We present a method to make optical mosaic gratings that uses the exposure beams and the latent grating created by the previous exposure to adjust the lateral position and readjust the attitude of the substrate for the current exposure. As thus, it is a direct method without using any auxiliary reference grating(s) and it avoids the asynchronous drifts between otherwise independent exposure and alignment optical sub-systems. In addition, the method uses a red laser wavelength in the plane-mirror interferometers for the multi-dimensional attitude adjustment, so the adjustment can be done at leisure. The mosaic procedure is described step by step, and the principles to minimize substrate alignment errors are explained in detail. Experimentally we made several mosaics of (50 + 30) × 50 mm2 final grating area. The typical peak-valley and root-mean-square values of the measured −1st-order diffraction wavefront errors are 0.036 λ and 0.006 λ, respectively.
© 2009 Optical Society of America
Large-size diffraction gratings are key optical components of pulse compressors in contemporary chirped-pulse-amplified (CPA) high-power laser systems . Fabricating a monolithic large grating by a single holographic exposure is subject to the attainment of stable and uniform large-aperture laser beams with collimated and aberration-free wavefronts. To acquire a greater than 1 m size exposure system is both technologically and financially impractical.
To obviate the above difficulty mechanical and optical grating mosaics have been proposed. Mechanical mosaic joins gratings made on separate substrates to substitute for a single large grating with as low wavefront aberration as possible. For instance, researchers at the University of Rochester developed an interferometric tiling technique and realized a 1.5 m tiled grating assembly using three half-meter grating tiles . Optical mosaic employs multiple-exposure approaches to fabricating monolithic large gratings with low wavefront aberration. One of its representatives is the scanning beam interference lithography (SBIL) technique originally developed at MIT , which has delivered gratings of size 910 × 420 mm2 . Another is the phase synthesis technique proposed by Turukhano et al. . Many high-accuracy control techniques are required to monitor and adjust positions and attitudes of grating substrates in the repeated and prolonged application processes of mechanical mosaic. In contrast, high-accuracy control and adjustments are needed only once in the fabrication process of optical mosaic. Although in the final consideration mechanical mosaic may be inevitable, optical mosaic can enlarge the size of the monolithic gratings to be used, thus reducing the complexity of the pulse compressor.
For exposing a large grating area the SBIL uses two beams of small sizes on the order of 1 mm radius and scans the substrate; whereas the phase synthesis technique uses two expanded and collimated beams in a small number of consecutive exposures. The SBIL is very complex incorporating many high-accuracy control techniques; the phase synthesis technique is simpler, using only several reference gratings for phase alignment. Two years ago, we made an optical mosaic grating by consecutive exposures using diffraction from the latent grating (exposed but undeveloped grating in photoresist) [6,7] rather than from reference gratings as in  for alignment. The use of a separate red laser to generate phase alignment beams does not harm the latent grating, but suffers from having different phase drifts between the exposure beams and the alignment beams, which tends to result in additional mosaic errors.
Here we report an improved optical mosaic technique. It uses the latent grating as core adjustment object and the exposure beams as adjustment beams to in situ adjust the lateral position of the substrate. A dual-beam interferometer is employed to achieve high attitude precision. The optical mosaic conditions, the whole mosaic method, and the experimental results showing the controllability of the attitude and phase are presented. Some important issues of extending the present work to fabricate large mosaic gratings are also discussed.
2. Mosaic conditions
We set up a Cartesian coordinate system whose x, y, and z axes are along the directions of the grating vector (the “lateral” direction), the grating groove, and the normal of the substrate (the “longitudinal” direction), respectively. The x-z plane is considered horizontal and the y axis is considered vertical. In optical mosaic, after one area of the photoresist-coated substrate has been exposed, another area is moved laterally into place for the next exposure (In this paper we only consider mosaic along the x direction). Between different exposures, the attitude and lateral position of the substrate should be adjusted to guarantee that after the mosaic grating is made the diffraction wavefronts from all exposure areas are coplanar and in phase.
