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Polarization insensitive linear intensity modulation in a Sagnac fiber-optic loop with an electro-optic cell and optoelectronic feedback
Optics and Lasers in Engineering 39 (2003) 567–579
Polarization insensitive linear intensity modulation in a Sagnac fiber-optic loop with an electro-optic cell and optoelectronic feedback A. Garcia-Weidner*, C. Torres Torres, A.V. Khomenko, M.A. Garcia-Zarate ! Cient!ıfica y de Estudios Superiores de Ensenada, CICESE, Centro de Investigacion Carretera Tijuana-Ensenada Km. 107, Ensenada, Baja California, C.P. 22860, Mexico
Abstract We present a modulation scheme consisting of a fiber-optic Sagnac interferometer, which has an electro-optic modulator (Pockels cell) in the loop of the interferometer. In order to diminish the nonlinearity of the Sagnac interferometer transmittance function, we monitor a very small part of the output light intensity by means of a photodiode and then amplify this signal by a constant factor x: This amplified signal is added to the electrical modulating signal that is being applied to the electro-optical modulator. The consequence of introducing this feedback loop is that by properly choosing the amplifying factor x; we can increase the linearity of the modulation. The device is insensitive to the polarization state of the input light, so it should be used for intensity modulation of nonpolarized light with up to 100% modulation depth. The theoretical and experimental examination of the device is presented. r 2002 Elsevier Science Ltd. All rights reserved. Keywords: Sagnac interferometer; Electro-optic modulators; Optical fiber devices
1. Introduction Electro-optic light intensity modulators root their behavior on interference phenomena, and then they tend to have nonlinear squared sinusoidal intensity transfer functions, i.e. the dependence of the intensity modulated light beam on the applied voltage. In order to maintain linearity in the modulation, these devices are *Corresponding author. Tel.: +52-646-175-0551; fax: +52-646-175-0553. E-mail address: [email protected] (A. Garcia-Weidner). 0143-8166/03/$ - see front matter r 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 1 4 3 - 8 1 6 6 ( 0 2 ) 0 0 0 4 6 - 5
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biased to the linear portion of the sine curve and used with small modulating signals. Then the sine transfer function results in a limited linear dynamic range for the device. Some schemes for enhancing the linearity using several modulators in tandem have been proposed [1–3]. Electro-optic modulators based on a Pockels cell sandwiched between polarizers have the disadvantage that up to 50% of input light intensity is lost. Although Sagnac interferometer modulators also exhibit a nonlinear transfer function [4–6], in this work we analyze the use of a Sagnac fiber-optic interferometer with optoelectronic feedback to improve the linearity of the modulator’s response. Since the device does not need the use of polarizers, it is simpler in it’s construction, and by properly choosing the amplifying factor x; close to 100% of output light intensity can be achieved. Also, due to the lack of the need of polarizers, the device is able to modulate nonpolarized light, making it more suitable for fiber optics communication systems. This is important since optical fibers tend to introduce random fluctuations on light polarization due to imperfections on the fiber, as well as due to bending or twisting of the fiber. A solution for diminishing these fluctuations, could be the use of hibirrefringence (HB) polarization preserving fibers [7,8], but then another problem arises: the use of HB fibers in the Sagnac loop makes it highly sensitive to environmental temperature variations, and the scheme should be used as a temperature sensor rather than as a light intensity modulator. Also, in order to properly operate the Sagnac loop, the fast axis of the birrefringent fiber needs to be rotated by an angle of 901 [7,8] by means of a mechanical twist or by the use of a quarter wave-plate, but in our scheme, a DC voltage applied to the electro-optic modulator solves the problem, making it’s optical set up more simple.
2. Description of the Sagnac modulator 2.1. Optical set up The experimental set up is shown in Fig. 1. In our description, light fields impinging on the coupler are represented by Jones vectors Ei (i ¼ 1; 2; 3; 4), whereas optical fields emerging from the coupler are represented by primed vectors Ei 0 . Light emitted by an unpolarized He–Ne laser that has a light intensity I1 ¼ 10 mW is launched by a lens L into an optical fiber that is connected to port (1) of a bidirectional fiber optic 2 2 coupler C: This coupler has an intensity coupling coefficient k ¼ 0:5: Ports (3) and (4) of the coupler are connected into a Sagnac loop, which is formed by two optical fibers F1 and F2 of equal length of 25 cm (drawing is not to scale) and one electro-optical modulator MOD. The modulator MOD consists of a Pockels cell with its induced fast F and slow S axes oriented at an angle c ¼ 451 relative to the plane defined by the Sagnac loop. The modulator MOD has a half-wave voltage of Vp E230 V. Light intensity that impinges on port (1) of coupler C is split by 50% and directed towards ports (3) and (4) and then both light beams travel in opposite directions in the closed loop. Due to the bending of the Sagnac’s loop, the horizontal ‘‘x’’ axis of the laboratory coordinate system is rotated by an angle of 1801, as indicated in Fig. 1. After traversing the closed loop, light beams
A. Garcia-Weidner et al. / Optics and Lasers in Engineering 39 (2003) 567–579 Y
569
F1
X
Y
E '3
3
LASER
S
E1
I1
45
F2 1 C
L
E '2
2x2
4
X
F MOD V (t)
E '4
2 I 'f
D PD f
I 'm
PDm
D ig. O sc.
