Optics Communications 238 (2004) 237–244 www.elsevier.com/locate/optcom
Self-mixing microscopic interferometer for the measurement of microprofile Ming Wang *, Guanming Lai Department of Physics, Nanjing Normal University, Ninghai Road 122, Nanjing 210097, PR China Department of Electrical and Electronic Engineering, Shizuoka University, Japan Received 24 February 2003; received in revised form 27 April 2004; accepted 30 April 2004
Abstract In this paper, we propose a simple, compact surface profilometer based on a fiber-coupled self-mixing interference (SMI) system and the confocal scanning principle, in which the surface microprofile is determined. Analysis of the confocal property in the measurement path of the proposed interference system is given, hence providing high horizontal resolution. The phase of SMI signal is demodulated by Fourier transform analysis technique, so that high vertical resolution can be obtained. After a brief discussion of the basis of SMI and the confocal scanning principle, we demonstrate some of the characteristics of this system and its application to measuring the surface microprofile with high resolution and with nanometer precision. The horizontal measuring range depends on the scanning range of the PZT actuator. Ó 2004 Published by Elsevier B.V. PACS: 07.60.L; 07.79; 42.25.H Keywords: Self-mixing interference; Confocal microscopy; Surface microprofile
1. Introduction Recently, various optical techniques have been proposed to accurately measure the surface microprofile of a variety of objects, e.g., interference microscope and scanning probe microscope can get high resolution surface profilometry. In addition, the confocal principle is also extended to the profilometry. However, all of them are usually
*
Corresponding author. Tel./fax: +86-25-83598685. E-mail address:
[email protected] (M. Wang).
0030-4018/$ - see front matter Ó 2004 Published by Elsevier B.V. doi:10.1016/j.optcom.2004.04.052
quite difficult to align. Consequently, there is a strong demand on developing simple, compact surface profilometer with very low alignment tolerance. Two basic schemes were developed in simple and compact confocal microscopy: using single mode optical fiber and direct laser feedback to detect confocal image signal [1,2]. Those can produce three-dimensional image in the same way as conventional confocal microscope, but their vertical resolution is insufficient in surface microprofile measurement. Interference microscopy methods are often used in microprofile measurement. In this case, the image signal is the coherent
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superposition of a measurement signal with a reference signal. It is important to use the phase modulation and demodulation for the analysis of the microscopy interference. In previous applications of self-mixing interference (SMI), we combined the principles of selfmixing interferometry with phase measurement method to build some micro-displacement sensing systems [3,4]. The sensors based on the self-mixing effect of the laser diode have many advantages that the optical system configuration is extremely simple and compact. No other optical elements like beamsplitter, reference mirror, and external photo detector are required. Meanwhile, phase measurement method is introduced to SMI signal analysis. However, these sensors only provide onedimensional information and have very low spatial resolution. In this paper, we combine the principles of self-mixing interferometry with the confocal scanning to configure of a self-mixing microscopic interferometer, and introduce the feedback from the phase sensor to control the object in focus. We use LD as light source and use injection current modulation to introduce the modulation phase. By using the current modulation, absolute distance and relative displacement were determined by the signal frequency and the initial phase, respectively. In this new way, threedimensional information can be obtained with high horizontal and vertical resolution even on a complex object. 2. Principle 2.1. Confocal principle with a single mode optical fiber The schematic diagram of the confocal scanning microscopic interferometer based on self-mixing
effect is shown in Fig. 1. The single mode fiber serves both to lunch the light into the microscope and to collect the reflected confocal field amplitude, U . Light from a laser diode is coupled by a coupling lens LN1 , the light beam propagates through the single-mode fiber F. The emergence beam is focused by a microscope objective lens LN2 . An object O with the amplitude reflectivity function Oðx1 ; y1 Þ is placed on the focal plane of the microscopic objective lens. The reflected field from a scanning point on the object is collected by the same microscopic objective lens and focused onto the tip of the single-mode optical fiber and delivered to the active cavity. Assume that (x; y) donates the variable on the plane perpendicular to the optical axis and z is the axial variable. Let Oðx1 ; y1 Þ be the amplitude distribution of the electric field of the object point. After passing through the imaging system, the measurement field amplitude at a point x2 on the input end of fiber F can be expressed [5] Z Z þ1 U2 ðx2 ; y2 Þ ¼ h1 ðx1 ; y1 ÞOðx2 x1 ; y2 y1 Þ 1
h2 ðx1 x2 ; y1 y2 Þ dx1 dy1 ;
ð1Þ
where (x2 ; y2 ) represents the scan position on the image, h1 and h2 are the amplitude point-spread functions of the illuminating and receiving system, respectively. Confocal amplitude signal traveled back along the fiber, and these were allowed to interfere with incident light on active cavity. Considering the propagation of field U2 ðxs ; x2 Þ through the singlemode fiber, an arbitrary electric field propagating in an optical fiber can be expanded as a superposition of a complete set of orthonormal modes. As to the special case of single-mode optical fiber, the propagation field U ðx; y; zÞ has only one term. In terms of Snyder and Love [6],
Fig. 1. Schematic diagram of the fiber-optic based confocal scanning microscopic interferometer. LD, laser diode; LN1 , coupling lens; LN2 , microscope objective lens; F, single-mode fibers; O, object to be measured; PD, detector.