In optical mosaic, the substrate has five degrees of freedom: the in-plane rotation angle Δθz (rotation about the z axis), the tilt angle Δθy (rotation about the y axis), the lateral shift Δx (movement along the x axis), the tip angle Δθx (rotation about x axis), and the longitudinal shift Δz (displacement along the z axis). Among them, Δθz, Δθy, and Δx (shown in Fig. 1 ) are mosaic errors that give rise to wavefront aberrations of the optical mosaic. Because the exposure interference field is longitudinally invariant, Δθx and Δz (not shown in Fig. 1) do not directly cause any wavefront aberration.
2.1 Attitude condition
The in-plane rotation angle Δθz leads to an equal amount of error in grating groove direction, which in turn gives the diffraction wavefront a rotation Δγv = m λ Δθz /d in the vertical direction, where m is the diffraction order number, λ is wavelength, and d is grating period. So, the peak-valley wavefront error that Δθz introduces to the first-order diffraction beam is
2.2 Phase condition
We define the interference fringes generated by the exposure beams as exposure fringes. Between two consecutive exposures the phase of the exposure fringes may have changed by an amount Δβ during the substrate repositioning. If the distance Δx that the substrate has moved is incommensurate with Δβ, i.e., Δx′ = Δx – Δβ d /(2π) is not an integer multiple of the grating period, then the wavefront between the two exposure areas of the grating mosaic will have a step error that is given by
A perfect optical mosaic grating should meet both the attitude and phase conditions defined above; therefore, the main work of optical mosaic is to adjust the in-plane rotation angle Δθz and the relative distance between the latent grating and the exposure fringes to minimize δv and δp, respectively.
3. Mosaic method
Our optical setup for making an optical mosaic grating is shown in Fig. 2 . It contains three subsystems for making holographic exposure, attitude adjustment, and phase adjustment. In this section we will take a procedural description approach to explaining how each subsystem works. The names and functions of the optical elements in the figure will be accounted for along the way. At the end, we will summarize the whole mosaic procedure.
3.1 Holographic exposure
As shown in the lower near side of Fig. 2, when the shutter K is open, a vertically polarized light beam of wavelength λb = 413.1 nm from a Kr+ ion laser is divided and directed into the left and right beams by beamsplitter BS0 and mirrors M0, M1, and M2. The two beams are cleaned up by spatial filters SF1 and SF2 and limited to have rectangular cross sections by optical diaphragms D1 and D2. After going through lenses L1 and L2, they become collimated beams I1 and I2 and form an interference field of width w in the plane containing the front surface of the photoresist-coated substrate G0. The height of the rectangular exposure area covers the full height of G0 and extends beyond its lower edge. G0 is mounted in a holder H sitting on a translation stage equipped with fine attitude adjustment mechanisms. A holographic exposure is made by opening the shutter K for an appropriate amount of time. As a result, a weak volume grating (latent grating) is written in the photoresist layer. After the first exposure, G0 is moved laterally a distance Δx = w – w′ and prepared for the second exposure, where w′ << w is the width of the overlap region between two consecutive exposure areas which is greater than the widths of the projections of the optical elements WBS and A0 on G0. During the first exposure WBS and A0, whose functions are to be described later, are not in the path of the exposure beams.
Below G0 a small reference grating Gr is fixed on the optical table, intercepting the lower portions of I1 and I2. Gr has a period different from that of the grating to be made and it is placed so that the −1st-order diffraction of I1 and the 0th-order diffraction of I2 create interference fringes of convenient period on screen Sr (henceforth referred to as Sr fringes) for fringe locking. Before each exposure, we record a set of instantaneous fringes as the reference Sr fringes. In the duration of exposure, the real-time Sr fringes are electronically monitored and kept locked on the reference Sr fringes by moving M2 along its normal with a piezoelectric transducer PZTr.
