershtein, 1954; Wagner & Spicer, 1987). The shot noise is the instability of the photodetector current caused by statistical fluctuation of the number of received photons. Its level is primarily defined by the average number of photons and it is given as:

$$
\left\langle i\_{\rm shot}^2 \right\rangle = 2 \frac{e^2 \eta}{h \nu} \left\langle P\_D \right\rangle \Delta f \tag{7}
$$

where *e* is the electron's charge, *η* is the quantum efficiency of the photo-detector, *hν* is the energy of the photon, and Δ*f* is the frequency bandwidth of the detection electronics.

Multi-Channel Adaptive Interferometers Based on Dynamic Hologram Multiplexing 111

Performance of any adaptive interferometer, which is schematically shown in Fig. 2(b), is conveniently compared with the performance of the classical interferometer by introducing

> *min A C*

phase difference in the adaptive interferometer and *SNRA* is the signal-to-noise ratio

hand, the relative detection limit, *δrel*, is the ratio of the detection limits of both interferometers. On the other hand, it shows how the response of the classical

Correct comparison of the adaptive and classical interferometers can be carried out if we use the same power of the object beam, the same photodetector and electronic circuit, and the same phase difference between the interfering beams. If the performance of the adaptive interferometer is analytically described, the relative detection limit can be calculated theoretically. Alternatively, *δrel* can be measured in the experiment as follows. Suppose that

1

0 *D*

*D*

change of the light power at the photodetector by Δ*PD* but the mean power, <*PD*0>, remains

*C A*

*SNR SNR*

, (11)

*<sup>A</sup>* is the minimal detectable

is larger than that of the adaptive

in the adaptive interferometer results in a

. (12)

. (13)

is applied to

*<sup>A</sup>* for the

On the one

*rel min*

achieved in the adaptive interferometer by introducing the phase shift of Δ

the same. In this case the signal-to-noise ratio of the adaptive interferometer is

*rel*

visibility of an oscilloscope trace when the small phase excursion of Δ

Moreover, it also allows us to estimate the minimal detectable phase shift *min*

 *A rel C* .

In following Sections we consider different schemes for multichannel adaptive interferometers based on spatial, angular and spectral multiplexing of dynamic holograms

*A*

*<sup>P</sup> SNR Q <sup>P</sup>*

The light power *PD*0 is certainly related to the incident power *PS* as *P P D S* <sup>0</sup> , where is the system transmission, which takes into account the optical losses in the crystal and all other optical elements ( 1 ), which is readily measured in the experiment. By introducing

<sup>0</sup> 2 *<sup>D</sup>*

The relative detection limit is always larger than unity and it increases for higher optical losses (when decreases). The ratio *P P D D*<sup>0</sup> is readily measured because it is equal to the ratio of the modulated part of the photodetector current to its non-modulated part (i.e., to

interferometer. Other parameters of equation (13) can be also readily estimated in the experiment. Therefore, Eq. (13) provides easy way to measure the relative detection limit *δrel*. This parameter is very convenient for comparison of different adaptive interferometers.

*D P P*

which can be called the relative detection limit. In Eq. (11) *min*

interferometer to a known phase difference Δ

introduction of the small phase difference of Δ

Eqs. (9) and (12) into Eq. (11), we get for *β* >> 1:

particular adaptive interferometer as *min min*

in a photorefractive crystal.

the parameter

interferometer.

Fig. 2. Combining the signal (*PS*) and the reference (*PR*) waves using the conventional beamcombiner in the classical interferometer (a) and the dynamic hologram of an adaptive interferometer (b): 1 – the beam combiner; 2 – the photodetector; 3 – the photorefractive crystal

Let us now calculate the signal-to-noise ratio for phase demodulation in the classical interferometer, which is schematically shown in Fig. 2(a). As is known, the highest sensitivity of the interferometer is achieved at the quadrature condition, when the average phase difference between the interfering beams is equal to *π*/2 (Osterberg, 1932). When we introduce a small phase difference, Δ, the light power arriving at the photo-detector is

$$P\_D = \frac{1}{2}(P\_S + P\_R) + \sqrt{P\_S P\_R} \sin \Delta \phi \approx P\_S \frac{1+\beta}{2} + P\_S \sqrt{\beta} \cdot \Delta \phi \tag{8}$$

where *PR* and *PS* are the power of the reference and object wave, respectively, *β* is their ratio. Here we neglected all optical losses and assumed that the cross-sections of the interfering beam are the same. The informative signal in the interferometer is the electric current of the photodetector proportional to ΔAssuming linear response of the photodetector to the incoming light power *PD*, the signal is directly proportional to the second term in Eq. (8), while the first term is responsible for the shot noise given by Eq.(7). Therefore, the signal-tonoise ratio achieved in the classical interferometer is

$$\text{SNR}\_{\text{C}} = \frac{\Lambda \phi}{Q} \sqrt{\frac{2P\_{\text{S}}}{1 + \beta^{-1}}}, \quad \text{with} \quad Q = \sqrt{\frac{4h\nu\Delta f}{\eta}} \,. \tag{9}$$

