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Displacement measurement by synthesized light source based on fiber Bragg gratings
15 September 1998
Optics Communications 154 Ž1998. 261–267
Displacement measurement by synthesized light source based on fiber Bragg gratings Lih-Wuu Chang a , Ching-Ting Lee
a,)
, Pie-Yau Chien
b
a
b
Institute of Optical Sciences, National Central UniÕersity, Chung-Li, Taiwan Material Research Center, Chung-Shan Institute of Science and Technology, P.O. Box 90008-8-10, Lung-Tan, Tao-Yuan, Taiwan Received 29 December 1997; revised 13 April 1998; accepted 10 June 1998
Abstract This work presents a novel two-wavelength interferometer based on fiber Bragg gratings. Two light sources with different wavelengths are generated from separate fiber Bragg gratings. The high wavelength accuracy of fiber Bragg gratings allows us to enhance the measured accuracy of the interferometer. In addition, a Michelson interferometer is used to reliably measure the path-length difference signal, thereby demonstrating the proposed interferometer’s effectiveness. q 1998 Elsevier Science B.V. All rights reserved. Keywords: Fiber Bragg grating; Broadband light source; Er-doped fiber
1. Introduction Optical interferometers are conventional devices for accurately measuring the displacement and roughness of an object. The measurement range is limited to one wavelength of the light source owing to the ambiguous problem of sinusoidal curve phase output. However, a two-wavelength interferometer w1–5x can be employed to extend the measurement range for a longer period, depending on the synthetic wavelength technique of light sources. Light sources with different wavelengths used in the interferometer can be obtained either by driving varying currents during a period in a single laser diode w2x or by employing two independent laser diodes with different wavelengths w3x. Notably, controlling the wavelength of each laser diode at a fixed wavelength is extremely difficult when two independent laser diodes are employed as light sources on a two-wavelength interferometer. In addition, the
)
Corresponding author.
mode-jumping in a laser diode limits the use of the frequency modulation method on a single laser diode with different driving currents. Moreover, the driving current and environmental temperature must be precisely controlled to obtain a high stability of the synthesized wavelength on a laser diode. The fiber Bragg grating ŽFBG. device has recently been under intense development, having been extensively applied in the fields of fiber sensor systems w6,7x. This device has also been adopted to construct a fiber laser w8x. Moreover, the fact that the reflective index in FBG can be induced, or written, in the optical fiber by interfering beams of an UV laser accounts for why the Bragg wavelength of FBG can be accurately obtained. Therefore, FBG devices can be more easily used than a laser diode in terms of setting up two-wavelength interferometer repeatability. In this paper, we adopt the FBG as a high Q-value optical band-pass filter to generate a light source with high wavelength accuracy with respect to line-width and wavelength. Two light sources with high wavelength accuracy are generated after a broadband fiber source passes through two FBGs; those wavelengths are independent of the power
0030-4018r98r$ - see front matter q 1998 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 Ž 9 8 . 0 0 3 2 3 - X
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L.-W. Chang et al.r Optics Communications 154 (1998) 261–267
fluctuation of the broadband light source. Thus, it is unnecessary to accurately control the wavelengths of FBG used in the interferometer while measuring the optical path difference.
superposition of the two sets of individual interference fringes. It can be expressed as Iout s ² E1 E1) : q ² E2 E2) : ,
Ž1.
where E1 and E2 are the electrical fields at the detector of the two lights, respectively, and they are given by 2. Principle Fig. 1 schematically depicts the experimental setup of the two-wavelength interferometer based on FBGs. A broadband light source ŽBLS., which is generated after a pumping laser diode passing into an Er-doped fiber, used as a source passes into FBGs, and a fiber isolator ŽFI. is used to prevent the return light from backing into a broadband light source which would ultimately cause light lasing. Next, a broadband fiber coupler ŽBFC. is employed to obtain the lights reflected from the two FBGs. The two wavelengths of FBGs are denoted as l1 and l2 , respectively; a temperature stabilizer ŽTS. is used to control the wavelengths of FBGs. The lights reflected from the two FBGs are virtually incoherent with each other, and then when they are sent into an optical interferometer ŽOI., the output intensity Iout of the interferometer is the intensity
E1 s A1 exp w i Ž v 1 t q 2p L1rl1 . x q A1 exp w i Ž v 1 t q 2p L2rl1 . x , and E2 s A 2 exp w i Ž v 2 t q 2p L1rl2 . x q A 2 exp w i Ž v 2 t q 2p L 2rl2 . x ,
Ž2.
where v 1,2 represents the angular frequency of the two lights, and L1,2 denotes the two individual optical paths in the interferometer. The output signal Iout of the optical interferometer can be written as Iout s K c1 P1 w 1 q cos Ž f 1 . x q K c2 P2 w 1 q cos Ž f 2 . x , Ž 3 . where K c1 denotes the coherent function of light reflected from FBG1; K c2 represents the coherent function of light reflected from FBG2; P1 is the output power of light reflected from FBG1; P2 denotes the output power of light reflected from FBG2; f 1 s 2p Ž L1 y L 2 .rl1 represents the phase delay of wavelength l1 in the optical interferometer; f 2 s 2p Ž L1 y L2 .rl2 is the phase delay of wavelength l 2 in the optical interferometer; L s L1 y L 2 denotes the unbalanced path-length difference of the optical interferometer. In our system, the measured distance is significantly smaller than the coherence lengths of both light sources. Therefore, we can assume that K c1 s K c2 s K, and P1 s P2 s P. The AC component IAC of the output signal Iout can be expressed as IAC s KP cos Ž fL . cos Ž fl . ,
Ž4.