M. Wang, G. Lai / Optics Communications 238 (2004) 237–244
U ðx; y; zÞ ¼ f1 ðx; yÞ expðib1 zÞ;
ð2Þ
where b1 is the propagating constant, f1 ðxÞ is the single-mode amplitude profile and can be approximated to be " 2 # xþy f1 ðx; yÞ ¼ exp ð1=2Þ ; ð3Þ r0 where r0 is the modal spot-size. We hence derive the measurement field amplitude U3 ðx3 ; y3 Þ at point ðx3 ; y3 Þ on the output end of fiber F as Z þ1 U3 ðxs ; x3 Þ ¼ f1 ðx3 ; y3 Þ expðib1 z2 Þ hðx1 ; y1 Þ 1
Oðx2 x1 ; y2 y1 Þ dx1 dy1
Here r1 and r2 are the amplitude reflectivity of LD facets, r3 and r4 are the amplitude reflectivity of the fiber F, r5 is the amplitude reflectivity of the external target. The optical beam is back-scattered into the LD active cavity (of geometrical length l) by the target. The laser operation is affected, causing a substantial variation of the optical output power. Under the condition of weak feedback, (r3 , r4 and r5 r2 Þ, neglecting intensity noise as well as multiple reflection within the external cavity, the complex reflectivity of equivalent cavity is expressed as r20 ¼ r2 þ ð1 jr2 j2 Þr3 expðiU1 Þ þ ð1 jr2 j2 Þ ð1 jr3 j2 Þr4 expðiU2 Þ þ ð1 jr2 j2 Þ
¼ f1 ðx3 ; y3 Þ expðib1 z2 Þhðx1 ; y1 Þ Oðx1 ; y1 Þ; ð4Þ where ‘’ denotes the convolution operation, z2 is the length of optical fiber F, and hðx1 ; y1 Þ is given by hðx1 ; y1 Þ ¼ h1 ðx1 ; y1 Þh2 ðx1 ; y1 Þ:
ð5Þ
The high horizontal measuring resolution in this interferometer can be deduced from the effective point-spread function hðx1 ; y1 Þ given by Eq. (5), which is characteristic of a confocal configuration. The term hðx1 ; y1 Þ Oðx1 ; y1 Þ includes the surface height information. 2.2. A fiber-coupled self-mixing interferometry The self-mixing effect of the external cavity has been extensively analyzed [7,8]. Usually, we use the three-mirror cavity approach and schematize the laser diode with an external target. The LD with a fiber and an external target is shown as Fig. 2(a) (the five-mirror cavity approach). Its equivalent model is a Fabry–Perot cavity, shown as Fig. 2(b).