The minimum detectable phase difference (the classical homodyne detection limit), *min <sup>C</sup>* , is usually defined as the phase shift, which leads to *SNR* = 1. For large beam intensity ratio (*β* >> 1) Eq. (9) yields

$$
\Delta \phi\_{\mathbb{C}}^{\text{min}} = \sqrt{\frac{2\hbar \nu \Delta f}{\eta P\_S}} \,\,\,\,\,\tag{10}
$$

*PS* 

*PR*

(a) (b)

*PD* 

<sup>1</sup> <sup>2</sup>

crystal

*PS* 

*PR*

is

we introduce a small phase difference, Δ

photodetector proportional to Δ

>> 1) Eq. (9) yields

Fig. 2. Combining the signal (*PS*) and the reference (*PR*) waves using the conventional beamcombiner in the classical interferometer (a) and the dynamic hologram of an adaptive interferometer (b): 1 – the beam combiner; 2 – the photodetector; 3 – the photorefractive

Let us now calculate the signal-to-noise ratio for phase demodulation in the classical interferometer, which is schematically shown in Fig. 2(a). As is known, the highest sensitivity of the interferometer is achieved at the quadrature condition, when the average phase difference between the interfering beams is equal to *π*/2 (Osterberg, 1932). When

where *PR* and *PS* are the power of the reference and object wave, respectively, *β* is their ratio. Here we neglected all optical losses and assumed that the cross-sections of the interfering beam are the same. The informative signal in the interferometer is the electric current of the

incoming light power *PD*, the signal is directly proportional to the second term in Eq. (8), while the first term is responsible for the shot noise given by Eq.(7). Therefore, the signal-to-

> 1 2 1 *S*

The minimum detectable phase difference (the classical homodyne detection limit), *min*

*min* 2 *C*

*S h f P* 

usually defined as the phase shift, which leads to *SNR* = 1. For large beam intensity ratio (*β*

, the light power arriving at the photo-detector

*PD* 

<sup>3</sup> <sup>2</sup>

 

Assuming linear response of the photodetector to the

. (9)

*<sup>C</sup>* , is

. (10)

, (8)

, with 4*h f <sup>Q</sup>*

 1 1 sin 2 2 *P P P PP P P D S R SR S S*

noise ratio achieved in the classical interferometer is

*C <sup>P</sup> SNR Q* 

Performance of any adaptive interferometer, which is schematically shown in Fig. 2(b), is conveniently compared with the performance of the classical interferometer by introducing the parameter

$$\mathcal{S}\_{rel} = \frac{\Delta \phi\_A^{\text{min}}}{\Delta \phi\_\mathbb{C}^{\text{min}}} = \frac{SNR\_\mathbb{C}}{SNR\_A} \,\,\,\,\,\tag{11}$$

which can be called the relative detection limit. In Eq. (11) *min <sup>A</sup>* is the minimal detectable phase difference in the adaptive interferometer and *SNRA* is the signal-to-noise ratio achieved in the adaptive interferometer by introducing the phase shift of ΔOn the one hand, the relative detection limit, *δrel*, is the ratio of the detection limits of both interferometers. On the other hand, it shows how the response of the classical interferometer to a known phase difference Δ is larger than that of the adaptive interferometer.

Correct comparison of the adaptive and classical interferometers can be carried out if we use the same power of the object beam, the same photodetector and electronic circuit, and the same phase difference between the interfering beams. If the performance of the adaptive interferometer is analytically described, the relative detection limit can be calculated theoretically. Alternatively, *δrel* can be measured in the experiment as follows. Suppose that introduction of the small phase difference of Δ in the adaptive interferometer results in a change of the light power at the photodetector by Δ*PD* but the mean power, <*PD*0>, remains the same. In this case the signal-to-noise ratio of the adaptive interferometer is

$$\text{SNR}\_A = \text{Q}^{-1} \frac{\Delta P\_D}{\sqrt{P\_{D0}}} \cdot \tag{12}$$

The light power *PD*0 is certainly related to the incident power *PS* as *P P D S* <sup>0</sup> , where is the system transmission, which takes into account the optical losses in the crystal and all other optical elements ( 1 ), which is readily measured in the experiment. By introducing Eqs. (9) and (12) into Eq. (11), we get for *β* >> 1:

$$
\delta\_{rel} = \sqrt{\frac{2}{\mathfrak{T}}} \frac{P\_{D0}}{\Delta P\_D} \Delta \phi \,\tag{13}
$$

The relative detection limit is always larger than unity and it increases for higher optical losses (when decreases). The ratio *P P D D*<sup>0</sup> is readily measured because it is equal to the ratio of the modulated part of the photodetector current to its non-modulated part (i.e., to visibility of an oscilloscope trace when the small phase excursion of Δ is applied to interferometer. Other parameters of equation (13) can be also readily estimated in the experiment. Therefore, Eq. (13) provides easy way to measure the relative detection limit *δrel*. This parameter is very convenient for comparison of different adaptive interferometers. Moreover, it also allows us to estimate the minimal detectable phase shift *min <sup>A</sup>* for the particular adaptive interferometer as *min min A rel C* .

In following Sections we consider different schemes for multichannel adaptive interferometers based on spatial, angular and spectral multiplexing of dynamic holograms in a photorefractive crystal.

Multi-Channel Adaptive Interferometers Based on Dynamic Hologram Multiplexing 113

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

(a) (b)

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

level of a channel (– 11 dB).

between channels
