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Abstract
We propose a new low-coherence interferometry system for dual-wavelength off-axis digital holography. By utilizing diffraction gratings, two beams with narrower bandwidths and different center wavelengths could be filtered in a single light-emitting diode. The characteristics of the system are analytically determined to extend the coherence length and field-of-view enough for off-axis configuration. The proposed system enables the fast and accurate measurement of the surface profile with more than a micrometer step height and less noise. The performance of the system is verified by the experimental results of a standard height sample.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Digital holography (DH) can provide researchers with a practical method of performing high-resolution three-dimensional (3D) measurements [1–3]. Based on an interferometric technique, DH uses both the amplitude and phase information of light scattered by a target to reconstruct a large field-of-view (FOV) 3D image, and phase quantitative analysis of the object's 3D data can then allow one to obtain measurements with nanometer-scale axial resolution using a non-contact and label-free acquisition [4]. Moreover, since the acquisition and reconstruction processes are performed numerically, and DH is based on a wide-field optical system with conventional and affordable components, a cheap and portable 3D measurement system can be readily established for application in various fields [5,6].
While conventional DH captures interferograms using high-coherence illumination such as that provided by lasers, the resolution and quality of the resulting image can be degraded by a large amount of speckle noise. Recently, alternative DH systems utilizing low-coherence light sources such as light-emitting diodes (LEDs) have been proposed in several studies [7–13]. In comparison to conventional laser-based configurations, such LED-based DH systems can suppress the overall speckle noise to achieve a better image quality. Moreover, since LEDs are relatively small and cheaper than a high-coherence light source, LED-based DH systems are far more suited for use as a compact, portable, affordable, and practical measurement device. Hence, as low-coherence DH also inherits the main advantages of conventional DH, it can be applied in various fields that require fast and non-invasive 3D measurement systems, e.g., measuring biological properties [14], or sample thicknesses and surface profiles [15].
However, although adapting DH systems to use a low-coherence light source can result in improved image quality, there is a trade-off in terms of the associated optical characteristics. While general LED lights have a coherence length of less than 20 μm, the light is typically assumed to be both spatially and temporally incoherent [16]. It limits the range of measurable samples in both the axial and lateral dimensions. Moreover, although off-axis DH can be used to achieve fast 3D measurements by single-shot hologram acquisition [17], the relatively short coherence length of LED light makes it more difficult to use the off-axis setup with a sufficient FOV for practical use. While recent studies have introduced the fast, large FOV DH system including in-line configuration [9], grating-based optical path modulation [12,13,18], or pulse front tilt method [19], these approaches have not considered to increase the temporal characteristics of initial beam, which limits its applications to the measurement of objects with small amount of optical path difference (OPD). Therefore, to apply low-coherence light sources to DH in terms of constructing a practical 3D measurement system, extending both coherence length and axial range are required.
In order to resolve these problems, an enhanced DH system using a single low-coherence light source was proposed in a previous study [20]. The two main concepts behind the system are as follows. First, both a bandpass filter and a spatial filter are used to increase the coherence of the initial light source. Second, by changing the bandpass filter for differently centered wavelengths, a dual-wavelength DH system can be established using a single LED, which extends the axial range of the system. These mechanisms enable the possibility of high-quality 3D measurement within a compact DH configuration. However, since each bandpass filter has a fixed center wavelength and bandwidth, more than one filter is needed to utilize the system. Moreover, using off-axis DH to perform a large height measurement requires the use of a light source with an even greater coherence length than that required in the phase-shifting method, and thus cannot be readily achieved using commercial fixed specification bandpass filters. Therefore, improved temporal filtering with an adjustable center wavelength and bandwidth, as well as further analysis of the performance of the system is necessary to establish a practical measurement system.
