Projection Moiré Profilometry using Liquid Crystal Digital Gratings

Transcription

1 0th IMEKO TC4 ymsium n Braunschweig, GERMAY, 20, etember 2-4 Prjectin Miré Prfilmetry using iquid Crystal Digital Gratings Fumi Kbayashi, Center fr Otical Research and Educatin, Utsunmiya University; Yukitshi Otani, chl f Bimedical Engineering; Tru Yshizawa, aitama Medical University Abstract: A rjectin miré rfilmetry using a air f liquid crystal digital gratings (CDG) is alied t determine a ste height using tw liquid crystal digital gratings (CDGs). Miré cnturs are filtered ver t remve the images f the riginal grating frm the miré cunter when a airs f grating images are traveled t the same directin in-hase electrically. The new idea f this rert is rsed t vercme the 2π ambiguity f miré cntur by changing the fringe interval f miré cnturs which are adjusted the erid f CDGs. The measured accuracy has been achieved t ±0.7mm in the range f 96 mm f stee height by rjecting tw different f grating erids as 6.7 and 3.3 time f riginal grating.. Intrductin In this rert, we rse the rjectin miré rfilmetry by means a hase-shifting methd using a air f liquid crystal digital gratings (CDG) t measure a stee height r searated areas. Miré cnturs are filtered ver t remve the images f the riginal grating frm the miré cunter when a airs f grating images are traveled t the same directin inhase electrically. The new idea f this rert is rsed that we vercme the 2π ambiguity f miré cntur by dual grating erid methd by changing the fringe interval f miré cnturs which are adjusted the erid f CDGs. Recently, there are requirements fr three-dimensinal rfile with measuring seed and high satial reslutin. One f the useful methds is sterescic analysis. A light-sectining methd and a grating-rjectin methd are ne f the ular three-dimensinal rfilmetry. A shadw miré tgrahy is smart t remve the image f the riginal grating frm the miré cunters by mving the grating[]. We have succeeded t measure a ste height shae by means the frequency mdulatin methd fr shadw miré tye f tgrahy by rtating the grating [2]. A rjectin tye f miré tgrahy requires nly a small system which is made f a rjectr and a camera with gratings. Hwever there are rblems f measurement time and accuracy because f the effect f riginal gratings. We have rsed a structured C digital grating, esecially fr the grating rjectin methd[3].

2 0th IMEKO TC4 ymsium n Braunschweig, GERMAY, 20, etember 2-4 We rsed the three-dimensinal shae measurement by a hase-shifting grating rjectin methd by the structured C digital grating. In this aer, we rse a rjectin miré rfilmetry by hase-shifting using a air f liquid crystal digital gratings (CDG) t measure stee height areas and searated areas. 2. Thery and exerimental setu Prjectin Miré Prfilmetry Figure shws an exerimental setu a rjectin miré rfilmetry by means the hase-shifting using CDGs. A air f liquid-crystal digital gratings (CDGs) and imaging lenses are set arallel t the tical axis. One CDG wrks as a rjectin grating and the ther is reference grating. A defrmed grating image that is illuminated by light surce t s samle is verlaed t the reference grating and is rduced miré cnturs. The intensity Ix f the miré cntur at the bject with height h frm the reference lane n the CCD camera is written as ), Fig. : Otical cnfiguratin f rjectin miré rfilmetry I x I 0 2π 2π ah = + γ cs x+ γ cs x+ 4 m 2 2π + γ cs 2x+ ah 2 2π + γ cs ( + h ) ( ) + ( + ) m h m h ah, () where, the maximum f intensity I 0,the visibility f miré cntur, the distance between imaging lenses a, the itch f CDG, the distance between the imaging lens and reference lane, the distance between the imaging lens and CDG b, and the magnitude f the imaging lens at the reference lane m (= /b ), resectively. The height h is exressed by the gemetrical relatin by the fringe rder as, h = m. (2) a m In Eq. (), the first term is the bias, and the secnd t furth terms are the fundamental and secnd harmnic frequencies f the riginal gratings, the s-called nise terms, and the last

