Microsoft Word _ _614_ 이혁교

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1 한국정밀공학회지제 3 권 7 호 pp J. Korean Soc. Precis. Eng., Vol. 3, No. 7, pp ISSN 5-907(Print), ISSN (Online) July 04 / 자유곡면형상측정을위한백색광주사간섭계의정확도향상및시스템오차분석 Accuracy Improvement and Systematic Bias Analysis of Scanning White Light Interferometry for Free-form Surfaces Measurements 김영식, Angela Davies, 이혁교, Young-Sik Ghim, Angela Davies, and Hyug-Gyo Rhee, 한국표준과학연구원우주광학센터 (Center for Space Optics, Korea Research Institute of Standards and Science) 노스캐롤라이나대학교물리 / 광과학과 (Department of Physics and Optical Science, University of North Carolina at Charlotte) Corresponding author: hrhee@kriss.re.kr, Tel: Manuscript received: / Revised: / Accepted: Scanning white-light interferometry is an important measurement option for many surfaces. However, serious profile measurement errors can be present when measuring free-form surfaces being highly curved or tilted. When the object surface slope is not zero, the object and reference rays are no longer common path and optical aberrations impact the measurement. Aberrations mainly occur at the beam splitter in the interference objective and from misalignment in the optical system. Both effects distort the white-light interference signal when the surface slope is not zero. In this paper, we describe a modified version of white-light interferometry for eliminating these measurement errors and improving the accuracy of white-light interferometry. Moreover, we report systematic errors that are caused by optical aberrations when the object is not flat, and compare our proposed method with the conventional processing algorithm using the random ball test. Key Words: Scanning White Light Interferometry ( 백색광주사간섭계 ), Free-form Surface ( 자유곡면 ), Frequency Domain Analysis ( 주파수영역분석법 ), Self-calibration Method ( 자가보정법 ), System Bias Analysis ( 시스템오차분석 ). 서론 휴대폰, 디지털카메라, 인공위성, 광통신, 평판디스플레이등의최첨단산업에서정밀광학부품은핵심부품으로널리사용되고있다. 특히고성능과다기능이동시에요구됨에따라이러한광학부품의형상은나날이복잡해지고있다. 한예로제조기술의발달로인해등장한비구면렌즈는기존구면렌즈의단점을보완함과동시에여러장의구면렌즈들을한장으로대체하는효과 를가져와광학계의소형화및경량화를가능하게해주었다. 이러한제조기술의발달은뒷받침해주는측정기술과함께더불어성장해왔다. 현재측정기술은최종생산품을비롯하여제조과정중각단계마다의중간생산품에이르기까지전공정에걸쳐광범위하게활용됨으로써생산공정의비용과시간, 그리고효율을높이는효과를가져오고있다. 삼차원표면형상측정기술로는삼차원측정기 (three-dimensional coordinate measuring machine), 원자