In this work, we propose a novel DH system for precise quantitative 3D measurement with fast acquisition and extended axial range, by utilizing dual-wavelength off-axis DH system with a single low-coherence light source. Instead of filtering the LED using a bandpass filter, diffraction gratings are applied to disperse the beam with respect to the wavelength. After dispersion, an aperture then filters the beam to control the center wavelength and bandwidth. This method allows us to increase both the coherence length and intensity of the filtered beams sufficiently to enable the off-axis acquisition, and consequently requires fewer holograms and results in a shorter acquisition time than the phase-shifting method. In order to achieve a FOV large enough to measure millimeter-sized objects with reasonable quality, the relations between the system's characteristics and the coherent length of the filtered beam must be defined. Aperture locations and sizes suited for this purpose were determined based on the resulting analysis. Moreover, by selecting different center wavelengths, it is possible that dual-wavelength off-axis DH with a single low-coherence light source could be applied to high-stepped samples.
The remainder of the manuscript is organized as follows. In Section 2, the experimental setup for the proposed system is presented and described. To ensure the implementation of low-coherence light source in off-axis DH system, the available coherence length and inclination angle is analyzed. In order to show the effectiveness of the proposed system, the results of an experimental measurement of a sample with a step height of ~1.8 μm are presented and discussed in Section 3. Finally, our conclusions are presented in Section 4.
2. Dual-wavelength off-axis digital holography
2.1 Experimental setup
The optical setup of the proposed system is shown in Fig. 1. The surface of the object is measured with a lens-free Michelson interferometer setup using a single LED with center wavelength λC = 631 nm and a bandwidth of 17 nm (Luminus, CBT-90) as the light source. While the collimated LED beam is regarded as both spatially and temporally incoherent, diffraction-grating-based spectral and spatial filters are located before the interferometer in order to increase the coherence of the light. While in the reference arm of the interferometer, the mirror is attached to an angular stage in order to tilt the reference beam relative to the object wave. The off-axis hologram is captured by a charge-coupled device (CCD) with a 2048 (H) × 2448 (W) pixel array and a 3.45 μm (H) × 3.45 μm (W) pixel pitch (Sony XCL-C500).
Fig. 1 Sketch of the experimental setup. DG: diffraction gratings, BS: beamsplitter, OBJ: object. The labels O and R appearing in the inset represent the object and reference beams, respectively.
In order to control the temporal coherence in such a low-coherence DH setup, we have proposed the use of a diffraction grating-based spectral filter. Two reflection-type diffraction gratings with 2400 gr / mm (Thorlab GH25-24V) are used to disperse and select suitable wavelengths from the initial light source. After the collimated beam is diffracted by DG1, the diffraction angle varies with respect to the incident wavelength. After reflection, the relation between the center wavelength λC and the 1st diffraction angle β is defined by [21]:
where d is the grating spacing, and α is the input angle of the light source with respect to DG1. Since the collimator is applied before the dispersion, α can be assumed to be uniform throughout the beam profile. The diffracted beam is then reflected onto DG2 to halt the dispersion, which makes it easier to then filter the beam using a slit. In order to facilitate the alignment of the system and match the intended wavelength to the fixed slit and interferometer, DG1 is mounted on a rotating stage. The filtering process can thus be considered as comprising two steps. First, by adjusting the input angle α with respect to the fixed slit location, the center wavelength of the filtered beam can be determined. Second, the spacing of the slit can be modulated to select the bandwidth. In fact, if the distance D between DG1 and DG2 is sufficiently long, the spacing of the slit, t, for a given wavelength range can be expressed aswhere and are the diffraction angles for wavelengths of and , respectively. The spatial filter positioned behind the slit then acts to provide the beam with increased spatial coherence, enough to be utilized for interferometric imaging. While the monochromator using grating and concave mirrors to filter the spatially focused beam has analogous procedures [22], the proposed configuration does not require additional spatial filtering or entrance slit before grating, which could be utilized easily for low-coherence light source.Figure 2 presents a comparison of the resulting filtered beams as modulated by bandwidth and center wavelength. In Fig. 2(a), we see that a bandwidth as narrow as 3 nm could be obtained for a center wavelength of 617 nm by adjusting the spacing of slit. In addition, by changing the angle of DG1 i.e., the location of the slit, the center wavelength of the beam could be adjusted while maintaining the narrow bandwidth, as evident in Fig. 2(b). Although the full-width half-maximum (FWHM) of the filtered beam from the grating is larger than that from a high-coherence light source such as a laser [23], the temporal coherence length is sufficient for off-axis implementation, as will be discussed in the following.