r1
LD
r2
ð1 jr3 j2 Þð1 jr4 j2 Þr5 expðiU3 Þ;
L1
The equivalent reflectivity is given by, 0 r ¼ r2 ½1 þ n1 cos U1 þ n2 cos U2 þ n3 cos U3 : 2 ð8Þ Below, we deal with the fact that a stable laser operation can be obtained only if the factors, n1 , n2 and n3 , are much smaller than unity. By solving the equation of stationary laser oscillation for the equivalent LD, the optical frequency m and the emitted optical power I can be expressed
L2 L3
l
(a)
ð6Þ
Defining the coupling coefficients from the external reflections back into the LD cavity external cavities, n1 , n2 and n3 , as follow: r3 n1 ¼ ð1 jr2 j2 Þ; r2 r4 n2 ¼ ð1 jr2 j2 Þð1 jr3 j2 Þ; ð7Þ r2 r5 n3 ¼ ð1 jr2 j2 Þð1 jr3 j2 Þð1 jr4 j2 Þ: r2
r4
r3
239
r5
Ui
Ur
r2’
r1
l
(b)
Fig. 2. The equivalent model of an LD with a fiber and an external target.
240
m0 m ¼
M. Wang, G. Lai / Optics Communications 238 (2004) 237–244
C1 sinð2pms1 þ arctan aÞ 2ps1 C2 þ sinð2pms2 þ arctan aÞ 2ps2 C3 sinð2pms3 þ arctan aÞ; þ 2ps3
IðtÞ ¼ I0 ðtÞ½1 þ m1 ðtÞ cos U1 þ m2 ðtÞ cos U2 þ m3 ðtÞ cos U3 ;
ð9Þ
ð10Þ
where m0 and I0 are the optical power and the optical frequency without feedback light, a is the linewidth enhancement factor, mi is the undulation coefficient. s1 , s2 , s3 stand for the round-trip times in the external cavities formed with the two fiber facets as well as the target. L1 , L2 , L3 are lengths between the laser cavity facet and two fiber facets as well as the target surface. The external feedback strength parameter of there external cavities Ci is defined by si pffiffiffiffiffiffiffiffiffiffiffiffiffi Ci ¼ ni 1 þ a2 ; i ¼ 1; 2; 3; sl where l is the active cavity length of LD, sl denotes the flight times within the active cavity. The parameters, a and Ci , are important parameters affecting the dynamics of the laser diode as well as its output power. At weak feedback ðCi 6 1Þ, the LD is operating as a single mode laser diode and the optical power of LD as a single valued function of the external cavity length exhibits no hysteresis. As the target displaced along the beam axis, the laser intensity varies with a period of k=2. Solving this equation; we got the numerical solution. The output variation with the external phase is shown graphically in Fig. 3, corresponding to the simulation parameters: r1 ¼ r2 ¼ 0:36, r3 ¼ 0:2, r4 ¼ 0:07 and r5 ¼ 0:01, 0.02, 0.03. l ¼ 0:3 mm, L1 ¼ 5 cm, L2 ¼ 15 cm, L3 ¼ 20 cm. In the values given above, we may make the external feedback strength parameter Ci < 1. This is the effective self-mixing regime. From the Fig. 3, we can see that the output of a fiber-coupled SMI is similar to that of single double external cavity. FFT phase detection method was used for the analysis of the SMI signal, in which SMI signal was modulated by varying the optical frequency
Fig. 3. Optical output versus the external phase for a fibercoupled self-mixing system.
and demodulated by FFT method. We can express SMI signal as the carrier frequency and the initial phase of the fringe. Ui ¼ 2p
m0 2cLi 2Li þ 2p t ¼ 2pfi t þ /i : c c
ð11Þ
The carrier frequency depends on the external cavity Li and the modulating coefficient c, fi ¼
2cLi : c
ð12Þ
The initial phase term is determined by the optical frequency m0 and original external cavity Li . m0 /i ¼ 2p 2Li : ð13Þ c In terms of the carrier frequency and the initial phase, output intensity becomes IðtÞ ¼ I0 ðtÞf1 þ m1 ðtÞ cos½2pf1 t þ /1 þ m2 ðtÞ cos½2pf2 t þ /2 þ m3 ðtÞ cos½2pf3 t þ /3 g; ð14Þ where m1 ðtÞ, m2 ðtÞ and m3 ðtÞ are undulaion coefficient of interference signal components. The carrier frequency and initial phase of the interference fringes can be extracted by using the first harmonic component of the Fourier transform. Computing the Fourier transform of Eq. (14), we can obtain the Fourier spectrum components and phase terms.
