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Dispersive white-light spectral interferometry including the effect of thin-film for distance measurement
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Optik
Optics
Optik 118 (2007) 319–324 www.elsevier.de/ijleo
Dispersive white-light spectral interferometry including the effect of thin-film for distance measurement P. Hlubina, D. Ciprian, J. Lunˇa´cˇek Department of Physics, Technical University Ostrava, 17. listopadu 15, 708 33 Ostrava-Poruba, Czech Republic Received 6 February 2006; accepted 15 April 2006
Abstract A spectral-domain white-light interferometric technique is used for measuring distances in a Michelson interferometer with a mirror represented by a thin-film structure (TFS) on a substrate. A fibre-optic spectrometer is employed for recording spectral interferograms that include wide wavelength range effects of dispersion in a cube beam splitter and multiple reflection within the TFS. Knowing the effective thickness of the beam splitter, its dispersion and parameters of the TFS and substrate, the positions of the second interferometer mirror are determined precisely by a least-squares fitting of the theoretical spectral interferograms to the recorded ones. We apply the technique to the beam splitter made of BK7 optical glass and to a uniform SiO2 thin film on a silicon wafer. We compare the results of the processing that include and do not include the effect of the TFS. r 2006 Elsevier GmbH. All rights reserved. Keywords: White light; Spectral interferometry; Thin-film structure; Nonlinear phase function; Distance
1. Introduction Spectral interferometric techniques that employ white-light sources and are based on channelled spectrum detection have been widely used for measuring distances and displacements [1–6], in optical profilometry [7–9], for dispersion characterization of optical specimens [10–12] and/or for measuring their thicknesses [6,11,13]. An optical configuration with whitelight channelled spectrum detection operates in a limited distance range with the minimum distance given by the spectral bandwidth of a white-light source and the
Corresponding author. Tel.: +420 59 732 3134; fax: +420 59 732 3139. E-mail address: [email protected] (P. Hlubina).
0030-4026/$ - see front matter r 2006 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2006.04.002
maximum distance given by the spectrometer resolving power [2]. Recently, we extended the use of white-light spectral interferometry for precise determination of the group dispersion in fused silica [14] and for measuring distances and displacements in substantially larger range [15]. The technique employs a low-resolution spectrometer to resolve spectral interference fringes in a narrow wavelength range in the vicinity of the so-called equalization wavelength. We have demonstrated that processing of the recorded spectral interferograms using a least-squares method gives distances with resolution comparable to vertical resolutions of standard spatialdomain white-light profilometers [16]. More recently, we have included the effect of transparent thin films present in the Michelson interferometer configuration and specified the corresponding measurement errors [17].
ARTICLE IN PRESS 320
P. Hlubina et al. / Optik 118 (2007) 319–324
The aim of the paper is to extend the application of white-light spectral interferometry for measuring distances in a Michelson interferometer with a mirror represented by a thin-film structure (TFS) on a substrate. We employed a fibre-optic spectrometer for recording spectral interferograms that include wide wavelength range effects of dispersion in a cube beam splitter and multiple reflection within the TFS. Knowing the effective thickness of the beam splitter, its dispersion and parameters of the TFS and substrate, we determined precisely the positions of the second interferometer mirror by a least-squares fitting of the theoretical interferograms to the recorded ones. The technique was applied to the beam splitter made of BK7 glass and to a thin SiO2 film on a silicon wafer. We evaluated for two thicknesses of SiO2 thin film the deviations resulting from the processing that included and did not include the effect of the TFS.
2. Experimental configuration The experimental set-up used in the application of spectral-domain white-light interferometry to measure distances in a slightly dispersive Michelson interferometer is shown in Fig. 1. It consists of a white-light source, a halogen lamp with launching optics, an optical fibre and a collimating lens, a bulk-optic Michelson interferometer with a cube beam splitter made of BK7 optical glass, a thin-film structure on a substrate as mirror 1, metallic mirror 2 connected to a micropositioner, a microscope objective, micropositioners, a read optical fibre, a miniature fibre-optic spectrometer S2000, an A/D converter and a personal computer. The thinfilm structure is represented by a uniform SiO2 thin film and the substrate by a silicon wafer. The fibre-optic spectrometer S2000 (Ocean Optics, Inc.), the design of which was reported previously [18], has the spectral operation range from 350 to 1000 nm and uses a 2048-element linear CCD-array detector. The Thin-film structure (Mirror 1)
Light source Optical fibre
Micropositioner
Collimator Beam splitter
Mirror 2
Objective Optical table Micropositioners
PC
wavelength of the spectrometer is calibrated so that a third-order polynomial relation between pixel number and wavelength is used. The spectrometer resolution is in our case given by the effective width of the light beam from a core of the read optical fibre: we used the read optical fibre of a 50 mm core diameter to which a Gaussian response function of the width DlR ¼ 3 nm corresponds [18]. Spectrometer sensitivity is adjusted by the spectrometer integration time of 5 ms.
