Research Article Study of a Linear Acoustooptic...

Research ArticleStudy of a Linear Acoustooptic Laser Modulator Based onAll-Fibre Sagnac Interferometer

G. Trinidad García,1 E. Molina Flores,2 B. A. Ramírez Solís,2 and H. Azucena Coyotecatl2

1Facultad de Ciencias Fısico Matematicas, Posgrado en Fısica Aplicada, Benemerita Universidad Autonoma de Puebla,Boulevard 18 Sur y Avenida San Claudio, San Manuel, 72570 Puebla, PUE, Mexico2Facultad de Ciencias de la Electronica, Benemerita Universidad Autonoma de Puebla, Boulevard 18 Sur y Avenida San Claudio,San Manuel, 72570 Puebla, PUE, Mexico

Correspondence should be addressed to E. Molina Flores; [email protected]

Received 9 February 2016; Revised 11 May 2016; Accepted 24 May 2016

Academic Editor: Somenath N. Sarkar

Copyright © 2016 G. Trinidad Garcıa et al. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

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

𝑖

was observed. It is discoveredthat, through mathematical analysis, two operation points in the spectrum, 𝑇SI(𝜆), operate in a linear region, and expressions forspectral gain and sensibility are also discovered. AOM has a bandwidth from 0.1Hz to 20 kHz, and its dynamic range is from 0.0to 43.5 dB.

1. Introduction

Photonic crystal fibres (PCFs) are a class of optical fibresthat have generated a lot of research interest in recentyears [1]. The flexibility in the design of PCFs distinguishesthem from conventional fibres, and various PCFs have beendeveloped targeting different applications such as fibre-opticbased sensing [2, 3]. Polarization-maintaining photonic crys-tal fibres (PM-PCFs) have become commercially availablenowadays; these PCFs have important features such as highbirefringence and low temperature sensitivity. Up to now, thePM-PCFs implemented in loops of Sagnac interferometers(SI) have worked properly for purposes of detecting mechan-ical stress and temperature variations [4]. Particularly, thisacoustooptic modulator offers an input port to the constantpower CW beam 𝑃in(𝜆0) and an output port, through whichthe modulated beam emerges, 𝑃out(𝜆0, 𝑡), ready to get tothe end of the fibre-optic line to reach the receiver stage(PD). In this work mainly, we present an all-optical fibreacoustooptic 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, underspecific conditions, is able to work as a linear acoustoopticlaser modulator.

2. Operation of AOM

2.1. 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, Thorlabs�) which interconnects the twooutput ports of coupler forming the loop Sagnac of length𝐿, as shown in Figure 1(a). To find the spectral response ofSI, an infrared source spectrum is used with a flat intensitydistribution in the vicinity of 1550 nm. This produces anoutput spectrum with cosine profile, which is measured withan optical spectrum analyser (OSA). Transmittance functionof SI is as follows [5–8]:

𝑇SI =𝑃out (𝜆)

𝑃in (𝜆)=1

2[1 + cosΔ𝜑

𝑛V] , (1)

Hindawi Publishing CorporationAdvances in OptoElectronicsVolume 2016, Article ID 5606417, 9 pageshttp://dx.doi.org/10.1155/2016/5606417

2 Advances in OptoElectronics

Broadspectrum

source

OSA

L

Coupler

Pin(𝜆)

Pout(𝜆)

3dB

(a)

0.0

0.2

0.4

0.6

0.8

1.0

Nor

mal

ized

tran

smitt

ance

1542 1544 1546 1548 1550 1552 1554 1556 15581540Wavelength (nm)

𝜆2 = 1553.12nm𝜆1 = 1544.2nm

(b)

Figure 1: (a) All-fibre Sagnac interferometer with PM-PCF loop, 𝐿. (b) Typical transmittance spectrum of SI.

where Δ𝜑𝑛V is represented by

Δ𝜑𝑛V =

2𝜋

𝜆Δ𝑛𝑛V𝐿, (2)

where Δ𝑛𝑛V = 𝑛

𝑥𝑛V − 𝑛𝑦𝑛V, where 𝑛𝑥𝑛V and 𝑛𝑦𝑛V are the

corresponding refractive index of each birefringent fibre axis.Subindex 𝑛Vmeans that the variable is valuated in the absenceof mechanical stress and at laboratory temperature, 21∘C.Thus, it is necessary to know Δ𝑛

𝑛V of PM-PCF to make aspecific characterization of SI beforehand. This is possible byusing

Δ𝑛𝑛V =

𝜆2

Δ𝜆

1

𝐿, (3)

where Δ𝜆 = 𝜆2

− 𝜆1

, where 𝜆1

and 𝜆2

are wavelengths cor-responding to a couple of adjacent maximum (or minimum)of the transmittance spectrum 𝑇SI(𝜆) shown in Figure 1(b). 𝜆is the average value between 𝜆

1

and 𝜆2

. Likewise, (4) is usefulfor calculating the beat length of the birefringence fibre:

𝐿𝐵

=𝜆

Δ𝑛𝑛V. (4)

