Refractive index and thickness determination of thin

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This equation can be used to determine a wide range of ... light coming from the collimating lens. ... emerging from the lens to reach the screen directly ... boarder tangent to the optical axis and the edge of ... second beam, part of it passes through the substrate ... sample is inserted in the path of the parallel beam to cause a.
Optics Communications 225 (2003) 341–348 www.elsevier.com/locate/optcom

Refractive index and thickness determination of thin-films using LloydÕs interferometer A.A. Hamza a, M.A. Mabrouk b, W.A. Ramadan b

b,*

, A.M. Emara

b

a Physics Department, Faculty of Science, Mansoura University, Mansoura, Egypt Physics Department, Faculty of Science (Damietta), Mansoura University, Damietta, Egypt

Received 15 May 2003; received in revised form 30 July 2003; accepted 2 August 2003

Abstract Determination of the refractive index and the thickness of thin-films using light interference have been presented. This has been done, for the first time, with the use of LloydÕs interferometer. The mean idea is based on using the sample in two different positions in the same interferometer. The method has been applied for four different samples with different thickness. The thickness is measured using a reflection regime in a separate step. In case of determining both of refractive index and thickness of the thin films, the needed data have been obtained directly from a comparison between two interference fields in the same interferogram for each case. In this way we avoid mistakes that could be produced from the direct experimental measurements. The method with its simplicity, in experimental mounting and mathematical interpretation, can offer quite accurate results. An investigation of the uncertainty in the measured values demonstrates that, for the measured thickness we have a tolerance of 3% and for refractive index the tolerance is 0.002, which is the reasonable for two-beam interference systems.  2003 Elsevier B.V. All rights reserved. PACS: 07.60.Ly; 42.25.Hz; 78.66.Qn; 78.20.Ci Keywords: Interference; Thin films; Refractive index; Thickness

1. Introduction Thin films are very important in many applications such as integrated optical circuits, polarizers, low-pass filter, beam splitter and antireflection film. For these applications, it is important to know the * Corresponding author. Tel.: +20-57-403866; fax: +20-57403868. E-mail address: [email protected] (W.A. Ramadan).

optical properties of the thin film, especially the refractive index and the thickness. Many attempts were made to determine these two parameters accurately. Lee et al. [1] measured the refractive indices when using light vibrating parallel and perpendicular to the film surface from the reflectivity. This technique gives both values of the refractive indices and thickness simultaneously. Chiu et al. [2] used another method for measuring the refractive index. Beginning with measuring phase difference between parallel and perpendicular

0030-4018/$ - see front matter  2003 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2003.08.003

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polarizations of the light used at the total internal reflection, this was done by the heterodyne interferometric technique, then it is substituted into FresenelÕs equations, and the refractive index of the test thin film is obtained. The measurable range of this technique is limited by the refractive index of the prism. To expand the measurable range it is better to use a prism with a high refractive index. Zheng and Kikuchi [3] measured refractive index and the extinction coefficient of a weakly absorbing thin film using analytical method, which is based on measurements of the reflectance extreme and corresponding transmittance of the film at normal incidence. One of the most important techniques is ellipsometry, Jenkins [4]. It is a technique that has been studied for several centuries. However, it has recently found much favor in the non-destructive characterization of solids, particularly semiconductors. The modern era of measurement of the ellipticity of light arguably began in 1945 with the work of Rothen [5]. He devised a simple form of rotating analyzer ellipsometer, consisting of a light source, polarizer, sample and an analyzer in the form of a quarter-wave plate and another polarizer. By rotating the analyzer, it was possible to determine the ellipticity of the light reflected from the surface of the sample. But in this technique the refractive index is given via the reflection measurement, which make it quite complicated. Many authors used reflectance dependent techniques [6,7] to determine the thickness of a thin film. Davazoglou [8] used a transmission technique to study the optical dispersion and film thickness of a semiconductor layer. Interference dependent techniques [9–12] are used as well to measure the thin-film thickness. Hernandez et al. [11] used Michelson interferometer to measure the film thickness. They replaced one of the MichelsonÕs interferometer mirrors by a substrate containing partially (as step) the film. Although this method gives good results at certain wavelengths, the measured thickness depends on the wavelength used, and the obtained fringes have a poor resolution. Liu et al. [12] used a non-contact, non-destructive and simple method for measuring the thickness, refractive index and extinction coefficient of a dip coated film by using p-polarized laser