3.2 Attitude adjustment
As shown in the upper right corner of Fig. 2, a He-Ne laser of wavelength λr = 632.8 nm serves as the light source of two independent interferometers for attitude adjustment. The laser beam is collimated and expanded to have a diameter D. The beam transmitted through BS1 is further divided by BS2 and M3 into two parallel beams with a vertical separation distance L. They are the incident beams of a dual-beam interferometer consisting of the beamsplitter BS3, the reference mirror M4 that can be driven by PZT1, and the test mirror M that is fixed on the grating holder H with its normal along the x axis. The interference fringes of the upper and lower branches of the interferometer can be observed on screen S1 (henceforth referred to as S1 fringes) and captured by a CCD camera (not shown in Fig. 2). The use of this high-low, dual-beam interferometer is to achieve the desired Δθz measurement accuracy, which could be obtained with a much greater beam diameter D at a much higher cost. We relegate the discussion of its measurement principle to Appendix, and only describe its application procedure below.
Prior to the first exposure the S1 fringes are tuned to be horizontal, so later their period and orientation can be used to adjust Δθz and Δθy, respectively. The image of the initial S1 fringes is saved as a reference, as shown in Fig. 3(a) , where the upper and lower images are for the upper and lower beams, respectively. They are assumed to represent the attitude of G0 during the first exposure. After G0 is repositioned for the next exposure, its attitude is in general altered, so both the period and the orientation of the S1 fringes are different from before, as shown in Fig. 3(b), where the left half frame is the same as that in Fig. 3(a) and the right half contains Δθz and Δθy errors. Note that the left and right frames are taken on the same screen and by the same camera, but at different times; the left frame is always the reference frame. (We use this convenient and effective scheme, which we call half-image pair, to inspect and display measurement errors throughout the rest of the paper.) Then, the real-time S1 fringes are made horizontal by adjusting Δθy. Because the test mirror M has moved with G0, the left and right fringes are in general misaligned, as shown in Fig. 3(c). Moving the reference mirror M4 by PZT1 can change the phase of the S1 fringes without changing its orientation. In this way the lower left and right fringes are aligned, but the upper fringe pair is not aligned due to a small error in Δθz, as shown in Fig. 3(d). Finally, by tuning Δθz and the position of M4 both the upper and lower fringe pairs are aligned, as shown in Fig. 3(e) which is almost indistinguishable from Fig. 3(a), indicating that Δθy and Δθz have been minimized to the experimental limits.
The beam reflected from BS1 provides the incident light for another interferometer that is composed of the beamsplitter BS4, the reference mirror M5, and the back surface of the grating substrate G0 as the test mirror. When the interference fringes observed on screen S2 (henceforth referred to as S2 fringes) are set to be vertical just before the first exposure, their later orientation can be used to adjust Δθx, just as how Δθy is adjusted with the dual-beam interferometer. Although in principle the Δθx error does not lead to any mosaic error, minimizing it facilitates the phase adjustment to be described next.
3.3 Phase adjustment
As shown in Fig. 2, the phase adjustment system consists of the substrate G0, the wedged beamsplitter WBS (one of its surfaces is anti-reflection coated), the attenuator A0, the CCD camera C, and the PZT driven mirror M2. When the shutter K is open and WBS and A0 have been inserted into I1 and I2, respectively, as shown in Fig. 2, the portion of I1 passing through WBS creates a −1st-order diffraction beam, and that of I2 passing through A0 creates a 0th-order diffraction beam, both from the same area of the latent grating. The two diffracted beams are reflected by WBS and their interference fringes are recorded by C (henceforth referred to as C fringes). The wedge direction of WBS makes the C fringes horizontal and the small wedge angle renders a convenient fringe period (neither null nor too fine) for phase adjustment; the attenuation factor of A0 improves the fringe contrast. The precise positioning of WBS and A0 will be specified shortly.