2.2. Description and Operation of the AOM System. Theexperimental setup to demonstrate the functionality of AOMis shown in Figure 2. One can see that the AOM systemconsists of a sine wave generator (Gen) to excite a speakerof 100 watts RMS with tones from 0.1Hz to 25 kHz at auniform intensity of 10 dB. The orthogonal distance betweenthe speaker and the birefringent loop is 30 cm. The AOMsystem has SI as an important part, which is irradiated withCW-ILD 1550 nm by its input port with constant power𝑃in(𝜆0) (1–5mw). 𝑃in(𝜆0) will be modulated by the spectralshifts of 𝑇SI(𝜆) dependent on induced vibration modes in the

PM-PCF segment of length𝐿. It should bementioned that thebirefringent segment, 𝐿, is placed within a soundboard and istense at the ends by using a couple of twisters, with strength of200 gr, enough to keep it in straight line, so that only a portion𝐿V is exposed to vibration emitted by the speaker, 𝐿V < 𝐿.As a result of laser modulation, the output signal 𝑃out(𝜆0, 𝑡) isgenerated, which affects the InGaAs photodetector (PD) to beconverted into an electrical signal.This electrical signal entersthe audio amplifier, whose output is recognized as the outputof the AOM system. Finally, this signal is sent for its recovery,measurement, and analysis together with Gen signal to thesame oscilloscope, as shown in Figure 2.

The PM-PCF employed in this SI had the followingcharacteristics: pitchΛ (spacing between holes): 4.4𝜇m, largehole diameter: 4.5 𝜇m, small hole diameter: 2.2𝜇m, diameterof holey region: 40.0 𝜇m, and outside diameter: 125 𝜇m (seeFigure 3).

A physical response of the birefringent fibre is that itsbirefringence Δ𝑛

𝑛V is modified when the fibre experiencesmechanical stress or variations of temperature, and then itsresulting birefringence, Δ𝑛

𝑟

, can be described like Δ𝑛𝑛V ±

Δ𝑛V [9]. Δ𝑛𝑖

represents the induced birefringence on fibre.Consequently, this response produces a spectral shifting oftransmittance function of Sagnac interferometer. The trans-mittance shifting starts from its initial position towards theshort wavelengths, in proportion of the excitation energy,and returns to its initial position, when the excitation energydisappears. Taking into account these conditions, a model ofthe phase function, Δ𝜑[𝑖

𝑚

(𝑡)], dependent on the mechanicalvibration of the birefringent segment, 𝑖

𝑚

(𝑡), which is inducedby the incident acoustic waves can be formulated.

2.3. Frequency Response of AOM: 𝐻(𝜔). The mathematicalmodel of the operation of the acoustoopticmodulator (AOM)starts considering that the incident acoustic signals are

Advances in OptoElectronics 3

Audioamplifier PD

Oscilloscope

Sine wavegenerator

Axialtwister

Axialtwister

PM-PCF Speaker

Soundingboard

Longitudinalstrength

Coupler3dB

L L�

ii(t)

CW-ILD𝜆0 = 1550nm

Pout(𝜆0, t)

Pin(𝜆0)

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

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

changes of pressure, represented by 𝑖𝑖

(𝑡) = 𝑖𝑐

+ 𝑖V(𝑡), where𝑖𝑐

is the constant intensity component and 𝑖V(𝑡) is the time-variant intensity component. In this application, 𝑖

𝑖

(𝑡) belongsto the intensity of a mechanical wave in the acoustic range.When this wave hits the Sagnac loop, 𝑖

𝑖

(𝑡) becomes as shownin

𝑖𝑚

(𝑡) = 𝛽𝑖𝑖

(𝑡) , (5)

where 𝛽 is the fraction of energy 𝑖𝑖

(𝑡), which is converted to𝑖𝑚

(𝑡). 𝑖𝑚

(𝑡) represents energy intensity modulation inducedin the Sagnac interferometer loop. By the Fourier transform,𝑖𝑚

(𝑡) has a spectral representation 𝐼𝑚

(𝜔), as follows:

𝑖𝑚

(𝑡)F←→ 𝐼𝑚

(𝜔) . (6)

SIHSI(𝜔)

PhotodetectorHPH(𝜔)

Audio amplifier

H(𝜔) of acoustic-optic modulator

Io(𝜔)Im(𝜔)

HG(𝜔)

Figure 4: The resulting transfer function acoustooptic modulator,𝐻(𝜔), is represented by the multiplication of the transfer functionsof the Sagnac interferometer, the photodiode, and the audio ampli-fier.