beams. In this method, the oblique sample is illuminated with a p-polarized laser beam to measure the intensity ratio of the two reflected beams, from the front and back surfaces after receiving them with a detector, versus the angle of incidence. The parameters of the film were obtained by means of data fitting. Recently, LloydÕs interferometer has been used by Bertolotti and co-workers [13] to determine directly the refractive index profile of a planar graded index waveguides. Moreover, the same group [14,15] has used this technique, for the first time, to detect experimentally the birefringence in the used Kþ –Naþ ion exchange samples. Here, we underline that LloydÕs interference technique was applied on graded index samples. For step index planar waveguide, the problem was not solved yet. In the present work, we provide the first trial to measure the refractive index and thickness of polymeric thin films using LloydÕs interference. This interference system can offer some advantage in comparison with the previously nominated ones. In case of thickness investigation, the measurement is reflection dependent, so, the transparency and the refractive index of the investigated film do not affect our measurements. Moreover, due to use of the light source as a line parallel to the sample surface, the variation of the film thickness (on macro-scale) can be detected.

2. Theory In this work the substrate carrying the film will be hold in two different positions. Position A (described in Fig. 1); in which the sample itself is used instead of the mirror in LloydÕs interferometer. In this position the thickness of the thin film can be determined. Position B (described in Fig. 3); in which the sample is inserted in the beam path to cause a shift in the observed fringes. The refractive index of the thin film can be determined measuring that shift. 2.1. Position A Interference takes place in LloydÕs mirror between the direct and reflected rays on the mirror. The substrate containing the film is used as a

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Fig. 1. The experimental set-up describing position A. The sample is used as a LloydÕs mirror.

LloydÕs mirror as shown in Fig. 1, which shows the set-up representing LloydÕs interferometer used to determine the film thickness. The line source sends light on the film and substrate surfaces at the same time. Two field patterns are produced. The first is due to reflection on the substrate only and the second is due to reflection on the film surface as shown in Fig. 2(a). The optical path difference producing such fringes can be found considering the theoretical interpretation mentioned in [16].

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According to this theory, we have to consider that the distance between the two sources d is very small compared with the distance between the source and the screen D. In our case we are satisfying this condition (typical values are d  80 lm and D  25  104 lm). Also, according to the used set-up (see Fig. 1), the incidence angles on the reflected surfaces are found to be in the range of 84.8–89.8, which give high value of reflected light intensity. In this case the contribution of phase change, due to the reflection on the thin film, is minimized and could be neglected [16]. On the screen, two pattern fields can be observed; substrate fringes field and film with substrate fringes field, as shown in Fig. 2(b). The first pattern is characterized by a small fringe width b1 , while the other has a wider fringe width b2 . This is because the vertical distance between the source and the reflecting surface, in case of substrate only is d, and is greater than that between the source and the film surface ðd  tÞ; t is the thickness of the film. The ratio between b1 and b2 can be given by b1 d  t : ¼ d b2

ð1Þ

This ratio can be calculated from the microinterferograms because the zero-order fringes of both patterns coincide with each other. If x is the distance between the zero-order fringes of the two patterns and any point on the micro-interferogram as shown in Fig. 2(b), the ratio between the fringes width is given by b1 m  a ; ¼ m b2

ð2Þ

where a is the fractional number of fringe order. It can be expressed as Dh=h at the mentioned distance x or at a certain order m, where h is the interfering spacing of the substrate and Dh is the shift due to the film. From Eqs. (1) and (2), the thickness can be expressed in the following form: t¼ Fig. 2. A schematic diagram representing LloydÕs two-beam interference. (a) The substrate with and without the thin film works as two mirrors and shows different optical paths. (b) A schematic diagram representing the produced fringe patterns.

ad : m

ð3Þ

This equation can be used to determine a wide range of thin films thickness regardless to their material transparency and refractive index.