Fig. 2 Normalized spectral distribution of the proposed filtered beam modulated by (a) bandwidths with constant center wavelength (617 nm), and (b) center wavelengths with constant bandwidth (FWHM = 3 nm).
While the bandwidth and center wavelength could be adjusted freely using the proposed method, the resulting light intensity is reduced when compared with the emitted LED light, on account of the slit and diffraction gratings. This reduction in intensity could be alleviated, however, by selecting imaging wavelengths close to the center wavelength of original beam, where most of the optical power is included. We also note that the intensity of filtered beam is reduced from collimation, dispersion, and filtering procedures. While the diffraction efficiency of the grating at the given wavelength range of the light source is around 50%, the dispersed beam reflected from DG2 maintains its intensity about 25% of the initial power. Compared with the previous work [20], adopting general commercial high-power LEDs could help to achieve the intensity and coherence required for the proposed DH system.
2.2 Coherence lengths in off-axis digital holography
While a LED-based light source can be used to achieve a final DH image of better quality than a high-coherence light source, the low coherence of the beam also limits the FOV and axial measurement range, which are critical to the implementation of the off-axis configuration. Therefore, analyzing and establishing sufficient coherence lengths in terms of both temporal and spatial coherence is crucial to the establishment of a practical DH system. Figure 3 describes the relation between the characteristics of the light source and the proposed system. As shown in Fig. 3(a), the spatial filter—comprising an objective lens, pinhole, and tube lens—increases the spatial coherence of the initial light source. After filtering, the radius of the spatially coherent region LSC can be defined as [24]
where w is the radius of the pinhole and z is the focal length of the tube lens, respectively. Since the hologram image is obtained from diffracted light, illuminated object waves outside of this coherent region cannot be properly reconstructed. Therefore, the valid beam radius i.e., the maximum FOV is defined by the spatial coherence length and can be adjusted by altering the parameters of the spatial filter.Fig. 3 Sketch of the (a) spatial and (b) temporal coherence lengths in the proposed low-coherence off-axis DH system.
Compared with the phase-shifting method commonly used in many LED-based DH systems, off-axis DH requires the use of a reference beam with linear-phase modulation in order to separate the real image from the zero-order and twin images in the Fourier domain [17]. In general, this phase profile is obtained by inclining the angle of the reference arm relative to the object arm. During the subsequent measurement process, the off-axis beam generates an optical path difference (OPD), as outlined in Fig. 3(b). Although the maximum OPD and inclination angle of the DH system is defined by the pixel pitch of the imaging sensor [8], the low temporal coherence of the light source additionally limits the OPD. In continuous-wave light source, the temporal coherence is defined from the wavelength characteristic of the beam. For a given FWHM Δλ, the resulting temporal coherence length LTC for a light source with Gaussian spectral distribution is given by [25]
From Fig. 3(b), one can see that the maximum OPD hmax cannot exceed LTC, which leads to the following relation between the maximum inclination angle θmax and the spatial and temporal coherence lengths of the light source:Equation (5) implies that both temporal and spatial coherence lengths are significant in low-coherence off-axis DH. While the temporal coherence defines the possible axial range and OPD of the system, enough spatial coherence has to be provided to provide a suitable FOV and prevent the degradation of the temporal coherent length [7].3. Results and discussion
In order to utilize the proposed dual-wavelength off-axis DH system using a low-coherence light source, two main prerequisites must be satisfied. First, two beams with different center wavelengths need to illuminate the same optical path of the system. Second, sufficiently large coherence lengths to cover the OPD between the object and inclined reference beams are required. As was discussed in Section 2, the proposed system includes mechanisms to increase both the temporal and spatial coherence lengths.
