Study of a Linear Acoustooptic Laser Modulator Based on All-Fibre Sagnac Interferometer

The feasibility of polarization-maintaining photonic crystal fibre (PM-PCF) strategy for acoustooptic modulation using all-fibre Sagnac interferometer is demonstrated. The principal constraint to apply the strategy is defined by a linear laser acoustooptic modulator (AOM) for 1550 nm. The intensity of incident acoustic waves over the PM-PCF loop segment affected the signal interference transmission; here, modulation by birefringence variation around 7.6 × 10−4 ± Δn


Introduction
Photonic crystal fibres (PCFs) are a class of optical fibres that have generated a lot of research interest in recent years [1].The flexibility in the design of PCFs distinguishes them from conventional fibres, and various PCFs have been developed targeting different applications such as fibre-optic based sensing [2,3].Polarization-maintaining photonic crystal fibres (PM-PCFs) have become commercially available nowadays; these PCFs have important features such as high birefringence and low temperature sensitivity.Up to now, the PM-PCFs implemented in loops of Sagnac interferometers (SI) have worked properly for purposes of detecting mechanical stress and temperature variations [4].Particularly, this acoustooptic modulator offers an input port to the constant power CW beam  in ( 0 ) and an output port, through which the modulated beam emerges,  out ( 0 , ), ready to get to the end of the fibre-optic line to reach the receiver stage (PD).In this work mainly, we present an all-optical fibre acoustooptic modulator based on SI with PM-PCF loop, perfectly adjusted to operate in the C band (1530-1570 nm) or in the L band (1570-1620 nm) for optical communications.Now, we will demonstrate that SI of PM-PCF loop, under specific conditions, is able to work as a linear acoustooptic laser modulator.

Theory of Sagnac Interferometer with PM-PCF Loop.
Sagnac interferometer (SI) is made up of a single mode fibre (SMF), 3 dB coupler for 1550 nm, and a segment of PM-PCF (PM-1550-01, Thorlabs5) which interconnects the two output ports of coupler forming the loop Sagnac of length , as shown in Figure 1(a).To find the spectral response of SI, an infrared source spectrum is used with a flat intensity distribution in the vicinity of 1550 nm.This produces an output spectrum with cosine profile, which is measured with an optical spectrum analyser (OSA).Transmittance function of SI is as follows [5][6][7][8]: where Δ V is represented by where Δ V =  V −  V , where  V and  V are the corresponding refractive index of each birefringent fibre axis.Subindex V means that the variable is valuated in the absence of mechanical stress and at laboratory temperature, 21 ∘ C. Thus, it is necessary to know Δ V of PM-PCF to make a specific characterization of SI beforehand.This is possible by using where Δ =  2 −  1 , where  1 and  2 are wavelengths corresponding to a couple of adjacent maximum (or minimum) of the transmittance spectrum  SI () shown in Figure 1(b). is the average value between  1 and  2 .Likewise, ( 4) is useful for calculating the beat length of the birefringence fibre:

Description and Operation of the AOM System.
The experimental setup to demonstrate the functionality of AOM is shown in Figure 2. One can see that the AOM system consists of a sine wave generator (Gen) to excite a speaker of 100 watts RMS with tones from 0.1 Hz to 25 kHz at a uniform intensity of 10 dB.The orthogonal distance between the speaker and the birefringent loop is 30 cm.The AOM system has SI as an important part, which is irradiated with CW-ILD 1550 nm by its input port with constant power  in ( 0 ) (1-5 mw). in ( 0 ) will be modulated by the spectral shifts of  SI () dependent on induced vibration modes in the PM-PCF segment of length .It should be mentioned that the birefringent segment, , is placed within a soundboard and is tense at the ends by using a couple of twisters, with strength of 200 gr, enough to keep it in straight line, so that only a portion  V is exposed to vibration emitted by the speaker,  V < .
As a result of laser modulation, the output signal  out ( 0 , ) is generated, which affects the InGaAs photodetector (PD) to be converted into an electrical signal.This electrical signal enters the audio amplifier, whose output is recognized as the output of the AOM system.Finally, this signal is sent for its recovery, measurement, and analysis together with Gen signal to the same oscilloscope, as shown in Figure 2.
The PM-PCF employed in this SI had the following characteristics: pitch Λ (spacing between holes): 4.4 m, large hole diameter: 4.5 m, small hole diameter: 2.2 m, diameter of holey region: 40.0 m, and outside diameter: 125 m (see Figure 3).
A physical response of the birefringent fibre is that its birefringence Δ V is modified when the fibre experiences mechanical stress or variations of temperature, and then its resulting birefringence, Δ  , can be described like Δ V ± Δ V [9].Δ  represents the induced birefringence on fibre.Consequently, this response produces a spectral shifting of transmittance function of Sagnac interferometer.The transmittance shifting starts from its initial position towards the short wavelengths, in proportion of the excitation energy, and returns to its initial position, when the excitation energy disappears.Taking into account these conditions, a model of the phase function, Δ[  ()], dependent on the mechanical vibration of the birefringent segment,   (), which is induced by the incident acoustic waves can be formulated.