1x2
A Vm (t) Vs
Vs Vf
Fig. 1. Experimental optical set up.
impinging in ports (3) and (4) are recombined inside the bi-directional coupler and directed to ports (1) and (2). Light intensity I20 emerging from port (2) is connected to the input port of a 1 2 directional coupler D which has a coupling coefficient kf ¼ 0:1: The output modulated light intensity Im0 is given by Im0 ¼ km I20 ;
ð1Þ
where km ¼ 1 kf : This Im0 emerging from coupler D is captured by a photo detector PDm, converted into a time varying voltage signal Vm ðtÞ and then connected to the digital oscilloscope Dig Osc. On the other output port of coupler D the emerging light intensity If0 will be used as the feedback signal to control the behavior of the modulator. The light intensity If0 is given by If0 ¼ kf I20 :
ð2Þ
This light intensity If0 is captured by a photo detector PDf and converted into an electrical feedback signal Vf given by Vf ¼ bIf0 ¼ bkf I20 ;
ð3Þ
where b is the conversion factor of the photo detector given in [V/W]. The modulating electrical signal Vs and the feedback signal Vf are connected to an electronic amplifier A; and then the resulting voltage signal V ðtÞ is given by V ðtÞ ¼ aðVs þ Vf Þ ¼ VDC þ VAC cosð2pftÞ;
ð4Þ
where subscripts of VDC and VAC correspond to the DC and AC components of the modulating voltage, f is the frequency, and a is the gain factor of the amplifier. Since the modulator is a Pockels cell, voltage V ðtÞ induces a phase retardation G between the fast and slow ray components traveling inside of the modulator. The induced
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retardation is given by pV ðtÞ G¼ : Vp
ð5Þ
For convenience, using Eq. (5) we can express the half value of V ðtÞ in angular units [deg] or [rad] as p V ðtÞ ¼ Vq þ Vy cosð2pftÞ; VY ðtÞ ¼ ð6Þ Vp 2 where subscripts of Vq and Vy correspond to the DC and AC components of VY ðtÞ: In practice, the values of voltage are limited to 0oVy oVp =4 or 0oVAC oVp =2: 2.2. Polarization independent intensity transfer function of the modulator As we will see, although we are using a polarization Pockels cell modulator, the output modulated light beam Im0 is independent of the polarization state of the input ~1 : For calculating the transfer function of the light beam represented by E modulating system we will use the Jones calculus and the diagram of its equivalent ~0 optical circuit [9] for the experimental setup which is shown in Fig. 2. The field E 2 0 that emerges from coupler D and its corresponding light intensity I2 are given by ~1 ; ~0 ¼ ½J12 E ð7Þ E 2
~0 ~0 j2 ¼ E ~0w E I20 ¼ jE 2 2 2;
ð8Þ
where [J12 ] is the Jones matrix that represents the optical system connecting ports (1) and (2) of coupler C and the superscript w denotes the transpose of the complex conjugate. Then the intensity transfer function between these two ports is ~w ½J12 w ½J12 E ~1 ~0 j2 E I 0 jE T12 ¼ 2 ¼ 2 2 ¼ 1 : ð9Þ 2 ~ ~ I1 j E 1 j jE 1 j OUTPUT
3’
4
[Mi] [JC]
[K14] V(t)
+
2
Kf
A [K32]
4’
3
[JA] [Mi]
[K24]
E 'm
2’
[K41]
[K23]
'
1’ E 1
E 'f
Vi
+
[K31] S
E2 = 0
Km
+
1
+
E1
+
INPUT
[K42]
FEEDBAC K LOOP
[K13]
Fig. 2. Equivalent optical circuit of the experimental set up.
Vf
β
If
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Following the diagram of the equivalent optical circuit sketched in Fig. 2, the Jones matrix [J12 ] can be expressed as ½J12 ¼ ½K42 ½Mi ½JC ½K13 þ ½K32 ½JA ½Mi ½K14 ;
ð10Þ
where [JC ] and [JA ] are the Jones matrices of the interferometer loop in the clockwise and anticlockwise sense respectively, [Mi ] is a coordinate conversion matrix due to the folded fiber in the Sagnac loop (similar to a mirror reflection as can be seen from Fig. 1), and [Kpq ] represents the directional coupler C matrix between ports p and q. These matrices are given by " # 1 0 ½Mi ¼ ; ð11Þ 0 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 kÞ½I ; ð12Þ pffiffiffi ð13Þ ½K32 ¼ ½K14 ¼ i k½I ; pffiffiffiffiffiffiffi where i ¼ 1 and [I] is the identity matrix. If matrices [JC ] and [JA ] represent ideal lossless linear optical components, which also are reciprocal on light propagation direction (such as wave plates do), then the Jones matrices are unitary, and therefore we can write # " # " # " JXX JYX A þ iB C þ iD JXX JXY T ; ð14Þ ½JC ¼ ½JA ¼ ¼ ¼ C þ iD A iB JYX JYY JYX JXX ½K42 ¼ ½K13 ¼
with jJxx j2 þ jJyx j2 ¼ A2 þ B2 þ C 2 þ D2 ¼ 1: Substituting Eqs. (10)–(14) in Eq. (9) we obtain T12 ¼ 4D2 kð1 kÞ þ ð2k 1Þ2 ¼ D2 :
ð15Þ
From this equation, we can see that T12 is independent from any input ~1 : The Jones matrix [JC ] for an electron optical modulator is polarization state E given by the Jones matrix of a linear retardation wave plate. Then the Jyx component is given by Jyx ¼ i sinð2cÞsinðG=2Þ: Since in our case c ¼ 451; the light intensity Im0 emerging from port (2) can be written as G pV ðtÞ I20 ¼ T12 I1 ¼ I1 sin2 ¼ I1 sin2 : ð16Þ 2 2Vp If we define the following expressions for VA and the feedback factor x [W1] by pa VA ¼ ð17Þ VS ; 2Vp pa bkf ; ð18Þ x¼ 2Vp km and taking into account that Im0 ¼ km I20 ; then we can write an expression for this output modulated light intensity as Im0 ¼ km I1 sin2 fVA þ xIm0 g:
ð19Þ
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It is important to note from Eq. (17) that VA is expressed in angular units rather than in volts. From Eq. (19) we can see that the output light intensity Im0 can reach almost 90% of input light intensity I1 ; depending on the chosen value of km E1:
3. The effect of feedback on the modulator’s performance In the case that we have no feedback signal (x ¼ 0), the intensity transfer function of a Sagnac modulator is reduced to Eq. (16) which is the same as the intensity transfer function of a conventional electro-optic modulator [1], i.e. a Pockels Cell sandwiched between crossed polarizers. But there are two main advantages of using a Sagnac modulator: (1) a Sagnac modulator becomes insensitive to the input polarization state and it does not require the use of polarizers, making the optical setup more simple, and (2) output light intensity can reach almost 100% of input light intensity compared to 50% of the conventional modulator. Using Eq. (19), in Fig. 3 we show the theoretical plot of the normalized output light intensity Im0 vs. VA (with Km I1 ¼ 1) for different values of x within the interval 1oxo1: For jxj > 1 the device shows a multistable behavior. In order to operate this device appropriately, we need to apply a DC bias voltage so that Vq ¼ p=4; which lead us to the operating point Q at the central part of the transmittance curve (T ¼ 50%). As can be seen from Fig. 3, when xa0 the intensity transfer function apparently becomes more linear. To examine this linearity we make a Taylor series expansion of Eq. (19) around the operating point Q: In the case for
Fig. 3. Theoretical plot of normalized transmittance T ¼ ðIm0 =km I1 Þ vs. VA [deg]. (a) x ¼ 1; (b) x ¼ 0; (c) x ¼ 0:5; (d) x ¼ 1:
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x ¼ 0 we obtain Im0
N X km I1 ð1Þð2Nþ1Þ ð2VA Þð2Nþ1Þ 1þ ¼ 2 ð2N þ 1Þ! N¼0
E
!