where
fL s 2p LrL ,
fl s 2p LrG ,
L s l1 l2r Ž l1 y l2 . ( l12r Ž l1 y l2 . , G s l1 l2r Ž l1 q l2 . ( 2 l12r Ž l1 q l2 . ,
Fig. 1. Experimental setup of the measured system. The broadband light source ŽBLS. is generated after a pumping source passes into an Er-doped fiber ŽEDF.. The synthesized sources derived from fiber Bragg gratings ŽFBG1 and FBG2. are sent into a Michelson interferometer ŽMI. and meanwhile are phase modulated after applying a sawtooth signal to the PZT. The signal output ŽOrP. of path-length difference is demodulated using an electric demodulation system ŽEDS..
where L and G represent the synthesized and averaged wavelength of the two-wavelength interferometer, respectively. In Eq. Ž4., the two optical phase delays, fL and fl , can be used to detect the unbalanced path-length difference L. In this paper, the synthesized optical phase delay fL is adopted for measuring the unbalanced path-length difference L and, meanwhile, the optical phase delay fl of average wavelength is used as a phase modulation of the interferometer for coherence function detection. In general, there are several techniques of deriving the coherence function of the light source, such as peak detection, lock-in demodulation, etc. However, we can also use the methods of detection technique mentioned above to evaluate the envelope of the coherence function comprised of two light
L.-W. Chang et al.r Optics Communications 154 (1998) 261–267
263
Fig. 2. Optical interfering output signal of the synthesized wavelength with various path-length differences.
sources. The optical phase of average optical phase delay fl is phase modulated with a sawtooth waveform with modulation index of 2p , and its output signal is sinusoidal. Similar to the lock-in detection technique, the output signal can be used as a reference signal, and are then mixed together. While the mixed output signal is passed through a low pass filter, the envelope of the coherence function is derived. Fig. 2 displays the output signal of Eq. Ž4.. According to this figure, G ( l1 ( l 2 and L 4 l1, l 2 , for l1 ( l2 s l. Meanwhile, fl ( f 1 ( f 2 . The path-length difference of the optical interferometer is modulated by a sawtooth modulation signal Sm ŽSMS., and then the optical phase modulation can be expressed as
fm s a t , 0 F t F T ,
Ž6.
form. Next, the square waveform signal used as a reference signal is mixed with the output signal IAC and then the carrier frequency vm is eliminated after a low pass filter. The output signal Iout2 , indicated as the envelope of the coherent function, of the low pass filter can be expressed as Iout2 s KPK 0 cos Ž fL . s KPK 0 cos Ž 2p LrL . ,
Ž8.
where K 0 denotes the transfer gain of the electric circuit. The advantages of the synthesized wavelength based on FBG are discussed in the following. 2.1. WaÕelength stability For our synthesized wavelength interferometer, the optical phase delay is expressed as fL s 2p LrL. Moreover,
where T and a are the period and slope of the sawtooth modulation signal Sm , respectively. The peak modulation index Žs a T . is selected herein to be 2p , and an AC coupling optical receiver is used to transfer the interfering output of the phase modulated two-wavelength interferometer into the electric signal. The output signal IAC can be rewritten as IAC s KP cos Ž fL . cos Ž fm q fl . s KP cos Ž fL . cos Ž vm t q fl . ,
Ž7.
where vm s 1rT denotes the angular frequency of the modulation signal Sm . Fig. 3 illustrates the relationship between the interferometric output signal and the applied sawtooth modulation signal Sm . This figure clearly indicates that the output signal is purely sinusoidal with 2p modulation index. When the output signal IAC is passed through an electrical demodulation system ŽEDS., mentioned above, its waveform is shaped into a square wave-
Fig. 3. Electric output signal of the synthesized wavelength after sawtooth modulation with frequency vm s1r T. The modulation index is 2p rad.
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the change of optical phase delay dfL induced by both changes of path-length difference and synthesized wavelength can be expressed as
dfLrfL s d LrL y dLrL .
Ž9.
In addition, the synthesized light source also has better path-length difference detection resolution than that of the laser diode. 2.2. System reproducibility
By differential 1rL Žs 1rl1 y 1rl2 ., we have
dLrL s 2 dlrl q d Ž l1 y l2 . rl = Lrl .
Ž 10.
When the FBGs are chosen with similar temperature dependent coefficients, and those are also temperature stabilized within the same temperature controller, then d Ž l1 y l2 . approximately approaches zero. The influenced factor, shown in Eq. Ž10., of the synthesized wavelength is then ignored, and Eq. Ž10. can be rewritten by
dLrL ( 2 dlrl ,
Ž 11.
The wavelength change dl, which depends on both temperature change d T and driving current change d I, can be expressed as
dlrl s w Ž dlrl . rd T x d T q w Ž dlrl . rd I x d I.
Ž 12 .