𝐼𝑚

(𝜔) is a continuous spectrum because any induced 𝜔produce expansions and contractions, to a greater or lesserextent, to the structure of the birefringent fibre loop, soΔ𝑛 values are modulated. Characteristics of 𝐼

𝑚

(𝜔) influencethe modulator bandwidth. 𝐼

𝑚

(𝜔) is the input continuousspectrum of AOM, which produces a continuous outputoptical spectrum, 𝐼

𝑜

(𝜔). The relationship between them isshown in

𝐼𝑜

(𝜔) = 𝐻 (𝜔) ⋅ 𝐼𝑚

(𝜔) , (7)

where 𝐻(𝜔) is the resulting transference function or filterresponse of acoustoopticmodulator. Before 𝐼

𝑚

(𝜔) reaches theoutput, the system has to go through the transfer functionsof 𝐻SI(𝜔) and 𝐻PH(𝜔) and then through the audio amplifier𝐻𝐺

(𝜔), as shown in Figure 4.The transference function of linear modulator, 𝐻(𝜔), is

defined in

𝐻(𝜔) = 𝐻SI (𝜔) ⋅ 𝐻PH (𝜔) ⋅ 𝐻𝐺 (𝜔) . (8)


The term𝐻SI(𝜔) is defined according to

𝐻SI (𝜔) = 𝑎0 exp (−𝛼1𝜔2

) sin2 [ 2𝜋Δ𝜔𝜔] ⋅ exp (−𝑗𝜔𝑡

0

) , (9)

which represents the frequency response of SI. 𝑎0

is themaximum value reached at the IS output. 𝛼

1

allows us toadjust the extinction ratio of resonancemaximums of𝐻SI(𝜔).The quadratic sinusoidal behaviour appears due to the per-mitted oscillations of birefringent fibre segment tensed bythe extremes, 𝐿V. Δ𝜔 represents resonance mode separation.The term −𝑗𝜔𝑡

0

represents the phase shift between the outputand the input signals, and 𝑡

0

represents the delay betweenthe output and the input signals. The term𝐻PH(𝜔) is definedaccording to

𝐻PH (𝜔) =𝐾

𝑗𝑎1

𝜔 + 1, (10)

which represents the spectral response of the photodetector,which corresponds to a first-order filter. 𝐾 represents thegain of the filter, and 𝑎

1

regulates the degree of phase shiftbetween input and output signals, 𝜃PH. Moreover, 𝑎

1

allowsus calculate the cut-off frequency 𝜔

𝑐

= 1/𝑎1

; the photodioderise time is 2 𝜇s; then 𝜔

𝑐

= 1MHz-rad; we can consider that𝐻PH(𝜔) has a predominantly planar response, so that𝐻PH(𝜔)does not present notable dependence on𝜛, thus contributinglike a constant to 𝐻PH(𝜔), only. 𝐻𝐺(𝜔) represents a planarfrequency response of audio amplifier gain in the acousticalspectral region, as is shown in

𝐻𝐺

(𝜔) = 𝑎2

exp (−𝑗𝜔𝑡1

) , (11)

where 𝑎2

is the gain in the audio range. The term −𝑗𝜔𝑡1

represents the phase shift between the output and the inputsignals, and 𝑡

1

represents the delay between the output andthe input signals. The maximum resonant modes can becalculated up to ±𝑚th harmonic, so that 𝜔

1

= 2𝜋𝑓1

= Δ𝜔,which represents the first vibrationmode of birefringent fibretensed by the ends. 𝑓

1

is compatible with the mode of PM-PCF tensed by the extremes, through

𝑓𝑚

=𝑚

2𝐿VV, (12)

where V = (𝑇/𝜌𝐿

)1/2 is wave propagation speed in the

length segment 𝐿V of PM-PCF with tension 𝑇 and linearmass density 𝜌

𝐿

. Finally, the transference function of thecomplete system𝐻(𝜔), which contains the information aboutthe magnitude and phase response of AOM, is shown in

𝐻(𝜔) = 𝑎0

exp (−𝛼1

𝜔2

) sin2 [ 2𝜋Δ𝜔𝜔]

⋅ exp [−𝑗 (𝜔 (𝑡0

+ 𝑡1

) − tan−1 (𝑎1

𝜔))] .

(13)

Consequently,𝐻(𝜔) expresses both the proportion in whichthe intensity of the vibrationmodes is induced in the loop andhow these contribute to the modulation of the birefringencevalue Δ𝑛, and it also shows the filter response of AOM.

2.4. Acoustooptic Modulator Transmittance Model. To obtainthe function of transmittance in dependence on the inducedtransversal vibration modes in the segment, 𝐿V, of thebirefringence fibre of length 𝐿, we must consider the phasechanges, Δ𝜑, produced through the loop segments: 𝐿 − 𝐿Vand 𝐿V. On the basis of our experimental results, the resultingchange rapidity of refraction index of both axes 𝑛

𝑟𝑥

and 𝑛𝑟𝑦

in the birefringent fibre with regard to the induced vibration,𝑖𝑚

(𝑡), is predominantly linear, as is expressed in

𝑛𝑟𝑥

= 𝑛𝑥𝑛V +

𝜕𝑛𝑥V

𝜕𝑖𝑚

⋅ 𝑖𝑚

,

𝑛𝑟𝑦

= 𝑛𝑦𝑛V +

𝜕𝑛𝑦V

𝜕𝑖𝑚

⋅ 𝑖𝑚

,

(14)

where 𝑛𝑥𝑛V and 𝑛𝑦𝑛V are the refraction indexes in 𝑥 and 𝑦 in

the absence of vibration and 𝑛𝑥V and 𝑛𝑦V are the refraction

indexes induced by 𝑖𝑚

(𝑡) to each of the axes. It is emphasizedthat this analysis is realized to constant temperature oflaboratory. Thus, given that Δ𝑛