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2.2. Position B In this position, we used the parallel beam of light coming from the collimating lens. When this beam is incident on the cylindrical lens, half of it is emerging from the lens to reach the screen directly while the other half is incident on a mirror to be reflected toward the screen. In this case two-beam normal LloydÕs interference can be observed, the interference fringes are in the form of parallel lines, which are parallel to the line source and are characterized by good visibility and equal fringe width.

Fig. 3. The experimental set-up describing position B. The sample is inserted in the path of the parallel beam to cause a shift in normal LloydÕs fringes.

In the path of this parallel beam, we put a rectangular optical cell perpendicular to the optical axis and containing a suitable matching liquid, see Fig. 3. The liquid is chosen to make the path length difference between the light passing through the substrate (Fig. 4(b)) and the film-substrate (Fig. 4(c)) is not more than one wavelength of the light used. So, we can avoid the doubt in defining the fringe shift order. The sample has been immersed in this liquid and is adjusted to make its boarder tangent to the optical axis and the edge of the thin film dividing the half of light spot (see Fig. 4(a)). In this case one of the interfering beams passes through the liquid only, reflected beam. The second beam, part of it passes through the substrate while the other part passes through the substrate and the film together. These beams will constitute the direct beam, which interfere with the reflected one. Due to this optical configuration, we will get on the screen two patterns of interference. They have the same fringe width but one of these patterns is suffering a shift with respect to the other. Now let us follow the optical path length difference for each case. In the first case, we have a reflected beam traversing only the liquid and the direct beam traversing the substrate material, see Fig. 4(b). So the optical path difference can be given by D 1 ¼ j ns  nL j T s ;

ð4Þ

Fig. 4. This figure illustrate, how the edge of the sample divided the laser parallel spot into two half and the light passing through the sample is divided by the edge of the thin film into two parts (a). The light paths in case of substrate and liquid interference and in case of substrate with thin-film and liquid interference are shown in (b) and (c), respectively.

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where ns and nL are the substrate the liquid refractive indices, respectively. Ts is the substrate thickness. In the second case, the reflected beam traverses only the liquid and the direct beam will traverse the substrate and the film materials, see Fig. 4(c). So the optical path difference can be written as D2 ¼ jns  nL j Ts þ jnf  nL j t;

ð5Þ

where nf and t are the refractive index and the thickness of the film, respectively. From Eqs. (4) and (5) one can notice that the presence of the substrate or of the substrate with the film, in the path of one of the interfering beams, will cause a constant change in the phase. In other words, all the fringes will suffer a constant shift in their order. This shift is directly proportional to the values of D1 and D2 . To determine directly the film refractive index let us subtract Eq. (4) from Eq. (5) Df ¼ D2  D1 ¼ jnf  nL j t:

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3. Experimental set-up and results In this set-up a laser beam passes through a spatial filter and a collimating lens to get a wide spot of a parallel beam. A cylindrical lens focuses this parallel beam on a line, which is used as a line source. The line source is located in front of the sample (position A) or a mirror (position B). In position A, the specular face of the substrate and the film is used to do the mirror job in LloydÕs interferometer, Fig. 1. Adjusting the position of the sample, one can observe the formation of the fringes shown in Figs. 5(a)–(d). In position B, an optical cell containing a matching liquid is inserted perpendicular in the path of the parallel beam after the collimating lens, Fig. 3. The substrate carrying

ð6Þ

This simply means that the displacement between the two obtained patterns is coming only from the presence of the thin film. Thus the problem of determining the refractive index of the film is reduced to determine experimentally the path difference Df . The path difference can be recovered from the interferogram, considering that the distance between any successive bright or dark fringes on the screen is equivalent to the path difference between the interference beams, which equal to one wavelength of the light used. So, Df can be calculated from the relation Df ¼ k

Dhb ; hb

ð7Þ

where hb is the interfering spacing and Dhb is the shifted distance of the film fringes. From Eqs. (6) and (7) we can get the refractive index nf as nf ¼ nL 

k Dhb : t hb

ð8Þ

The positive or the negative sign will be chosen according to the direction of the shift. When the shift is toward or far from the optical axis the sign will be negative or positive, respectively.

