As an alternative to the conventional optical frequency comb technique, a spatial frequency comb technique is proposed for dispersionfree optical coherence depth sensing. Instead of generating an optical frequency comb over a wide range of time spectrum, we generate a spatial frequency comb by modulating the incident angle of a monochromatic plane wave with a spatial light modulator (SLM). The use of monochromatic light combined with the SLM enables dispersion-free depth sensing that is free from mechanical moving components.
©2006 Optical Society of America
Optical coherence tomography and profilometry using an optical frequency comb (OFC) has the marked advantage of fast and direct measurements being free from mechanical moving components as well as being free from complicated signal processing [1–3]. By generating the OFC with stepwise frequency modulation of the light from a tunable laser and synchronous phase modulation faster than the integration time of the image sensor, Hotate et al. have successfully demonstrated the synthesis of a temporal coherence function that has high coherence peaks at arbitrarily specified depth locations [4, 5], and more recently Choi et al. have proposed improved OFC-based interferometry . These methods presented a new concept of coherence synthesis in which a temporal coherence function was controlled by time modulation of the source spectrum. However, because OFC is composed of multiple spectral components whose spectral range must be extended for higher depth resolution, the technique suffers from spectral absorption and/or index dispersion problems, particularly when the object and/or the propagation medium have inhomogeneous spectral response as in the case of biological samples submerged in a liquid medium. This is a fundamental problem inherent to many other techniques making use of a light source with a broadband spectrum, among which are white-light interferometry [7–9] or low-coherence interferometry [10–12] and spectral interferometry [13–18].
A hint for a solution for this dispersion problem can be obtained from the analogy between space and time that can be found in many principles of optical metrology [19–21]. As an alternative to the use of OFC, we propose the use of a spatial frequency comb (SFC), in which the angular spectrum of quasi-monochromatic light is tailored to have a comb shape in the spatial frequency domain with a spatial-frequency-tunable source made of a spatial light modulator (SLM). The proposed technique enables spatial coherence depth sensing that is completely free from dispersion problems and mechanical moving components. Experimental results are presented that demonstrate the validity of the proposed principle.
The technique for generating SFC may be regarded as a natural extension of the idea of the angular spectrum scanning technique recently proposed by these authors  and also the tilt scanning technique proposed by Ruiz and Huntley . For convenience of explanation, let us first briefly review the principle of the angular spectrum control for two-beam interferometry with a Michelson interferometer illustrated in Fig. 1. A point source S is placed on the focal plane of a lens L1 whose optical axis is normal to the surface of a reference mirror MR. One of the collimated rays (shown in red) exiting from lens L1 is reflected by beam splitter BS and reaches an observation point A on the surface of the object Obj; the point A is imaged onto a point Ã on an image sensor by lens L2. Another ray (shown in blue) comes to the same point Ã on the image sensor after being reflected at point B on the surface of a reference mirror. In Fig. 1, point A’ is the mirror image of point A with respect to the virtual reference mirror M’R. The propagation vector k (which will be referred to as the k-vector for short) of the collimated beam and the height vector h are in the direction of the vectors and , respectively. The phase difference Δφ between these two rays is given by
where kh =k cos θ is the height component of the vector k, and θ is the angle of incidence to the reference surface defined by the angle between the vector k and the height vector h.
It is common practice in optical tomography and profilometry using OFC to adjust the k-vector k to be parallel to the height vector h so as to maximize the fringe sensitivity such that Δφ=-2hk with θ=0. The OFC is formed with equally-spaced multiple line spectra corresponding to wavenumbers k=nΔω/c, with n and Δω being an integer and the modefrequency separation, respectively. This characteristic of the OFC is illustrated in the k-vector space of Ewald sphere shown in Fig. 2.