3. Experimental method Experimental method utilizes the mutual interference of two beams from a broadband source at the output of a slightly dispersive Michelson interferometer with a cube beam splitter and a thin-film structure on a substrate representing mirror 1 as is shown in Fig. 1. We assume that the geometrical path lengths of the light rays in dispersive glass of the beam splitter are not the same for both interferometer arms so that the beam splitter can be represented by an ideal beam splitter and a plate of the same dispersion and of the effective thickness tef [13] (see Fig. 1). The spectrum IðlÞ recorded at the output of the interferometer by a fibre-optic spectrometer of a Gaussian response function can be expressed as [19] IðlÞ ¼ I ð0Þ ðlÞf1 þ V ðlÞ expfðp2 =2Þ½Dg ðlÞDlR =l2 2 g cos½ð2p=lÞDðlÞg,
ð1Þ
ð0Þ
where I ðlÞ is the reference spectrum, V ðlÞ is the wavelength-dependent visibility term, DðlÞ and Dg ðlÞ are the wavelength-dependent optical path difference (OPD) and group OPD between interfering beams, respectively, and DlR is the width of the response function of the spectrometer. When the thin-film structure is characterized by a complex reflection coefficient pffiffiffiffiffiffiffiffiffiffi rðlÞ ¼ RðlÞ exp½idr ðlÞ, (2) where RðlÞ is the wavelength-dependent reflectivity and dr ðlÞ is the wavelength-dependent phase change on reflection, the visibility term is given by pffiffiffiffi 2 RðlÞ , (3) V ðlÞ ¼ V I 1 þ RðlÞ where V I is the visibility term including the effect of the spatial integration of the read optical fibre on the interference fringes. The OPD DðlÞ in Eq. (1) is given by
Spectrometer
DðlÞ ¼ 2L þ 2nðlÞtef ldr ðlÞ=ð2pÞ,
S2000
where 2L is the difference of path lengths between the interfering beams in the air, nðlÞ is the wavelengthdependent refractive index of the beam splitter material. If the TFS consists of a uniform thin film and if its thickness and wavelength-dependent refractive index are
Read optical fibre
Fig. 1. Experimental set-up of a slightly dispersive Michelson interferometer with a thin-film structure on a substrate as mirror 1 to measure distances of mirror 2.
(4)
ARTICLE IN PRESS P. Hlubina et al. / Optik 118 (2007) 319–324
denoted d and n1 ðlÞ, respectively, the wavelengthdependent phase change dr ðlÞ can be represented as the sum of two contributions (5)
where fnl ðlÞ is the wavelength-dependent nonlinear phase function due to the multiple reflection within the TFS [4,20]. In the case of a thick film, dr ðlÞ 2n1 ðlÞd and Eqs. (4) and (5) give the OPD which corresponds to two different media in the individual arms of the interferometer [13,19]. The group OPD Dg ðlÞ in Eq. (1) can be approximated by Dg ðlÞ 2L þ 2NðlÞtef ,
(6)
where NðlÞ is the wavelength-dependent group refractive index of the beam splitter material. When the slightly dispersive Michelson interferometer is considered, there is a possibility to use a simple procedure to measure the effective thickness tef [13] and to calculate the complex reflection coefficient rðlÞ or the reflectivity RðlÞ and the phase function dr ðlÞ [21]. Once these quantities are known together with dispersions of refractive indices nðlÞ and n1 ðlÞ, it results from Eqs. (4) to (6) that it is possible to fit the theoretical spectral interferogram (1) to the recorded one and to determine precisely the mirror position L or displacement DL.