After the first exposure and insertion of WBS and A0 but before G0 is moved, a set of C fringes are recorded as the reference C fringes. Then the following sequence of actions takes place: the shutter K is closed (the fringe locking system is automatically turned off when the shutter is closed); G0 is repositioned for the second exposure and its attitude is adjusted as described above; the shutter is reopened. With the left half of the reference C fringes and the right half of the real-time C fringes displayed in their respective half frames of the half-image pair, the real-time wavefront is compared with the reference one. If the attitude of G0 has been accurately adjusted, the two set of C fringes should be both horizontal and have the same fringe period, but they may be mutually shifted. The latter is because Δx' is in general not an integer multiple of the grating period. To adjust Δx', we move the exposure fringes rather than the latent grating, because moving the mirror M2 is much easier than moving the grating substrate G0. We can also use the period of the real-time C fringes to check the in-plane rotation angle Δθz adjusted in the attitude adjustment step. If for some reason Δθz is out of tolerance, for example, due to asynchronous drifts of the exposure system and the attitude adjustment system, a final readjustment can be made. When the reference C fringes and the real-time C fringes are aligned, the phase condition Eq. (2) is satisfied. At this very moment, a new set of reference Sr fringes is saved, immediately followed by closing of the shutter K and then insertion of beam blocks B1 and B2, as shown in Fig. 2, to prevent the overlap region of width w′ from being exposed for the second time. The system is now ready for the second exposure. In some of the above steps opening the shutter K is required; however, the unwanted exposure of the photoresist film is negligible because the opening time can be kept less than a few seconds (see more discussion in Subsection 3.4).
The relative positions of the projections of all relevant optical elements mentioned above in the directions of beams I1 and I2 onto the grating substrate G0 are depicted in Fig. 4 .
If there is a measurable amount of Δθx error remaining after attitude adjustment, the whole picture frame of the real-time C fringes will be moved up or down relative to that of the reference C fringes. The frame shift may interfere with the detection and measurement of the fringe shift originated from Δx'. This is why we include the adjustment of Δθx in the attitude adjustment step.
3.4. Summary of mosaic procedure
We summarize the complete mosaic steps as follows.
- (1) Adjust the initial position and attitude of G0. Save the reference S1 fringes and the reference S2 fringes.
- (2) Open the shutter. Save the initial reference Sr fringes. Turn on the fringe locking system. Make the first exposure. Close the shutter. [Fig. 4(a)]
- (3) Insert WBS and A0. [Fig. 4(b)]
- (4) Open the shutter. Turn on the fringe locking system. Save the reference C fringes. Close the shutter.
- (5) Move G0 to the position as shown in Fig. 4(c) for the next exposure.
- (6) Adjust the attitude of G0 with the aid of the S1 and S2 half-image pairs.
- (7) Open the shutter. Adjust the phase of I2 and readjust the attitude parameter Δθz of G0 with the aid of the C half-image pair. Save the current reference Sr fringes. Close the shutter.
- (8) Insert B1 and B2. [Fig. 4(d)]
- (9) Open the shutter. Turn on the fringe locking system. Take the next exposure. Close the shutter. [Fig. 4(d)]
- (10) If the third and later exposures are to be taken, withdraw B1 and B2, and repeat Steps 5 through 10. Otherwise, stop.
Among the above ten steps, except for taking exposure whose time length depends on the output power of the Kr+ laser and the size of exposure area, Step 6 takes the most time; however, the latent grating formed by the previous exposure is not harmed in any way because photoresist is insensitive to the red He-Ne laser light. This step can take as long as required. Saving the reference C fringes and adjusting the phase of I2 by using the real-time C fringes with the exposure shutter open in Steps 4 and 7, respectively, are harmful to the latent grating theoretically, but Step 4 takes almost no time and Step 7 takes little time thanks to the convenience of adjusting the phase of I2 and the preceding high-accuracy attitude adjustment. The total time taken by these two steps is much less than the time taken by one exposure; therefore, the harm to the latent grating is negligible.