Frequency Response of AOM: 𝐻(𝜔
where  is the fraction of energy   (), which is converted to   ().  () represents energy intensity modulation induced in the Sagnac interferometer loop.By the Fourier transform,   () has a spectral representation   (), as follows: Audio amplifier H() of acoustic-optic modulator   () is a continuous spectrum because any induced  produce expansions and contractions, to a greater or lesser extent, to the structure of the birefringent fibre loop, so Δ values are modulated.Characteristics of   () influence the modulator bandwidth.  () is the input continuous spectrum of AOM, which produces a continuous output optical spectrum,   ().The relationship between them is shown in where () is the resulting transference function or filter response of acoustooptic modulator.Before   () reaches the output, the system has to go through the transfer functions of  SI () and  PH () and then through the audio amplifier   (), as shown in Figure 4.
The transference function of linear modulator, (), is defined in The term  SI () is defined according to which represents the frequency response of SI.  0 is the maximum value reached at the IS output. 1 allows us to adjust the extinction ratio of resonance maximums of  SI ().
The quadratic sinusoidal behaviour appears due to the permitted oscillations of birefringent fibre segment tensed by the extremes,  V .Δ represents resonance mode separation.The term − 0 represents the phase shift between the output and the input signals, and  0 represents the delay between the output and the input signals.The term  PH () is defined according to where  2 is the gain in the audio range.The term − 1 represents the phase shift between the output and the input signals, and  1 represents the delay between the output and the input signals.The maximum resonant modes can be calculated up to ±th harmonic, so that  1 = 2 1 = Δ, which represents the first vibration mode of birefringent fibre tensed by the ends. 1 is compatible with the mode of PM-PCF tensed by the extremes, through where V = (/  ) 1/2 is wave propagation speed in the length segment  V of PM-PCF with tension  and linear mass density   .Finally, the transference function of the complete system (), which contains the information about the magnitude and phase response of AOM, is shown in Consequently, () expresses both the proportion in which the intensity of the vibration modes is induced in the loop and how these contribute to the modulation of the birefringence value Δ, and it also shows the filter response of AOM.

Acoustooptic Modulator Transmittance Model.
To obtain the function of transmittance in dependence on the induced transversal vibration modes in the segment,  V , of the birefringence fibre of length , we must consider the phase changes, Δ, produced through the loop segments:  −  V and  V .On the basis of our experimental results, the resulting change rapidity of refraction index of both axes   and   in the birefringent fibre with regard to the induced vibration,   (), is predominantly linear, as is expressed in where  V and  V are the refraction indexes in  and  in the absence of vibration and  V and  V are the refraction indexes induced by   () to each of the axes.It is emphasized that this analysis is realized to constant temperature of laboratory.Thus, given that Δ  =   −   and Δ V =  V −  V , the expression of Δ  is shown in where where  disp is the induced birefringence dispersion, which is proportional to the acoustooptic modulator gain, and represents the birefringence changes rapidity of PM-PCF with regard to contained oscillation modes in   ().From (2), the total phase difference reached across  is obtained by adding algebraically the phase differences Δ V and Δ  in their respective segments  −  V and  V , as shown in which, by replacing (15) and ( 16) in (17) and regrouping, becomes phase changes due to Δ[  ()] are hardly amplified in the measure where  V increases.Equation (18) will serve to adjust the experimental response of modulator towards operation point, gain, and appropriated sensibility.Thus, [  ()] of AOM for  0 is described through  and spectral shifting rapidity of [  ()] due to   () is calculated through The modulation capacity of the interferometer depends on the frequency of the induced vibration modes,   ().This capacity depends proportionally on the birefringence dispersion,  disp .Shifting rapidity can be amplified, when the measure of relationship ( V /) is adjusted, propitiating an increase in the sensibility of acoustooptic modulator for each vibration frequency of the tensed PM-PCF by the ends.