M. Wang, G. Lai / Optics Communications 238 (2004) 237–244
^IðlÞ ¼ ^ aðlÞ þ
3 X
^ i ðl fi Þ expði/i Þ; m
ð15Þ
241
By performing a two-dimensional scanning of the object, surface topography can be reconstructed.
i¼1
where ^ aðlÞ ¼
Z
3. Experimental setup and result
1
I0 expð2piltÞ dt; 1 Z 1 ^ i ðlÞ ¼ 1=2 m I0 mi ðtÞ expð2piltÞ dt; 1
^ ^ i ðlÞ are zero-order and first-order aðlÞ, and m spectra components. In order to get the carrier frequency and phase value, we make the carrier frequency high enough, so that the first order is much larger than the other orders at its peak position. Then, we can determine the carrier fre^ i ðlÞ, and quency from the peak of the spectra m obtain the phase of fringe signal at the peaks of power spectrum from its real parts and imaginary parts of the Fourier coefficient /i ¼ arctan
Im½^Iðf Þ : Re½^Iðf Þ
ð16Þ
where ‘Re[ ]’ and ‘Im[ ]’ represent the operation of fetching real part and imaging part, respectively. Because we only consider the height variation, the frequency and phase is extracted for the U3 . On the one-dimensional case, the surface height is obtained by L3 ðx1 Þ ¼
k0 / ðx1 Þ; 4p 3
ð17Þ
The arrangement of the experimental system is shown in Fig. 4. Optical system only consists of the LD (Sharp LT021MD, 780 nm, 10-mw), a single mode fiber equipped with coupling lenses, a microscopic objective lens and the object to be tested. This measuring and control system includes a personal computer (PC), a data acquisition card (PCI-6024E, 12-bits analog inputs and analog outputs, 200 kS/s sampling rate), and a piezoelectric transducer actuator with a driver. The injection current is a modulation of sawtooth wave current at a frequency of 10 Hz and an amplitude of 3.5 mA on a DC bias of 50 mA. The periodical signal is converted into binary data by A/D converter. Then, the interference signal is analyzed by FFT initial phase algorithm to obtain the frequency and initial phase of the signal in every sampling period. Experimental results by a fiber-coupled SMI are shown in Fig. 5. Fig. 5(a) depicts the fringe signal obtained by a fiber-coupled SMI. The fiber-coupled SMI signal contains multi-frequency components; hence, the FFT method is easy to extract the resonance frequencies corresponding to the object cavity used in the measurement. Fig. 5(b) is Fourier spectra of SMI signal. The higher harmonic component is
Fig. 4. Schematic diagram of the experimental system.
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M. Wang, G. Lai / Optics Communications 238 (2004) 237–244 2.5 2.0 1.5
Output (V)
1.0 0.5 0.0 -0.5 -1.0 -1.5 -2.0 0.0
10.0
20.0
30.0
40.0
50.0
Time (ms)
(a)
Spectrum Intensity (Arb.u.)
0.25
0.20
f2 0.15
f1
0.10
0.05
0.00 0
(b)
500
1000
1500
2000
2500
3000
3500
Frequency (Hz)
Fig. 5. (a) The fringe signal. (b) Fourier spectra. The signal and analysis of a fiber-coupled SMI.
corresponding to the resonance frequency of the object external cavity. The lower harmonic components are corresponding to the cavity resonance frequency formed by two facets, and the higher harmonic component is corresponding to the resonance frequency of the object external cavity formed by object surface. The resonance frequencies of the object external is measured at f3 ¼ 1230 Hz. The actual length of object external cavity are L3 ¼ 20:0 cm. Object horizontal scanning was achieved by a PZT actuator scanning in x and y directions. The scanning signals from the PC are sent through the
digital-to-analog converter (D/A) to the PZT driver, and then to the PZT actuator (PZT, P762). The output signal from the PD was converted into binary data by the digital-to-analog converter (A/D) and then fed to the PC for phase detection by FFT algorithm. The surface height information is then calculated by using LabView 6.0 software from Eq. (17). The detected phase is also used as the feedback signal to control another PZT actuator (PZT, P-841) displacing in z directions, so that the object can always be kept within the focus range of the microscopic objective lens.