Since the FBG is a passive device, the driving current change d I can be ignored. Typically, the coefficient dlrd T equals 0.01 nmr8C, and for a temperature stabilized FBG with wavelength of 1550 nm, we obtain a wavelength stability for d T of FBG stabilized within 10y2 8C, then
dlrl s 10y7 , for single FBG and
dLrL s 2 = 10y7 , for the synthesized FBG.
The wavelength variation D L of a synthesized light source used in an interferometer profoundly influences the measured path-length difference accuracy of an interferometer. The relationship between the minimum detectable path-length difference and the wavelength variation of the light source with a fixed optical phase delay Ž D f s 0. can also be expressed as D LrL s D LrL .
Ž 16.
The fact that FBG is a passive device accounts for why Ža. its wavelength variation relies on the wavelength accuracy of UV beams for masking and Žb. the wavelength variation of FBG can be derived within 0.1 nm. However, the wavelength variation of a laser diode with wavelength of 1550 nm is about 10 nm due to the characteristics of the material and the fabrication process. Thus we have D LrL
laser diode s100 = D LrL FBG .
Ž 17.
The above equation indicates that FBG system reproducibility is superior than that of the laser diode, thereby requiring no further efforts for system calibration. Meanwhile, the measurement accuracy of the two-wavelength interferometer generated by the FBG is far better than that of the laser diode.
Ž 13.
However, for a laser diode, the coefficients of dlrd T and dlrd I are approximately 0.3 nmr8C and 0.024 nmrmA with the same operating wavelength, respectively. When the stability of temperature and driving current is respectively controlled at 0.018C and 0.01 mA, the wavelength stability is given by
2.3. Output power of the light source
and
For a commercial laser diode, its output power and line-width are 1 mW and 1 nm, respectively. However, for FBGs with 99% reflection used to back reflect the Er-doped broadband light source with the respective output power and bandwidth being 40 mW and 40 nm, the output power and line-width of the light source reflected from FBG are 0.1 mW and 0.1 nm, respectively. Thus, we have
dLrL s 4 = 10y6 , for the synthesized laser diode. Ž 14 .
Po
The minimum detectable path-length difference for a synthesized wavelength L with a fixed optical phase delay Ž fL s 0. can be expressed as
Nevertheless, the system signal to noise ratio ŽSrN. relies on the output power of the light source; the SrN ratio of FBG is smaller than that of the laser diode as well. The fundamental limit of the measured optical phase is set by the photon shot noise in optical interferometers. Moreover, the amount of the noise depends on the optical power received by a photodetector. The corresponding root mean square value of the phase noise dfrms can be expressed as
dlrl s 2 = 10y6 , for single laser diode
d LrL s dLrL .
Ž 15.
Eq. Ž13. and Eq. Ž14. indicates that dLrL equals 2 = 10y7 for FBGs, and meanwhile it is 4 = 10y6 or larger for two laser diodes. If the path-length difference L s 100 mm, then the minimum detectable path-length differences d L are respectively 0.02 nm for FBG and 0.4 nm for laser diodes. Comparing the synthesized light source with a laser diode in terms of wavelength stabilization dlrl, it clearly reveals that the wavelength stabilization of a passive FBG device is better than that of an active laser diode.
FBG s 0.1
Po
laser diode
dfrms s Ž 2 hnrPo .
1r2
B,
Ž 18.
Ž 19.
where h and B are the Plank constant and bandwidth of the detection system, respectively. n and Po are the frequency and optical power of the light source, respectively. For a light source with output power of 100 mW, its
L.-W. Chang et al.r Optics Communications 154 (1998) 261–267
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Table 1 Source
Synthesized light source Laser diode
Item Wavelength stability Ž dlrl.rd T
Reproducibility Ž dl.
Minimum detectable phase delay Ž dfmin .
Output power
- 10y5 - 10y4
- 0.1 nm - 10 nm
10y6 radrHz 1r2 10y7 radrHz 1r2
- 0.1 mW - 1 mW
minimum detectable optical phase delay is approximately 10y7 radrHz 1r2 . However, the output power of FBG is one-tenth that of the laser diode, then its minimum optical phase delay df < min is up to 10y6 radrHz 1r2 . Table 1 compares the performance of a two-wavelength interferometer with a laser diode and synthesized light source reflected from FBGs. According to the comparison, light reflected from FBG can be more appropriately adopted as light source in two-wavelength interferometers than the laser diode.
3. Experimental setup and results Fig. 1 depicts the experimental setup. A high power laser diode ŽLD. with l s 0.98 mm was used to pump the Er-doped fiber ŽEDF. after passing through a wavelength division multiplexer ŽWDM. and, subsequently, a broadband light source ŽBLS. was generated. The output power of the BLS was kept constant by feedback controlling the driving current of the laser diode. The bandwidth of BLS measured by the optical spectrum analyzer is from 1.525 to 1.565 mm, Ži.e., the line-width is about 40 nm., and the output power of BLS is about 20 mW. A fiber isolator ŽFI. with l s 1.55 mm was used to prevent the return light backing into the Er-doped fiber loop from the FBGs; such a backing effect would cause the light within the pumping
Fig. 4. Optical spectrum of two light sources generated by fiber Bragg gratings. 1 and 2 indicate light of wavelength l s1527 and 1550 nm, respectively.