4. Experimental results
Following the above procedure we made several mosaic gratings of period d = 600 nm. Taking into account of our experimental conditions, we limited the exposure area to be w × h = 50 × 50 mm2. We used substrates G0 of size 110 × 80 mm2, so with an overlap width of w′ = 20 mm we could make mosaic gratings of 80 × 50 mm2 in final size. The dimension of the overlap region between the projections of WBS and A0 was 10 × 50 mm2. The beam diameter for the red-light interferometers was D = 15 mm, and the center distance between the high and low beams was L = 50 mm. The focal length of the camera C was f = 75 mm. The substrate surface was coated with a layer of no less than 50 nm thick chromium followed by a layer of around 100 nm thick photoresist (Rohm and Haas S9912NX-L). The purpose of the chromium layer was to enhance the latent grating diffraction and to facilitate the lithographic process. Under our experimental conditions, the exposure time and the time taken by Steps 4 and 7 are 90, less than 3 and 10 seconds, respectively. We used a Fizeau interferometer to measure the −1st-order diffraction wavefront quality of our mosaic gratings. The interferometer operating at the wavelength of λ = 632.8 nm had an aperture diameter of 100 mm and a wavefront measurement uncertainty of λ/10.
Given the interferometer wavefront measurement uncertainty, we set the upper limits of wavefront errors δv and δp of our mosaic gratings to be both less than λ/20. The δp target required us to be able to discern a relative fringe displacement as small as 1/20 of a fringe period. We achieved this by using the C half-image pair. To meet the δv target, from Eq. (1) it follows that for our d and h values Δθz had to be less than 0.6 μrad. For a beam diameter of 15 mm the angular alignment uncertainty of a single-beam interferometer operating at the 632.8 nm wavelength is about 1.0 μrad (assuming a 1/20 fringe alignment uncertainty). It clearly could not meet the requirement for Δθz. This is why we used the dual-beam interferometer (see Appendix). Our adjustments of Δθy and Δθx relied on detecting the fringe inclination angles, instead of period difference or fringe shift, in the S1 and S2 half-image pairs, respectively. Take the adjustment of Δθy as an example. If the reference S1 fringes in the left half image are adjusted to be horizontal (zero inclination angle) and have a period e r, then it is easy to show that Δθy = γ λ /(2e r), where γ is the fringe inclination angle in the right half image. For e r = 4.2 mm, λ = 632.8 nm, and γ = 7°, it gives Δθy = 10 μrad. Therefore, when a single-beam interferometer is used in this mode an angular uncertainty equal to or better than 10 μrad could be easily achieved, which far exceeded the requirements for both Δθy and Δθx. Since the period error is of the second order in Δθy, we only needed Δθy < 0.1 mrad. To ensure the accuracy of the phase adjustment, the vertical shift Δq of the C fringes due to a Δθx error should be smaller than the minimum detectable relative fringe shift due to a phase error, i.e., Δq < 0.05 e, where e is the fringe period of the C fringes on the image surface of the camera. Since Δq is related to Δθx by Δq = f Δθx, with e = 0.5 mm and f = 75 mm the requirement for Δq was easily met by Δθx = 10 μrad.
In the following to demonstrate the effectiveness of our proposed method we show several examples of mosaic gratings with controlled adjustment errors. Figure 5 shows measurement results of a mosaic grating with a Δθz error of 1μrad, which was intentionally set by having a relative fringe shift in the upper S1 half-image pair while the fringes in the lower pair were aligned, as shown in Fig. 5(a). This error led to a vertical rotation of the diffraction wavefront that appeared as a fringe period difference in the C fringe half-image pair [Fig. 5(b)] and in the Fizeau interferogram [Fig. 5(c)], and an abrupt phase variation across the vertical middle line in the wavefront map [Fig. 5(d)]. Note that the image in Fig. 5(c) is not a half-image pair; it was taken by the Fizeau interferometer with one shot. The wavefront map was automatically derived by the interferometer from a set of phase-shifted interferograms. Along the middle vertical line in Figs. 5(c) and (d) lies the boundary between two exposures. The peak-valley wavefront error reported by the interferometer was 0.040 λ, which agrees with the theoretical δv value of 0.083 λ predicted by Eq. (1) with Δθz = 1 μrad and h = 50 mm (The interferometer outputs equivalent surface form errors; hence the factor of 2 difference).