As shown in Fig. 2, the object is illuminated with the beam whose k-vector k(0) is parallel to the height vector h such that θ=0. Under this illumination, the OFC corresponds to a set of collinear k-vectors aligned parallel to the vector k(0) with their arrow tips equally spaced at an interval Δk=Δω/c on the line through the center of the Ewald sphere. These radially distributed k-vectors inside the Ewald sphere cause the dispersion problems as they correspond to multiple optical frequencies. If we take a closer look at Eq. (1), we note an alternative solution in which we change the angle θ while keeping the optical frequency constant. In the k-space shown in Fig. 2, this operation corresponds to changing the cone angle θ of the k(θ) vector while keeping the radius of the k-sphere unchanged. The projected height component -kh of the k(θ) vector plays the role of the k(0) vector in the OFC. For example, if one can change θ over 0~30 degrees for the wavelength of 633nm, one can in principle realize the dispersion-free measurement with the performance comparable to the OFC with the wavelength range as wide as 98nm. We use a set of collimated monochromatic beams with different k-vectors angles whose longitudinal spatial frequency components kh are equally spaced in the direction of h, and refer to this technique as the spatial frequency comb (SFC) technique to differentiate it from the conventional optical frequency comb (OFC) technique. To generate the equally spaced SFC spectra equivalent to those of the OFC, the lateral component of the k-vector, k ⊥(θ), which is the projection of k(θ) onto the plane normal to the k(0) vector, should have a concentric Fresnel zone-plate structure as shown in Fig. 2(b). Just as the OFC generates a temporal coherence comb function with periodic coherence peaks with a time interval Δτ=π/Δω, the SFC generates a longitudinal spatial coherence comb function with periodic coherence peaks with an axial distance interval Δh=π/Δkh (where the double optical path in the reflection interferometer has been taken into account). The Fourier-transform relation of Wiener-Khinchin theorem between the temporal coherence function and the optical frequency spectrum is now replaced by the Fourier-transform relation of the generalized van Cittert-Zernike theorem between the longitudinal spatial coherence function and the longitudinal component of the spatial frequency spectrum, similarly to the McChutchen theorem . To show this, let the fields at point A and point A’ in Fig. 1 be uA and uA′ , respectively. They are created as a superposition of the angular spectra of collimated beams from an extended quasimonochromatic source placed on the focal plane of the collimator lens:
where rA and rA′ are position vectors pointing at point A and point A’ from an arbitrarily chosen origin, and the integration is taken over the surface of the Ewald sphere assuming a quasi-monochromatic source. The mutual intensity between the two points A and A’ are given by
where we have assumed that the each elementary point source constituting the extended source is perfectly incoherent such that 〈A(k)A*(k′)〉=|A(k)|2 Δ(k-k′) with 〈〉 being an ensemble average, and have used the relation dk=2πksin θkdθ=2πkdkh about the ring surface element on the Ewald sphere. The complex degree of coherence is therefore given by
For the SFC with the maximum angular spatial frequency k hmax=k and the comb spatial frequency interval Δkh
the complex degree of coherence becomes
The modulus of the complex degree of coherence has a comb shape with multiple coherence peaks separated by π/Δkh as in the Hotate scheme, and the depth resolution will be given by Δh=2π/(NΔkh ), which is inversely proportional to the total comb frequency bandwidth.
The practical implementation of the SFC is a straightforward extension of the technique we developed for angular spectrum scanning . Because tuning angular spectrum is equivalent to controlling the angle of the illumination beam, the SFC can be generated by a set of thin ring sources, with appropriate radii, placed coaxially on the focal plane of a collimator lens L1, which are created, for example, with a spatial light modulator (SLM). Suppose the optical axis of the collimator lens L1 is normal to the reference mirror surface MR and is parallel to the height vector by virtue of reflection at beam splitter BS. Then the h-component of the angular spectrum kh can be written as
where r is the radius of the ring source, and in arriving at the last expression, the paraxial assumption has been made that f≫r. One can generate a SFC with the maximum angular spatial frequency k hmax=k and the comb spatial frequency interval Δkh by adjusting the radius of the n-th ring source as
Note that these ring sources correspond to the concentric Fresnel zone-plate structure shown in Fig. 2(b). In common with the angular spectrum scanning technique , the use of the ring source has the advantages in the amount of usable light and also in the robustness to the shading problem that occurs when a high and/or deep object is illuminated at a large incidence angle. The shadow-free illumination from the circular source solves the shadowing problem intrinsic to the SFC technique. To avoid the spurious interference fringe noises arising from the interference between the beams from different points on the ring source, the ring source should be a spatially incoherent (and yet temporally coherent) source. Such a source can be realized by placing a rotating ground glass on the source plane to destroy the spatial coherence (but preserve the temporal coherence).
