Fiber-optic interferometer for surface profile measurement with vibration suppression

We describe an improved design of fiber-optic interferometer intended to measure surface profiles with enhanced capability of vibration suppression. The reference wavefront is generated directly from the measurement wave using a multi-mode fiber that eliminates only the spatial wavefront distortion by means of bend loss. The temporal fluctuation caused by vibration is consequently cancelled out in the process of interference since it becomes to exist in both the measurement and reference waves. Further, an injection locking technique is incorporated to stabilize the reference wave intensity and hence make stable the interferometric fringe intensity. Experimental result proves that the proposed fiber-optic interferometer is capable of producing sub-wavelength measurement precision even in the presence of severe vibration with 100-μm amplitude. ©2011 Optical Society of America OCIS codes: (120.3180) Interferometry; (120.3930) Metrological instrument; (140.3520) Lasers, injection-locked. References and links 1. M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. 72(1), 156–160 (1982). 2. R. Smythe, and R. Moore, “Instantaneous phase measuring interferometry,” Opt. Eng. 23, 361–364 (1984). 3. O. Y. Kwon, “Multichannel phase-shifted interferometer,” Opt. Lett. 9(2), 59–61 (1984). 4. J. E. Millerd, N. J. Brock, J. B. Hayes, M. B. North-Morris, M. Novak, and J. C. Wyant, “Pixelated phase-mask dynamic interferometer,” Proc. SPIE 5531, 304–314 (2004). 5. C. Zhao, and J. H. Burge, “Vibration-compensated interferometer for surface metrology,” Appl. Opt. 40(34), 6215–6222 (2001). 6. M. B. North-Morris, J. VanDelden, and J. C. Wyant, “Phase-shifting birefringent scatterplate interferometer,” Appl. Opt. 41(4), 668–677 (2002). 7. H. Elfström, A. Lehmuskero, T. Saastamoinen, M. Kuittinen, and P. Vahimaa, “Common-path interferometer with diffractive lens,” Opt. Express 14(9), 3847–3852 (2006). 8. R. N. Smartt, and W. H. Steel, “Theory and application of point-diffraction interferometers,” Jpn. J. Appl. Phys. 14, 351–356 (1975). 9. H. Kihm, and S.-W. Kim, “Fiber-diffraction interferometer for vibration desensitization,” Opt. Lett. 30(16), 2059–2061 (2005). 10. O. Wallner, P. J. Winzer, and W. R. Leeb, “Alignment tolerances for plane-wave to single-mode fiber coupling and their mitigation by use of pigtailed collimators,” Appl. Opt. 41(4), 637–643 (2002). 11. J. P. Koplow, D. A. V. Kliner, and L. Goldberg, “Single-mode operation of a coiled multimode fiber amplifier,” Opt. Lett. 25(7), 442–444 (2000). 12. J. M. Fini, “Bend-resistant design of conventional and microstructure fibers with very large mode area,” Opt. Express 14(1), 69–81 (2006). 13. R. T. Schermer, “Mode scalability in bent optical fibers,” Opt. Express 15(24), 15674–15701 (2007). 14. D. Marcuse, “Curvature loss formula for optical fibers,” J. Opt. Soc. Am. 66(3), 216–220 (1976). 15. G. R. Hadley, “Injection locking of diode lasers,” IEEE J. Quantum Electron. 22(3), 419–426 (1986). 16. F. Mogensen, H. Olesen, and G. Jacobsen, “Locking conditions and stability properties for a semiconductor laser with external light injection,” IEEE J. Quantum Electron. 21(7), 784–793 (1985). 17. J. Jin, J. W. Kim, C.-S. Kang, and J.-A. Kim, “Visibility enhanced interferometer based on an injection-locking technique for low reflective materials,” Opt. Express 18(23), 23517–23522 (2010). 18. I.-B. Kong, and S.-W. Kim, “General algorithm of phase-shifting interferometry by iterative least-square fitting,” Opt. Eng. 34(1), 183–188 (1995). #139786 $15.00 USD Received 21 Dec 2010; revised 16 Feb 2011; accepted 16 Feb 2011; published 17 Feb 2011 (C) 2011 OSA 28 February 2011 / Vol. 19, No. 5 / OPTICS EXPRESS 4223