𝑟

= 𝑛𝑟𝑥

− 𝑛𝑟𝑦

and Δ𝑛V =

𝑛𝑥V − 𝑛𝑦V, the expression of Δ𝑛

𝑟

is shown in

Δ𝑛𝑟

= Δ𝑛𝑛V +

𝜕Δ𝑛V

𝜕𝑖𝑚

⋅ 𝑖𝑚

, (15)

where

𝜕Δ𝑛V

𝜕𝑖𝑚

= 𝛿disp, (16)

where 𝛿disp is the induced birefringence dispersion, whichis proportional to the acoustooptic modulator gain, andrepresents the birefringence changes rapidity of PM-PCFwith regard to contained oscillation modes in 𝑖

𝑚

(𝑡). From(2), the total phase difference reached across 𝐿 is obtainedby adding algebraically the phase differences Δ𝜑

𝑛V and Δ𝜑𝑖in their respective segments 𝐿 − 𝐿V and 𝐿V, as shown in

Δ𝜑 [𝑖𝑚

(𝑡)] = Δ𝜑𝑛V + Δ𝜑𝑖

=2𝜋

𝜆Δ𝑛𝑛V ⋅ (𝐿 − 𝐿V) +

2𝜋

𝜆Δ𝑛𝑖

⋅ 𝑖𝑚

(𝑡) ⋅ 𝐿V,(17)

which, by replacing (15) and (16) in (17) and regrouping,becomes

Δ𝜑 [𝑖𝑚

(𝑡)] =2𝜋

𝜆(Δ𝑛𝑛V𝐿 + 𝛿disp𝑖𝑚 (𝑡) 𝐿V) ; (18)

phase changes due to Δ𝜑[𝑖𝑚

(𝑡)] are hardly amplified in themeasure where 𝐿V increases. Equation (18) will serve to adjustthe experimental response of modulator towards operationpoint, gain, and appropriated sensibility. Thus, 𝑇[𝑖

𝑚

(𝑡)] ofAOM for 𝜆

0

is described through

𝑇 [𝑖𝑚

(𝑡)]SI =1

2

+1

2cos(2𝜋

𝜆0

[Δ𝑛𝑛V𝐿 + 𝛿disp𝑖𝑚 (𝑡) 𝐿V])

(19)


0 1 2 3 4 5 60.0

0.2

0.4

0.6

0.8

1.0

Nor

mal

ized

tran

smitt

ance

m1 = −0.46

Δ𝜑n�H

Δ𝜑 (rad)Δ𝜑n�L

m2 = +0.46

Figure 5: Two operation points: (Δ𝜑𝑛V𝐿, 0.2) or (Δ𝜑𝑛V𝐻, 0.8) with

their tangent lines whose slopes are𝑚1,2

= ±0.46, respectively.

and spectral shifting rapidity of 𝑇[𝑖𝑚

(𝑡)] due to 𝑖𝑚

(𝑡) iscalculated through

𝜕𝜆 (𝑖𝑚

)

𝜕𝑖𝑚

=

𝜆0

𝛿disp

Δ𝑛𝑛V⋅𝐿V

𝐿. (20)

The modulation capacity of the interferometer depends onthe frequency of the induced vibration modes, 𝑖

𝑚

(𝑡). Thiscapacity depends proportionally on the birefringence dis-persion, 𝛿disp. Shifting rapidity can be amplified, when themeasure of relationship (𝐿V/𝐿) is adjusted, propitiating anincrease in the sensibility of acoustooptic modulator for eachvibration frequency of the tensed PM-PCF by the ends.

2.5. Operation Point Adjustment of AOM. The variable Δ𝜑0

represents the phase value from which the phase values ofthe Sagnac interferometer transmittance will be increased.The incident of the wave acoustics produces a phase increaseΔ𝜑V(𝑖𝑚), which implies a spectral shifting for 𝑇SI[𝑖𝑚(𝑡)]towards the short wavelengths. In phase terms, there exist twomain zones to make linear modulation in the transmittancecurve, as shown in Figure 5. Two possible values exist forΔ𝜑𝑛V as operation point: Δ𝜑

𝑛V𝐻 or Δ𝜑𝑛V𝐿. So, the value of

Δ𝜑𝑛V must be adjusted to one of these values, at 21∘C, for

𝜆0

= 1550 nm, and in the absence of induced vibration,𝑖𝑚

(𝑡) = 0. Δ𝜑𝑛V must reach one of these phase values, like

an operation point. Numerical values of these phases areΔ𝜑𝑛V𝐻 = (9/25)𝜋+𝑘⋅2𝜋 andΔ𝜑𝑛V𝐿 = (13/10)𝜋+𝑘⋅2𝜋, where

𝑘 is an entire number. These values should be located strictlyin the wavelength of operation, 𝜆

0

, just at the beginning ofone of the linear modulation regions (see Figure 5). Thus, itis said that 𝑇SI[𝑖𝑚(𝑡)] is in its operation point, as is shown in

𝑇SI [𝑖𝑚 (𝑡)] =1

2+1

2cos (Δ𝜑

𝑛V + Δ𝜑V (𝑖𝑚 (𝑡))) . (21)

It is important to make a good choice of 𝐿 value, in order toproduce an appropriated operation point in 𝑇SI[𝑖𝑚(𝑡)].

