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OPTICS LETTERS / Vol. 29, No. 23 / December 1, 2004
Fiber interferometer for simultaneous multiwavelength phase measurement with a broadband femtosecond laser Catherine E. Towers, David P. Towers, Derryck T. Reid, William N. MacPherson, Robert R. J. Maier, and Julian D. C. Jones School of Engineering and Physical Sciences, Heriot-Watt University, Edinburgh EH14 4AS, UK Received June 3, 2004 We present a fiber interferometer for the simultaneous measurement of phase at multiple wavelengths from a single broadband femtosecond laser. Narrow-bandwidth fiber Bragg gratings isolate a particular frequency from the broad-bandwidth laser pulse produced. The multiwavelength phase data permit the unambiguous measurement range to be significantly increased compared with the wavelengths used in the interferometer. Preliminary experimental results are presented for a two-frequency sensor with an absolute range of 0.13 mm and associated dynamic range of 43,000:1. © 2004 Optical Society of America OCIS codes: 060.2370, 120.3180, 120.2650.
Optical interferometry has become a primary method for measurements of absolute length. A number of techniques have been developed that permit the unambiguous measurement range (UMR) to be extended beyond the magnitude of the wavelength used in the interferometer. Wavelength-scanning methods require continuous tuning from an external-cavity diode laser.1 The number of samples recorded across the wavelength-tuning range determines the resolution of the measurement. White-light interferometers produce absolute data by scanning the reference arm to locate the position of maximum fringe visibility.2 Rapid-scanning optical delay lines are required for in vivo measurement in optical coherence tomography; currently3 volume measurement systems achieve single-point depth scans at 29 kHz over several millimeters. Superheterodyne interferometry replaces the basic unit of measurement, the optical wavelength, with a synthetic wavelength generated electronically and applied to the optical signal by acousto-optic modulators.4 This method produces a measurement range that is more relevant to engineering applications but requires phase-locked detection at the superheterodyne frequency, typically 100 kHz, and hence a finite time to obtain the data. All these approaches are based on acquiring a temporal data sequence and hence are not applicable to fast transient events, e.g., surface velocities up to 1000 ms21 and with data sampling rates .1 MHz. Alternative, nonscanning techniques include time of f light5; however, insuff icient temporal resolution means that the depth resolution is at best a few millimeters. Surface velocities can be measured by the Doppler effect with a Fabry –Perot interferometer; however, the reconstruction of surface position can be obtained only by assuming data continuity.6 In this Letter we present a f iber interferometer for simultaneous phase measurement at multiple wavelengths generated from a broadband femtosecond laser pulse. This approach is applicable to high-speed transient motion analysis. The femtosecond pulsed source,7 together with a number of fiber Bragg gratings, allows narrow bands of the laser spectrum to be 0146-9592/04/232722-03$15.00/0
isolated with a suitable coherence length for multiwavelength interferometry (MWI). With a suitable number of measurement wavelengths, resolutions of a few nanometers over an UMR up to tens of millimeters can be achieved. Preliminary experimental results are reported for a two-wavelength sensor demonstrating the principle of operation and yielding a dynamic range of 43,000:1. A major consideration of the sensor design was minimizing complexity while producing the maximum possible UMR for the number of wavelength bands selected. We recently reported a time-eff icient generalized optimum multifrequency (GOMF) approach to maximize the performance of a MWI based on phase coincidence and the formation of a geometric series of synthetic wavelengths.8 This technique requires a wavelength range of typically 50 nm and is compatible with the bandwidth available from a femtosecond laser. Furthermore, from this method the performance of interferometers with increasing numbers of measurement wavelengths can be predicted.9 The absolute, unwrapped phase can be expressed as a sum of integer fringe order components and wrapped phase values: fuk 苷 2pmk 1 fwk ,
k 苷 1, . . . , s ,
(1)
where fuk is the unwrapped phase at the kth measurement wavelength, fwk is the wrapped phase, and mk is the integer fringe order. With two measurement wavelengths, l1 and l2 , the UMR is given by the synthetic beat wavelength L12 苷 l1 l2 兾共l2 2 l1 兲. Optical path difference z is given by z 苷 共2pm1 1 fwl 兲l1 苷 共2pm2 1 fw2 兲l2 苷 共2pm12 1 fw12 兲L12 ,