Operation Point Adjustment of AOM.
The variable Δ 0 represents the phase value from which the phase values of the Sagnac interferometer transmittance will be increased.The incident of the wave acoustics produces a phase increase Δ V (  ), which implies a spectral shifting for  SI [  ()] towards the short wavelengths.In phase terms, there exist two main zones to make linear modulation in the transmittance curve, as shown in Figure 5. Two possible values exist for Δ V as operation point: Δ V or Δ V .So, the value of Δ V must be adjusted to one of these values, at 21 ∘ C, for  0 = 1550 nm, and in the absence of induced vibration,   () = 0. Δ V must reach one of these phase values, like an operation point.Numerical values of these phases are Δ V = (9/25)+⋅2 and Δ V = (13/10)+⋅2, where  is an entire number.These values should be located strictly in the wavelength of operation,  0 , just at the beginning of one of the linear modulation regions (see Figure 5).Thus, it is said that  SI [  ()] is in its operation point, as is shown in It is important to make a good choice of  value, in order to produce an appropriated operation point in  SI [  ()].

Laser Modulation Equation.
The modulator is mainly constituted by Sagnac interferometer with high birefringence PM-PCF, in which transmittance function, before acoustic stimulus, responds with spectral shifting forward short wavelengths.Thus, the input and output optical power have the following relationship: where modulating function is the transmittance  SI ( 0 ,   ()).
where  1 =  SI /Δ V = −0.46 is the negative slope in the middle region between the crest and valley of function cosine.

Positive Modulation.
For the behaviour of  SI ( 0 , Δ V ,   ()) in the region  0 < , its cosine profile can be approximated with tangent linear equation at the operation point (Δ V , 0.2), where positive slope, , is expressed in where  2 =  SI /Δ V = +0.46 is the positive slope in the middle region between the crest and valley of function cosine.
In both cases, in linear region, we have the following: where  can be  or .Thus, laser modulation in nonlinear regions is avoided.Phase relationship between   () and  out ( 0 , ) is  rad for negative modulation and 0.0 rad for positive modulation.
From the spectral point of view, linear modulation occurs because  SI (Δ) undergoes a proportional shift towards short wavelengths, to the magnitude of the induced acoustic signal,   (), returning to its initial position, as this disappears.This modulation process is represented in the diagrams shown in Figures 6(a) and 6(b).Note that operation points are  SI (Δ V ) = 0.8 with Δ V < Δ 1 < Δ 2 for case (a) and  SI (Δ V ) = 0.2 with Δ V < Δ 1 < Δ 2 for case (b).The linear modulation process happens when the linear region of the  SI (Δ) cosine profile regulates the intensity of the CW-ILD laser beam with its shifting and this effect is reflected in the reproduction of the form of   () in  out ( 0 , ) values.

Sensibility of AOM.
The laser linear modulator, as a linear time-invariant system, is admitted as input signal to resonant signal,   (), which is induced in the segment of PM-PCF, and as an output signal to  out ( 0 , ), as a result of spectral displacement of  SI ( 0 ,   ), which modulates  in .This modulation is proportional to shifting rapidity of  SI ( 0 ,   ), due to the changes of Δ V (  ).Thus, sensibility of AOM can be defined, as expressed in Replacing ( 22) in (26) yields where  in ( 0 ) appears as a constant term which is modulated by Sagnac interferometer transmittance.The expression  SI ( 0 )/Δ V characterizes the change rapidity of the transmittance due to the induced phase changes by acoustic signal.Because of the trigonometric properties of  SI () function, its linear region of modulation has an intrinsic corresponding slope , which is described by where the sign of slope depends on the side of the crest which is employed to realize the modulation.The expression Δ V /  represents the sensibility to the induced phase change by induced signal   , in the birefringent fibre: Let us observe that both  disp and  V perform a relevant role in the adjustment of sensibility and denote that sensibility to the phase change also depends on intensity and vibration frequencies,   ().This final result provokes the notion that the sensibility of AOM is high in accordance with the acoustic noise and with the external vibration of the workbench.

Experimental Results and Discussion
3.1.Required Characteristics for   ().Δ V and   parameters are determined by using (3) and (4) [9].Consider that we intend to design and build a SI, with loop PM-PCF, whose  SI () without mechanical excitation and at 21 ∘ C offers a spectral distribution, such that  SI ( 0 ) = 0.2 to make positive modulation.Moreover, consider that the shifting ratio of  SI () towards shorter wavelengths, with acoustical excitation, is 1 nm/dB for a component of 1 kHz, when ( V /) = 1 typically.Consequently,  SI () should have Δ = 10 nm to have at least 4 nm of linear modulation region.In consideration of the characterization of the PM-PCF, Table 1 shows the experimentally measured optical parameters and Table 2 shows the spectral characteristics of the built SI.We can show as a result of the spectral measurements that the overlapping spectra of SI transmittance and of emission of ILD ( 0 = 1550 nm) are shown in Figure 7, observing that the location of the spectrum of ILD is right where  SI ( 0 ) = 0.2, guaranteeing the right conditions to perform positive acoustooptic modulation.