km I1 ð1 þ a1 VA a3 VA3 þ a5 VA5 a7 VA7 þ a9 VA9 ?Þ; 2
ð20Þ
where a1 ¼ 2; a3 ¼ 4=3; a5 ¼ 4=15; a7 ¼ 8=315; a9 ¼ 4=2835: In the case of 1oxo0; the Taylor series expansion of Eq. (19) around the operating point Q is given by Im0 ¼
k m I1 km I1 4km I1 VA3 þ VA þ 2 1 k m I1 x ðkm I1 x 1Þ 3! 16km I1 ð1 þ 9km I1 xÞ VA5 þ 5! ðkm I1 x 1Þ7 þ
2 2 2 64km I1 ð1 þ 54km I1 x þ 225km I1 x Þ VA7 10 5! ðkm I1 x 1Þ
2 2 2 3 3 3 256km I13 ð1 þ 243km I1 x þ 4131km I1 x þ 11025km I1 x Þ VA9 ? 9! ðkm I1 x 1Þ13 k m I1 E ð1 þ b1 VA b3 VA3 þ b5 VA5 b7 VA7 þ b9 VA9 ?Þ: 2
þ
ð21Þ
Now we can compare the bi coefficients of this series with coefficients ai which correspond to the Taylor series regarding the case of a modulator without feedback. This comparison leads us to 4 1 b3 b1 ¼ ; ; 1 k m I1 x a3 a1 7 b1 ¼ ð1 þ 9km I1 xÞ; a1 10 b1 2 2 2 ¼ ð1 þ 54km I1 x þ 225km I1 x Þ; a1 13 b1 2 2 2 3 3 3 ¼ ð1 þ 243km I1 x þ 4131km I1 x þ 11025km I1 x Þ: a1
b1 ¼ a1 b5 a5 b7 a7 b9 a9
ð22Þ
In Figs. 4a, 5a and 6a, we can see the three-dimensional plots of the ratios (bi =ai ) vs. (x; km I1 ) for i ¼ 1; 3; 5: In Figs. 4b, 5b and 6b, we can see the bi-dimensional plots of (bi =ai ) vs. x for our particular case where km I1 ¼ 0:9: From these plots, and taking into account that we are dealing within the interval 1oxo0; we can see that bi oai (i ¼ 1; 3; 5; y), i.e. the higher order terms of the series that contribute to the nonlinearity become smaller when xa0 compared to those corresponding to the modulator without feedback (x ¼ 0).
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0 _ 0.2
0.6
_ 0.4 _ 0.6 ξ
0.7
0.8
b1/a1
b1/a1
0.9
0.8 0.6 0.4 0.5
0.7 0.6
_ 0.8
0.8
KmI1
0.9
0.5
1
_1
(a)
_ 0.8
_ 0.6
_ 0.4
_ 0.2
0
ξ
(b)
0.4 0.3 0.2 0.1 0 0.5
0.6
_
0.7
KmI1
0.8
_
0.9
0.6 0.5
0 _ 0.2 _ 0.4 0.6
0.8
ξ
b3/a3
b3/a3
Fig. 4. Theoretical plot of (a) coefficient ratio (b1 =a1 ) vs. (Km I1 ; x) and (b) theoretical plot of ratio (b1 =a1 ) vs. x for Km I1 ¼ 0:9 [W]. In both plots data is calculated using a normalized I1 ¼ 1 [W] and x is given in [W1].
0.4 0.3 0.2 0.1 0 _
1
(a)
1
_
0.8
_ 0.6
(b)
_
ξ
0.4
_
0.2
0
Fig. 5. Theoretical plot of (a) coefficient ratio (b3 =a3 ) vs. (Km I1 ; x) and (b) theoretical plot of ratio (b3 =a3 ) vs. x for Km I1 ¼ 0:9 [W]. In both plots data is calculated using a normalized I1 ¼ 1 [W] and x is given in [W1].
0.5
0.6
0.7
KmI1 (a)
0.8
0 _ 0.2 _ 0.4 _ 0.6 ξ _ 0.8 0.9
b5/a5
b5/a5
0.05
0 0.1
0
_ 0.05 _ 0.1
1
_1
(b)
_ 0.8
_ 0.6
_ 0.4
_ 0.2
0
ξ
Fig. 6. Theoretical plot of (a) coefficient ratio (b5 =a5 ) vs. (Km I1 ; x) and (b) theoretical plot of ratio (b5 =a5 ) vs. x for Km I1 ¼ 0:9 [W]. In both plots data is calculated using a normalized I1 ¼ 1 [W] and x is given in [W1].