Unlike the OFC made of aligned collinear k-vectors, the SFC has a freedom in the choice of the direction of k-vectors. This can be used for the compensation of misalignment of the optical system. Let us consider a misaligned optical geometry as shown in Fig. 3, in which reference mirror MR has a tilt with the normal vectors n, and n′ are not aligned to the optical axis. For this geometry, k·(rA-rA′) in Eq. (4) becomes k·2h′=2kh′h′, with h′ being a new vector shown Fig. 3. Since the k-vectors are now projected onto the new direction of the height vector h′, the projected spatial frequency spectrum |A(kh′ )|2 no longer has the spectral form of an equally spaced comb function. As the result of projection of the k-vectors of the beams from the ring sources, |A(kh′ )|2 can even have a continuous broadband spectrum, as shown in green color in Fig. 4(a), and gives rise to low spatial coherence for all points except for the point h′=0. In this way, the SFC generated by ring sources is extremely sensitive to a tilt alignment error of the reference mirror. This sensitivity to tilt may first look a disadvantage but our experience in experiments has revealed that this is in fact a great advantage in monitoring and adjusting the optical alignment. This characteristic of a ring source has also been used successfully for reducing the effects of coherent artifacts in interferometry by Kuechel . Furthermore, this influence of mirror tilt can easily be compensated for by rotating the Ewald sphere to align its axis to the new height-vector h′, as shown in Fig. 4(b). To generate the equally spaced comb spectra in the direction of the h’- spatial frequency component, the lateral component of the k-vector k ⊥(θ), which is the projection of k(θ) onto the plane normal to the k(0) vector, should now have a decentered ellipse structure as shown in Fig. 4(b); the center of the shifted ellipse for the n-th SFC spectrum is given by (k-nΔkh′ )n′⊥, with n′⊥ being the lateral component of the unit normal vector of the virtual reference mirror MR’. It should be noted that by combining the tilt of the reference mirror with the adequately tailored spatially incoherent elliptical sources generated by SLM, we can control the direction of the depth sensing. This is one of the unique characteristics of the SFC technique as compared with the optical frequency comb technique.
3. Relation to other techniques
Although the proposed SFC technique is inspired by the OFC technique and also by the analogy between space and time, it naturally has close relationship to several techniques that have been proposed for spatial coherence control. In this section, we clarify the relation of our technique to some of the relevant techniques.
Let us consider a special case where the SFC spectrum is replaced by a non-negative sinusoidal function of the same period Δkh and the same bandwidth Kh =(N-1) Δkh such that
We call this as a sinusoidal spatial frequency comb (SSFC). One can easily see from Eq. (5) that this will give a longitudinal coherence function
which has three coherence peaks at h=0 and h=±π/Δkh . When projected onto the lateral plane normal to the h vector as in Fig. 2(b), the SSFC becomes a zone plate in terms of the lateral component of the k-vector ;
It has now become clear that the SSFC technique corresponds to the technique for longitudinal coherence control proposed by Rosen and Takeda . However, there are several important differences in their characteristics between the previously proposed SSFC and the newly proposed SFC. The SSFC requires the precise gray level control of SLM, and has only a pair of side coherence peaks, with their maximum modulus value being reduced down to a half, as seen in Eq. (11). The SFC, on the other hand, needs only simple binary control of SLM, and has equally spaced multiple high coherence peaks, with their modulus value being kept unity; this gives the possibility of the extending the dynamic range of depth sensing by making use of the higher order coherence peaks. Recently Gokhler and Rosen  proposed an alternative technique to generate multiple coherence peaks. They divided the ring sources into several sectors and assigned SSFCs of Eq. (10) with different spatial frequency intervals Δkh to these sectors, so that each sector generates coherence peaks at different locations. The advantage of the Gokhler-Rosen scheme is that the multiple coherence peaks need not be equally spaced as in the case of the SFC technique. However, since their technique adheres to the sinusoidal comb, the modulus value of the coherence peaks is halved as compared with the central peak, as seen in Eq. (11), and requires gray level control of SLM for its ideal performance. Furthermore, because of the sectored structure of the multiple zone plates, directional anisotropy is introduced to the sensitivity to tilt and the immunity to shading.
4.1. Experimental setup
The schematic diagram of the depth-sensing system based on the SFC is illustrated in Fig. 5. Light from a He-Ne laser (10mW) is expanded and collimated by a beam expander EX and lens L1. A spatial light modulator (DMD Discovery TM 1100 Starter Kit, Texas Instruments) based on DMD (Digital Micromirror Device) is used to project the light source pattern generated by a personal computer PC. A diaphragm D1 is placed at the entry side of DMD to adjust the proper area of the illumination. The spatially-modulated light reflected from DMD enters a confocal lens pair consisting of lens L2, L3 and a diaphragm D2, which functions as a spatial filter to remove the effect of discrete pixels of DMD and to adjust the illumination level. A SFC displayed on DMD is relayed by the confocal lens pair and imaged onto a rotating ground glass GG placed on the front focal plane of lens L4, to generate a spatially incoherent source. The light from the source created on this rotating ground glass is introduced into a Michelson interferometer composed of a beam splitter BS, a reference mirror MR, and an object GB made of a pair of gauge brocks. Lens L5 images the interference fringe pattern on the object surface onto CCD. A quarter-wave plate QWP is placed directly behind the laser to generate a circularly polarized light for uniformity of reflection at DMD. The object is a pair of gauge blocks with heights h1=1.600mm and h2=2.000mm, which are placed on an optical plate. Lens L5 is focused on the surfaces of the gauge blocks. The magnification of the imaging lens L5 is adjusted to ~1×. The focal length of lens L2, L3 and L4 are f2 =120mm, f3 =250mm and f4 =150mm, respectively. Before starting the measurement, we first performed alignment by placing the center of the ring sources on the axis of lens L4 so as to ensure that the angular spectrum from each point source on the ring gives the same spatial comb-frequency component kh =kcosθ. The detail of the alignment procedure has been described in our previous paper , and, as mentioned in Sec. 2, we made use of the fact that the ring source is highly sensitive to misalignment.