loop to generate a lasing effect. Next, a broadband fiber coupler ŽBFC. was employed to obtain the lights reflected from the two FBGs. Fig. 4 displays the optical spectrum of the combined light sources. This figure clearly indicates that only two wavelengths l s 1.527 and 1.550 mm existed simultaneously. Owing to that the line-width of the wavelengths l s 1.527 and 1.550 mm were 0.1 nm and 0.2 nm, respectively, it could be theoretically calculated that the synthesized wavelength, L s l1 l2r< l1 y l 2 <, was 110 mm. Those corresponding coherence lengths were 16 and 8 mm, respectively. Fig. 5 summarizes the results of the measured visibility. The solid and dashed lines shown the visibility curves of wavelength of 1.527 and 1.550 mm, respectively. Obviously, both coherence lengths were longer than the synthesized wavelength, and the visibility of the two light sources could be considered constant during the measurement. The two light sources with optical power of about 26 mW combined together through the broadband fiber coupler were first collimated using a lens ŽL1. and then sent into an optical Michelson interferometer ŽMI.; those sources were then separated into two parts using beam splitter ŽBS.. One of the light sources was reflected by a mirror ŽM1., and the other light source, was reflected by a mirror ŽM2., both were modulated by a sawtooth modulation signal with a frequency of 100 Hz employed to the
Fig. 5. Measured visibility of the two individual light sources. The dashed and solid curves represent the wavelengths of 1527 nm and 1550 nm, respectively.
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L.-W. Chang et al.r Optics Communications 154 (1998) 261–267
Fig. 8. Measured output with a path-length difference step change of 10 mm. The bandwidth of the low pass filter ŽLPF. is 100 Hz. Fig. 6. Block diagram of the self-mixing demodulation system. The electric waveform is shaped into a square waveform through ZD, and the electric signal is demodulated using an EM demodulation system. The signal output is derived after a low pass filter.
piezo-electric transducer ŽPZT., which was used to linearly scan the optical phase of the Michelson interferometer. The sawtooth modulation signal with a frequency of 100 Hz was generated from a function generator and sent into a power amplifier. The modulation index of the applied sawtooth modulation signal was selected to be 2p rad for averaged wavelength G . Next, the light sources of phase modulation were back reflected by a mirror ŽM2. and, then, the interfering signals were collimated through a lens ŽL2. and received by a photo-detector ŽPD.. The fact that the visibilities of the two wavelengths were maintained constant within the measured region accounts for why the output of the optical interferometer after modulation was a pure sinusoidal. Moreover, its magnitude could be derived by self-mixing with demodulation technique. The interfering lights were initially transferred into an electric signal, which was later demodulated by the selfmixing demodulation system shown in Fig. 6. When the electric signal of the photodetector ŽPD. output initially
went through a zero crossing comparator ŽZD., its waveform was shaped into a square waveform; that was used as a reference signal for the electric mixer ŽEM.. Meanwhile, the electric signal of the photodetector output was also sent to the input port of the electric mixer. When the output of the electric mixer went through a low pass filter ŽLPF., the visibility of the synthesized light source with various path-length differences, L1, L2 , L3 , . . . , was derived. Fig. 7 summarizes the results obtained from various path-length differences. This figure indicates that the measurable range was about 110 mm, i.e. approximately equal to the synthesized wavelength. During the whole experimental process, various disturbances, such as the fluctuation of pumping diode intensity, mechanical vibrations of the fiber caused by polarization changes, temporal changes of the orientation of the interferometer mirrors, etc., influenced the experimental results. The optical output power was controlled by feedback controlling the driving current of the pumping laser diode, and a polarization controller was used to minimize the effect of the polarization changes. The experimental setup was also isolated to avoid environmental disturbance. Fig. 8 depicts the experimental output when the path-length difference step changes 10 mm. The results reveal that the minimum detection path-length difference of the synthesized light sources is 6 nmrHz 1r2 .
4. Conclusion
Fig. 7. Measured visibility of the synthesized light source. The measured range is approximately 110 mm.
This study presents a novel two-wavelength interferometer based on a light source reflected from fiber Bragg gratings. The light sources are generated by the broadband light source through fiber Bragg gratings; the stability of the wavelengths is easily controlled using a temperature stabilizer. The synthesized wavelength is longer than those of conventional light sources, thereby allowing it to extend the measured range of path-length difference up to several hundreds times. Because a modulation index of 2p rad of the modulation signal is employed, the waveform of the interfering signal after a receiving photodetector has then a pure sinusoidal form. In addition, the demodulation be-
L.-W. Chang et al.r Optics Communications 154 (1998) 261–267
comes easier using a self-mixing lock-in detection technique. The two-wavelength interferometer based on FBG proposed herein has the following merits: 1. The wavelength stability of FBG can be controlled to better than dlrl F 10y7; 2. The FBG is a passive device and, thus, the reproducibility is better than that of an active device such as a laser diode, in which the wavelength uncertainty is less than 0.1 nm; 3. Because FBG is a compact device, the environmental temperature can be easily stabilized within 0.018C using a temperature controller; and 4. The wavelength of the FBG can be easily tuned by changing the wavelength of a UV laser.