2.6. Laser Modulation Equation. The modulator is mainlyconstituted by Sagnac interferometer with high birefringencePM-PCF, in which transmittance function, before acousticstimulus, responds with spectral shifting forward short wave-lengths. Thus, the input and output optical power have thefollowing relationship:

𝑃out (𝜆0, 𝑡) = 𝑇SI (𝜆0, 𝑖𝑚 (𝑡)) ⋅ 𝑃in (𝜆0) , (22)

where modulating function is the transmittance 𝑇SI(𝜆0,𝑖𝑚

(𝑡)). Thus, 𝑇SI(Δ𝜑𝑛V) must have a point operation with thenext specifications: Δ𝜑

𝑛V = Δ𝜑𝑛V𝐻 or Δ𝜑

𝑛V = Δ𝜑𝑛V𝐿 at

21∘C, for 𝜆0

= 1550 nm laser with constant power, 𝑃in(𝜆0),and in the absence of induced vibration, 𝑖

𝑚

(𝑡). Consequently,𝑇SI(Δ𝜑𝑛V) has to be equal to 0.8 or 0.2, respectively. Operationpoints have coordinate points (Δ𝜑

𝑛V𝐿, 0.2) or (Δ𝜑𝑛V𝐻, 0.8) andare represented by circles, as shown in Figure 5. The straightlines with 𝑚

1

and 𝑚2

slopes represent the linear modulationregions. Thus, two kinds of modulation exist.

2.6.1. Negative Modulation. For the behaviour of 𝑇SI(𝜆0,Δ𝜑𝑛V𝐻, 𝑖𝑚(𝑡)) in the region 𝜆

0

< 𝜆, its cosine profile can beapproximated with tangent linear equation at the operationpoint 𝑃(Δ𝜑

𝑛V𝐻, 0.8), where positive slope,𝑚, is expressed in


2+1

2cos (Δ𝜑

𝑛V + Δ𝜑V (𝑖𝑚))

≈𝜕𝑇SI𝜕Δ𝜑V

⋅ (Δ𝜑V − Δ𝜑𝑛V𝐻) +8

10,

(23)

where 𝑚1

= 𝜕𝑇SI/𝜕Δ𝜑V = −0.46 is the negative slope in themiddle region between the crest and valley of function cosine.

2.6.2. Positive Modulation. For the behaviour of 𝑇SI(𝜆0,Δ𝜑𝑛V𝐿, 𝑖𝑚(𝑡)) in the region 𝜆

0

< 𝜆, its cosine profile can beapproximated with tangent linear equation at the operationpoint 𝑃(Δ𝜑

𝑛V𝐿, 0.2), where positive slope,𝑚, is expressed in


2+1

2cos (Δ𝜑

𝑛V + Δ𝜑V (𝑖𝑚))

≈𝜕𝑇SI𝜕Δ𝜙V

⋅ (Δ𝜑V − Δ𝜑𝑛V𝐿) +2

10,

(24)

where 𝑚2

= 𝜕𝑇SI/𝜕Δ𝜑V = +0.46 is the positive slope in themiddle region between the crest and valley of function cosine.In both cases, in linear region, we have the following:

0 < Δ𝜑V − Δ𝜑𝑛V𝑥 ≤4

3rad, (25)

where 𝑥 can be 𝐻 or 𝐿. Thus, laser modulation in nonlinearregions is avoided. Phase relationship between 𝑖

𝑚

(𝑡) and𝑃out(𝜆0, 𝑡) is 𝜋 rad for negative modulation and 0.0 rad forpositive modulation.

From the spectral point of view, linearmodulation occursbecause𝑇SI(Δ𝜑) undergoes a proportional shift towards shortwavelengths, to themagnitude of the induced acoustic signal,𝑖𝑚

(𝑡), returning to its initial position, as this disappears. Thismodulation process is represented in the diagrams shown


1546 1550 1556Wavelength (nm)

0.2

0.4

0.6

0.8

1.0

0.0

Laserspectrum

(𝜆0)

norm

aliz

ed

m1 = −0.46

TSI(Δ𝜑n�H)

Δ𝜑1

Δ𝜑2 Δ𝜑n�H

Pou

t

(a)

Wavelength (nm)1546 1550 1556

0.0

0.2

0.4

0.6

1.0

0.8Laser

spectrum

(𝜆0)

norm

aliz

ed

m2 = +0.46

TSI(Δ𝜑n�L)

Δ𝜑1Δ𝜑2

Δ𝜑n�L

Pou

t

(b)

Figure 6: (a) Operation point of negative modulation; (b) operation point of positive modulation.

in Figures 6(a) and 6(b). Note that operation points are𝑇SI(Δ𝜑𝑛V𝐻) = 0.8 with Δ𝜑𝑛V𝐻 < Δ𝜑1 < Δ𝜑2 for case (a) and𝑇SI(Δ𝜑𝑛V𝐿) = 0.2 with Δ𝜑𝑛V𝐿 < Δ𝜑1 < Δ𝜑2 for case (b). Thelinear modulation process happens when the linear region ofthe 𝑇SI(Δ𝜑) cosine profile regulates the intensity of the CW-ILD laser beam with its shifting and this effect is reflected inthe reproduction of the form of 𝑖

𝑚

(𝑡) in 𝑃out(𝜆0, 𝑡) values.