(2)
with m12 arbitrarily set to 0 as L12 is the UMR. Hence the fringe order at l1 is given by ∂ µ L12 fw12 fw1 , m1 苷 NINT 2 (3) l1 2p 2p © 2004 Optical Society of America
December 1, 2004 / Vol. 29, No. 23 / OPTICS LETTERS
where NINT denotes the nearest integer. The ratio L12 兾l1 def ines a scaling factor that amplif ies the noise in the wrapped phase measurement at the synthetic wavelength. Previously, we def ined an expression for the scaling factor as9 L12 2p , # p l1 6 2 sf
(4)
where sf is the standard deviation phase noise at l1 and l2 for 6s reliability in the fringe order calculated. Hence for a practical single-point interferometer with sf 苷 2p兾500 the scaling factor is L12 兾l1 # 58.9. Figure 1 shows the experimental setup for a two-wavelength f iber interferometer. The system is centered around three 50兾50 couplers: C1 combined with the Bragg gratings provides frequency selection, C2 forms the interferometer, and C3 separates the two wavelengths for detection. A femtosecond-pulsed fiber laser source with a mean wavelength of 1530 nm, a pulse repetition rate of 30 MHz, a pulse duration of 400 fs, and an average output power of 80 mW is employed to provide the broad spectral bandwidth and short pulse length needed to freeze high-speed transient motions.7 Quasi-monochromatic light suitable for interferometric measurement is obtained by means of narrow-bandwidth fiber Bragg gratings10 (FBGs) of 50 and 70 pm FWHM, yielding a coherence length of $30 mm. The two gratings labeled FBG1 are at 1534 nm, and FBG2 is centered at 1552 nm, yielding a synthetic wavelength of 0.132 mm and a scaling factor of 85.7. Angle cleaving the distal fiber ends after C1 renders the backref lections negligible compared with the intensity ref lected by the FBGs. To accommodate noncooperative, lowref lectivity targets, an erbium-doped f iber amplif ier (EDFA) is incorporated into the sample arm of the interferometer. The EDFA produces an optical gain in excess of 104 from a single pass with a pump power of 150 mW. The amplif ier incorporated here is used in a double-pass configuration, f irst, to amplify the initial narrowband signal going out and, second, to amplify the returned scattered radiation from the object collected by the f iber collimator. The return loss of the fiber collimator was ,260 dB. The interferometer output signals at the two wavelengths are separated optically by means of another FBG that is matched spectrally with one of the original gratings. Light ref lected from this grating propagates back into the interferometer but adds incoherently, and therefore contributes only to a dc offset of this signal. Figure 2 shows the phase resolution achievable by temporal phase stepping in the interferometer. A noncooperative object was coated with a retroref lective material consisting of 25-mm spheres and located at ⬃25 mm from the sample beam collimator. The pump diode to the EDFA was adjusted to ⬃15 mW to maximize the interference fringe contrast. The upper plot shows the phase modulator excitation waveform, and the center section shows the corresponding interference intensity at one of the InGaAs photodiodes. Phase stepping was achieved with a f iber-wound piezo generating sequences of four intensities with equal
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phase steps. Wrapped-phase measurements, shown in the lower plot of Fig. 2, were obtained with the Carré algorithm to accommodate for an unknown but constant phase step.11 Analysis of the signal shows that a phase resolution of ⬃2p兾310 is attainable, and hence for 6s reliability the maximum scaling factor is ⬃36. This compares with a phase resolution of ,2p兾100 with no optical amplif ier present. For noncooperative surfaces, in this case a rough metal surface, the EDFA pump diode was operated at ⬃45 mW such that a phase resolution of ⬃2p兾30 was obtained. The bandwidth of the detection system was to be ⬃1 MHz. A preliminary evaluation of the two-wavelength fiber interferometer was performed by temporal phase stepping with a cooperative target. The phase measurements for the two wavelength channels were obtained simultaneously. The phase at the beat frequency was determined over a range of optical path
Fig. 1. Schematic of a two-wavelength, fiber interferometer. FI, Faraday isolator; AC, angle cleave; IPD, InGaAs photodiode; LD, 980-nm 150-mW laser diode; WDMC, wavelength division multiplexing 980兾1550 coupler; EDF, erbium-doped f iber; PLMR, path-length matching in the reference arm; PC, polarization controller; FC, fiber collimator; PZT, piezoelectric transducer.
Fig. 2. Single-wavelength, phase-stepped interferometer performance showing standard deviation phase resolution of ⬃2p兾310 rad.
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Fig. 3.