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3.1. Frequency spectrum and distortion of the modulator In order to analyze the frequency spectrum of the modulator, we need to assume a given applied voltage signal. If we apply a modulating signal such as the one given by Eq. (6) around the operating point Q; we can find its spectrum by substituting Eq. (6) into Eq. (21), which lead us to Im0 ¼
k m I1 f1 þ b1 Vy cosðotÞ b3 Vy3 cos3 ðotÞ þ b5 Vy5 cos5 ðotÞ 2 b7 Vy7 cos7 ðotÞ þ ?g 1 b1 Vy 3b3 Vy3 5b5 Vy5 35b7 Vy7 þ þ þ ? cosðotÞ Ekm I1 2 2 8 16 128 b3 Vy3 5b5 Vy5 21b7 Vy7 þ þ þ ? cosð3otÞ 8 32 128 b5 Vy5 7b7 Vy7 b7 Vy7 þ þ ? cosð5otÞ þ þ ? cosð7otÞ þ ? ; 32 128 128
ð23Þ
where o ¼ 2pf : This equation is of the form Im0 ¼ I1 ðr0 þ r1 ðVy ÞcosðotÞ þ r3 ðVy Þcosð3otÞ þ yÞ;
ð24Þ
and we can calculate the percentage of distortion %di as
PN
jr ðVy Þj n¼1 r2nþ1 ðVy Þ %di ¼ 100E 3 100: jr1 ðVy Þj jr1 ðVy Þj
ð25Þ
ξ 1 0.9 0.8
0 -0.4 -1
0.7
T
0.6 0.5 0.4 0.3 0.2 0.1 0 0
45
90
135
180
VA [deg] Fig. 7. Theoretical and experimental plots of the intensity transfer function T vs. VA [deg], for x ¼ 1; x ¼ 0:4 and x ¼ 0: Solid lines correspond to theory whereas the markers correspond to experiment.
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4. Experimental results In our experiments a modulating signal Vs of low frequency (f E6:5 Hz) was applied to the modulator. In Fig. 7 we show the theoretical and experimental plots of the intensity transfer function T vs. VA for different values of the feedback factor x; i.e. for x ¼ 1; x ¼ 0:4 and x ¼ 0: The graphs for T ¼ ðIm0 =km I1 Þ were recorded by a digital oscilloscope and normalized using a value of km I1 ¼ 1 [W], while the factor x is given in [W1]. Solid lines correspond to theory whereas the markers correspond to experiment. Figs. 8 and 9 correspond to the experimental plots when the modulator is operating with a feedback factor x ¼ 1: In Fig. 8 we show the applied voltage VA ðtÞ vs. time t for a value of Vy ¼ 21:51; and in Fig. 9 we show the
110
VA [deg]
100
90
80
70
60 0
0.2
0.4
0.6
0.8
1
t [sec.] Fig. 8. Applied voltage VA [deg] vs. time t [s], for a feedback factor x ¼ 1:
0.8
[a.u.]
I m'
0.7
0.6
0.5
0.4 0
0.2
0.4
0.6
0.8
1
t [sec.] Fig. 9. Output light intensity Im0 [a.u.] vs. time t [s], for a feedback factor x ¼ 1:
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0.07 0.06
M [a.u.]
0.05 0.04 0.03 0.02 0.01 0
20
40
60
80
100
f [Hz] Fig. 10. Frequency spectrum (magnitude M [a.u.] vs. frequency f [Hz]) of the output light intensity Im0 for x ¼ 1 and Vy ¼ 21:51: Values for the 3rd, 5th, 7th and 9th harmonics are amplified by a factor of 20, so %di ¼ 0:9%:
0.08 0.07 0.06
M [a.u.]
0.05 0.04 0.03 0.02 0.01 0
20
40
60
80
100
f [Hz] Fig. 11. Frequency spectrum (magnitude M [a.u.] vs. frequency f [Hz]) of the output light intensity Im0 for x ¼ 0:4 and Vy ¼ 21:51: Values for the 3rd, 5th, 7th and 9th harmonics are amplified by a factor of 20, so %di ¼ 1:6%:
corresponding plot of the output light intensity Im0 vs. time t: Figs. 10–12 show the frequency spectrum (magnitude M vs. frequency f [Hz]) of the output light intensity Im0 for x ¼ 1; x ¼ 0:4 and x ¼ 0; respectively. All these spectra correspond to the same value of the modulating voltage signal Vy ¼ 21:51E55 V, and the values of the 3rd, 5th, 7th and 9th harmonics are amplified by a factor of 20 times. From Figs. 10–12 we can estimate the values for the percentage of distortion %di ¼ 0:9%; %di ¼ 1:6% and %di ¼ 2:3%; respectively. This means that in this case the feedback loop helps to reduce the distortion from 2.3% to 0.9%, but if higher modulating voltage signal Vy is used, a more dramatic improvement can be achieved.
578
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M [a.u.]
0.08
0.06
0.04
0.02
0
20
40
60
80
100
f [Hz] Fig. 12. Frequency spectrum (magnitude M [a.u.] vs. frequency f [Hz]) of the output light intensity Im0 for x ¼ 0 and Vy ¼ 21:51: Values for the 3rd, 5th, 7th and 9th harmonics are amplified by a factor of 20, so %di ¼ 2:3%:
5. Conclusions We have presented a simple modulation scheme that does not require polarization optical components, whose output light intensity can be close to 100% of input light intensity, and that shows a more linear response than the conventional interferometric light intensity modulators. The device is insensitive to the input polarization state of light, making it more suitable for communications systems. Our experimental results are in good concordance with the theoretical predictions.
Acknowledgements This work was carried out under Federal Research Projects 35208-A and 32249-E of CONACYT, Mexico. The authors want to thank L. Mazzola for reviewing the manuscript.
References [1] Garcia-Weidner A. Design of an ultralinear Pockels effect modulator: a successive approximation method based on Poincare sphere analysis. Appl Opt 1993;32(36):7313–25. [2] Farwell ML, Lin Z-Q, Ed Wooten, Chang WSC. An electrooptic intensity modulator with improved linearity. IEEE Photon Tech Lett 1991;3:792–5. [3] Burns WK. Linearized optical modulator with fifth order correction. J Lightwave Tech 1995;13: 1724–7. [4] Birks TA, Morkel PR. Jones calculus analysis of single-mode fiber Sagnac reflector. Appl Opt 1988;27:3107–13.
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[5] Dennis ML, Moeller RP, Burns WK. Bias-drift-free intensity modulator for low-frequency operation. CLEO’97 Proceedings, 1997. p. 54–5. [6] Frankel MY, Esman RD. Optical single-sideband supressed-carrier modulator for wide-band signal processing. J Lightwave Tech 1998;16:859–63. [7] Tapia-Mercado J, Khomenko AV, Garc!ıa-Weidner A. Precision and sensitivity optimization for whitelight interferometric fiber-optic sensors. J Lightwave Tech 2001;19(1):70–4. [8] Khomenko AV, Garc!ıa-Weidner A, Tapia Mercado J, Garcia-Zarate MA. High-resolution fast response fiber-optic laser calorimeter using a twin interferometric temperature sensor. Opt Commun 2001, in press. [9] Yu A, Siddiqui AS. Systematic method for the analysis of optical fibre circuits. IEE Proc Optoelectron 1995;142:165–75.