2 한국정밀공학회지제 3 권 7 호 pp July 04 / 606 Fig. White-light interferogram (red line) at a single pixel and its corresponding envelope signal (black line) Fig. Profile measurement errors called ghost step errors in white-light interferometry when measuring a steel ball 힘현미경 (atomic force microscope), 위상천이간섭계 (phase-shifting interferometry), 공초점주사현미경 (confocal scanning microscope), 그리고백색광간섭계 (white-light interferometry) 등이있다. 이러한삼차원표면형상측정기술은측정방법에따라크게접촉식측정법과비접촉식측정법으로나눌수있다. 비접촉식측정법은빛을이용한광학적측정방법으로접촉식측정법에비해시편에물리적손상을주지않으면서도빠른속도로측정이가능하여산업전반에걸쳐널리활용되고있다. 이중백색광간섭계는지난수십년간정밀미세부품의삼차원표면형상측정을비롯하여박막두께측정에이르기까지광범위하게활용되어왔다. -6 백색광간섭계는간섭무늬를획득하는방식에따라 PZT 나모터와같은기계적인이송구동장치를이용해간섭무늬를얻는백색광주사간섭계 (scanning white-light interferometry) -3 와회절격자나프리즘과같은광분산장치를이용해간섭무늬를분광시켜얻는분산백색광간섭계 (dispersive white-light interferometry) 4-7 로나눌수있다. 분산백색광간섭계는백색광의넓은분광대역폭을이용해측정물체와기준면과의광경로차 (optical path difference) 에의해생기는간섭무늬를파장별로분광시켜실시간측정이가능하고외부진동이나환경에둔감한장점이있다. 하지만아직관련연구가미미하여백색광주사간섭계에비해상용화가이루어지고있지않은실정이다. 이미국내외의많은회사들에의해상용화가되어널리쓰이고있는백색광주사간섭계는백색광의낮은결맞음성 (coherence) 으로인한국부화된간섭무늬를획득하고분석하여위상모호성 (π-ambiguity) 없이수밀 리미터 (mm) 이상의큰단차도나노미터 (nm) 이하의높은수직분해능으로표면형상을정확하게측정할수있다. 측정원리는 Fig. 에서보듯이획득된백색광간섭무늬의가시도정점 (envelope peak) 또는위상정점 (phase peak) 의위치를분석함으로써측정물체의삼차원높이정보를얻게된다.. 자유곡면형상측정시발생하는백색광주사간섭계의측정오차분석 백색광주사간섭계의알고리즘은간섭무늬를분석하는방법에따라크게가시도정점추출법 (envelope peak detection method) 8-0 과위상정점추출법 (phase peak detection method) 이있다. 고속측정을위해가시도정점추출법이위상정점추출법에비해많이사용되고있지만백색광원의결맞음길이 (coherence length) 보다낮은단차를측정할경우모서리부근에서배트윙효과 (batwing effect) 가발생하고수직분해능이상대적으로떨어지는단점이있다. 따라서이러한문제점을극복하고자위상정점추출법을가시도정점추출법에적용한연구가진행되어왔다.,3 하지만자유곡면과같이측정물체의국부기울기가심할경우위상정점의절대차수계산시오차가발생할뿐만아니라광학수차도증가하여 Fig. 와같이심각한측정오차를유발하게된다. 4,5 이는 Fig. 3에서보듯이측정물체의국부기울기 (α) 가심할경우측정물체로부터반사되어오는파면 (W obj ) 과기준면으로부터반사되어오는파면 (W ref ) 이광학계를통과하는광경로에있어심

한국정밀공학회지제 3 권 7 호 pp. 605-63 July 04 / 607 (a) White-light interference signals when α is 0. (in the upper) and.9 (in the lower) (a) (b) Fig.

3 한국정밀공학회지제 3 권 7 호 pp July 04 / 607 (a) White-light interference signals when α is 0. (in the upper) and.9 (in the lower) (a) (b) Fig. 3 (a) Optical path of the object and reference waves when measuring highly curved surfaces, (b) illumination area of wave reflected from reference, and (c) illumination area of wave reflected from object (θ is the half angle of the numerical aperture of the objective and α is the surface slope angle with respect to the reference surface) 한불일치가발생하여측정파면 (W obj ) 과기준파면 (W ref ) 이공통경로를겪지않아수차가서로상쇄되는효과가없어지기때문이다. 6 따라서, 광학수차에더욱민감하게반응하여 Fig. 4(a) 에서보는바와같이표면의국부기울기 (α) 가커질수록백색광간섭무늬의왜곡으로인하여간섭무늬의비대칭성이커지고간섭무늬의길이가길어지게된다. 이와같은현상은표면의국부기울기의증가로인해유효개구수 (effective numerical aperture) 가작아지게되어 Fig. 4(b) 에서보듯이광원의중심주파수가짧은파장쪽으로이동하고광원의폭이좁아지게되어나타난다. (c) (b) Spectral distributions of effective wavelength corresponding to white-light interference signals of (a) Fig. 4 White-light interference signal and its spectral distribution when a 0.4 N.A. objective (θ=3.6 ) is used 이로인하여자유곡면과같은표면의국부기울기가심한측정물체의경우에는백색광간섭무늬의가시도정점과위상정점의불일치가심하고광학수차도많이발생되어 Fig. 에서보듯이 ghost step error 와같은심각한측정오차가유발된다. 이는위상정점의절대차수가가시도정점과의위치를기준으로계산되는데측정물체의국부기울기가커짐에따라이러한두정점의불일치가커져위상정점의차수가잘못계산되기때문이다. 본논문에서는측정물체의국부기울기가심한자유곡면과같은형상을측정할경우발생하는측정오차를줄이기위해새로운측정알고리즘을제안하고자가보정방법중의하나인 Random Ball Test (RBT) 를이용해시스템오차를분석하고자한다.