267
References w1x P.D. Groot, S. Kishner, Appl. Optics 30 Ž1991. 4026. w2x K. Seta, B.K. Ward, Optics Comm. 77 Ž1990. 275. w3x C.C. Williams, H.K. Wickramasinghe, J. Appl. Phys. 60 Ž1986. 1900. w4x O. Sasaki, T. Yoshida, T. Suzuki, Appl. Optics 30 Ž1991. 3618. w5x E. Fischer, E. Dalhoff, S. Heim, U. Hofbauer, H.J. Tiziani, Appl. Optics 34 Ž1995. 5589. w6x Y.J. Rao, D.A. Jackson, L. Zhang, I. Bennion, Optics Lett. 21 Ž1996. 1556. w7x T.A. Berkoff, A.D. Kersey, IEEE Photon. Technol. Lett. 8 Ž1996. 1677. w8x S.M. Melle, A.T. Alavie, S. Karr, T. Coroy, K. Liu, R.M. Measures, IEEE Photon. Technol. Lett. 5 Ž1993. 263.
Optics Communications 154 Ž1998. 261–267
Displacement measurement by synthesized light source based on fiber Bragg gratings Lih-Wuu Chang a , Ching-Ting Lee
a,)
, Pie-Yau Chien
b
a
b
Institute of Optical Sciences, National Central UniÕersity, Chung-Li, Taiwan Material Research Center, Chung-Shan Institute of Science and Technology, P.O. Box 90008-8-10, Lung-Tan, Tao-Yuan, Taiwan Received 29 December 1997; revised 13 April 1998; accepted 10 June 1998
Abstract This work presents a novel two-wavelength interferometer based on fiber Bragg gratings. Two light sources with different wavelengths are generated from separate fiber Bragg gratings. The high wavelength accuracy of fiber Bragg gratings allows us to enhance the measured accuracy of the interferometer. In addition, a Michelson interferometer is used to reliably measure the path-length difference signal, thereby demonstrating the proposed interferometer’s effectiveness. q 1998 Elsevier Science B.V. All rights reserved. Keywords: Fiber Bragg grating; Broadband light source; Er-doped fiber
1. Introduction Optical interferometers are conventional devices for accurately measuring the displacement and roughness of an object. The measurement range is limited to one wavelength of the light source owing to the ambiguous problem of sinusoidal curve phase output. However, a two-wavelength interferometer w1–5x can be employed to extend the measurement range for a longer period, depending on the synthetic wavelength technique of light sources. Light sources with different wavelengths used in the interferometer can be obtained either by driving varying currents during a period in a single laser diode w2x or by employing two independent laser diodes with different wavelengths w3x. Notably, controlling the wavelength of each laser diode at a fixed wavelength is extremely difficult when two independent laser diodes are employed as light sources on a two-wavelength interferometer. In addition, the
)
Corresponding author.
mode-jumping in a laser diode limits the use of the frequency modulation method on a single laser diode with different driving currents. Moreover, the driving current and environmental temperature must be precisely controlled to obtain a high stability of the synthesized wavelength on a laser diode. The fiber Bragg grating ŽFBG. device has recently been under intense development, having been extensively applied in the fields of fiber sensor systems w6,7x. This device has also been adopted to construct a fiber laser w8x. Moreover, the fact that the reflective index in FBG can be induced, or written, in the optical fiber by interfering beams of an UV laser accounts for why the Bragg wavelength of FBG can be accurately obtained. Therefore, FBG devices can be more easily used than a laser diode in terms of setting up two-wavelength interferometer repeatability. In this paper, we adopt the FBG as a high Q-value optical band-pass filter to generate a light source with high wavelength accuracy with respect to line-width and wavelength. Two light sources with high wavelength accuracy are generated after a broadband fiber source passes through two FBGs; those wavelengths are independent of the power
0030-4018r98r$ - see front matter q 1998 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 Ž 9 8 . 0 0 3 2 3 - X
262
L.-W. Chang et al.r Optics Communications 154 (1998) 261–267
fluctuation of the broadband light source. Thus, it is unnecessary to accurately control the wavelengths of FBG used in the interferometer while measuring the optical path difference.
superposition of the two sets of individual interference fringes. It can be expressed as Iout s ² E1 E1) : q ² E2 E2) : ,
Ž1.
where E1 and E2 are the electrical fields at the detector of the two lights, respectively, and they are given by 2. Principle Fig. 1 schematically depicts the experimental setup of the two-wavelength interferometer based on FBGs. A broadband light source ŽBLS., which is generated after a pumping laser diode passing into an Er-doped fiber, used as a source passes into FBGs, and a fiber isolator ŽFI. is used to prevent the return light from backing into a broadband light source which would ultimately cause light lasing. Next, a broadband fiber coupler ŽBFC. is employed to obtain the lights reflected from the two FBGs. The two wavelengths of FBGs are denoted as l1 and l2 , respectively; a temperature stabilizer ŽTS. is used to control the wavelengths of FBGs. The lights reflected from the two FBGs are virtually incoherent with each other, and then when they are sent into an optical interferometer ŽOI., the output intensity Iout of the interferometer is the intensity
E1 s A1 exp w i Ž v 1 t q 2p L1rl1 . x q A1 exp w i Ž v 1 t q 2p L2rl1 . x , and E2 s A 2 exp w i Ž v 2 t q 2p L1rl2 . x q A 2 exp w i Ž v 2 t q 2p L 2rl2 . x ,
Ž2.