2.7. Sensibility of AOM. The laser linear modulator, as alinear time-invariant system, is admitted as input signal toresonant signal, 𝑖

𝑚

(𝑡), which is induced in the segment ofPM-PCF, and as an output signal to 𝑃out(𝜆0, 𝑡), as a resultof spectral displacement of 𝑇SI(𝜆0, 𝑖𝑚), which modulates𝑃in. This modulation is proportional to shifting rapidity of𝑇SI(𝜆0, 𝑖𝑚), due to the changes of Δ𝜑V(𝑖𝑚). Thus, sensibilityof AOM can be defined, as expressed in

𝑆 =𝜕𝑃out𝜕𝑖𝑚

. (26)

Replacing (22) in (26) yields

𝑆 = 𝑃in ⋅𝜕𝑇SI (𝜆0)

𝜕Δ𝜑V⋅𝜕Δ𝜑V

𝜕𝑖𝑚

, (27)

where 𝑃in(𝜆0) appears as a constant term which is modu-lated by Sagnac interferometer transmittance.The expression𝜕𝑇SI(𝜆0)/𝜕Δ𝜑V characterizes the change rapidity of the trans-mittance due to the induced phase changes by acoustic signal.Because of the trigonometric properties of𝑇SI(𝜆) function, itslinear region of modulation has an intrinsic correspondingslope𝑚, which is described by

𝑚1,2

=𝜕𝑇SI (𝜆0)

𝜕Δ𝜑V= ∓0.46, (28)

where the sign of slope depends on the side of the crestwhich is employed to realize the modulation. The expression𝜕Δ𝜑V/𝜕𝑖𝑚 represents the sensibility to the induced phasechange by induced signal 𝑖

𝑚

, in the birefringent fibre:𝜕Δ𝜑V

𝜕𝑖𝑚

=2𝜋

𝜆0

𝛿disp𝐿V. (29)

Table 1: Optical parameters of PM-PCF.

Measured parameters experimentallyΔ𝑛𝑛V 𝐿

𝐵

, mm 𝛿𝜆/𝛿𝑖𝑚

, nm/dB7.6 × 10−4 2.0 1

Let us observe that both 𝛿disp and 𝐿V perform a relevant rolein the adjustment of sensibility and denote that sensibilityto the phase change also depends on intensity and vibrationfrequencies, 𝐼

𝑚

(𝜔). This final result provokes the notion thatthe sensibility of AOM is high in accordancewith the acousticnoise and with the external vibration of the workbench.

3. Experimental Results and Discussion

3.1. Required Characteristics for 𝑇𝑆𝐼

(𝜆). Δ𝑛𝑛V and 𝐿𝐵 parame-

ters are determined by using (3) and (4) [9]. Consider thatwe intend to design and build a SI, with loop PM-PCF,whose𝑇SI(𝜆)withoutmechanical excitation and at 21∘Coffersa spectral distribution, such that 𝑇SI(𝜆0) = 0.2 to makepositive modulation. Moreover, consider that the shiftingratio of 𝑇SI(𝜆) towards shorter wavelengths, with acousticalexcitation, is 1 nm/dB for a component of 1 kHz, when(𝐿V/𝐿) = 1 typically. Consequently, 𝑇SI(𝜆) should have Δ𝜆 =10 nm to have at least 4 nm of linear modulation region. Inconsideration of the characterization of the PM-PCF, Table 1shows the experimentally measured optical parameters andTable 2 shows the spectral characteristics of the built SI.

We can show as a result of the spectral measurements thatthe overlapping spectra of SI transmittance and of emissionof ILD (𝜆

0

= 1550 nm) are shown in Figure 7, observing thatthe location of the spectrum of ILD is right where 𝑇SI(𝜆0) =0.2, guaranteeing the right conditions to perform positiveacoustooptic modulation.

3.2. Modulator Response in Time and Frequency Domain.Time response of acoustooptic modulator was determined byusing the setup procedure shown in Figure 2. An electricalsine wave, 𝑖

𝑖

(𝑡), is applied in the speaker by using wave


Table 2: Physical and spectral characteristics of SI.