4 한국정밀공학회지제 3 권 7 호 pp July 04 / 주파수영역분석법을이용한백색광주사간섭계의정확도향상 P. de Groot 와 L. Deck 에의해제안된주파수영역분석법 (frequency domain analysis) 을통해위상값과해당하는위상정점의절대차수를계산하여측정물체의높이를구하면식 () 과같다. 7,8 d h λ λ m, m Int φ φ = φ + = 0 4π π λ dk 0 () 여기서, λ 는중심파장, φ 는중심파장에해당 0 0 하는위상값, k 는파수, m 은위상정점의절대차수, 그리고 Int[] 는함수안의인자값과가장가까운정수값을도출해낸다. 광학수차로인해 m 이부정확하게계산되면이로인해 ghost step error가유발된다. 따라서이를해결하고자많은연구가진행되어왔고 9- 본논문에서는 m 의부정확성을주파수영역분석법과두파장위상측정법을이용하여해결하고자한다. 본방법의기본아이디어는백색광의넓은대역에걸친파장의위상정보를푸리에변환 (Fourier transform) 을거쳐획득한후이중임의의두개파장을선택한다. 그리고선택된파장을기준으로주파수영역분석법 (frequency domain analysis) 을이용해높이를각각계산한후계산된높이값을서로비교분석함으로써 ghost step error를제거하는것이다. 획득된백색광간섭무늬를푸리에변환을하게되면 Fig. 5(a) 에서보는바와같이백색광원의모든파장과그에해당하는위상값을얻을수있다. 그리고위상정점의절대차수를결정하기위해파수 (k) 에따른위상기울기를구하면 Fig. 5(b) 와같다. 따라서임의의선택된두개의다른파장인 λ 과 λ 를기준으로식 () 을이용해측정물체의높이를각각구하면식 () 와같이표현된다. h = λ φ λ m, h λ φ λ m 4π + = 4π + () 여기서, φ 과 φ 는 λ 과 λ 에각각대응하는위 상값이다. 그리고 m 과 m 는계산된위상정점 의절대차수값이된다. 만약측정값에오차가없다고가정을하면식 () 를통해구한 h 과 h 는 이론적으로동일한값을가져야한다. 하지만자유곡면과같이국부기울기가급한면을측정할 경우광학수차로인해 h 과 h 는상당히틀린 값을가지게되고이두높이값의차이는식 (3) 으로표현된다. λm λ m Δ h= h h = + δ (3) 이때 δ 는일반적으로작은값으로 ( λφ /4π λφ / 4 π ) 로주어지고각파장 ( λ, λ ) 과그에대응하는위상값 ( φ, φ ) 을알기때문에미리계산할수있다. 따라서이를반영한 Δh' 는식 (4) 와같이표현된다. Δ h =Δh δ = (4) ' (a) (b) Fig. 5 (a) White-light interference signal and its corresponding Fourier transformed signal and (b) unwrapped phase information of white-light interferogram and phase slope at k (=π/λ ) and k (=π/λ ) λm λ m