Figure 6 shows measurement results of a mosaic grating with a lateral position error. The error was intentionally set by holding a fringe shift of about 0.4 time of the fringe period in the C half-image pair [Fig. 6(a)]. The same amount of phase shift was reproduced in the interferogram [Fig. 6(b)], and half of that amount was reported by the wavefront map [Fig. 6(c)]. The attitudes of the substrate were well adjusted to meet the attitude condition, so very little rotational error can be seen in Fig. 6(c).
Figure 7 shows measurement results of a good mosaic grating with all mosaic errors reduced as much as possible during fabrication. All fringes in Figs. 7(a), 7(b), and 7(c) are aligned well and the wavefront shown in Fig. 7(d) is flat. The faint vertical narrow stripe in Fig. 7(c) is the joint area between two exposures. It is slightly brighter because it was exposed twice, so it diffracted differently from the normal area. The measured peak-valley wavefront error over the whole mosaic grating area is 0.036 λ, while that over the area of 20 × 50 mm2 centered along the joint is 0.024 λ. Figure 7(e) shows the focal intensity image of the −1st-order Littrow diffraction from the mosaic grating. A well-collimated laser beam of 632.8 nm wavelength and 60 mm diameter was used in the measurement. The ratio of the energy in the white circle to that of the entire focal spot is 0.97. Comparing Figs. 7(d) and 7(f), we conclude that the wavefront error of the mosaic grating was mainly contributed by the surface flatness error of the substrate on which the mosaic grating was made.
In Fig. 8 we show measurement results of a good mosaic grating made on a not so good substrate. The peak-valley wavefront error over the whole mosaic area is 0.115 λ [Fig. 8(b)], while that over the joint area (24 × 50 mm2) is only 0.027 λ. Comparing the wavefront error of the mosaic grating with that of the substrate, which is 0.118 λ as shown in Fig. 8(d), and noticing that their distributions are similar, we conclude that the former came mainly from the latter.
5.1 Mosaic method
The most prominent feature of our grating mosaic method is to use the exposure beams and the latent grating created by the previous exposure to adjust the lateral position and readjust the attitude of the substrate for the current exposure. Using the same exposure beams as the alignment beams avoids the problem of asynchronous drifts between the two otherwise independent optical sub-systems. Using the latent grating in the phase adjustment makes the adjustment direct and removes the need for an intermediate reference grating that might be a source of error. The second feature of our method is to use the red line of a He-Ne laser source in the plane-mirror interferometers for the attitude adjustment. The separation of phase and attitude adjustments makes it possible to take advantages of each sub-system. The phase adjustment using the exposure beams is harmful to the latent grating theoretically, but it is simple and can be done quickly (It is possible to automate this step to cut down the adjustment time further). The attitude adjustment includes multiple degrees of freedom and needs much more time, but it can be done leisurely because photoresist is insensitive to the He-Ne laser wavelength. Using a dual-beam interferometer reduces the cost of the attitude adjustment system without sacrificing adjustment accuracy. We also find the left-right half-image pair a simple, intuitive and effective tool for performing the various alignment steps summarized at the end of Section 3.
5.2 Joint between two exposures
In a mosaic grating a joint region between two exposure areas is inevitable. For optical mosaic two types of joint are possible. In the overlap type the photoresist film is exposed twice in the joint area; whereas in the gap type no exposure takes place in the area. After ion-beam etching, the overlap area diffracts differently from the normal area and the gap area does not diffract at all. Experimentally, we control the width of the joint by precisely inserting the beam blocks B1 and B2 into the exposure beam paths (See Figs. 2 and 4). Over insertion leads to a gap type joint, and under insertion leads to an overlap type joint. Ideally, one should aim at a zero joint width; in practice, it is difficult to achieve. The effect of straight-edge diffraction also contributes to the joint. At the present we can easily keep the width of joint region less than 1 mm, but we have not focused on minimizing the width, nor have we had time to consider reducing the straight-edge diffraction effect. Studying effects of the two joint types on the diffraction characteristics of a mosaic monolithic grating in the end-use is beyond the scope of this paper. However, from a practical point of view it can be expected that at any rate the detrimental effects are less than those of a physical gap in a mechanical mosaic grating assembly.