Introduction
Optical interferometers are widely being used with various system configurations for testing optics such as lenses and mirrors.At the same time, many efforts have been made to achieve sub-wavelength measurement precision even in the presence of external vibration and air fluctuation.An example is the approach of single-shot fringe measurement, which was successfully demonstrated by means of Fourier-transform analysis [1], spatial phase shifter [2], diffractive optics [3] and pixilated phase mask [4].Another approach was direct suppression of vibration by implementing closed loop control in real time [5].Cancelling out vibration effects by adopting the common-path configuration was also another solution as verified by employing a scatter plate [6], diffractive optics [7] and pinhole [8].
The concept of fiber-optic interferometer was recently proposed as an effective means of vibration desensitization [9].The interferometer adopted a single-mode fiber to generate the reference wavefront directly from the measurement wave by means of spatial mode filtering.The result is that the reference wavefront becomes to share the same level of temporal fluctuation caused by vibration with the measurement wavefront, thus the vibration effect is cancelled out in the interference fringe.Now, in this paper we propose an improved scheme of fiber-optic interferometer with the intention of enhancing the vibration-suppression capability.For the purpose, the single-mode fiber is replaced with a multi-mode fiber of a large core diameter to produce better immunity to large vibration.In addition, an injection locking scheme is incorporated to stabilize the reference wave intensity and hence sub-wavelength measurement precision can be achieved even in the presence of severe vibration with 100-μm amplitude.

Interferometer system design
Figure 1 shows the fiber-optic interferometer system designed in this investigation to measure the surface profile of concave mirrors.The light source is a semiconductor laser emitting a beam of 636-nm wavelength with a 25-mW power.The source beam is linearly polarized after collimation and then made incident on the target mirror through a polarizing beam splitter (PBS1).The reflected beam from the target mirror constitutes the measurement wave whose wavefront is affected not only by the surface profile of the target mirror but also by the presence of vibration.The former appears in the measurement wavefront as a spatial distortion, while the latter as a temporal fluctuation of the wavefront as a whole.The reference wave is generated directly from the measurement wave so that both the waves are affected equally by the temporal fluctuation due to the vibration effect.For this, the measurement wave is divided into two waves by a beam splitter (BS); one is focused on a multi-mode fiber to generate the reference wave and the other is used as the measurement wave to be interfered with the reference wave later on.
The used multi-mode fiber performs spatial mode filtering, which removes all higher modes but the fundamental mode.For this spatial mode filtering, a predetermined amount of bent is deliberately given to the fiber so that higher modes become dissipated through the fiber cladding by means of bent loss.The filtered beam as the reference wave has a spatially uniform wavefront with its distortion caused by the target surface profile being removed.The reference wave is then power-stabilized by being injected into a laser diode through a Faraday rotator (FR) to yield a constant output intensity even in the presence of vibration.Now, the reference wave is delivered through a single-mode fiber to a polarization beam splitter (PBS2), after which it is recombined with the measurement wave to create interference fringes.In the process of interference, the temporal vibration effect is cancelled out since it exists in both the reference and measurement waves.On the other hand, the spatial distortion induced by the target surface profile is preserved only in the measurement wave and hence it remains in the interference fringes.For fringe analysis, phase shifting is provided with a PZT actuator varying the optical path length of the reference wave.Finally, the resulting interference fringes are captured using a digital CCD camera of 640 x 480 pixels.Figure 2 shows the interferometer system actually built in this investigation to test the vibration suppression capability explained so far.