5 한국정밀공학회지제 3 권 7 호 pp July 04 / 609 Table Fringe order determination to remove ghost step errors (l = 0, ±, ±, ) Δ h' Fringe Order New Height h' 0 ( λ / ) l λ / ( λ / ) l [ λ / ( λ / ) l] λ / ( λ / ) l [ λ / ( λ / ) l] γλ / ( λ / ) l γ [ γλ / ( λ /) l] ± m m h' = h ± m m h' = h λ / ± m m + h' = h + λ / ± m m h' = h λ / ± m m + h' = h + λ / ± m m ± m m γ h' = h γλ / + γ h' = h + γλ / (a) Surface profiles, h and h calculated by Eq. () 따라서식 (4) 로표현된 Δh' 의값을기준으로측정된높이값 h 또는 h 에측정오차가존재하는지를 파악할뿐만아니라보상해줄수도있게된다. 이는 Table 에서보듯이측정된높이값 h 을 Δh' 의값을기준으로위상정점의차수를정확히계산하여새로운측정값인 h ' 을도출하게된다. 예를들어임의의두개의파장인 λ 과 λ 를각 각 604 nm와 680 nm로선택을하여구의형상을 λ λ 측정하게, λ λ h = φ + m h = φ + m 되면 4π 4π Fig. 6(a) 와같이 h 과 h 를얻게된다. 이때, h 에 서 ghost step error가발생하는위치의 Δh' 를기준으로분석해보면 Table 와같이총 6군데에서발생하게된다. 따라서위의정보를바탕으로 ghost step error가발생하는곳에간섭무늬의차수 (fringe order) 를다시계산하여보상값을적용하면 Fig. 6(d) 와같은 ghost step error가없는새로운높이측정값인 h' 을 구할수있게된다. (b) Height difference, Δh (h -h ) (black line) determined with Eq. (3) and its phase difference Δδ (red line) (c) New height difference, Δh determined with Eq. (4) 4. 자가보정법 (Self-calibration Method) 을이용한백색광간섭계의시스템오차분석 시스템오차를분석할때많이쓰이는자가보정방법중의하나인 Random Ball Test (RBT) 3-5 를이용하여제안된측정알고리즘의성능을분석해보았다. RBT 는시스템의오차를평가및분석하기위해가장널리쓰이는방법중의하나로임의로회전된구의표면을측정한다음또다시구를임의로회전시켜구의표면을측정하는작업을반복수행하는방법이다. 이때반복수행하여얻은측정값의평균치를구하게되면구면수차와시 (d) Final measurement result h' after ghost step errors correction of h Fig. 6 A series of signal procedures of our proposed method

_ 한국정밀공학회지제 3 권 7 호 pp. 605-63 July 04 / 60 Table Ghost step errors and their corresponding compensation values Pixel No.

(a) Systematic bias error map with a conventional coherence peak detection method nm W = W + W + W _, _ ball spherical ball error sys bias W = W + W + W _, _ i ball spherical ball error sys bias i W

6 _ 한국정밀공학회지제 3 권 7 호 pp July 04 / 60 Table Ghost step errors and their corresponding compensation values Pixel No. Δ h' (nm) h (nm) h ' (nm) nm 스템오차성분만이남게된다. 따라서구면수차제거하게되면시스템오차성분만을쉽게구할수있게된다. 예를들어구를임의로회전시킨다음구의표면을측정하는작업을 N 번반복하게되면식 (5) 와같이각측정값이표현된다. (a) Systematic bias error map with a conventional coherence peak detection method nm W = W + W + W _, _ ball spherical ball error sys bias W = W + W + W _, _ i ball spherical ball error sys bias i W = W + W + W (5) _, _ N ball spherical ball error sys bias N N N W = N W + i ball _ spherical W + N W ball, error sys _ bias i i= i= 이때 W 는 i번째측정된구의표면형상값, i W 은구의구면수차값, W 는이상 ball spherical ball, errori 적인구의형상으로부터벗어난 i번째구의표면형상오차값, 그리고 W 는측정시발생하는 sys _ bias 시스템오차값이다. 구의표면형상오차는반복측정하여평균값을구하게되면형상오차값들이백색잡음 (white noise) 과같이서로상쇄되어없어지기때문에 W = 0 이된다. 따라서시스 ball, error i 템오차는아래식과같이표현된다. N = sys _ bias i ball _ spherical N i = W W W (6) 본실험에서는직경 6.35 mm 의구를이용해총 65 번의측정을수행하였다. 그리고측정된값의평균값을구한후구면수차값을계산하여제거하게되면식 (6) 과같이시스템의오차값만남게된다. Fig. 7 은본논문에서제안한측정알고리즘을이용해구의표면을측정한후 RBT 를통해시스템오차값을얻은결과이다. 기존의가시도정점 (b) Systematic bias error map with the proposed method Fig. 7 The RBT measurement results 추출법과비교를한결과본논문에서제시된방법을이용할경우시스템오차가 PV(Peak to Valley) 로는 7.05 nm, RMS(Root Mean Square) 로는 3.9 nm 가나왔다. 이는기존의측정방법보다시스템오차의크기가 /6 이하로작음이확인되었다. 이는구의경우측정점의위치에따라표면의국부기울기가틀리기때문에측정영역에걸쳐서로다른유효개구수를가지게되어광원의중심파장이위치에따라달라지게된다. 이로인해 Fig. 8(a) 에서보듯이측정점이구의중심점 ( 번위치 ) 에서가장자리 ( 번위치 ) 로갈수록표면의국부기울기가증가하고이로인해광원의중심파장이짧은쪽으로이동을하게된다. 즉, Fig. 8(b) 에서보듯이광원의중심파장이짧아짐과동시에광원의폭이점차좁아지게되어위치에따라간섭무늬의왜곡정도가달라지게된다. 따라서측정알고리즘의민감도에따라측정오차