Characteristic values obtained

𝐿, m 𝐿V, m 𝜆1

, nm 𝜆2

, nm Δ𝜆, nm 𝑇SI(𝜆0

)𝛿𝜆/𝛿𝑖

𝑚

,nm/dB

Δ𝜑𝑛V𝐿,

rad0.34 0.19 1544.20 1553.12 8.9 0.2 1 4.0

1540 1542 1544 1546 1548 1550 1552 1554 1556 1558 1560

0.0

0.2

0.4

0.6

0.8

1.0

Nor

mal

ized

tran

smitt

ance

Wavelength (nm)

AOMILD

Figure 7: Overlapping of the normalized transmittance spectrumwith the laser beam spectrum at 1550 nm, just on the operation pointΔ𝜑𝑛V𝐿.

generator (Gen), at a frequency, 𝑓. 𝑖𝑖

(𝑡) at 10 dB is inducedin the segment, 𝐿V, like 𝐼𝑚(𝑡). According to (22), 𝑃out(𝜆0, 𝑡) isproportionallymodulated and detected by PD. Photodetectoroutput is amplified and represents the AOM output. Gen andAOM signals aremeasured and analysed in amplitude, phase,and frequency by using the oscilloscope. Some significantexamples, at different frequencies, are shown in Figures 8(a),8(b), 8(c), and 8(d).The small phase shifting is presented dueto the phase shifting provided both by photodetector and byaudio amplifier. Analysis on frequency is made by using (7)and (13), determining theoretically and experimentally theBode diagram.

Frequency response of acoustoopticmodulatorwas deter-mined by using the same setup shown in Figure 2. Magnitudeand phase characteristics of 𝐻SI(𝜔) are hardly influenced bymagnitude and phase characteristics of permitted vibrationmodes in the tense fibre by the ends. For this reason, gainspectrum of acoustooptic modulator shows maximum andminimum resonant peaks in audible region. Superposition oftheoretical and experimental frequency response evaluatedin the range from 0.1Hz to 20 kHz is shown in Figure 9.Theoretical spectral gain is reached considering a photode-tector as a first-order filter, with a cut-off frequency of175 kHz and planar audio amplifier bandwidth of 20 kHz.We are considering that extinction ratios of resonant modesintensity are lower than the filter response of photodetec-tor and amplifier. Separation modes were calculated forΔ𝑓 = 7.0 kHz. Experimental gain spectrum has a lobular

Table 3: AOM elements.

Elements SpecificationsILD InGaAs 1550 ± 0.5 nm, fibre pigtail, stabilizedPhotodetector InGaAs low power, BW = 175 kHzAmplifier 1 < gain < 150, BW = 1MHzModulation frequency 0.1 Hz up to 21 kHzSensibility 2.39mW/dB to 1 kHzMinimum power 1mWMaximum power 5mWInsertion loss <0.4 dBDynamic range 0–43.5 dBTemperature 21∘ ± 19∘C

distribution profile, whose separation between maximums isapproximately 7.0 kHz. The values of the maximums dependon the extinction ratio of the intensities of the vibrationmodes permitted in the segment PM-PCF, 𝐿V. The valuesof the minimums depend on the sensitivity of the AOM tovibrate at frequencies not permitted.

Superposition of theoretical and experimental phaseresponse is shown in Figure 10. It can be seen that the-oretically a linear and proportional shift with respect to𝜔 is expected, whereas the experimental result shows ashift asymptotically towards 90 degrees. The fluctuationsare characteristic of the interferometer instabilities due toexternal vibrations outside the experiment: the ventilationsystem of equipment and vibration of ballast, which is veryweak but existent.Thedifferences between the theoretical andexperimental response are as follows: (1) theoretical responsedoes not include the dissipative forces of the segment 𝐿V(force of restitution). (2) 𝐻SI damping coefficient dependson the operation frequency. (3) Bandwidth of every lobeis slightly different due to the fact that the losses of thevibration depend on the operation frequency value. Theseloss values cannot be known a priori, due to the fact thatthis one depends on the viscosity forces not only of thesurrounding media to 𝐿V (Stokes law) but especially also ofthe friction forces at the inside of the fibre segment, 𝐿V, forwhich analytical expressions do not exist. Nevertheless, thisdemonstrates that the optical fibre acoustooptic modulator iscapable of reproducing with enough intensity the frequenciesthat are contained in these lobes of the gain spectrum. Let usappreciate also that the intensity of these maximums declinesin the measure where the frequency increases; thus, the filterresponse of the modulator is similar to a low pass filter.

In spite of the fact that the optical fibre interferometersaremore stable than the classic ones, themeasurements of thegain spectrum were made in the absence of any other sourceof acoustic noise and/or vibration of the workbench. Finally,specifications of elements used for the integration of AOM tooperate in the audible region, at 21∘C, are specified in Table 3.