4.2. Spatial frequency comb of fireworks
For convenience of explanation, we have so far assumed that the SFC is implemented in the form of concentric thin ring sources displayed on SLM. However, if the thin ring sources have the same thickness and the same irradiance, the projected spatial frequency spectra |A(kh′ )|2 cannot create such an ideal SFC with uniform comb heights as modeled in Eq. (6), but it will generate a tapered comb whose comb height varies in proportion to the peripheral length of the ring sources. One could solve this problem by adjusting the irradiance of the ring sources by introducing the gray level control of SLM, but this would compromise the advantage of SFC over SSFC. One could also control the thickness of the ring sources, but inner rings would become too thick to function as the ideal sources for comb spectra. To get around this problem, we propose a new implementation scheme which we call a spatial frequency comb of fireworks (SFCF). As its name stands for, the source structure looks like a firework as shown in Fig. 6(a). In stead of a continuous ring structure, we adopted an array structure of discrete point sources of the same irradiance on a polar mesh. Specifically, we placed 18 discrete point sources at equal azimuthal angular intervals on the periphery of the ring. In Fig. 6(b) shows the radial distribution of the irradian ce on the source plane, and (c) shows the the corresponding comb spectrum in the spatial frequency domain. We excluded the central point source at r 0 and its spectrum k 0, because 18 point sources on the ring degenerate into a single point source at the center, and the comb spectrum for this point is reduced down to 1/18.
The full bandwidth of the SFCF can be calculated from the predefined experimental parameters. We generated the SFCF on a M×M pixel matrix of the DMD, where M=600, and the pixel size of the DMD is ΔxΔy=13.7µm×13.7µ. Considering the magnification of the confocal lens system composed of L2 (f 2=120mm) and L3 (f 3=250mm), and the field angle of the Fourier transform lens L4 (f 4=150mm), the full spatial frequency bandwidth of the SFCF can be calculated as k(f 3/f 2)2 (MΔx/2)2/(2). For the SFCF generated by a set of discrete point sources on the polar mesh composed of N rings, the spatial comb frequency interval is given by
which is equivalent to an OFC with the optical frequency interval of
where c is light velocity. The corresponding coherence comb interval becomes
In the following proof-of-principle experiments, we use the number of rings N in the firework source as a parameter specifying SFCF.
4.3. Characteristic of longitudinal coherence comb function
To demonstrate the characteristic of the longitudinal coherence comb function predicted from Eq. (7), we observed the fringe contrast on one of the gauge block surfaces for the varying height h by scanning the reference mirror. We carried out experiments for two different SFCFs with the ring parameters N=8 and N=16. The results are shown in Fig. 7. As predicted from Eq. (15), high contrast fringes were observed when the height becomes integer multiple of Δh, which means h=1.6mm for the SFCF with N=8, and h=3.2mm for the SFCF with N=16.
4.4. Depth ssensing by spatial comb frequency scanning
Next we fixed the offset height of the gauge blocks and scanned the spatial comb frequency parameter N of SFCF from 1 to 32. Several examples of fringe patters selected from the total 32 steps are shown in Fig. 8. During 32-step scanning, high-contrast fringes appeared at NL=2, 4, 8, and 16 on the left surface, and at NR=7 and 14 on the right surface of the gauge block set. From Eq. (15), the heights of the left and right surfaces, hL and hR , relative to the reference mirrors are given by the integer multiple of the coherence comb intervals for the largest SFCF parameters NL =16 and NR =14, such that hL =3.2mL [mm] and hR =2.8mR [mm], with mL and mR being integer numbers. Our a priori knowledge that the heights do not exceed 6.4mm and their difference hL -hR is less than 0.5mm permits us to uniquely determine the integers as mL =1 and mR =1. We therefore have hL =3.2mm and hR =2.8mm, and their height difference 0.4mm is in agreement to the nominal height-difference (400 µm) of the block gauge set.