where v 1,2 represents the angular frequency of the two lights, and L1,2 denotes the two individual optical paths in the interferometer. The output signal Iout of the optical interferometer can be written as Iout s K c1 P1 w 1 q cos Ž f 1 . x q K c2 P2 w 1 q cos Ž f 2 . x , Ž 3 . where K c1 denotes the coherent function of light reflected from FBG1; K c2 represents the coherent function of light reflected from FBG2; P1 is the output power of light reflected from FBG1; P2 denotes the output power of light reflected from FBG2; f 1 s 2p Ž L1 y L 2 .rl1 represents the phase delay of wavelength l1 in the optical interferometer; f 2 s 2p Ž L1 y L2 .rl2 is the phase delay of wavelength l 2 in the optical interferometer; L s L1 y L 2 denotes the unbalanced path-length difference of the optical interferometer. In our system, the measured distance is significantly smaller than the coherence lengths of both light sources. Therefore, we can assume that K c1 s K c2 s K, and P1 s P2 s P. The AC component IAC of the output signal Iout can be expressed as IAC s KP cos Ž fL . cos Ž fl . ,
Ž4.
where
fL s 2p LrL ,
fl s 2p LrG ,
L s l1 l2r Ž l1 y l2 . ( l12r Ž l1 y l2 . , G s l1 l2r Ž l1 q l2 . ( 2 l12r Ž l1 q l2 . ,
Fig. 1. Experimental setup of the measured system. The broadband light source ŽBLS. is generated after a pumping source passes into an Er-doped fiber ŽEDF.. The synthesized sources derived from fiber Bragg gratings ŽFBG1 and FBG2. are sent into a Michelson interferometer ŽMI. and meanwhile are phase modulated after applying a sawtooth signal to the PZT. The signal output ŽOrP. of path-length difference is demodulated using an electric demodulation system ŽEDS..
where L and G represent the synthesized and averaged wavelength of the two-wavelength interferometer, respectively. In Eq. Ž4., the two optical phase delays, fL and fl , can be used to detect the unbalanced path-length difference L. In this paper, the synthesized optical phase delay fL is adopted for measuring the unbalanced path-length difference L and, meanwhile, the optical phase delay fl of average wavelength is used as a phase modulation of the interferometer for coherence function detection. In general, there are several techniques of deriving the coherence function of the light source, such as peak detection, lock-in demodulation, etc. However, we can also use the methods of detection technique mentioned above to evaluate the envelope of the coherence function comprised of two light
L.-W. Chang et al.r Optics Communications 154 (1998) 261–267
263
Fig. 2. Optical interfering output signal of the synthesized wavelength with various path-length differences.
sources. The optical phase of average optical phase delay fl is phase modulated with a sawtooth waveform with modulation index of 2p , and its output signal is sinusoidal. Similar to the lock-in detection technique, the output signal can be used as a reference signal, and are then mixed together. While the mixed output signal is passed through a low pass filter, the envelope of the coherence function is derived. Fig. 2 displays the output signal of Eq. Ž4.. According to this figure, G ( l1 ( l 2 and L 4 l1, l 2 , for l1 ( l2 s l. Meanwhile, fl ( f 1 ( f 2 . The path-length difference of the optical interferometer is modulated by a sawtooth modulation signal Sm ŽSMS., and then the optical phase modulation can be expressed as
fm s a t , 0 F t F T ,
Ž6.
form. Next, the square waveform signal used as a reference signal is mixed with the output signal IAC and then the carrier frequency vm is eliminated after a low pass filter. The output signal Iout2 , indicated as the envelope of the coherent function, of the low pass filter can be expressed as Iout2 s KPK 0 cos Ž fL . s KPK 0 cos Ž 2p LrL . ,
Ž8.
where K 0 denotes the transfer gain of the electric circuit. The advantages of the synthesized wavelength based on FBG are discussed in the following. 2.1. WaÕelength stability For our synthesized wavelength interferometer, the optical phase delay is expressed as fL s 2p LrL. Moreover,
where T and a are the period and slope of the sawtooth modulation signal Sm , respectively. The peak modulation index Žs a T . is selected herein to be 2p , and an AC coupling optical receiver is used to transfer the interfering output of the phase modulated two-wavelength interferometer into the electric signal. The output signal IAC can be rewritten as IAC s KP cos Ž fL . cos Ž fm q fl . s KP cos Ž fL . cos Ž vm t q fl . ,
Ž7.
where vm s 1rT denotes the angular frequency of the modulation signal Sm . Fig. 3 illustrates the relationship between the interferometric output signal and the applied sawtooth modulation signal Sm . This figure clearly indicates that the output signal is purely sinusoidal with 2p modulation index. When the output signal IAC is passed through an electrical demodulation system ŽEDS., mentioned above, its waveform is shaped into a square wave-
Fig. 3. Electric output signal of the synthesized wavelength after sawtooth modulation with frequency vm s1r T. The modulation index is 2p rad.
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the change of optical phase delay dfL induced by both changes of path-length difference and synthesized wavelength can be expressed as
dfLrfL s d LrL y dLrL .
Ž9.
In addition, the synthesized light source also has better path-length difference detection resolution than that of the laser diode. 2.2. System reproducibility
By differential 1rL Žs 1rl1 y 1rl2 ., we have
dLrL s 2 dlrl q d Ž l1 y l2 . rl = Lrl .
Ž 10.
