We proposed and demonstrated a low-cost optical system for surface profilometry with nanometer-resolution. The system is based on a composite interferometer consisting of a Michelson interferometer and a Mach-Zehnder interferometer. With the proposed phase compensating mechanism, the phase deviation due to the instability of the optical delay system and environmental perturbation can be compensated simultaneously. The system can perform a wide-field imaging in the millimeter range and a measurement with the axial resolution within ±5 nm without special shielding and protection of the system as well as any special preparation of the sample.
© 2007 Optical Society of America
Traditionally, techniques for imaging of surface profiles of materials such as atomic force microscopy (AFM) and scanning electron microscopy (SEM) can perform high-resolution imaging at the nanometer scale in both of the axial and lateral directions. However, these methods are time-consuming and lack the ability to perform wide range scannings. Furthermore, some intrinsic limitations such as special preparation of sample and environmental shielding are needed in these techniques. On the other hand, optical methods possess many advantages, as they are noninvasive, sensitive, highly stable and can perform wide-range imagings making them suitable for the in vivo study on the dynamical behavior of biological cells and tissues. However, the lateral resolution of an optical system is usually limited by the diffraction nature of light and is unable to achieve nanometer scale. Optical microscopies such as interference microscopy and confocal microscopy are widely used for imaging of biological cells and tissues. Conventional interference microscopy using monochromatic or incoherent light offers excellent vertical resolution . However, the difficulty for the measurement of surface with an extended range of height and the entanglement of amplitude and phase information become its inherent problems. Confocal microscopy is a mature technique which utilizes a confocal aperture to improve the resolution of the image. However, the axial resolution of a confocal microscope is difficult to be within a hundred nanometers.
Recently, many optical techniques have been proposed for the imaging or measurement of surface profiles of materials. Included in these techniques, interferometry using phase shifting can achieve three dimensional profiles of materials with high resolution [2–10]. However, these techniques usually need a phase-shifter that can perform a precise phase orientation, and it is difficult to achieve real-time imaging. Optical coherence tomography (OCT) was also used for the imaging of surface profiles [11, 12]. However, such techniques have difficulty to reach to nanometer scale. The projection interferometer with digital analysis was proved to be a viable nondestructive method for contouring optically rough surfaces . Although such method can achieve high-resolution imaging with high speed, its accuracy depends on the uniformity of surface reflectivity and speckle noise due to the roughness of the surface. Other techniques such as wavelength-shifting interferometry  and white-light interferometry [15, 16] were also used for three dimensional profilometry. However, these techniques usually suffer from limitations in scanning speed and depth range.
In this paper, we propose and demonstrate an optical system for measurement and imaging of the surface profiles of materials. Although interferometers were commonly utilized for the measurements of surface profiles, it is the first time to our knowledge that a composite interferometer is used for surface profilometry with axial resolution at nanometer scale. This system possesses the advantages of optical methods and can perform accurate measurement of the longitudinal position of the sample. Furthermore, this system is a low-cost system which can image the surface profile in a large area without any special preparation of the sample and environmental shielding. Therefore, it is useful in the application for the in vivo imaging of different types of biological cells and tissues.
2. Experimental setup
Figure 1 represents a schematic of the composite interferometer, which is a modification of our previous configuration  to improve the performance and simplicity of the system. The system consists of a typical Michelson interferometer where the rays propagating in it are represented with red lines and a Mach-Zehnder interferometer where the rays propagating in it are represented with blue lines. A laser diode with wavelength of 532 nm and output power of 5 mW is used as the light source. The reference arm of the Michelson interferometer is designated with path BDE, where a reflection mirror is fixed on a stage driven by a piezoelectric transducer that serves as an optical delay component. In the sample arm designated with path BC, an objective is introduced to focus the probe beam onto the surface of the sample. The sample is placed on a two-axis translation stage synchronized with the piezoelectric transducer for the two dimensional lateral scanning. The phases of the interference signals detected with the two photodetectors are extracted and a subtraction is performed to obtain the surface height of the sample at that point.