Polarization insensitive linear intensity modulation in a Sagnac fiber-optic loop with an electro-optic cell and optoelectronic feedback A. Garcia-Weidner*, C. Torres Torres, A.V. Khomenko, M.A. Garcia-Zarate ! Cient!ıfica y de Estudios Superiores de Ensenada, CICESE, Centro de Investigacion Carretera Tijuana-Ensenada Km. 107, Ensenada, Baja California, C.P. 22860, Mexico
Abstract We present a modulation scheme consisting of a fiber-optic Sagnac interferometer, which has an electro-optic modulator (Pockels cell) in the loop of the interferometer. In order to diminish the nonlinearity of the Sagnac interferometer transmittance function, we monitor a very small part of the output light intensity by means of a photodiode and then amplify this signal by a constant factor x: This amplified signal is added to the electrical modulating signal that is being applied to the electro-optical modulator. The consequence of introducing this feedback loop is that by properly choosing the amplifying factor x; we can increase the linearity of the modulation. The device is insensitive to the polarization state of the input light, so it should be used for intensity modulation of nonpolarized light with up to 100% modulation depth. The theoretical and experimental examination of the device is presented. r 2002 Elsevier Science Ltd. All rights reserved. Keywords: Sagnac interferometer; Electro-optic modulators; Optical fiber devices
1. Introduction Electro-optic light intensity modulators root their behavior on interference phenomena, and then they tend to have nonlinear squared sinusoidal intensity transfer functions, i.e. the dependence of the intensity modulated light beam on the applied voltage. In order to maintain linearity in the modulation, these devices are *Corresponding author. Tel.: +52-646-175-0551; fax: +52-646-175-0553. E-mail address: [email protected] (A. Garcia-Weidner). 0143-8166/03/$ - see front matter r 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 1 4 3 - 8 1 6 6 ( 0 2 ) 0 0 0 4 6 - 5
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biased to the linear portion of the sine curve and used with small modulating signals. Then the sine transfer function results in a limited linear dynamic range for the device. Some schemes for enhancing the linearity using several modulators in tandem have been proposed [1–3]. Electro-optic modulators based on a Pockels cell sandwiched between polarizers have the disadvantage that up to 50% of input light intensity is lost. Although Sagnac interferometer modulators also exhibit a nonlinear transfer function [4–6], in this work we analyze the use of a Sagnac fiber-optic interferometer with optoelectronic feedback to improve the linearity of the modulator’s response. Since the device does not need the use of polarizers, it is simpler in it’s construction, and by properly choosing the amplifying factor x; close to 100% of output light intensity can be achieved. Also, due to the lack of the need of polarizers, the device is able to modulate nonpolarized light, making it more suitable for fiber optics communication systems. This is important since optical fibers tend to introduce random fluctuations on light polarization due to imperfections on the fiber, as well as due to bending or twisting of the fiber. A solution for diminishing these fluctuations, could be the use of hibirrefringence (HB) polarization preserving fibers [7,8], but then another problem arises: the use of HB fibers in the Sagnac loop makes it highly sensitive to environmental temperature variations, and the scheme should be used as a temperature sensor rather than as a light intensity modulator. Also, in order to properly operate the Sagnac loop, the fast axis of the birrefringent fiber needs to be rotated by an angle of 901 [7,8] by means of a mechanical twist or by the use of a quarter wave-plate, but in our scheme, a DC voltage applied to the electro-optic modulator solves the problem, making it’s optical set up more simple.
2. Description of the Sagnac modulator 2.1. Optical set up The experimental set up is shown in Fig. 1. In our description, light fields impinging on the coupler are represented by Jones vectors Ei (i ¼ 1; 2; 3; 4), whereas optical fields emerging from the coupler are represented by primed vectors Ei 0 . Light emitted by an unpolarized He–Ne laser that has a light intensity I1 ¼ 10 mW is launched by a lens L into an optical fiber that is connected to port (1) of a bidirectional fiber optic 2 2 coupler C: This coupler has an intensity coupling coefficient k ¼ 0:5: Ports (3) and (4) of the coupler are connected into a Sagnac loop, which is formed by two optical fibers F1 and F2 of equal length of 25 cm (drawing is not to scale) and one electro-optical modulator MOD. The modulator MOD consists of a Pockels cell with its induced fast F and slow S axes oriented at an angle c ¼ 451 relative to the plane defined by the Sagnac loop. The modulator MOD has a half-wave voltage of Vp E230 V. Light intensity that impinges on port (1) of coupler C is split by 50% and directed towards ports (3) and (4) and then both light beams travel in opposite directions in the closed loop. Due to the bending of the Sagnac’s loop, the horizontal ‘‘x’’ axis of the laboratory coordinate system is rotated by an angle of 1801, as indicated in Fig. 1. After traversing the closed loop, light beams
A. Garcia-Weidner et al. / Optics and Lasers in Engineering 39 (2003) 567–579 Y
569
F1
X
Y
E '3
3
LASER
S
E1
I1
45
F2 1 C
L
E '2
2x2
4
X
F MOD V (t)
E '4
2 I 'f
D PD f
I 'm
PDm
D ig. O sc.