4. Experimental results and discussion First, we used the standard Michelson interferometer configuration with two identical metallic mirrors to determine the effective thickness tef of the cube beam splitter made of BK7 optical glass. One of the spectral interferograms recorded for the slightly dispersive Michelson interferometer was processed using the Fourier-transform method [22] and a simple procedure was used [13] to determine the slope of the linear dependence of the OPD DðlÞ on the refractive index nðlÞ of BK7 optical glass, which is given by the Sellmeier dispersion relation [23]. The slope of the linear function yielded the effective thickness tef ¼ 10:10 mm. The negative value means that in our case the dispersive plate is in the other arm of the interferometer than is depicted in Fig. 1. Next, we replaced mirror 1 of the Michelson interferometer with a structure consisting of a uniform SiO2 thin film on silicon wafer. We used two structures with two different thicknesses d 1 ¼ 395 nm and d 2 ¼ 450 nm, which were determined by a new method presented in a previous paper [21]. In the same paper are also described fundamentals of the multiple reflection of light within the thin-film structure that enable us to calculate the wavelength-dependent nonlinear phase function fnl ðlÞ. Fig. 2 shows the function for both thicknesses of SiO2 thin film. It should be noted from
0.6 0.4 Nonlinear Phase (rad)
dr ðlÞ ¼ 2n1 ðlÞd þ fnl ðlÞ,
321
0.2 0 − 0.2 − 0.4 − 0.6 450
500
550
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650
700
750
800
850
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Fig. 2. Nonlinear phase as a function of wavelength calculated for two different thicknesses d 1 ¼ 395 nm (solid) and d 2 ¼ 450 nm (dashed) of SiO2 thin film on a silicon wafer.
Fig. 2 that the nonlinear phase function has an apparent oscillatory behavior in the wavelength domain and that positions of its maxima and minima depend on the thin film thickness. In the measurement of the distances of mirror 2 in the Michelson interferometer with the first structure of SiO2 thin film having thickness d 1 ¼ 395 nm we adjusted mirror 2 to resolve the spectral interference fringes in the spectral range as wide as possible. Mirror 2 was displaced manually by using of the micropositioner with a constant 10 mm step and the spectral region from 450 to 850 nm was chosen as that in which the spectral interference fringes should be fully observable. The knowledge of the effective thickness tef of the beam splitter, the dispersions of the refractive indices n ¼ nðlÞ and n1 ¼ n1 ðlÞ [21], respectively, and the nonlinear phase function fnl ðlÞ enabled us to determine precisely the corrected position L ¼ Lc of mirror 2 by using of Eq. (4) and a least-squares fit of the theoretical spectral interferogram (1) to the recorded one. Thus Fig. 3 shows by the dots one of the recorded spectral interferograms, corresponding to the displacement of 30 mm, which is compared with the theoretical spectral interferogram (1) represented by the solid line and characterized by V I ¼ 0:946 and Lc ¼ 7:718 mm. The reference spectrum I ð0Þ ðlÞ in Eq. (1) we took was the spectrum of one of the recorded spectral interferograms not including the spectral interference fringes and Lc was determined with a resolution of 1 nm. We can see very good agreement between theory and experiment. Using the same procedure, the mirror positions Lc corresponding to the other recorded spectral interferograms were determined precisely. The procedure was applied to 11 spectral interferograms and Fig. 4 shows the value of Lc determined as a function of the adjusted mirror
ARTICLE IN PRESS P. Hlubina et al. / Optik 118 (2007) 319–324
4000
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Fig. 3. Comparison of recorded (dots) and theoretical (solid line) spectral interferogram for the displacement of 30 mm and SiO2 thin-film thickness d 1 ¼ 395 nm. 0
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100 0.596 0.594
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Fig. 4. The mirror position as a function of the adjusted mirror displacement (points) determined from the processed spectral interferograms (d 1 ¼ 395 nm) and the deviations for dr ðlÞ ¼ 0.
displacement. We clearly see that this dependence is well fitted to the linear function characterized by a correlation factor as high as 0.99963 so that 10 mm mirror displacements were adjusted manually with high precisions. The method measures mirror positions with resolution comparable to vertical resolutions of standard spatial-domain white-light profilometers [16]. It is interesting to determine precisely the uncorrected mirror positions L ¼ Luc when no effect of the thin-film structure is taken into account and the phase function dr ðlÞ ¼ 0. Fig. 4 shows the corresponding deviations eðLÞ ¼ Lc Luc of the mirror positions thus determined from the corrected mirror positions. We can see that the deviations eðLÞ are of the order of 0.58–0:60 mm. Moreover, Fig. 4 shows that the deviation depends on the mirror position in such a manner that it has the
highest values for the interferograms with largest OPDs, that is, at the edges of the depicted mirror displacement range. Finally, we processed the spectral interferograms recorded for the Michelson interferometer including the second structure of SiO2 thin film having thickness d 2 ¼ 450 nm. Once again we displaced mirror 2 manually by using of the micropositioner with a constant 10 mm step and choose the spectral region from 450 to 850 nm as that in which the spectral interference fringes should be fully observable. We determined precisely the corrected position L ¼ Lc of mirror 2 by using of Eq. (4) and a least-squares fit of the theoretical spectral interferogram (1) to the recorded one. Fig. 5 shows by the dots one of the recorded spectral interferograms, corresponding to the displacement of 30 mm, which is compared with the theoretical spectral interferogram (1) represented by the solid line and characterized by V I ¼ 0:966 and Lc ¼ 6:832 mm, which was determined with a resolution of 1 nm. We can see again very good agreement between theory and experiment. Using the same procedure, the mirror positions Lc corresponding to the other recorded spectral interferograms were determined precisely. The procedure was applied to 11 spectral interferograms and Fig. 6 shows the value of Lc determined as a function of the adjusted mirror displacement. We clearly see that this dependence is well fitted to the linear function characterized by a correlation factor as high as 0.99988 so that 10 mm mirror displacements were adjusted manually with high precisions. Once again we determined precisely the uncorrected mirror positions L ¼ Luc when the effect of the thin-film structure was not taken into account. Fig. 6 shows the corresponding deviations eðLÞ ¼ Lc Luc of the mirror positions, which are of the order of 0.65–0:67 mm. As in previously
4500 4000 Spectral Power (A. U.)