OPTICS LETTERS / Vol. 29, No. 23 / December 1, 2004
Wrapped-phase values at the beat wavelength.
differences in the interferometer, and the results are plotted in Fig. 3. The main uncertainty in these data originates from the resolution of the traverse employed at 610 mm. Analysis of the data showed that a phase resolution of 2p兾495 rad was achieved. Therefore, for the wavelengths used, the reliability of fringe order calculation was ⬃4s or 95%. The wrapped phase at L12 shows a good correlation with the expected beat wavelength of 0.132 mm. The ratio of the synthetic to the measurement wavelength represents a scaling factor of 85.7, and therefore a substantial increase in the UMR is evident for the two-wavelength setup. A dynamic range of 43,000:1 was achieved. The GOMF technique shows that the ratio of the UMR to the optical wavelength increases by the scaling factor for each additional measurement wavelength. The performance of the two-wavelength sensor showed that a scaling factor of ⬃59 can be applied with 6s reliability at sf 苷 2p兾500. Hence, for example, to obtain absolute measurements over 300 mm with a cooperative surface, four terms in the geometric series of synthetic wavelengths must be used with appropriate selection of measurement wavelengths.8 For a noncooperative, retroref lective-coated surface, our initial results indicate that four measurement wavelengths will produce an UMR of .50 mm. Calculating the measurement wavelengths required8 shows that the minimum wavelength separation is ⬃50 pm with a total wavelength range of 50 nm. The f iber system demonstrated herein included FBGs of suitable bandwidth,10 and the wavelength range is available from current ultrashort-pulse laser systems. The speed of the fiber interferometer presented is limited by the temporal response of the phase modulator. Solid-state modulators, e.g., lithium niobate, would allow phase stepping at frequencies in excess of 1 MHz. For instantaneous phase measurement each wavelength channel can be extended to include quadrature detection by means of a pair of photodi-
odes and a polarization-based passive phase-shifting arrangement.12 An alternative set of measurement wavelengths can be used with processing by the method of excess fractions or a recently presented modif ied Chinese remainder theorem algorithm.13 For these techniques the UMR is maximized by selection of wavelengths that are pairwise relatively prime when expressed as integers. A number of options are available from within the 1530–1575-nm wavelength range with the wavelength separation being greater than for the GOMF approach. A f iber-based MWI with the potential for ultrahigh-speed absolute ranging utilizing narrowbandwidth f iber Bragg gratings for wavelength selection has been reported. Our preliminary experiments have demonstrated that a dynamic range of 43,000:1 is achieved from a two-wavelength fiber sensor. Therefore a measurement range of tens of millimeters may be achieved for four measurement wavelengths, depending on the optical signal obtained from the object. An EDFA has been incorporated in a double-pass configuration to compensate for varying scattering characteristics of different object surfaces. With an optical gain in excess of 104 it has been shown that satisfactory interference signals can be obtained from retroref lective-coated targets at a range of over 25 mm. The authors thank the Atomic Weapons Establishment, Aldermaston, UK, for provision of funds in support of this research. C. E. Tower’s e-mail address is
[email protected]. References 1. H. J. Tiziani, B. Fanze, and P. Haible, J. Mod. Opt. 44, 1485 (1997). 2. L. Deck and P. de Groot, Appl. Opt. 33, 7334 (1994). 3. N. A. Nassif, B. Cense, B. H. Park, M. C. Pierce, S. H. Yun, B. E. Bouma, G. J. Tearney, T. C. Chen, and J. F. de Boer, Opt. Express 12, 367 (2004), http:// www.opticsexpress.org. 4. R. Dandliker, Y. Salvade, and E. Zimmermann, J. Opt. 29, 105 (1998). 5. S. Pellegrini, G. Buller, J. Smith, A. Wallace, and S. Cova, Meas. Sci. Technol. 11, 712 (2000). 6. D. R. Goosman, Appl. Opt. 30, 3907 (1991). 7. K. Tamura, C. R. Doerr, L. E. Nelson, E. P. Ippen, and H. A. Haus, Opt. Lett. 19, 46 (1994). 8. C. E. Towers, D. P. Towers, and J. D. C. Jones, Opt. Lett. 29, 1348 (2004). 9. C. E. Towers, D. P. Towers, and J. D. C. Jones, Opt. Lett. 28, 887 (2003). 10. I. Bennion, J. A. R. Williams, L. Zhang, K. Sugden, and N. J. Doran, Opt. Quantum Electron. 28, 93 (1996). 11. P. Carré, Metrologia 2, 13 (1966). 12. K. Creath, Prog. Opt. 36, 349 (1988). 13. C. E. Towers, D. P. Towers, and J. D. C. Jones, Opt. Express 12, 1136 (2004), http://www.opticsexpress.org.
