1. Introduction

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 [1]. 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 [13]. 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 [14] 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 [17] 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.

 

Fig. 1. Schematic of the composite interferometer. LD, laser diode; BS1, BS2, BS3 ,BS4, beamsplitters; Obj, objective; RM, reflection mirror; TS, two-axis translation stage; PZT, piezoelectric transducer; PD1, PD2, photodetectors; I, iris. Insets: The definitions of the surface height, h, and the starting position of the axial scanning of the optical delay component, d, where SR denotes the scanning range.

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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 ℓ_R and ℓ_S 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).

 

Fig. 2. Measurement of the accuracy of the system by evaluating the surface height of a stationary reflection mirror in one thousand continuous scannings: (a) The evaluated surface height from the interference signals detected in photodetector 1, (b) the relative displacement of the axial scanning ranges detected by photodetector 2, (c) the axial resolution of the system after the phase compensation mechanism.

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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.

 

Fig. 3. The time-lapse images of portion of an onion cell in the process of dehydration. The image sizes are 12.75 μm × 12.75 μm and the time interval between images is about ten minutes.

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4. Conclusions

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.