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Fig. 5. Comparison of recorded (dots) and theoretical (solid line) spectral interferogram for the displacement of 30 mm and SiO2 thin-film thickness d 2 ¼ 450 nm.
ARTICLE IN PRESS P. Hlubina et al. / Optik 118 (2007) 319–324
50
0
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100 0.666
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40 30
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Fig. 6. The mirror position as a function of the adjusted mirror displacement (points) determined from the processed spectral interferograms (d 2 ¼ 450 nm) and the deviations for dr ðlÞ ¼ 0.
mentioned case, Fig. 6 shows that the deviation is higher at the edges of the depicted mirror displacement range.
5. Conclusions We applied a white-light spectral interferometric technique for measuring distances in a Michelson interferometer with a mirror represented by a thin-film structure on a substrate. We employed a fibre-optic spectrometer for recording spectral interferograms that included wide wavelength range effects of dispersion in a cube beam splitter and multiple reflection within the thin-film structure. The knowledge of the effective thickness of the beam splitter, its dispersion and parameters of the thin-film structure and substrate enabled us to determine precisely the positions of the second interferometer mirror using a least-squares fitting of the theoretical spectral interferograms to the recorded ones. The technique was applied to the beam splitter made of BK7 optical glass and to a uniform SiO2 thin film on a silicon wafer. We also confirmed by this measurement technique that a systematic error affects the determination of the mirror positions when the effect of the thin-film structure is not taken into account. The results of the experiment demonstrate the applicability of the technique for measuring distances and displacements with high resolution.
Acknowledgements This research was partially supported by the Grant Agency of the Czech Republic (project No. 202/06/0531) and by the Grant MSM6198910016.
[1] L.M. Smith, C.C. Dobson, Absolute displacement measurement using modulation of the spectrum of white light in a Michelson interferometer, Appl. Opt. 28 (1989) 3339–3342. [2] U. Schnell, E. Zimmermann, R. Da¨ndliker, Absolute distance measurement with synchronously sampled whitelight channelled spectrum interferometry, Pure Appl. Opt. 4 (1995) 643–651. [3] P. Hlubina, Experimental demonstration of the spectral interference between two beams of a low-coherence source at the output of a Michelson interferometer, J. Mod. Opt. 44 (1997) 1049–1059. [4] U. Schnell, R. Da¨ndliker, S. Gray, Dispersive white-light interferometry for absolute distance measurement with dielectric multilayer systems on the target, Opt. Lett. 21 (1996) 528–530. [5] I. Verrier, G. Brun, J.P. Goure, SISAM interferometer for distance measurements, Appl. Opt. 36 (1997) 6225–6230. [6] P. Hlubina, I. Gurov, V. Chugunov, Slightly dispersive white-light spectral interferometry to measure distances and displacements, Optik 114 (2003) 389–393. [7] J.E. Calatroni, P. Sandoz, G. Tribillon, Surface profiling by means of double spectral modulation, Appl. Opt. 32 (1993) 30–37. [8] J. Schwider, L. Zhou, Dispersive interferometric profilometer, Opt. Lett. 19 (1994) 995–997. [9] A.F. Zuluaga, R. Richards-Kortum, Spatially resolved spectral interferometry for determination of subsurface structure, Opt. Lett. 24 (1999) 519–521. [10] C. Sa´inz, P. Jourdain, R. Escalona, J. Calatroni, Real time interferometric measurements of dispersion curves, Opt. Commun. 110 (1994) 381–390. [11] V.N. Kumar, D.N. Rao, Using interference in the frequency domain for precise determination of the thickness and refractive indices of normal dispersive materials, J. Opt. Soc. Am. B55 (1995) 1559–1563. [12] Y. Liang, C.H. Grover, Modified white-light Mach– Zehnder interferometer for direct group-delay measurements, Appl. Opt. 37 (1998) 4105–4111. [13] P. Hlubina, White-light spectral interferometry to measure the effective thickness of optical elements of known dispersion, Acta Phys. Slov. 55 (2005) 387–393. [14] P. Hlubina, White-light spectral interferometry with the uncompensated Michelson interferometer and the group refractive index dispersion in fused silica, Opt. Commun. 193 (2001) 1–7. [15] P. Hlubina, Dispersive white-light spectral interferometry to measure distances and displacements, Opt. Commun. 212 (2002) 65–70. [16] P. De Groot, L. Deck, Surface profiling by analysis of white-light interferograms in the spatial frequency domain, J. Mod. Opt. 42 (1995) 389–401. [17] P. Hlubina, Spectral interferometry including the effect of transparent thin films to measure distances and displacements, Acta Phys. Slov. 54 (2004) 213–220. [18] P. Hlubina, I. Gurov, V. Chugunov, White-light spectral interferometric technique to measure the wavelength dependence of the spectral bandpass of a fibre-optic spectrometer, J. Mod. Opt. 50 (2003) 2067–2074.