In the Michelson interferometer, the electric field reflected from the surface of the sample can be represented as
where E 0S is the amplitude of the reflected field in sample arm and ω is the angular frequency of the light source. The phase shift, ϕ, due to the surface height with respect to some reference height, is defined as
where h is the surface height with respect to the reference height, as shown in the inset of Fig. 1, and λ is the wavelength of the light source. The electric field reflected from the reference mirror can be represented as
where E 0R is the amplitude of the reflected field in the reference arm and the time delay between the sample and reference arms is
where ℓR and ℓS are the optical path lengths in the reference and sample arms, respectively, and c is the speed of light. The intensity received by photodetector 1 is proportional to the square of the resultant field and therefore
The interference signal can then be represented by the cross-correlation function
In order to acquire the two dimensional distribution of the height of the surface of a sample, a lateral scanning in two dimensions is needed. However, due to the instability of the system and environmental perturbations, the axial scanning ranges of the optical delay component are in principle not exactly the same at different lateral positions. This will cause a deviation of the evaluation of the surface height of the sample. When the deviation of the starting position in each axial scanning is taken into account, the electric field reflected from the reference mirror becomes
where the phase shift, δ, due to deviation of the starting position in each axial scanning, is defined as
where d is the starting position of the axial scanning of the optical delay component, as shown in the inset of Fig. 1. Therefore, the cross-correlation function which represents the interference signal becomes
In the Mach-Zehnder interferometer, the optical path length through AF is constant, while the optical path length through ABDEDF is changed during the axial scanning of the optical delay component. Similarly, the cross-correlation function which represents the interference signal in the Mach-Zehnder interferometer can be represented as
where E′0 and E′0R are the amplitudes of the electric fields received by photodetector 2 from the two different paths, and the time delay between the two paths is
where ℓ′R and ℓ′ are the optical path lengths through ABDEDF and AF, respectively. In order to ensure the values of the phase shifts to be the same in both interferometers, the optical path length between the reflection mirror in the optical delay component and photodetector 1 through DB should be the same as the optical path length between the reflection mirror and photodetector 2 through DF to eliminate the deviation resulting from systematical errors and environmental perturbations. It is obvious that when the phases of the interference signals are extracted in each axial scanning that corresponds to a certain lateral position, the difference between the two phases is exactly proportional the surface height of the sample at that point. Therefore, a two dimensional surface profile of a sample can be obtained with high axial resolution.
3. Results and discussions
In order to measure the instability of the piezoelectric transducer as well as the effect due to environmental perturbations, and to demonstrate the capability of our phase compensation mechanism, a reflection mirror was placed statically on the sample stage and continuous axial scannings of the piezoelectric transducer were performed. Figures 2(a) and 2(b) show the surface heights of the reflection mirror evaluated from the phase shifts in the interference signals detected by photodetector 1, and the relative displacement of the axial scanning ranges detected by photodetector 2, respectively. There were one thousand successive scannings performed. In spite of the influence of environmental perturbations, the accuracy and thus the axial resolution without compensation was about 20 nm. However, as indicated by the arrows, the environmental perturbation can seriously cause instability of the system. Furthermore, there is a steady climbing of the signal in the repeated measurements. This phenomenon was supposed to results from the hysteresis and thermal effect of the piezoelectric transducer without a closed-loop feedback mechanism. When using the compensation mechanism by subtracting the phases detected in both photodetectors, the instabilities and systematical errors as well as the effect due to the environmental perturbation were eliminated, and the axial resolution of the system was significantly improved to be within ±5 nm, as shown in Fig. 2(c).