Table 1 compares the coherence lengths and inclination angles for LED-only and filtered light source experiments with the proposed DH system. The characteristics of the filtered beams are selected from Fig. 2(b). As both optical and numerical errors including wavefront aberration and edge diffraction in the image's marginal regions can arise during the acquisition and reconstruction stages of the DH process, these errors can significantly affect the image quality of the dual-wavelength DH system since the wavefront noise is amplified while combining the two sources of phase data [20]. In order to minimize such degradations, a partial region of 1000 (H) × 1000 (W) pixels was selected from the center of the CCD image, corresponding to a 3.45 mm (H) × 3.45 mm (W) FOV Therefore, the spatial filter for obtaining larger coherent region is required. By selecting a pinhole with a 100 μm radius and a tube lens with a focal length of 500 mm, the resulting LSC was calculated to be greater than 3.5 mm, which is sufficient to cover the desired FOV. In the case of an LED light source without any temporal filtering process, the temporal coherence length was calculated to be ~21 μm. In contrast, by applying the proposed filtering method, the filtered beam increases the temporal coherence length to over 105 μm, which is more than 5 times longer. Therefore, as may be calculated from Eq. (5), the proposed DH system can extract the real image utilizing a reference beam with an off-axis angle of just 1.5 degrees.
Table 1. Optical Properties of the Utilized Light Sources
Since the proposed setup could be used to select the center wavelength and bandwidth of the filtered beams, dual-wavelength DH using a single LED light source could be established that avoided any phase ambiguity and extended the axial measurement range. As was previously noted, the dual-wavelength method reconstructs the object data using two measurement images obtained with different wavelengths λ1 and λ2, which expands the range of step-height to beat wavelength Λ, measurements that can be resolved, where
Figure 4 shows the results of a stepped measurement using filtered beams with center wavelengths of 620.7 and 632.4 nm, using the proposed DH configuration. In order to verify the results arising from our proposed system, a standard height sample (VSLI, SHS-1.8QC) was selected as the object to be measured. By inclining the reference beam with the angle calculated in Table 1, the real image could be separated from the zero-order and twin images, as shown in Fig. 4(b). We note that adopting numerical processes to suppress the zero-order data can result in the extraction of a larger amount of real image data [26]. However, even after aberration compensation to remove the wavefront error [27], the single wavelength phase profiles could not resolve the step height below the wavelength limit. In order to calculate the real height, the dual-wavelength profile could be obtained as follows:
where and are single-wavelength phase data obtained from Figs. 4(c) and 4(d), and phase ambiguity calculations were also applied to reduce the amplified noise during the dual-wavelength DH process [28]. As shown in Figs. 4(e)–4(h), we recorded a height of 1.815 μm with a standard deviation of 2.30 nm. Although the phase-shifting DH method is considered to yield the best image quality [17], the method proposed here inherits the advantages of low-coherence DH systems, which can achieve reduced amounts of speckle noise and improved results over conventional laser-based systems. Indeed, the standard deviation, when compared with the phase-shifting system reported in previous work [20], shows that the proposed system performs sufficiently well for use in practical applications, requiring less hologram acquisitions and which completes measurements more quickly than the phase-shifting method.Fig. 4 Dual-wavelength reconstruction process using the proposed system. (a) Acquired hologram; (b) its Fourier spectrum for λ = 620.7 nm; (c) & (d) dual-wavelength phase images with aberration compensation for λ = 620.7 nm and λ = 632.4 nm, respectively; (e) & (f) 3D and 2D phase profiles obtained by dual-wavelength restoration, respectively; (g) cross-sectional profile of the solid line in (f); and (h) detailed profile of dashed region in (g).
4. Conclusions
In this work, we have demonstrated a dual-wavelength off-axis DH system that uses a single low-coherence light source. Expanding on the idea that the coherence of a wide-bandwidth light source can be increased by filtering, the proposed method utilizes two diffraction gratings to modulate the center wavelength and bandwidth of the filtered beam, without the restrictions imposed by bandpass filter characteristics. By selecting the appropriate properties of filtered beams based on the analytics of the system, the coherence length and FOV could be extended enough to utilize an LED light source in the off-axis configuration, with better image quality than in conventional high-coherence laser-based systems. Experimental results with standard step sample shows the standard deviation less than 3 nm. Although filtering the LED to achieve a higher coherence limits the optical power of the beam compared with a general light source, the lower noise level and single-shot measurement by off-axis hologram acquisition yield significant advantages for high-stepped sample measurement.
Funding
National Research Foundation of Korea (NRF) (2015R1A5A1037668).
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