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[19] P. Hlubina, Dispersive spectral-domain two-beam interference analysed by a fibre-optic spectrometer, J. Mod. Opt. 51 (2004) 537–547. [20] S.W. Kim, G.H. Kim, Thickness-profile measurement of transparent thin-film layer by white-light scanning interferometry, Appl. Opt. 38 (1999) 5968–5973. [21] P. Hlubina, D. Ciprian, J. Lunˇa´cˇek, M. Lesnˇa´k, Thickness of SiO2 thin film on silicon wafer measured by
dispersive white-light spectral interferometry, Appl. Phys. B 84 (2006) in press (doi: 10/1007/s00340-006-2282-2). [22] I. Gurov, P. Hlubina, V. Chugunov, Evaluation of spectral modulated interferograms using a Fourier transform and the iterative phase-locked loop method, Meas. Sci. Technol. 14 (2003) 122–130. [23] Schott Computer Glass Catalog 1.0, Schott Glasswerke Mainz, 1992.
Optik
Optics
Optik 118 (2007) 319–324 www.elsevier.de/ijleo
Dispersive white-light spectral interferometry including the effect of thin-film for distance measurement P. Hlubina, D. Ciprian, J. Lunˇa´cˇek Department of Physics, Technical University Ostrava, 17. listopadu 15, 708 33 Ostrava-Poruba, Czech Republic Received 6 February 2006; accepted 15 April 2006
Abstract A spectral-domain white-light interferometric technique is used for measuring distances in a Michelson interferometer with a mirror represented by a thin-film structure (TFS) on a substrate. A fibre-optic spectrometer is employed for recording spectral interferograms that include wide wavelength range effects of dispersion in a cube beam splitter and multiple reflection within the TFS. Knowing the effective thickness of the beam splitter, its dispersion and parameters of the TFS and substrate, the positions of the second interferometer mirror are determined precisely by a least-squares fitting of the theoretical spectral interferograms to the recorded ones. We apply the technique to the beam splitter made of BK7 optical glass and to a uniform SiO2 thin film on a silicon wafer. We compare the results of the processing that include and do not include the effect of the TFS. r 2006 Elsevier GmbH. All rights reserved. Keywords: White light; Spectral interferometry; Thin-film structure; Nonlinear phase function; Distance
1. Introduction Spectral interferometric techniques that employ white-light sources and are based on channelled spectrum detection have been widely used for measuring distances and displacements [1–6], in optical profilometry [7–9], for dispersion characterization of optical specimens [10–12] and/or for measuring their thicknesses [6,11,13]. An optical configuration with whitelight channelled spectrum detection operates in a limited distance range with the minimum distance given by the spectral bandwidth of a white-light source and the
Corresponding author. Tel.: +420 59 732 3134; fax: +420 59 732 3139. E-mail address: [email protected] (P. Hlubina).
0030-4026/$ - see front matter r 2006 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2006.04.002
maximum distance given by the spectrometer resolving power [2]. Recently, we extended the use of white-light spectral interferometry for precise determination of the group dispersion in fused silica [14] and for measuring distances and displacements in substantially larger range [15]. The technique employs a low-resolution spectrometer to resolve spectral interference fringes in a narrow wavelength range in the vicinity of the so-called equalization wavelength. We have demonstrated that processing of the recorded spectral interferograms using a least-squares method gives distances with resolution comparable to vertical resolutions of standard spatialdomain white-light profilometers [16]. More recently, we have included the effect of transparent thin films present in the Michelson interferometer configuration and specified the corresponding measurement errors [17].