When the FBGs are chosen with similar temperature dependent coefficients, and those are also temperature stabilized within the same temperature controller, then d Ž l1 y l2 . approximately approaches zero. The influenced factor, shown in Eq. Ž10., of the synthesized wavelength is then ignored, and Eq. Ž10. can be rewritten by
dLrL ( 2 dlrl ,
Ž 11.
The wavelength change dl, which depends on both temperature change d T and driving current change d I, can be expressed as
dlrl s w Ž dlrl . rd T x d T q w Ž dlrl . rd I x d I.
Ž 12 .
Since the FBG is a passive device, the driving current change d I can be ignored. Typically, the coefficient dlrd T equals 0.01 nmr8C, and for a temperature stabilized FBG with wavelength of 1550 nm, we obtain a wavelength stability for d T of FBG stabilized within 10y2 8C, then
dlrl s 10y7 , for single FBG and
dLrL s 2 = 10y7 , for the synthesized FBG.
The wavelength variation D L of a synthesized light source used in an interferometer profoundly influences the measured path-length difference accuracy of an interferometer. The relationship between the minimum detectable path-length difference and the wavelength variation of the light source with a fixed optical phase delay Ž D f s 0. can also be expressed as D LrL s D LrL .
Ž 16.
The fact that FBG is a passive device accounts for why Ža. its wavelength variation relies on the wavelength accuracy of UV beams for masking and Žb. the wavelength variation of FBG can be derived within 0.1 nm. However, the wavelength variation of a laser diode with wavelength of 1550 nm is about 10 nm due to the characteristics of the material and the fabrication process. Thus we have D LrL
laser diode s100 = D LrL FBG .
Ž 17.
The above equation indicates that FBG system reproducibility is superior than that of the laser diode, thereby requiring no further efforts for system calibration. Meanwhile, the measurement accuracy of the two-wavelength interferometer generated by the FBG is far better than that of the laser diode.
Ž 13.
However, for a laser diode, the coefficients of dlrd T and dlrd I are approximately 0.3 nmr8C and 0.024 nmrmA with the same operating wavelength, respectively. When the stability of temperature and driving current is respectively controlled at 0.018C and 0.01 mA, the wavelength stability is given by
2.3. Output power of the light source
and
For a commercial laser diode, its output power and line-width are 1 mW and 1 nm, respectively. However, for FBGs with 99% reflection used to back reflect the Er-doped broadband light source with the respective output power and bandwidth being 40 mW and 40 nm, the output power and line-width of the light source reflected from FBG are 0.1 mW and 0.1 nm, respectively. Thus, we have
dLrL s 4 = 10y6 , for the synthesized laser diode. Ž 14 .
Po
The minimum detectable path-length difference for a synthesized wavelength L with a fixed optical phase delay Ž fL s 0. can be expressed as
Nevertheless, the system signal to noise ratio ŽSrN. relies on the output power of the light source; the SrN ratio of FBG is smaller than that of the laser diode as well. The fundamental limit of the measured optical phase is set by the photon shot noise in optical interferometers. Moreover, the amount of the noise depends on the optical power received by a photodetector. The corresponding root mean square value of the phase noise dfrms can be expressed as
dlrl s 2 = 10y6 , for single laser diode
d LrL s dLrL .
Ž 15.
Eq. Ž13. and Eq. Ž14. indicates that dLrL equals 2 = 10y7 for FBGs, and meanwhile it is 4 = 10y6 or larger for two laser diodes. If the path-length difference L s 100 mm, then the minimum detectable path-length differences d L are respectively 0.02 nm for FBG and 0.4 nm for laser diodes. Comparing the synthesized light source with a laser diode in terms of wavelength stabilization dlrl, it clearly reveals that the wavelength stabilization of a passive FBG device is better than that of an active laser diode.
FBG s 0.1
Po
laser diode
dfrms s Ž 2 hnrPo .
1r2
B,
Ž 18.
Ž 19.
where h and B are the Plank constant and bandwidth of the detection system, respectively. n and Po are the frequency and optical power of the light source, respectively. For a light source with output power of 100 mW, its
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Table 1 Source
Synthesized light source Laser diode
Item Wavelength stability Ž dlrl.rd T
Reproducibility Ž dl.
Minimum detectable phase delay Ž dfmin .
Output power
- 10y5 - 10y4
- 0.1 nm - 10 nm
10y6 radrHz 1r2 10y7 radrHz 1r2
- 0.1 mW - 1 mW
minimum detectable optical phase delay is approximately 10y7 radrHz 1r2 . However, the output power of FBG is one-tenth that of the laser diode, then its minimum optical phase delay df < min is up to 10y6 radrHz 1r2 . Table 1 compares the performance of a two-wavelength interferometer with a laser diode and synthesized light source reflected from FBGs. According to the comparison, light reflected from FBG can be more appropriately adopted as light source in two-wavelength interferometers than the laser diode.