5.3 Possibility of extending the method to make large gratings
Having a high-quality interference field is the prerequisite of making a good mosaic grating. Due to limitations of experimental conditions, the largest size of our single exposure area at the moment is only 50 × 50 mm2. We are currently acquiring better conditions so that we will be able to make larger mosaic gratings. Notwithstanding, based on the experience that we have gained so far, we would like to comment on some issues that may arise in extending our mosaic method to make large gratings.
Substrate flatness is as important to a mosaic grating as to a large monolithic grating made with a single large area exposure. The substrate flatness error is essentially a longitudinal error. For a high-quality exposure interference field, which is longitudinally invariant, the projections of the latent grating lines onto a transverse plane are equally spaced and straight, uninfluenced by a slight substrate curvature. On the other hand, the phases of the −1st- and 0th-order diffraction wavefronts are influenced by the flatness error, but by the same amount. So, after the real-time C fringes are adjusted well, the −1st-order diffraction wavefront near the joint area of a mosaic grating is continuous and that over the whole mosaic area is similar to the reflection wavefront of the substrate, as shown in Fig. 8. Therefore, substrate flatness determines the upper limit of the wavefront quality of a mosaic grating, as shown in Section 4.
There are obviously two ways to increase the total mosaic grating area: to increase the single exposure area and to increase the number of exposures. On the one hand, the size of single exposure area is limited by the total laser power and the optical quality of expanded and collimated beams. The quality requirement for a single exposure area as a sub-area in a grating mosaic is higher than that for a single exposure area to be used as is. This is because a slight diffraction efficiency drop or wavefront curvature near the edge of the area in the latter case can be neglected, but such non-uniformity cannot be tolerated in the former case because it results in a sharp discontinuity in the middle of the mosaic grating. Meanwhile, the exposure time is proportional to the exposure area; therefore, increasing exposure area means increasing exposure time, which in turn puts a more severe requirement on stability of the exposure system. After all, the very idea of optical mosaic is to avoid using a very large exposure system. On the other hand, one cannot increase the number of exposures without a limit. More exposures mean more joints, and joints are nothing but defects of the mosaic grating. So, a compromise between size of single exposure area and number of exposure has to be reached. We envisage that a mosaic grating consists of no more than four or six single exposure areas.
Different mosaic errors and difficulties of different mosaic steps depend differently on the increase of single exposure area. The phase error δp defined by Eq. (2) and the width of the joint between two exposures by their natures are essentially independent of the single exposure area size. It follows from Eq. (1) that δv is apparently proportional to h, the height of the single exposure area. However, for a dual-beam interferometer whose beam axis distance L matches h or a single-beam interferometer whose beam diameter D matches h, the angular uncertainty Δθz is inversely proportional to h. Therefore, the main attitude error δv is also independent of the single exposure area size. Nevertheless, to make a meter size mosaic grating does present formidable challenges. To get a larger exposure area the laser beam must be further expanded, so the exposure beam is weaker. As a result, the light intensity diffracted by the latent grating is also weaker and more difficult to detect. As the large grating substrate is translated from one exposure position to the next, the gravitational forces exerted on the optic table are redistributed. This redistribution may deform the table surface and change the exposure beam paths significantly. In addition to fringe phase locking, attitude locking may be also necessary.
We have presented and experimentally demonstrated a method to make optical mosaic gratings, described step by step the mosaic procedure, and explained in detail the principles to minimize the different substrate alignment errors. The method uses the exposure beams and the latent grating created by the previous exposure to adjust the lateral position and readjust the attitude of the substrate for the current exposure. As thus, it is a direct method without using any auxiliary reference grating(s) and it avoids the asynchronous drifts between otherwise independent exposure and alignment optical sub-systems. In addition, the method uses a red laser wavelength in the plane-mirror interferometers for the multi-dimensional attitude adjustment, so it can be done at leisure. Experimentally we made several mosaics of (50 + 30) × 50 mm2 final grating area. The typical peak-valley and root-mean-square values of the measured −1st-order diffraction wavefront errors are 0.036 λ and 0.006 λ, respectively. Furthermore, we commented on some issues that may arise in extending our mosaic method to make larger gratings. Although the work currently underway is to make mosaic gratings of only decimeter sizes, we hope that the method with some refinements will be used to make meter or sub-meter size gratings in the future.