Bend loss of multi-mode fiber
A single-mode fiber has inherent capability of spatial mode filtering due to its small core diameter, but it requires a tight alignment tolerance in coupling with the incident beam as the core diameter is less than 3 μm [10].The required alignment tolerance is usually too taut to be met in receiving the measurement wave being reflected from the target mirror subject to large vibration, thereby restricting the maximum level of vibration to be dealt with.This problem is coped with by adopting a multi-mode fiber having a large core diameter of 12 μm.The used multi-mode fiber offers a larger alignment tolerance, but it lacks in the spatial mode filtering capability.Thus the bend loss technique has to be adopted which allows a large propagation loss by bending the fiber with a small curvature [11,12].In general the fundamental mode exhibits a minimum loss at a fixed curvature bending, making it possible to remove all higher modes by reducing the curvature to a certain threshold [13].
The bend loss technique relies on three control parameters; the core diameter D, bend radius R and fiber length L. The relationship between the parameters are given as [14] 2, 0 where , , and 1, 0 Other variables in Eq. ( 1) are: P is the input power, ΔP the power output flow per unit length, R the bend radius, a the fiber core radius, β g the propagation constant of each guided mode, n 1 the core refractive index, n 2 the clad refractive index, k the wave number and K ν the modified Henkel function of order of ν.
Table 1 shows several combinations of D, R and L considered in this investigation based on the theoretical prediction of Eq. ( 1).As for the core diameter (D), three different values of 10 μm, 12 μm and 20 μm were selected.For each core diameter, the fiber length (L) was varied from 20 to 200 cm.The bend radius (R) was also varied from 0.5 to 1.0 cm with the radius of 0.5 cm being the minimum value to avoid physical breakage of the fiber.For each combination, the fiber output spatial intensity profile was actually measured by using a beam profiler (SP-640, Spiricon) in terms of the Gaussian correlation factor that approaches to 1 as the spatial mode filtering becomes ideal.Finally, based on the data summarized in Table 1, a best combination was decided as D = 12 μm, R = 0.5 cm and L = 20 cm for the design parameters to be implemented in the interferometer built as in Fig. 2. The M 2 factor of the selected multi-mode fiber was measured as 1.353, which was comparable to 1.273 of the case of using a single-mode fiber.The selected combination leads to the shortest fiber length among those of the candidates, permitting minimizing the optical path delay that should be provided in the measurement arm for matching of coherence length with the reference arm.

Injection locking
The multi-mode fiber used for generating the reference wave has a relatively large alignment tolerance for coupling but the intensity of the generated reference wave tends to fluctuate if the target mirror is subject to large vibration.This intensity instability causes degradation of the visibility of interferometric fringes and consequently leads to measurement errors.In order to avoid the problem, as illustrated in Fig. 1, an injection locking scheme is adopted.The output beam of the multi-mode fiber is fed into a laser diode whose output power is being controlled to be stable all the time by regulating the input current and temperature.This injection locking method enables not only amplification but also stabilization of the reference wave to suppress vibration effects [15,16].
There is no change in the optical frequency of the reference wave between before and after injection locking.However, care is needed to take into consideration a certain amount of phase delay induced in the reference wave after injection locking.The phase delay Δφ is analytically given as [17]   where γ denotes a proportional constant, δI is the input current stability, Δν L is the overall injection locking range of the used laser diode, Δν is the frequency difference between the reference wave and the laser diode before injection locking.In practice, the phase delay Δφ of Eq. ( 2) is inevitable and becomes time-varying unless δI is kept to be zero absolutely.However, in the surface profile measurement by phase-shifting interferometry as attempted in this investigation, but the phase delay causes no serious measurement errors as it appears uniformly across the reference wavefront.Slave Laser (free running) Slave Laser (injection locked) Fig. 3. Spectra of the laser diode before and after injection locking.The master laser (blue) indicates the reference wave and the slave laser represents the laser diode before (red) and after (black) injection locking.
In our experiment, as the slave laser for injection locking, a microlens-coupled laser diode (PS028, Blue Sky Research) was used along with an isolator (IO-5-λ-PBS, Thorlabs) composed of two polarizing beam splitters and a Faraday rotator.Figure 3 shows an experimental result in which the spectrum of the output frequency of the laser diode was monitored using an optical spectrum analyzer (86142B, Agilent).The diode laser is in a free running state before injection locking, hopping over two consecutive modes.After injection locking, the slave laser is stabilized to a wavelength of 636.686 nm of the original reference wave.Time-varying fluctuation of the phase delay turns out to be 4•10 4 rad, which is small enough to be neglected.