ARTICLE IN PRESS 320
P. Hlubina et al. / Optik 118 (2007) 319–324
The aim of the paper is to extend the application of white-light spectral interferometry for measuring distances in a Michelson interferometer with a mirror represented by a thin-film structure (TFS) on a substrate. We employed a fibre-optic spectrometer for recording spectral interferograms that include wide wavelength range effects of dispersion in a cube beam splitter and multiple reflection within the TFS. Knowing the effective thickness of the beam splitter, its dispersion and parameters of the TFS and substrate, we determined precisely the positions of the second interferometer mirror by a least-squares fitting of the theoretical interferograms to the recorded ones. The technique was applied to the beam splitter made of BK7 glass and to a thin SiO2 film on a silicon wafer. We evaluated for two thicknesses of SiO2 thin film the deviations resulting from the processing that included and did not include the effect of the TFS.
2. Experimental configuration The experimental set-up used in the application of spectral-domain white-light interferometry to measure distances in a slightly dispersive Michelson interferometer is shown in Fig. 1. It consists of a white-light source, a halogen lamp with launching optics, an optical fibre and a collimating lens, a bulk-optic Michelson interferometer with a cube beam splitter made of BK7 optical glass, a thin-film structure on a substrate as mirror 1, metallic mirror 2 connected to a micropositioner, a microscope objective, micropositioners, a read optical fibre, a miniature fibre-optic spectrometer S2000, an A/D converter and a personal computer. The thinfilm structure is represented by a uniform SiO2 thin film and the substrate by a silicon wafer. The fibre-optic spectrometer S2000 (Ocean Optics, Inc.), the design of which was reported previously [18], has the spectral operation range from 350 to 1000 nm and uses a 2048-element linear CCD-array detector. The Thin-film structure (Mirror 1)
Light source Optical fibre
Micropositioner
Collimator Beam splitter
Mirror 2
Objective Optical table Micropositioners
PC
wavelength of the spectrometer is calibrated so that a third-order polynomial relation between pixel number and wavelength is used. The spectrometer resolution is in our case given by the effective width of the light beam from a core of the read optical fibre: we used the read optical fibre of a 50 mm core diameter to which a Gaussian response function of the width DlR ¼ 3 nm corresponds [18]. Spectrometer sensitivity is adjusted by the spectrometer integration time of 5 ms.
3. Experimental method Experimental method utilizes the mutual interference of two beams from a broadband source at the output of a slightly dispersive Michelson interferometer with a cube beam splitter and a thin-film structure on a substrate representing mirror 1 as is shown in Fig. 1. We assume that the geometrical path lengths of the light rays in dispersive glass of the beam splitter are not the same for both interferometer arms so that the beam splitter can be represented by an ideal beam splitter and a plate of the same dispersion and of the effective thickness tef [13] (see Fig. 1). The spectrum IðlÞ recorded at the output of the interferometer by a fibre-optic spectrometer of a Gaussian response function can be expressed as [19] IðlÞ ¼ I ð0Þ ðlÞf1 þ V ðlÞ expfðp2 =2Þ½Dg ðlÞDlR =l2 2 g cos½ð2p=lÞDðlÞg,
ð1Þ
ð0Þ
where I ðlÞ is the reference spectrum, V ðlÞ is the wavelength-dependent visibility term, DðlÞ and Dg ðlÞ are the wavelength-dependent optical path difference (OPD) and group OPD between interfering beams, respectively, and DlR is the width of the response function of the spectrometer. When the thin-film structure is characterized by a complex reflection coefficient pffiffiffiffiffiffiffiffiffiffi rðlÞ ¼ RðlÞ exp½idr ðlÞ, (2) where RðlÞ is the wavelength-dependent reflectivity and dr ðlÞ is the wavelength-dependent phase change on reflection, the visibility term is given by pffiffiffiffi 2 RðlÞ , (3) V ðlÞ ¼ V I 1 þ RðlÞ where V I is the visibility term including the effect of the spatial integration of the read optical fibre on the interference fringes. The OPD DðlÞ in Eq. (1) is given by
Spectrometer
DðlÞ ¼ 2L þ 2nðlÞtef ldr ðlÞ=ð2pÞ,
S2000
where 2L is the difference of path lengths between the interfering beams in the air, nðlÞ is the wavelengthdependent refractive index of the beam splitter material. If the TFS consists of a uniform thin film and if its thickness and wavelength-dependent refractive index are
Read optical fibre
Fig. 1. Experimental set-up of a slightly dispersive Michelson interferometer with a thin-film structure on a substrate as mirror 1 to measure distances of mirror 2.