4. Conclusions

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


−1.0

−0.5

0.0

0.5

1.0

Am

plitu

de (V

)

−0.005 0.000 0.005 0.010−0.010

Time (s)

AOMGen

(a) 𝑓𝑚

= 143Hz

AOMGen

Nor

mal

ized

ampl

itude

−1.5

−1.0

−0.5

0.0

0.5

1.0

1.5

−0.005 0.000 0.005−0.010 0.010

Time (s)

(b) 𝑓𝑚

= 200Hz

AOMGen

−0.6

−0.4

−0.2

0.0

0.2

0.4

0.6

Am

plitu

de (V

)

−0.0005 0.0000 0.0005 0.0010−0.0010

Time (s)

(c) 𝑓𝑚

= 4.16 kHz

Nor

mal

ized

ampl

itude

AOMGen

−0.6

−0.4

−0.2

0.0

0.2

0.4

0.6

0.00100.0000 0.0005−0.0005−0.0010

Time (s)

(d) 𝑓𝑚

= 4.9 kHz

Figure 8: Input (Gen) and output (AOM) signals overlapped in oscilloscope at different frequencies.

work as a linear acoustooptic laser modulator for frequenciesfrom 0.1Hz to 20 kHz, to 1550 nm. It has been demonstratedthat the realization of the modulation, at low frequencies, ofthe birefringence of the PM-PCF is achievable as well. Theprinciple of work of the AOM is that the induced shiftingsin 𝑇SI, by the incident mechanical waves on the birefringentfibre, tensed by the ends, modulate the laser beam intensity.We have demonstrated the ability to linearly modulate theintensity of aCW laser beambyusing theAOM.As a resonantsystem, it results in a gain spectrum distributed in adjacentlobes, which correspond to a response of low pass filter. We

have shown that AOMcanmake positive or negativemodula-tion. Phase relationship between acoustical signal (Gen) andoptical signal (output OAM) is 𝜋 rad for negative modulationand 0.0 rad for positive modulation. Modulator presents awide margin of immunity to the changes of environmentaltemperature due to thermal characteristic of the PM-PCFloop. The dynamic range of the linear modulator is in therange of 0.0 to 43.5 dB, enough to not reach the nonlinearregion of AOM.The sensibility of the modulator depends onthe frequency. Linear acoustooptic modulator can work alsoas a fibre microphone; we called it photonic phone.


−30

−25

−20

−15

−10

−5

0

Gai

n sp

ectr

um (d

B)

5 10 15 200Linear frequency (kHz)

ExperimentalTheoretical

Figure 9: Gain spectrums of AOM.

−100

−80

−60

−40

−20

0

Phas

e shi

fting

(∘)

5 10 15 200Linear frequency (kHz)

ExperimentalTheoretical

Figure 10: Theoretical and experimental phase shifting of AOM.

Competing Interests

The authors declare that they have no competing interests.

Acknowledgments

Thanks are due to Vicerrectorıa de Investigacion y Estudiosde Posgrado (VIEP) of BUAP, Consejo Nacional de Cienciay Tecnologıa (CONACyT), and Secretarıa de EducacionPublica (SEP) for their partial support to this project.

References

[1] P. St. J. Russell, “Photonic crystal fibres,” Science, vol. 299, no.5605, pp. 358–362, 2003.

[2] B. J. Eggleton, C. Kerbage, P. S. Westbrook, R. S. Windeler, andA. Hale, “Microstructured optical fiber devices,”Optics Express,vol. 9, no. 13, pp. 698–713, 2001.

[3] O. Frazao, J. L. Santos, F. M. Araujo, and L. A. Ferreira, “Opticalsensing with photonic crystal fibers,” Laser and PhotonicsReviews, vol. 2, no. 6, pp. 449–459, 2008.

[4] X. Y. Dong, H. Y. Tam, and P. Shum, “Temperature-insensitivestrain sensor with polarization-maintaining photonic crystalfiber based Sagnac interferometer,” Applied Physics Letters, vol.90, no. 15, Article ID 151113, 3 pages, 2007.

[5] M. L. V. Tse, H. Y. Tam, L. B. Fu et al., “Fusion splicing holeyfibers and single-mode fibers: a simple method to reduce lossand increase strength,” IEEE Photonics Technology Letters, vol.21, no. 3, pp. 164–166, 2009.

[6] H. Y. Fu, H. Y. Tam, L.-Y. Shao et al., “Pressure sensorrealized with polarization-maintaining photonic crystal fiber-based Sagnac interferometer,” Applied Optics, vol. 47, no. 15, pp.2835–2839, 2008.

[7] G. Kim, T. Cho, K. Hwang et al., “Strain and temperaturesensitivities of an elliptical hollow-core photonic bandgap fiberbased on Sagnac interferometer,” Optics Express, vol. 17, no. 4,pp. 2481–2486, 2009.

[8] E. Molina-Flores and E. Kuzin, “Thermically tuned erbium-doped fiber ring laser,” inProceedings of the Conference on Lasersand Electro Optics/Quantum Electronics and Laser Science ofEurope, Technical Digest, p. 620, Physical Society of Europe,June 2003.

[9] E. Molina-Flores and B. A. Ramırez-Solıs, “Tecnica de med-icion interferometrica espectral de la birrefringencia en fibrasde cristal fotonico,” Internet Electronic Journal. Nanociencia etMoletronica, vol. 11, no. 1, pp. 2057–2064, 2013, http://www.re-vista-nanociencia.ece.buap.mx/11nr1/Rev.%207Esteban%20.pdf.

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