한국정밀공학회지제 3 권 7 호 pp. 605-63 July 04 / 6 (a) (a) (b) Fig.

7 한국정밀공학회지제 3 권 7 호 pp July 04 / 6 (a) (a) (b) Fig. 8 (a) Spectral distribution of effective central wavelength according to the field of view when measuring a sphere and (b) white-light interference signals and their corresponding spectral distributions of position and of (a) 의정도가틀려지게된다. 본논문에서제안된알고리즘의정확도를좀더정확히분석하기위해지름이각각 mm, mm, 6.35 mm 인구의직경을마이크로미터를이용해비교측정을해보았다. Fig. 9(a) 는구의직경을삼차원형상을측정한다음 best-fitting 하여유추한결과로, 기존의알고리즘과개발된알고리즘의정확도를비교한결과이다. 측정결과에서보듯이본논문에서제시된알고리즘의성능이기존의방법보다마이크로미터로측정한결과와상당히유사함을알수있었다. Fig. 9(b) 에서보듯이기존의방법이 3% 이상의측정오차를보인반면에개발된알고리즘은 % 이하의측정오차를보였다. 5. 결론 본논문에서는백색광주사간섭계를이용하여자유곡면과같이표면의국부형상기울기가큰 (b) Fig. 9 (a) Measurement results when measuring three different steel balls with a diameter of mm, mm, and 6.35 mm using three different techniques (: micrometer, : our method, 3: conventional method), and (b) measurement error of our method and conventional method when the micrometer is the reference 측정물체에서발생하는측정오차에대해서알아보았다. 또한주파수영역분석법을응용한두파장위상측정법을이용하여백색광주사간섭계에서논란이되어왔던 ghost step error 와같은측정오차를효과적으로제거하였고자가보정방법중의하나인 RBT 를이용해시스템오차도분석해보았다. 분석결과본논문에서제안한방법의시스템오차 (7.05 nm PV, 3.9 nm RMS) 는기존의가시도정점추출법의시스템오차 (9.7 PV, 0.59 RMS) 보다 /6 이하로줄어들었고측정오차도 3% 에서 % 이하로 3 배이상향상됨을확인하였다. 본연구결과는백색광간섭계의성능을크게향상시켜향후자유곡면과같은복잡한형상측정에널리활용될것으로기대된다.

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Microsoft Word - 06 _ _ ÁÖ±â³² 한국정밀공학회지제 31 권 11 호 pp. 999-6 J. Korean Soc. Precis. Eng., Vol. 31, No. 11, pp. 999-6 ISSN 1225-9071(Print), ISSN 2287-8769(Online) November 2014 / 999 http://dx.doi.org/10.7736/kspe.2014.31.11.999 백색광주사간섭계의측정속도개선을위한서브샘플링기법연구