Fig. 5. The LloydÕs interferograms obtained when holding the sample in position A. The interferograms (a)–(c) show the coincidence between the double-field fringes for samples I–III, respectively. In interferogram (d) the film thickness of sample IV is too thin to provide fringes to coincide.

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the thin film is immersed in the liquid perpendicular to the parallel beam. The sample has been adjusted to accommodate the position illustrated in Fig. 4. Due to the optical path difference between the liquid and the film a shift can be noticed in the straight-line fringes, Figs. 6(a)–(d). The interferograms obtained from these configurations have been recorded in a cut-film and scanned to a computer. The parameters needed for the calculation a, hb and Dhb has been determined using a computer image program. The value of d has been measured directly using the micrometer of the precise translation stage with the observation of the light diffraction from the sample board. The wavelength of the light used was 632.8 nm and the liquid used has a refractive index of 1.462. Thin films of different thickness were prepared from the same polymer material on a flat substrate. The sample was adjusted in position A shown in Fig. 1. Two interference fringe fields are constructed in the same pattern. Fig. 5 shows

Fig. 6. Interferograms produced by LloydÕs interferometer with sample in position B. The interferograms, shown in (a)–(d), are used to determine the film refractive index of the samples I–IV, respectively.

the double-field interference patterns for four samples (labeled I, II, III and IV) of different thickness. The LloydÕs interference pattern of the sample-I is shown in Fig. 5(a). Because the film has a relatively thick value, first coincidence between the fringes in the double-field takes place at the sixth and the seventh fringes. In this image the measured value of d is 61:3 lm. Decreasing the film thickness, coincidence occurs at higher interference orders. For sample-II, the pattern is shown in Fig. 5(b). Coincidence takes place at the fringes of orders 11 and 12. For sample-III, with smaller film thickness, coincidence happens at the fringes of orders 22 and 23 as shown in Fig. 5(c). In these cases, the fractional order a is equal to one and m (order number) takes the values 7, 12 and 23, respectively. Also, the measured values of d used with samples I, II and III were 61.3, 66.3 and 81.3 lm, respectively. The fourth sample has the smallest thickness, no coincidence can be observed but a small shift between the fringes can be seen at higher interference orders as shown in Fig. 5(d). Herein, the value of a is dependent on the value of m. So different mÕs and their corespondent values of a have been taken to verify the measured thickness and substituted in Eq. (3). Table 1 shows the measured values of the film thickness for different m and a on the interferogram, or we can say at different locations on sample surface. These values are found to converge around an average value regardless to the used order. To determine the refractive index, the sample has been put in position B, Fig. 3. The obtained interferograms are shown in Figs. 6(a)–(d). One can notice that the fringes shift increases with increasing the sample thickness. The magnitude of the shift is a function of the refractive index of the liquid used. It is expected that for small thickness, the matching liquid could not be used any more, when the difference on the optical path between the light passing through the film material and the air does not exceed one order or k. Using Eq. (8), the refractive index of the samples has been determined. Table 2 shows the thickness and the refractive indices obtained for the different samples. Here, we underline that the thickness of the sample

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Table 1 The determined thickness t at different order m and their corespondent a Thickness of sample-I (lm)

Thickness of sample-II (lm)

Thickness of sample-III (lm)

Thickness of sample-IV (lm)

8.75  0.4 8.70  0.4 8.63  0.4 8.98  0.4 8.63  0.4 – –

5.52  0.2 5.34  0.2 5.33  0.2 5.36  0.2 5.28  0.2 5.33  0.2 5.57  0.2

3.53  0.1 3.45  0.1 3.55  0.1 3.49  0.1 3.42  0.1 3.50  0.1 3.47  0.1

0.91  0.03 0.92  0.03 0.88  0.03 0.89  0.03 0.88  0.03 0.87  0.03 0.88  0.03

8.74  0.4

5.39  0.2

3.49  0.1

0.89  0.03

The average values are listed at the end raw. Table 2 Refractive index for different samples; the used values of thickness, interfringe spacing hb , the shift values Dhb and the accuracy are shown Sample

t (lm)

hb (mm)