(4)
ARTICLE IN PRESS P. Hlubina et al. / Optik 118 (2007) 319–324
denoted d and n1 ðlÞ, respectively, the wavelengthdependent phase change dr ðlÞ can be represented as the sum of two contributions (5)
where fnl ðlÞ is the wavelength-dependent nonlinear phase function due to the multiple reflection within the TFS [4,20]. In the case of a thick film, dr ðlÞ 2n1 ðlÞd and Eqs. (4) and (5) give the OPD which corresponds to two different media in the individual arms of the interferometer [13,19]. The group OPD Dg ðlÞ in Eq. (1) can be approximated by Dg ðlÞ 2L þ 2NðlÞtef ,
(6)
where NðlÞ is the wavelength-dependent group refractive index of the beam splitter material. When the slightly dispersive Michelson interferometer is considered, there is a possibility to use a simple procedure to measure the effective thickness tef [13] and to calculate the complex reflection coefficient rðlÞ or the reflectivity RðlÞ and the phase function dr ðlÞ [21]. Once these quantities are known together with dispersions of refractive indices nðlÞ and n1 ðlÞ, it results from Eqs. (4) to (6) that it is possible to fit the theoretical spectral interferogram (1) to the recorded one and to determine precisely the mirror position L or displacement DL.
4. Experimental results and discussion First, we used the standard Michelson interferometer configuration with two identical metallic mirrors to determine the effective thickness tef of the cube beam splitter made of BK7 optical glass. One of the spectral interferograms recorded for the slightly dispersive Michelson interferometer was processed using the Fourier-transform method [22] and a simple procedure was used [13] to determine the slope of the linear dependence of the OPD DðlÞ on the refractive index nðlÞ of BK7 optical glass, which is given by the Sellmeier dispersion relation [23]. The slope of the linear function yielded the effective thickness tef ¼ 10:10 mm. The negative value means that in our case the dispersive plate is in the other arm of the interferometer than is depicted in Fig. 1. Next, we replaced mirror 1 of the Michelson interferometer with a structure consisting of a uniform SiO2 thin film on silicon wafer. We used two structures with two different thicknesses d 1 ¼ 395 nm and d 2 ¼ 450 nm, which were determined by a new method presented in a previous paper [21]. In the same paper are also described fundamentals of the multiple reflection of light within the thin-film structure that enable us to calculate the wavelength-dependent nonlinear phase function fnl ðlÞ. Fig. 2 shows the function for both thicknesses of SiO2 thin film. It should be noted from
0.6 0.4 Nonlinear Phase (rad)
dr ðlÞ ¼ 2n1 ðlÞd þ fnl ðlÞ,
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Fig. 2. Nonlinear phase as a function of wavelength calculated for two different thicknesses d 1 ¼ 395 nm (solid) and d 2 ¼ 450 nm (dashed) of SiO2 thin film on a silicon wafer.
Fig. 2 that the nonlinear phase function has an apparent oscillatory behavior in the wavelength domain and that positions of its maxima and minima depend on the thin film thickness. In the measurement of the distances of mirror 2 in the Michelson interferometer with the first structure of SiO2 thin film having thickness d 1 ¼ 395 nm we adjusted mirror 2 to resolve the spectral interference fringes in the spectral range as wide as possible. Mirror 2 was displaced manually by using of the micropositioner with a constant 10 mm step and the spectral region from 450 to 850 nm was chosen as that in which the spectral interference fringes should be fully observable. The knowledge of the effective thickness tef of the beam splitter, the dispersions of the refractive indices n ¼ nðlÞ and n1 ¼ n1 ðlÞ [21], respectively, and the nonlinear phase function fnl ðlÞ enabled us to determine precisely the corrected position L ¼ Lc of mirror 2 by using of Eq. (4) and a least-squares fit of the theoretical spectral interferogram (1) to the recorded one. Thus Fig. 3 shows by the dots one of the recorded spectral interferograms, corresponding to the displacement of 30 mm, which is compared with the theoretical spectral interferogram (1) represented by the solid line and characterized by V I ¼ 0:946 and Lc ¼ 7:718 mm. The reference spectrum I ð0Þ ðlÞ in Eq. (1) we took was the spectrum of one of the recorded spectral interferograms not including the spectral interference fringes and Lc was determined with a resolution of 1 nm. We can see very good agreement between theory and experiment. Using the same procedure, the mirror positions Lc corresponding to the other recorded spectral interferograms were determined precisely. The procedure was applied to 11 spectral interferograms and Fig. 4 shows the value of Lc determined as a function of the adjusted mirror
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Fig. 3. Comparison of recorded (dots) and theoretical (solid line) spectral interferogram for the displacement of 30 mm and SiO2 thin-film thickness d 1 ¼ 395 nm. 0
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Fig. 4. The mirror position as a function of the adjusted mirror displacement (points) determined from the processed spectral interferograms (d 1 ¼ 395 nm) and the deviations for dr ðlÞ ¼ 0.