Discussions
The overall capability of vibration suppression was verified while a target mirror of 25.4 mm diameter was intentionally excited using a PZT actuator with different amplitudes and frequencies.Figure 4 shows an experimental result for the output power of the reference wave, which was obtained when the excitation was given at 10 Hz with amplitude of 30 μm and 100 μm, respectively.For comparison, the same test was repeated for different system configurations; as depicted in the figure, SMF refer to the case when a single-mode fiber was used without injection locking, which is identical to the case of our previous experiment presented in Ref. 9.And MMF represents the case when only a multi-mode fiber was used with bend radius of 0.5 cm to relax the alignment tolerance.In this case, no injection locking was conducted.Lastly, MMF + IL denotes the case when the same bent multi-mode fiber was used together with injection locking.As the excitation amplitude increases, the intensity fluctuation amplitude grows up for the cases of SFM and MMF.On the contrary, in the case of MMF + IL, no significant fluctuation is monitored for both the 30 μm and 100 μm amplitudes, validating the stabilization capability of the proposed scheme of vibration suppression.In principle, the highest vibration frequency that can be suppressed is limited by the control bandwidth of injection locking for intensity stabilization.Even though the experiment for Fig. 4 was conducted at 10 Hz because of the power limitation of the PZT actuator used to excite the test mirror, the control bandwidth is fast enough to deal with high frequencies near a few kHz. Figure 5 shows actual interference fringes captured with the excitation conditions of Fig. 4. The video images confirm that our approach of utilizing a bent multi-mode fiber along with injection locking is effective in suppressing vibration.Table 2 shows experimental data obtained with 15 repeated measurements of the surface profile of a concave mirror of a 25.4-mm diameter.For each measurement, phase shifting was provided by controlling the PZT actuator (Fig. 1) with six steps with an increment of 90 degree.Then the A-bucket algorithm of Ref. 18 was used to extract phase information from captured interference fringes, which is insensitive to the phase delay caused by the injection locking process of the reference wave.Root-mean-square (RMS) values of measured profiles are listed for different excitation amplitudes.Standard deviations of measured RMS values are also presented.As expected, in the case of SMF, the standard deviation of RMS profile measurements increases to 34 nm as the vibration amplitude increases to 100 μm.On the contrary, in the case of MMF + IL, the standard deviation undergoes no significant variation even though the vibration amplitude grows up to 100 μm.The slight increase of the standard deviation to 3.5 nm in this case is attributed to the fact that our method is not capable of suppressing the tilt motion of the target mirror completely because it causes a non-uniform temporal fluctuation in the measurement wavefront that cannot be cancelled out in the process of interference with the reference wavefront.

Conclusions
Our design of fiber-optic interferometer offers a significant improvement in vibration desensitization capability by generating the reference wavefront directly from the measurement wave using a multi-mode fiber together with an injection locking scheme.The temporal fluctuation caused by vibration is cancelled out in the process of interference since it becomes to exist in both the measurement and reference waves.Experimental result proves that the proposed fiber-optic interferometer is capable of producing sub-wavelength measurement precision even in the presence of severe vibration with 100-μm amplitude,

Fig. 1 .
Fig.1.Interferometer system designed for vibration suppression by incorporating a multi-mode fiber for spatial mode filtering and also a laser diode for injection locking.The measurement wave is depicted in red while the reference wave in blue.HWP: half-wave plate, QWP: quarter-wave plate, LP: linear polarizer, PBS: polarizing beam splitter, BS: beam splitter, SMF: single-mode fiber, MMF: multi-mode fiber, L: lens, LD: laser diode, LP: linear polarizer, CL: collimating lens, FR: Faraday rotator, M: mirror, T: temperature, and i: electric current.

Fig. 2 .
Fig. 2. Interferometer system built to test vibration suppression in measuring the surface profile of a concave mirror of 25.4 mm diameter.

Fig. 5 .
Fig. 5. Singe-frame excerpts obtained by video recording the interference fringe of the target mirror with the same experimental conditions of Fig. 4. (a) Excitation amplitude of 30 μm for the case of MMF + IL (Media 1).(b) Excitation amplitude of 30 μm for the case of SMF (Media 2).(c) Excitation amplitude of 100 μm for MMF + IL (Media 3).(d) Excitation amplitude of 100 μm for SMF (Media 4).

Table 2 . Measurement Data of Surface Profile in RMS Value
139786 -$15.00USD Received 21 Dec 2010; revised 16 Feb 2011; accepted 16 Feb 2011; published 17 Feb 2011 (C) 2011 OSA 28 February 2011 / Vol. 19, No. 5 / OPTICS EXPRESS 4229 which is in fact an outstanding improvement by a factor of 10 in comparison with existing designs relying on single-mode fibers.