Dhb (mm)

nf

g

I II III IV

8.74 5.39 3.49 0.89

11.0 8.50 9.00 8.23

9.21 4.44 3.00 0.72

1.523 1.523 1.522 1.524

2.4  103 2.3  103 1.8  103 1.9  103

has been determined in a separate step. This is the mean difference between the presented interference method and the most of other interference techniques. The accuracy c, of the measured thickness can be determined using the following expression [11]: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2  2 ot ot c¼ ð9Þ ðDaÞ2 þ ðDd Þ2 ; oa od where Da and Dd are the error in the experimental measurement of the values a and d, respectively. The error in determining the refractive index is propagating due to the uncertainty of defining Dab and the error in determining t. So, this error could be evaluated from the following equation: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2  2 onf onf 2 2 ð10Þ g¼ ðDab Þ þ ðDtÞ ; oab ot where Dab ¼ DðDhb =hb Þ is the experimental error in determining ðDhb =hb Þ. The partial derivatives in Eqs. (9) and (10) represent the error accumulated from the dependence of the calculated parameter (t and nf ) on the experimentally measured values (a, d and ab ).

4. Conclusion This study demonstrates the capability of LloydÕs interferometer for investigating thin films with high efficiency. The method has been applied to determine the thickness and the refractive index of four thin-film samples. The method was sensitive to determine thickness up to 0.89 lm with an accuracy of 3% of the measured value. Due to the good determination of the thickness independently of the refractive index, the determination of the refractive index was quite accurate (0.002). Due to the simplicity of experimental preparation of the set-up and of the mathematical interpretation, the method result very promising for the following reasons: • The method is reflection dependent and does not depend on the transparency of the film material, so the thickness of both opaque and transparent samples can be measured. • Few parameters must be measured directly from the interferograms. This means increasing the speed and reducing the error in the measurements.

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• The method can be used to measure a wide range of thickness. One can notice from the described results that we are able to determine thin film thickness ranging from fractions of micrometers up to 9 lm. This range is suitable for studying the step index planar waveguides. • In case of position B, adapted to measure the refractive index, one cannot use an immersion liquid if the thickness of the sample is less than a certain value that depends on the wavelength of the light used. This means a complete determination of the film parameter in a nondestructive, non-contacting and direct way can be done. References [1] S.W. Lee, S.Y. Kwon, H. Lee, Macromol. Symp. 118 (1997) 451.

[2] M.H. Chiu, J.Y. Lee, D.C. Su, Appl. Opt. 36 (13) (1997) 2936. [3] Y. Zheng, K. Kikuchi, Appl. Opt. 36 (25) (1997) 6325. [4] T.E. Jenkins, J. Phys. D 32 (1999) R45. [5] A. Rothen, Rev. Sci. Instrum. 16 (1945) 26. [6] C. Cali, M. Mosca, G. Targia, Opt. Commun. 191 (2001) 295. [7] T. Kihara, K. Yokornori, Appl. Opt. 29 (1990) 5069. [8] D. Davazoglou, Appl. Phys. Lett. 70 (1997) 246. [9] S. Tolansky, Multiple-Beam Interference Microscopy of Metals, Academic Press, London, 1970, p. 95. [10] A.M. Nasr, A.M. Sadik, J. Opt. A 3 (2001) 200. [11] M. Hernandez, A. Juarez, R. Hernandez, Superficies y Vacio 9 (1999) 283. [12] X. Liu, P. Liang, W. Zhang, Y. Tang, Opt. Laser Technol. 30 (1998) 85. [13] W.A. Ramadan, E. Fazio, M. Bertolotti, Appl. Opt. 35 (31) (1996) 6173. [14] E. Fazio, W.A. Ramadan, M. Bertolotti, Opt. Lett. 21 (16) (1996) 1238. [15] M. Bertolotti, W.A. Ramadan, F. Monteleone, E. Fazio, S. Pelli, G.C. Righini, SPIE 2954 (1996) 111. [16] M. Born, E. Wolf, Principles of Optics, Pergamon Press, New York, 1964, p. 40.