displacement. We clearly see that this dependence is well fitted to the linear function characterized by a correlation factor as high as 0.99963 so that 10 mm mirror displacements were adjusted manually with high precisions. The method measures mirror positions with resolution comparable to vertical resolutions of standard spatial-domain white-light profilometers [16]. It is interesting to determine precisely the uncorrected mirror positions L ¼ Luc when no effect of the thin-film structure is taken into account and the phase function dr ðlÞ ¼ 0. Fig. 4 shows the corresponding deviations eðLÞ ¼ Lc Luc of the mirror positions thus determined from the corrected mirror positions. We can see that the deviations eðLÞ are of the order of 0.58–0:60 mm. Moreover, Fig. 4 shows that the deviation depends on the mirror position in such a manner that it has the
highest values for the interferograms with largest OPDs, that is, at the edges of the depicted mirror displacement range. Finally, we processed the spectral interferograms recorded for the Michelson interferometer including the second structure of SiO2 thin film having thickness d 2 ¼ 450 nm. Once again we displaced mirror 2 manually by using of the micropositioner with a constant 10 mm step and choose the spectral region from 450 to 850 nm as that in which the spectral interference fringes should be fully observable. We determined precisely the corrected position L ¼ Lc of mirror 2 by using of Eq. (4) and a least-squares fit of the theoretical spectral interferogram (1) to the recorded one. Fig. 5 shows by the dots one of the recorded spectral interferograms, corresponding to the displacement of 30 mm, which is compared with the theoretical spectral interferogram (1) represented by the solid line and characterized by V I ¼ 0:966 and Lc ¼ 6:832 mm, which was determined with a resolution of 1 nm. We can see again very good agreement between theory and experiment. Using the same procedure, the mirror positions Lc corresponding to the other recorded spectral interferograms were determined precisely. The procedure was applied to 11 spectral interferograms and Fig. 6 shows the value of Lc determined as a function of the adjusted mirror displacement. We clearly see that this dependence is well fitted to the linear function characterized by a correlation factor as high as 0.99988 so that 10 mm mirror displacements were adjusted manually with high precisions. Once again we determined precisely the uncorrected mirror positions L ¼ Luc when the effect of the thin-film structure was not taken into account. Fig. 6 shows the corresponding deviations eðLÞ ¼ Lc Luc of the mirror positions, which are of the order of 0.65–0:67 mm. As in previously
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Fig. 5. Comparison of recorded (dots) and theoretical (solid line) spectral interferogram for the displacement of 30 mm and SiO2 thin-film thickness d 2 ¼ 450 nm.
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Fig. 6. The mirror position as a function of the adjusted mirror displacement (points) determined from the processed spectral interferograms (d 2 ¼ 450 nm) and the deviations for dr ðlÞ ¼ 0.
mentioned case, Fig. 6 shows that the deviation is higher at the edges of the depicted mirror displacement range.
5. Conclusions We applied a white-light spectral interferometric technique for measuring distances in a Michelson interferometer with a mirror represented by a thin-film structure on a substrate. We employed a fibre-optic spectrometer for recording spectral interferograms that included wide wavelength range effects of dispersion in a cube beam splitter and multiple reflection within the thin-film structure. The knowledge of the effective thickness of the beam splitter, its dispersion and parameters of the thin-film structure and substrate enabled us to determine precisely the positions of the second interferometer mirror using a least-squares fitting of the theoretical spectral interferograms to the recorded ones. The technique was applied to the beam splitter made of BK7 optical glass and to a uniform SiO2 thin film on a silicon wafer. We also confirmed by this measurement technique that a systematic error affects the determination of the mirror positions when the effect of the thin-film structure is not taken into account. The results of the experiment demonstrate the applicability of the technique for measuring distances and displacements with high resolution.
Acknowledgements This research was partially supported by the Grant Agency of the Czech Republic (project No. 202/06/0531) and by the Grant MSM6198910016.
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