1x2
A Vm (t) Vs
Vs Vf
Fig. 1. Experimental optical set up.
impinging in ports (3) and (4) are recombined inside the bi-directional coupler and directed to ports (1) and (2). Light intensity I20 emerging from port (2) is connected to the input port of a 1 2 directional coupler D which has a coupling coefficient kf ¼ 0:1: The output modulated light intensity Im0 is given by Im0 ¼ km I20 ;
ð1Þ
where km ¼ 1 kf : This Im0 emerging from coupler D is captured by a photo detector PDm, converted into a time varying voltage signal Vm ðtÞ and then connected to the digital oscilloscope Dig Osc. On the other output port of coupler D the emerging light intensity If0 will be used as the feedback signal to control the behavior of the modulator. The light intensity If0 is given by If0 ¼ kf I20 :
ð2Þ
This light intensity If0 is captured by a photo detector PDf and converted into an electrical feedback signal Vf given by Vf ¼ bIf0 ¼ bkf I20 ;
ð3Þ
where b is the conversion factor of the photo detector given in [V/W]. The modulating electrical signal Vs and the feedback signal Vf are connected to an electronic amplifier A; and then the resulting voltage signal V ðtÞ is given by V ðtÞ ¼ aðVs þ Vf Þ ¼ VDC þ VAC cosð2pftÞ;
ð4Þ
where subscripts of VDC and VAC correspond to the DC and AC components of the modulating voltage, f is the frequency, and a is the gain factor of the amplifier. Since the modulator is a Pockels cell, voltage V ðtÞ induces a phase retardation G between the fast and slow ray components traveling inside of the modulator. The induced
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retardation is given by pV ðtÞ G¼ : Vp
ð5Þ
For convenience, using Eq. (5) we can express the half value of V ðtÞ in angular units [deg] or [rad] as p V ðtÞ ¼ Vq þ Vy cosð2pftÞ; VY ðtÞ ¼ ð6Þ Vp 2 where subscripts of Vq and Vy correspond to the DC and AC components of VY ðtÞ: In practice, the values of voltage are limited to 0oVy oVp =4 or 0oVAC oVp =2: 2.2. Polarization independent intensity transfer function of the modulator As we will see, although we are using a polarization Pockels cell modulator, the output modulated light beam Im0 is independent of the polarization state of the input ~1 : For calculating the transfer function of the light beam represented by E modulating system we will use the Jones calculus and the diagram of its equivalent ~0 optical circuit [9] for the experimental setup which is shown in Fig. 2. The field E 2 0 that emerges from coupler D and its corresponding light intensity I2 are given by ~1 ; ~0 ¼ ½J12 E ð7Þ E 2
~0 ~0 j2 ¼ E ~0w E I20 ¼ jE 2 2 2;
ð8Þ
where [J12 ] is the Jones matrix that represents the optical system connecting ports (1) and (2) of coupler C and the superscript w denotes the transpose of the complex conjugate. Then the intensity transfer function between these two ports is ~w ½J12 w ½J12 E ~1 ~0 j2 E I 0 jE T12 ¼ 2 ¼ 2 2 ¼ 1 : ð9Þ 2 ~ ~ I1 j E 1 j jE 1 j OUTPUT
3’
4
[Mi] [JC]
[K14] V(t)
+
2
Kf
A [K32]
4’
3
[JA] [Mi]
[K24]
E 'm
2’
[K41]
[K23]
'
1’ E 1
E 'f
Vi
+
[K31] S
E2 = 0
Km
+
1
+
E1
+
INPUT
[K42]
FEEDBAC K LOOP
[K13]
Fig. 2. Equivalent optical circuit of the experimental set up.
Vf
β
If
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Following the diagram of the equivalent optical circuit sketched in Fig. 2, the Jones matrix [J12 ] can be expressed as ½J12 ¼ ½K42 ½Mi ½JC ½K13 þ ½K32 ½JA ½Mi ½K14 ;
ð10Þ
where [JC ] and [JA ] are the Jones matrices of the interferometer loop in the clockwise and anticlockwise sense respectively, [Mi ] is a coordinate conversion matrix due to the folded fiber in the Sagnac loop (similar to a mirror reflection as can be seen from Fig. 1), and [Kpq ] represents the directional coupler C matrix between ports p and q. These matrices are given by " # 1 0 ½Mi ¼ ; ð11Þ 0 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 kÞ½I ; ð12Þ pffiffiffi ð13Þ ½K32 ¼ ½K14 ¼ i k½I ; pffiffiffiffiffiffiffi where i ¼ 1 and [I] is the identity matrix. If matrices [JC ] and [JA ] represent ideal lossless linear optical components, which also are reciprocal on light propagation direction (such as wave plates do), then the Jones matrices are unitary, and therefore we can write # " # " # " JXX JYX A þ iB C þ iD JXX JXY T ; ð14Þ ½JC ¼ ½JA ¼ ¼ ¼ C þ iD A iB JYX JYY JYX JXX ½K42 ¼ ½K13 ¼
with jJxx j2 þ jJyx j2 ¼ A2 þ B2 þ C 2 þ D2 ¼ 1: Substituting Eqs. (10)–(14) in Eq. (9) we obtain T12 ¼ 4D2 kð1 kÞ þ ð2k 1Þ2 ¼ D2 :
ð15Þ
From this equation, we can see that T12 is independent from any input ~1 : The Jones matrix [JC ] for an electron optical modulator is polarization state E given by the Jones matrix of a linear retardation wave plate. Then the Jyx component is given by Jyx ¼ i sinð2cÞsinðG=2Þ: Since in our case c ¼ 451; the light intensity Im0 emerging from port (2) can be written as G pV ðtÞ I20 ¼ T12 I1 ¼ I1 sin2 ¼ I1 sin2 : ð16Þ 2 2Vp If we define the following expressions for VA and the feedback factor x [W1] by pa VA ¼ ð17Þ VS ; 2Vp pa bkf ; ð18Þ x¼ 2Vp km and taking into account that Im0 ¼ km I20 ; then we can write an expression for this output modulated light intensity as Im0 ¼ km I1 sin2 fVA þ xIm0 g:
ð19Þ
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It is important to note from Eq. (17) that VA is expressed in angular units rather than in volts. From Eq. (19) we can see that the output light intensity Im0 can reach almost 90% of input light intensity I1 ; depending on the chosen value of km E1:
3. The effect of feedback on the modulator’s performance In the case that we have no feedback signal (x ¼ 0), the intensity transfer function of a Sagnac modulator is reduced to Eq. (16) which is the same as the intensity transfer function of a conventional electro-optic modulator [1], i.e. a Pockels Cell sandwiched between crossed polarizers. But there are two main advantages of using a Sagnac modulator: (1) a Sagnac modulator becomes insensitive to the input polarization state and it does not require the use of polarizers, making the optical setup more simple, and (2) output light intensity can reach almost 100% of input light intensity compared to 50% of the conventional modulator. Using Eq. (19), in Fig. 3 we show the theoretical plot of the normalized output light intensity Im0 vs. VA (with Km I1 ¼ 1) for different values of x within the interval 1oxo1: For jxj > 1 the device shows a multistable behavior. In order to operate this device appropriately, we need to apply a DC bias voltage so that Vq ¼ p=4; which lead us to the operating point Q at the central part of the transmittance curve (T ¼ 50%). As can be seen from Fig. 3, when xa0 the intensity transfer function apparently becomes more linear. To examine this linearity we make a Taylor series expansion of Eq. (19) around the operating point Q: In the case for
Fig. 3. Theoretical plot of normalized transmittance T ¼ ðIm0 =km I1 Þ vs. VA [deg]. (a) x ¼ 1; (b) x ¼ 0; (c) x ¼ 0:5; (d) x ¼ 1:
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x ¼ 0 we obtain Im0
N X km I1 ð1Þð2Nþ1Þ ð2VA Þð2Nþ1Þ 1þ ¼ 2 ð2N þ 1Þ! N¼0
E
!