The system was used for the measurements of the surface profiles of various samples. Figure 3 shows a series of time-lapse images of portion of an onion cell in the process of dehydration. The size of each image is 12.75 μm × 12.75 μm. The time interval between images is about ten minutes. One can obviously see the change of the surface profile of the cell. It was demonstrated that this system is suitable for the measurement of surface profiles of biological cells and tissues with the axial resolution at nanometer scale. Note that the length scales are different in the axial and lateral directions. Although this technique can perform the measurement of the surface profile of a material with axial resolution at nanometer scale, the process of point-scanning is time-consuming. For example, it took about 5 minutes for scanning each surface profile in Fig. 3 when the scanning rate of the piezoelectric transducer was set to be 10 Hz. Therefore, although it may well be stable against environmental perturbations, the result will not be accurate if the sample moves during the measurement. Furthermore, the lateral resolution is limited by the diffraction nature of light and cannot achieve nanometer scale.
We proposed and demonstrated a low-cost optical system for imaging of the surface profiles of materials with nanometer resolution. With the phase compensation mechanism, the phase deviation due to the different scanning range of the optical delay component during successive scannings and the environmental perturbations can be significantly reduced. The axial resolution of the system was demonstrated to be within ±5 nm. Although the lateral resolution is limited by the diffraction nature of light, the system can perform the imaging of the surface profile in a large area and large depth range without any special preparation of the sample. A series of time-lapse images was used to observe the morphological change of an onion cell in the process of dehydration. It is proved that this system is suitable for the measurement of the surface profiles of materials with the resolution at nanometer scale.
This research was supported by the National Science Council of Republic of China under the Grant No. NSC 95-2112-M-033-004 and The Center-of-Excellence Program on Membrane Technology, the Ministry of Education, Taiwan, R.O.C.
References and links
01. M. Roy, C. J. R. Sheppard, G. Cox, and P. Hariharan, “White-light interference microscopy: a way to obtain high lateral resolution over an extended range of heights,” Opt. Express 14, 6788–6793 (2006). [CrossRef] [PubMed]
05. E. Berger, W. von der Linden, V. Dose, M. Jakobi, and A. W. Koch, “Reconstruction of surfaces from phase-shifting speckle interferometry: Bayesian approach,” Appl. Opt. 38, 4997–5003 (1999). [CrossRef]
06. J. M. Kilpatrick, A. J. Moore, J. S. Barton, and J. D. C. Jones, “Measurement of complex surface deformation by high-speed dynamic phase-stepped digital speckle pattern interferometry,” Opt. Lett. 25, 1068–1070 (2000). [CrossRef]
07. M. Yokota, A. Asaka, and T. Yoshino, “Stabilized phase-shifting fringe analysis by use of current-induced frequency modulation of laser diodes,” Appl. Opt. 40, 5023–5027 (2001). [CrossRef]
08. Nava-Vega, L. Salas, E. Luna, and A. Cornejo-Rodríguez, “Correlation Algorithm to recover the phase of a test surface using Phase-Shifting Interferometry,” Opt. Express 12, 5296–5306 (2004). [CrossRef] [PubMed]
10. E. A. Barbosa, A. A. V. Filho, M. R. R. Gesualdi, B. G. Curcio, M. Muramatsu, and D. Soga, “Single-exposure, photorefractive holographic surface contouring with multiwavelength diode lasers,” J. Opt. Soc. Am. A 22, 2872–2879 (2005). [CrossRef]
11. Y. Yasuno, Y. Sutoh, M. Nakama, S. Makita, M. Itoh, T. Yatagai, and M. Mori, “Spectral interferometric optical coherence tomography with nonlinear b-barium borate time gating,” Opt. Lett. 27, 403–405 (2002). [CrossRef]
13. R. W. Wyant, S. P. Almeida, and O. D. D. Soares, “Surface inspection via projection interferometry,” Appl. Opt. 27, 4626–4630 (1988). [CrossRef]
14. K. Hibino, R. Hanayama, J. Burke, and B. F. Oreb, “Tunable phase-extraction formulae for simultaneous shape measurement of multiple surfaces with wavelength-shifting interferometry,” Opt. Express 12, 5579–5594 (2004). [CrossRef] [PubMed]
17. C. C. Lai, C. H. Cheng, W. C. Chiu, and I. J. Hsu, “Simultaneous measurement of the refractive index, thickness, and position of suspended thin film,” Proc. SPIE 6343, 634320-1–8 (2006). [CrossRef]