We have obtained data which demonstrate the utility of the instrument. Fig. 6 shows a silicon sample containing an etched cylinder pattern on its surface. Its surface profile is shown as Fig. 7, the height of cylinder is about 0.4 lm. The repeatability of the system was evaluated to be within 10 nm from four repeated measurements obtained by raster scanning cross the edge of the pattern, shown as in Fig. 8. The results clearly demonstrate a horizontal measuring resolution of less than 1 lm of the proposed system. The horizontal measuring range of this system depends only on the scanning range of the PZT actuator.
Height (nm)
M. Wang, G. Lai / Optics Communications 238 (2004) 237–244
243
375 350 325 300 275 250 225 200 175 150 125 100 75 50 25 0
B C D E
0
1
2
3
4
5
Scanning length (um) Fig. 8. The evaluation of repeatability accuracy by four repeated profile measurements.
6um
4. Discussion and conclusion
Scaning range 5×5um
Fig. 6. Test sample.
Fig. 7. Three profiles of the pattern on a silicon substrate.
The unwanted reflections from the two facets of the optical fiber will increase the external feedback strength, at the same time, the unwanted interference effects also affect the steady operation of the laser diode and cause some noises, thus limiting the measurement. The unwanted reflections could be diminished by using antireflecting film or optical isolator. The former is only effective for a single angle in the beam; and latter increase the price of instruments and also would block the backreflected light containing the SMI signal. Koelink et al. [9] has showed that a flat glass plate could considerably reduce the unwanted reflections and noises. We inserted a flat glass ND filter in front of the fiber facets and coupled with some indexmatching material. The incident laser will be focused onto the fiber facet, but reflections will be imaged back to the laser facet in an inefficient way. This enables us to reduce the unwanted noises and to construct a measurement system. The horizontal resolution of measuring system means the smallest resolvable object size. The horizontal resolution here is limited by two factors: the Airy disk for the diffraction limit of the optical system, and the mechanic scanning resolution driven by the PZT. The scanning range of the PZT (P-762) is 0–100 lm (corresponding to the drive voltage 0–100 V) and the smallest scanning resolution could be 1 nm. In our experiment, the
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sample was scanned in x–y direction with a step of 100 nm. Therefore, the horizontal resolution of our measuring system is mainly determined by the numerical aperture NA of the optical system. Although the low numerical aperture for the coupling lens can influence the horizontal resolution for the system, we can obtain a high horizontal resolution by the confocal property (Eq. (5)) of the interferometer. Further the microscopic objective lens in this system is implemented only for the imaging of axial-point. The measuring horizontal resolution, 1 lm, agrees with theoretical resolution of the 0.5-N.A. microscopic objective lens. A high vertical resolution of this measuring system mainly depends on fast Fourier transform (FFT) analysis method. The phase demodulation in this method is affected by some factors such as the laser frequency fluctuation, LD linewidth and electric noise. Because the phase noise of signal is mainly affected by the optical frequency (Eq. (13)), it is necessary to minimize frequency fluctuation of a laser diode. A temperature controller was used to maintain the temperature variation of DT < 0:02 K. Furthermore, the signal distortion increases with the value of a and C parameter through Eq. (10), which was explained by the phase changes of the second harmonic of the signal waveform. The signal waveform is sinusoidal with no fringe distortion for a ¼ 0. For nonzero a parameter, waveform shows a little inclination and causes the error for the phase extraction of intensity signal. By using FFT analysis method, a resolution of k=ð50Þ has been obtained. The vertical dynamic range (the maximum height variation of the sample under test) depends on the phase measurement range. Because the phase values retrieved from Eq. (14) must be in the range p–þp, which corresponds to the height variation range k=4–k=4, or )195–195 nm, the vertical dynamic range should be limited to the range )195–195 nm. If the phase variation exceeds
p, or the height variation was larger than k=4, the phase unwrapping process is necessary. Phase unwrapping is as difficult as in conventional interferometry, and its accuracy depends on the phase unwrapping process. We have built a simple, compact microscope based on the principle of SMI and the confocal scanning, which is able to provide measurement with high resolution and accuracy on surface microprofile. We demonstrate its application to measuring a mirror-like surface, e.g., a silicon surface. The measurement repeatability was evaluated to be within 10 nm and the horizontal resolution was confirmed to be within 1 lm. The measurement repeatability was evaluated to be within 10 nm.
Acknowledgements This study was supported by the National Natural Science Foundation of China, No. 50375074.
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