Modulator Response in Time and Frequency Domain.
Time response of acoustooptic modulator was determined by using the setup procedure shown in Figure 2.An electrical sine wave,   (), is applied in the speaker by using wave Frequency response of acoustooptic modulator was determined by using the same setup shown in Figure 2. Magnitude and phase characteristics of  SI () are hardly influenced by magnitude and phase characteristics of permitted vibration modes in the tense fibre by the ends.For this reason, gain spectrum of acoustooptic modulator shows maximum and minimum resonant peaks in audible region.Superposition of theoretical and experimental frequency response evaluated in the range from 0.1 Hz to 20 kHz is shown in Figure 9. Theoretical spectral gain is reached considering a photodetector as a first-order filter, with a cut-off frequency of 175 kHz and planar audio amplifier bandwidth of 20 kHz.We are considering that extinction ratios of resonant modes intensity are lower than the filter response of photodetector and amplifier.Separation modes were calculated for Δ = 7.0 kHz.Experimental gain spectrum has a lobular  10.It can be seen that theoretically a linear and proportional shift with respect to  is expected, whereas the experimental result shows a shift asymptotically towards 90 degrees.The fluctuations are characteristic of the interferometer instabilities due to external vibrations outside the experiment: the ventilation system of equipment and vibration of ballast, which is very weak but existent.The differences between the theoretical and experimental response are as follows: (1) theoretical response does not include the dissipative forces of the segment  V (force of restitution).(2)  SI damping coefficient depends on the operation frequency.(3) Bandwidth of every lobe is slightly different due to the fact that the losses of the vibration depend on the operation frequency value.These loss values cannot be known a priori, due to the fact that this one depends on the viscosity forces not only of the surrounding media to  V (Stokes law) but especially also of the friction forces at the inside of the fibre segment,  V , for which analytical expressions do not exist.Nevertheless, this demonstrates that the optical fibre acoustooptic modulator is capable of reproducing with enough intensity the frequencies that are contained in these lobes of the gain spectrum.Let us appreciate also that the intensity of these maximums declines in the measure where the frequency increases; thus, the filter response of the modulator is similar to a low pass filter.
In spite of the fact that the optical fibre interferometers are more stable than the classic ones, the measurements of the gain spectrum were made in the absence of any other source of acoustic noise and/or vibration of the workbench.Finally, specifications of elements used for the integration of AOM to operate in the audible region, at 21 ∘ C, are specified in Table 3.

Conclusions
In this research, we have demonstrated the feasibility of Sagnac interferometer composed of PM-PCF, which can

Figure 2 :
Figure 2: Acoustooptic modulator setup based on Sagnac interferometer with oscillating segment  V .

Figure 3 :
Figure 3: Cross-sectional SEM image of the PM-PCF.

Figure 4 :
Figure 4: The resulting transfer function acoustooptic modulator, (), is represented by the multiplication of the transfer functions of the Sagnac interferometer, the photodiode, and the audio amplifier.

Figure 6 :
Figure 6: (a) Operation point of negative modulation; (b) operation point of positive modulation.
).The mathematical model of the operation of the acoustooptic modulator (AOM) starts considering that the incident acoustic signals are Thus,  SI (Δ V ) must have a point operation with the next specifications: Δ V = Δ V or Δ V = Δ V at 21 ∘ C, for  0 = 1550 nm laser with constant power,  in ( 0 ), and in the absence of induced vibration,   ().Consequently,  SI (Δ V ) has to be equal to 0.8 or 0.2, respectively.Operation points have coordinate points (Δ V , 0.2) or (Δ V , 0.8) and are represented by circles, as shown in Figure5.The straight lines with  1 and  2 slopes represent the linear modulation regions.Thus, two kinds of modulation exist.2.6.1.Negative Modulation.For the behaviour of  SI ( 0 , Δ V ,   ()) in the region  0 < , its cosine profile can be approximated with tangent linear equation at the operation point (Δ V , 0.8), where positive slope, , is expressed in

Table 1 :
Optical parameters of PM-PCF.

Table 3 :
AOM elements.The values of the maximums depend on the extinction ratio of the intensities of the vibration modes permitted in the segment PM-PCF,  V .The values of the minimums depend on the sensitivity of the AOM to vibrate at frequencies not permitted.Superposition of theoretical and experimental phase response is shown in Figure