3 0th IMEKO TC4 ymsium n Braunschweig, GERMAY, 20, etember 2-4 term is the miré cnturs. We remve the secnd t furth terms in Eq. () and extract the miré cnturs. When the images f the grating atterns f the CDGs travel in the x- directin, synchrnized electrically with velcity V and integrating the exsure time t f the CCD camera, we can remve the nise grating images f the cmnents frm the secnd t furth terms in Eq. (). 3. te height and searated area measurement by the dual grating-erid methd The fringe rder is exressed using the hase f miré cnturs by the hase-shifting methd. It is easy t aly hase shifting by changing the hase f the CDG. Hwever, there is ambiguity, such as the integer f the fringe rder. It can be determined the fringe rder using the dual grating-erid methd and the recise hase by the shrt- and lnggrating erids because the sensitivity f miré cnturs deends un the grating erid f miré cnturs. Figure 2 shws the dual grating-erid methd, which cnsists f shrt- and lng-erids f grating. The lng erid f miré cntur cvers the dynamic range f the measurement in ne fringe, r a few erids in the case f an easy unwra area. First, the fringe rder f the shrt erid f miré cntur is calculated, and the hase φ f the shrt erid f miré cnturs can be easily unwraed using the fringe number as fllws: φ = 2 π + φ, (3) where indicates the hase f the shrt erid f miré cnturs, which is analyzed by the hase-shifting methd. The fringe rder f the shrt erid f miré cnturs can be determined as.5 One erid Reference 5 sitin Phase (r Fig. 2: Prsed technique using shrt- and lng-grating erids = 2 φ π + 4 ϕ ϕ, (4)

4 0th IMEKO TC4 ymsium n Braunschweig, GERMAY, 20, etember 2-4 where the hase f the lng erid f miré cnturs φ, the erid itch f grating, the hase data f the shrt-grating erid ϕ,, and Gaussian bracket [ ]. T cmensate fr the errr ε f the wraed hase within ± π 2 resulting frm the hase errr between the hase f the lng-grating erid and the hase t change the hase f the shrt-grating erid ϕ t ( φ ) between the hases f shrt- and lng-grating erids as fllws. φ =, we cmare the hase difference s ε = (5) φ s φ The determinatin arameter k is shwn as, k π = 2.. (6) If ε is larger than k, we need t crrect the wraed hase. We assume the next equatin with crrectin as j = and withut crrectin as j = 0. Finally, we btain the crrected hase as fllws. φ ε ϕ ' φ = j π + π + π ε 2 2 ϕ ϕ (7) Here, ε means the discriminant term, which gives the sign f ε. ε Figure 3 shws a simulated result in a case where randm errr at the lng-grating erid f the miré cnturs is added. The simulatin arameters used were: itch 6.08 mm er line fr the lng-grating erid, and mm er line fr the shrt-grating erid. The simulatin is used as a maximum hase errr with 7% fr the lng-grating erid and a maximum hase errr with 5% fr the shrt -grating erid. As the errr is much larger than the hase value f the shrt-erid grating erid, the unwraed errr is bserved. Hwever, there are n unwraed errrs using determinatin arameter k. Our rsed methd is a rbust enugh measurement t determine the fringe rder as a crrectin f the unwra errr.

5 0th IMEKO TC4 ymsium n Braunschweig, GERMAY, 20, etember 2-4 hase( = m m ) h ase ( = 6.08 mm) hase unw raed( = m m ) crrected hase unw raed( = m m ) = mm unwraed ith Prsed technique (crrected hase unwraed) 位相 (rad) 20 0 = 6.08 mm with errr Distance 高さ (mm) (mm) Fig. 3: imulatin f errr reductin by rsed technique 4. Exerimental results f ste height measurement Figure 4 shws the measured results f a stair mdel with three miré cnturs fr different heights. We calculated wraed miré cnturs with a high-grating erid by using the lwgrating erid f the miré data. The lw-grating erid has a itch f 3.04 mm. The highgrating erid has a itch f mm and a itch f mm. In the figure, 3.04 shws the individual result f a grating itch f 3.04 mm, and Grating itch (mm) shws the result using a lwgrating itch f 3.04 mm and 3.04 te a high-grating itch f mm, shws the result using a lw-grating itch f te mm and a high-grating itch f mm. We te 3 cmare each height in Table. Frm measurement 2 in this table, we estimate that the maximum height errr is less than 0.6 mm te 4 Pixels Fig. 4 Exerimental result f ste height

6 0th IMEKO TC4 ymsium n Braunschweig, GERMAY, 20, etember Cnclusin We have rsed hase-shifting rjectin miré tgrahy using a air f liquid crystal digital gratings (CDGs) t measure ste height r searated areas. Our nvel idea is that we vercme the 2π ambiguity f miré cnturs and determined the fringe number by changing the fringe interval f miré cnturs, which are adjusted t the CDG erid in this case by 6.7 and 3.3 times the itc.h he rsed methd is t determine the fringe number f miré cnturs by dual grating erid methd and the recise hase by lng and shrt grating erids because the sensitivity f miré cnturs deends n the grating erid f miré cnturs. Finally, we have checked that the measured accuracy has been achieved t ±0.7mm in the range f 96 mm f stee height by rjecting tw different f grating erids as 6.7 and 3.3 time f riginal grating. [] H.Takasaki ; Al. Ot. 9, 4 (970) [2].H.Jin, T.Yshizawa, Y.Otani ; Otical Engineering, 40, 7 (200) [3] K.Yamatani, Y.Otani, T.Yshizawa, H.Fujita, et.al : Prc.PIE Vl.3782 (999)