4.5. Relation of SFC to angular spectrum scanning
While the angular spectrum scanning technique in our previous paper was based on the sequential display of a single angular spectrum, the SFC and SFCF techniques proposed in this paper are based on a set of multiple spatial angular spectra displayed simultaneously. In order for the SFC or the SFCF to produce a high coherence peak for the specified height, mutually incoherent point sources constituting SFC or SFCF should individually produce inphase fringe signals that are superposed constructively on intensity basis at the specified position of observation, much in the same manner as a mode-locked laser produces high-peak pulses at a specified time interval as a result of in-phase superposition of multiple waves. We call this situation as SFC or SFCF being tuned to the height. On the other hand, when SFC or SFCF is detuned from the specified height, the fringes will no longer be phase-locked, and will vary with the position of the point source. Then the fringes are averaged out as the result of superposition.
To confirm this observation by experiment, we adjusted the height of the left surface of the gauge block to hL =3.2mm with respect to the reference mirror, and generated a SFCF with eight rings N=8, which is tuned to the left surface by its secondary coherence peak as shown in Fig. 7, but which is detuned to the right surface whose height is hR =2.8mm. As seen in Fig. 9(a), the contrast of fringes on the left surface is high, but the fringe contrast on the right surface is much lower. Instead of illuminating all the rings simultaneously, we illuminated only the point sources on a selected single ring at a time. The selected ring is shifted sequentially from the inner ring to the outer ring, and the fringes on the left and right surfaces are observed at each step of the sequential ring shift. In Fig. 9(b) and 9(c) show fringe patterns observed by the illumination with the first and third ring sources of the SFCF (N=8), respectively. Note the phase shift of the fringes on the right surface between (b) and (c). As shown in Fig. 9(d), the fringe intensity on the left surface does not change significantly with the shift of the ring source because the SFCF (N=8) is tuned on this surface; the slight decrease of the intensity is due to the vignetting of the light beam by the limited acceptance angle of the interferometer. On the other hand, the fringe intensity on the SFCF-detuned right surface varies significantly with the shift of the ring source, as seen in Fig. 9(e), serving as the evidence for the predicted large phase change for the SFCF-detuned height. Next we increased the comb parameter of N of the SFCF to 32, which is tuned to the distance 6.4mm by its first coherence peak, with 1/4 comb frequency interval. Fig. 9(g) shows the sinusoidal variation of the fringe intensity on the left surface with the shift of the ring source by the (four times finer) comb frequency interval for the SFCF (N=32); this operation is similar to the angular spectrum scanning technique in our previous paper. The red circles indicate the intensity at the ring position for the SFCF (N=8) tuned to the left surface. Note that the fringes are all in phase at the ring position of the tuned SFCF (N=8), which is in agreement to our theoretical observation.
4.6. Comparison between SFCF and SSFC
Next we filled the interval of the spatial frequency comb spectra with one period of sinusoidal function to generate SSFC with a Fresnel-zone-plate (FZP) light source distribution. We first illuminated the interferometer with a SFCF by varying N=4, 8, 16, and then switched DMD from a binary to a grey mode of operation using an ALP controlling board (ViALUX, GmbH) and sequentially generated a SSFC of the corresponding parameters N=4, 8, 16. The result is shown in Fig. 10. Whereas the SFCF produces the multiple coherence peaks (the primary coherence peak for N=8, and the secondary coherence peak for N=4), the SSFC produces only one coherence peak (the primary coherence peak for N=8), as predicted from Eq. (11).
Inspired by the optical frequency comb (OFC) technique and the analogy between the roles played by space and time in many optical systems, we have proposed a new technique of depth sensing based on the spatial frequency comb (SFC) and clarified its relation to other relevant techniques. We have presented experimental results that demonstrate the validity of the principle of the proposed SFC technique, which can serve an alternative to the conventional OFC technique. The use of monochromatic light combined with the SLM enables dispersion-free absolute interferometry that is free from mechanical moving components. The use of SFCF with a firework-like structure was proposed, which can improve the uniformity of the comb heights and solve the shadowing problem.
Part of this work was supported by Grant-in-Aid of JSPS B (2) No. 18360034, and The 21st Century Center of Excellence (COE) Program on “Innovation of Coherent Optical Science” granted to The University of Electro-Communications, from Japanese Government. Zhihui Duan gratefully acknowledges the scholarship given from Ministry of Education, Culture, Sports, Science and Technology of Japan.
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