Suppose in a single-beam interferometer two collimated laser beams contained in the x-y plane and forming a small crossing angle interfere. The interference fringes are observed on a screen situated perpendicular to the bisector of the two beams and through the interception point of the two beam axes. For a given crossing angle θ1, a set of fringes is recorded as the left half of a half-image pair. Then the test mirror is rotated a small angle Δθz about the z axis so that the new crossing angle is θ2, and the interference fringes are recorded as the right half of the half-image pair. We answer the question: what is the smallest |Δθz| [denoted by (Δθz)m below] one can experimentally measure by using the half-image pair?
When both θ1 and θ2 are small, the height of the interference field is approximately equal to the beam diameter D. Let e 1 and e 2 be the fringe periods and n 1 and n 2 be the integer numbers of fringes contained in the left and right halves, respectively. Then D = n 1 e 1 + ε1 = n 2 e 2 + ε2, where 0 ≤ ε1 < e 1 and 0 ≤ ε2 < e 2. If the center lines of the bottom dark fringes in the two halves are perfectly aligned, Δp = | n 2 e 2 – n 1 e 1 | gives the vertical shifts between the center lines of the two top dark fringes. To find the minimum measurable Δθz it is justified to assume that n 1 = n 2 and Δp < min(e 1, e 2)/2. Since e 1 = λ /θ1 and e 2 = λ /θ2, where λ is the wavelength, it follows that
In the above we have used e to approximate e 2 and D to approximate ne 1 because e 2 ~ e 1 and D ~ ne 1, where n is an integer and e is certain mean value of e 1 and e 2. If the minimum visually discernable normalized fringe shift is denoted by (Δp/e)m, then
For a dual-beam interferometer with the two beams parallel and in phase to each other, a similar consideration leads to
where L is the distance between the two beam axes, provided that near the middle height of the lower half-image pair two dark fringes are aligned and the fringe shift Δp is taken in the middle height of the upper pair. Therefore, the dual-beam interferometer works as a single-beam interferometer with an equivalent beam diameter L, except that the interference fringes in the space between the two beams are absent.
The condition n 1 = n 2 is crucial in the above derivations; without it the three equations are invalid. For a single-beam interferometer the fulfillment of this condition can be verified by visually matching the fringes in the two halves because all fringes contained in the vertical diameter D are observable. This method does not work for a dual-beam interferometer because some of the interference fringes are absent; an alternative has to be found. Suppose two dark fringes near the middle height of the lower image pair are aligned and they are numbered the 0th fringes. Suppose also that near the middle height of the upper image pair the n 1th fringe in the left half is close to the n 2th fringe in the right half, “close” meaning that Δp = | n 2 e 2 – n 1 e 1 | < min(e 1, e 2)/2. Then we have n 2 e 2 ≈n 1 e 1, i.e., n 2 / n 1 ≈e 1 / e 2. In practice we can adjust the interferometer to make e 1 and e 2 sufficiently large so that n 1 and n 2 are small. Since the ratio of two small and distinct integers is substantially different from 1, the difference between e 1 and e 2 can be easily noticed. Conversely, if the fringe periods e 1 and e 2 are sufficiently wide and have visually indistinguishable widths, then the condition n 1 = n 2 is satisfied.
We are grateful to Lan Xia of Shanghai Institute of Laser and Plasma for providing us the experimental result presented in Fig. 7(e). This work was supported by the National High Technology Research and Development Program (863 Program) of China, and by the National Natural Science Foundation of China under Project No. 60578001.
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