km I1 ð1 þ a1 VA a3 VA3 þ a5 VA5 a7 VA7 þ a9 VA9 ?Þ; 2
ð20Þ
where a1 ¼ 2; a3 ¼ 4=3; a5 ¼ 4=15; a7 ¼ 8=315; a9 ¼ 4=2835: In the case of 1oxo0; the Taylor series expansion of Eq. (19) around the operating point Q is given by Im0 ¼
k m I1 km I1 4km I1 VA3 þ VA þ 2 1 k m I1 x ðkm I1 x 1Þ 3! 16km I1 ð1 þ 9km I1 xÞ VA5 þ 5! ðkm I1 x 1Þ7 þ
2 2 2 64km I1 ð1 þ 54km I1 x þ 225km I1 x Þ VA7 10 5! ðkm I1 x 1Þ
2 2 2 3 3 3 256km I13 ð1 þ 243km I1 x þ 4131km I1 x þ 11025km I1 x Þ VA9 ? 9! ðkm I1 x 1Þ13 k m I1 E ð1 þ b1 VA b3 VA3 þ b5 VA5 b7 VA7 þ b9 VA9 ?Þ: 2
þ
ð21Þ
Now we can compare the bi coefficients of this series with coefficients ai which correspond to the Taylor series regarding the case of a modulator without feedback. This comparison leads us to 4 1 b3 b1 ¼ ; ; 1 k m I1 x a3 a1 7 b1 ¼ ð1 þ 9km I1 xÞ; a1 10 b1 2 2 2 ¼ ð1 þ 54km I1 x þ 225km I1 x Þ; a1 13 b1 2 2 2 3 3 3 ¼ ð1 þ 243km I1 x þ 4131km I1 x þ 11025km I1 x Þ: a1
b1 ¼ a1 b5 a5 b7 a7 b9 a9
ð22Þ
In Figs. 4a, 5a and 6a, we can see the three-dimensional plots of the ratios (bi =ai ) vs. (x; km I1 ) for i ¼ 1; 3; 5: In Figs. 4b, 5b and 6b, we can see the bi-dimensional plots of (bi =ai ) vs. x for our particular case where km I1 ¼ 0:9: From these plots, and taking into account that we are dealing within the interval 1oxo0; we can see that bi oai (i ¼ 1; 3; 5; y), i.e. the higher order terms of the series that contribute to the nonlinearity become smaller when xa0 compared to those corresponding to the modulator without feedback (x ¼ 0).
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0 _ 0.2
0.6
_ 0.4 _ 0.6 ξ
0.7
0.8
b1/a1
b1/a1
0.9
0.8 0.6 0.4 0.5
0.7 0.6
_ 0.8
0.8
KmI1
0.9
0.5
1
_1
(a)
_ 0.8
_ 0.6
_ 0.4
_ 0.2
0
ξ
(b)
0.4 0.3 0.2 0.1 0 0.5
0.6
_
0.7
KmI1
0.8
_
0.9
0.6 0.5
0 _ 0.2 _ 0.4 0.6
0.8
ξ
b3/a3
b3/a3
Fig. 4. Theoretical plot of (a) coefficient ratio (b1 =a1 ) vs. (Km I1 ; x) and (b) theoretical plot of ratio (b1 =a1 ) vs. x for Km I1 ¼ 0:9 [W]. In both plots data is calculated using a normalized I1 ¼ 1 [W] and x is given in [W1].
0.4 0.3 0.2 0.1 0 _
1
(a)
1
_
0.8
_ 0.6
(b)
_
ξ
0.4
_
0.2
0
Fig. 5. Theoretical plot of (a) coefficient ratio (b3 =a3 ) vs. (Km I1 ; x) and (b) theoretical plot of ratio (b3 =a3 ) vs. x for Km I1 ¼ 0:9 [W]. In both plots data is calculated using a normalized I1 ¼ 1 [W] and x is given in [W1].
0.5
0.6
0.7
KmI1 (a)
0.8
0 _ 0.2 _ 0.4 _ 0.6 ξ _ 0.8 0.9
b5/a5
b5/a5
0.05
0 0.1
0
_ 0.05 _ 0.1
1
_1
(b)
_ 0.8
_ 0.6
_ 0.4
_ 0.2
0
ξ
Fig. 6. Theoretical plot of (a) coefficient ratio (b5 =a5 ) vs. (Km I1 ; x) and (b) theoretical plot of ratio (b5 =a5 ) vs. x for Km I1 ¼ 0:9 [W]. In both plots data is calculated using a normalized I1 ¼ 1 [W] and x is given in [W1].