3. Experimental setup and results Fig. 1 depicts the experimental setup. A high power laser diode ŽLD. with l s 0.98 mm was used to pump the Er-doped fiber ŽEDF. after passing through a wavelength division multiplexer ŽWDM. and, subsequently, a broadband light source ŽBLS. was generated. The output power of the BLS was kept constant by feedback controlling the driving current of the laser diode. The bandwidth of BLS measured by the optical spectrum analyzer is from 1.525 to 1.565 mm, Ži.e., the line-width is about 40 nm., and the output power of BLS is about 20 mW. A fiber isolator ŽFI. with l s 1.55 mm was used to prevent the return light backing into the Er-doped fiber loop from the FBGs; such a backing effect would cause the light within the pumping
Fig. 4. Optical spectrum of two light sources generated by fiber Bragg gratings. 1 and 2 indicate light of wavelength l s1527 and 1550 nm, respectively.
loop to generate a lasing effect. Next, a broadband fiber coupler ŽBFC. was employed to obtain the lights reflected from the two FBGs. Fig. 4 displays the optical spectrum of the combined light sources. This figure clearly indicates that only two wavelengths l s 1.527 and 1.550 mm existed simultaneously. Owing to that the line-width of the wavelengths l s 1.527 and 1.550 mm were 0.1 nm and 0.2 nm, respectively, it could be theoretically calculated that the synthesized wavelength, L s l1 l2r< l1 y l 2 <, was 110 mm. Those corresponding coherence lengths were 16 and 8 mm, respectively. Fig. 5 summarizes the results of the measured visibility. The solid and dashed lines shown the visibility curves of wavelength of 1.527 and 1.550 mm, respectively. Obviously, both coherence lengths were longer than the synthesized wavelength, and the visibility of the two light sources could be considered constant during the measurement. The two light sources with optical power of about 26 mW combined together through the broadband fiber coupler were first collimated using a lens ŽL1. and then sent into an optical Michelson interferometer ŽMI.; those sources were then separated into two parts using beam splitter ŽBS.. One of the light sources was reflected by a mirror ŽM1., and the other light source, was reflected by a mirror ŽM2., both were modulated by a sawtooth modulation signal with a frequency of 100 Hz employed to the
Fig. 5. Measured visibility of the two individual light sources. The dashed and solid curves represent the wavelengths of 1527 nm and 1550 nm, respectively.
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Fig. 8. Measured output with a path-length difference step change of 10 mm. The bandwidth of the low pass filter ŽLPF. is 100 Hz. Fig. 6. Block diagram of the self-mixing demodulation system. The electric waveform is shaped into a square waveform through ZD, and the electric signal is demodulated using an EM demodulation system. The signal output is derived after a low pass filter.
piezo-electric transducer ŽPZT., which was used to linearly scan the optical phase of the Michelson interferometer. The sawtooth modulation signal with a frequency of 100 Hz was generated from a function generator and sent into a power amplifier. The modulation index of the applied sawtooth modulation signal was selected to be 2p rad for averaged wavelength G . Next, the light sources of phase modulation were back reflected by a mirror ŽM2. and, then, the interfering signals were collimated through a lens ŽL2. and received by a photo-detector ŽPD.. The fact that the visibilities of the two wavelengths were maintained constant within the measured region accounts for why the output of the optical interferometer after modulation was a pure sinusoidal. Moreover, its magnitude could be derived by self-mixing with demodulation technique. The interfering lights were initially transferred into an electric signal, which was later demodulated by the selfmixing demodulation system shown in Fig. 6. When the electric signal of the photodetector ŽPD. output initially
went through a zero crossing comparator ŽZD., its waveform was shaped into a square waveform; that was used as a reference signal for the electric mixer ŽEM.. Meanwhile, the electric signal of the photodetector output was also sent to the input port of the electric mixer. When the output of the electric mixer went through a low pass filter ŽLPF., the visibility of the synthesized light source with various path-length differences, L1, L2 , L3 , . . . , was derived. Fig. 7 summarizes the results obtained from various path-length differences. This figure indicates that the measurable range was about 110 mm, i.e. approximately equal to the synthesized wavelength. During the whole experimental process, various disturbances, such as the fluctuation of pumping diode intensity, mechanical vibrations of the fiber caused by polarization changes, temporal changes of the orientation of the interferometer mirrors, etc., influenced the experimental results. The optical output power was controlled by feedback controlling the driving current of the pumping laser diode, and a polarization controller was used to minimize the effect of the polarization changes. The experimental setup was also isolated to avoid environmental disturbance. Fig. 8 depicts the experimental output when the path-length difference step changes 10 mm. The results reveal that the minimum detection path-length difference of the synthesized light sources is 6 nmrHz 1r2 .
4. Conclusion
Fig. 7. Measured visibility of the synthesized light source. The measured range is approximately 110 mm.
This study presents a novel two-wavelength interferometer based on a light source reflected from fiber Bragg gratings. The light sources are generated by the broadband light source through fiber Bragg gratings; the stability of the wavelengths is easily controlled using a temperature stabilizer. The synthesized wavelength is longer than those of conventional light sources, thereby allowing it to extend the measured range of path-length difference up to several hundreds times. Because a modulation index of 2p rad of the modulation signal is employed, the waveform of the interfering signal after a receiving photodetector has then a pure sinusoidal form. In addition, the demodulation be-
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comes easier using a self-mixing lock-in detection technique. The two-wavelength interferometer based on FBG proposed herein has the following merits: 1. The wavelength stability of FBG can be controlled to better than dlrl F 10y7; 2. The FBG is a passive device and, thus, the reproducibility is better than that of an active device such as a laser diode, in which the wavelength uncertainty is less than 0.1 nm; 3. Because FBG is a compact device, the environmental temperature can be easily stabilized within 0.018C using a temperature controller; and 4. The wavelength of the FBG can be easily tuned by changing the wavelength of a UV laser.
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