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3.1. Frequency spectrum and distortion of the modulator In order to analyze the frequency spectrum of the modulator, we need to assume a given applied voltage signal. If we apply a modulating signal such as the one given by Eq. (6) around the operating point Q; we can find its spectrum by substituting Eq. (6) into Eq. (21), which lead us to Im0 ¼
k m I1 f1 þ b1 Vy cosðotÞ b3 Vy3 cos3 ðotÞ þ b5 Vy5 cos5 ðotÞ 2 b7 Vy7 cos7 ðotÞ þ ?g 1 b1 Vy 3b3 Vy3 5b5 Vy5 35b7 Vy7 þ þ þ ? cosðotÞ Ekm I1 2 2 8 16 128 b3 Vy3 5b5 Vy5 21b7 Vy7 þ þ þ ? cosð3otÞ 8 32 128 b5 Vy5 7b7 Vy7 b7 Vy7 þ þ ? cosð5otÞ þ þ ? cosð7otÞ þ ? ; 32 128 128
ð23Þ
where o ¼ 2pf : This equation is of the form Im0 ¼ I1 ðr0 þ r1 ðVy ÞcosðotÞ þ r3 ðVy Þcosð3otÞ þ yÞ;
ð24Þ
and we can calculate the percentage of distortion %di as
PN
jr ðVy Þj n¼1 r2nþ1 ðVy Þ %di ¼ 100E 3 100: jr1 ðVy Þj jr1 ðVy Þj
ð25Þ
ξ 1 0.9 0.8
0 -0.4 -1
0.7
T
0.6 0.5 0.4 0.3 0.2 0.1 0 0
45
90
135
180
VA [deg] Fig. 7. Theoretical and experimental plots of the intensity transfer function T vs. VA [deg], for x ¼ 1; x ¼ 0:4 and x ¼ 0: Solid lines correspond to theory whereas the markers correspond to experiment.
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4. Experimental results In our experiments a modulating signal Vs of low frequency (f E6:5 Hz) was applied to the modulator. In Fig. 7 we show the theoretical and experimental plots of the intensity transfer function T vs. VA for different values of the feedback factor x; i.e. for x ¼ 1; x ¼ 0:4 and x ¼ 0: The graphs for T ¼ ðIm0 =km I1 Þ were recorded by a digital oscilloscope and normalized using a value of km I1 ¼ 1 [W], while the factor x is given in [W1]. Solid lines correspond to theory whereas the markers correspond to experiment. Figs. 8 and 9 correspond to the experimental plots when the modulator is operating with a feedback factor x ¼ 1: In Fig. 8 we show the applied voltage VA ðtÞ vs. time t for a value of Vy ¼ 21:51; and in Fig. 9 we show the
110
VA [deg]
100
90
80
70
60 0
0.2
0.4
0.6
0.8
1
t [sec.] Fig. 8. Applied voltage VA [deg] vs. time t [s], for a feedback factor x ¼ 1:
0.8
[a.u.]
I m'
0.7
0.6
0.5
0.4 0
0.2
0.4
0.6
0.8
1
t [sec.] Fig. 9. Output light intensity Im0 [a.u.] vs. time t [s], for a feedback factor x ¼ 1:
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0.07 0.06
M [a.u.]
0.05 0.04 0.03 0.02 0.01 0
20
40
60
80
100
f [Hz] Fig. 10. Frequency spectrum (magnitude M [a.u.] vs. frequency f [Hz]) of the output light intensity Im0 for x ¼ 1 and Vy ¼ 21:51: Values for the 3rd, 5th, 7th and 9th harmonics are amplified by a factor of 20, so %di ¼ 0:9%:
0.08 0.07 0.06
M [a.u.]
0.05 0.04 0.03 0.02 0.01 0
20
40
60
80
100
f [Hz] Fig. 11. Frequency spectrum (magnitude M [a.u.] vs. frequency f [Hz]) of the output light intensity Im0 for x ¼ 0:4 and Vy ¼ 21:51: Values for the 3rd, 5th, 7th and 9th harmonics are amplified by a factor of 20, so %di ¼ 1:6%:
corresponding plot of the output light intensity Im0 vs. time t: Figs. 10–12 show the frequency spectrum (magnitude M vs. frequency f [Hz]) of the output light intensity Im0 for x ¼ 1; x ¼ 0:4 and x ¼ 0; respectively. All these spectra correspond to the same value of the modulating voltage signal Vy ¼ 21:51E55 V, and the values of the 3rd, 5th, 7th and 9th harmonics are amplified by a factor of 20 times. From Figs. 10–12 we can estimate the values for the percentage of distortion %di ¼ 0:9%; %di ¼ 1:6% and %di ¼ 2:3%; respectively. This means that in this case the feedback loop helps to reduce the distortion from 2.3% to 0.9%, but if higher modulating voltage signal Vy is used, a more dramatic improvement can be achieved.
578
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M [a.u.]
0.08
0.06
0.04
0.02
0
20
40
60
80
100
f [Hz] Fig. 12. Frequency spectrum (magnitude M [a.u.] vs. frequency f [Hz]) of the output light intensity Im0 for x ¼ 0 and Vy ¼ 21:51: Values for the 3rd, 5th, 7th and 9th harmonics are amplified by a factor of 20, so %di ¼ 2:3%:
5. Conclusions We have presented a simple modulation scheme that does not require polarization optical components, whose output light intensity can be close to 100% of input light intensity, and that shows a more linear response than the conventional interferometric light intensity modulators. The device is insensitive to the input polarization state of light, making it more suitable for communications systems. Our experimental results are in good concordance with the theoretical predictions.
Acknowledgements This work was carried out under Federal Research Projects 35208-A and 32249-E of CONACYT, Mexico. The authors want to thank L. Mazzola for reviewing the manuscript.
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[5] Dennis ML, Moeller RP, Burns WK. Bias-drift-free intensity modulator for low-frequency operation. CLEO’97 Proceedings, 1997. p. 54–5. [6] Frankel MY, Esman RD. Optical single-sideband supressed-carrier modulator for wide-band signal processing. J Lightwave Tech 1998;16:859–63. [7] Tapia-Mercado J, Khomenko AV, Garc!ıa-Weidner A. Precision and sensitivity optimization for whitelight interferometric fiber-optic sensors. J Lightwave Tech 2001;19(1):70–4. [8] Khomenko AV, Garc!ıa-Weidner A, Tapia Mercado J, Garcia-Zarate MA. High-resolution fast response fiber-optic laser calorimeter using a twin interferometric temperature sensor. Opt Commun 2001, in press. [9] Yu A, Siddiqui AS. Systematic method for the analysis of optical fibre circuits. IEE Proc Optoelectron 1995;142:165–75.