- Jaglarz
^{1}, K. Marszalek^{2}B. Jarzabek^{3}, R. Duraj^{4}and I. Noor^{5}AK Arof^{5}N. Wolska^{6}and B. Sahraoui^{7}

^{1,6}*Cracow University of Technology, Faculty of Mechanics M2, Al. Jana Pawła I37,31-864 Cracow, Poland*

^{2}*AGH University of Science and Technology, Mickiewicza av.30, 30-059 Cracow, Poland*

^{3}*Institute of Physics, Cracow University of Technology, ul. Podchorazych 1, 30‑084 Cracow, Poland*

^{4}Centre of Polimer Material PAN 41-819 Zabrze ul. *M. Curie-Skłodowskiej 34 Poland*

^{5}*University of Malaya, Ionic center, Dept Phys, Dept Phys, Kuala Lumpur 50603, Malaysia*

^{6}Elkom Trade S. A. Targowa 21 street, 27-400 Ostrowiec Świętokrzyski

^{7}*LUNAM Université, Université d’Angers, CNRS UMR 6200, Laboratoire MOLTECH-Anjou, 2 bd Lavoisier, 49045 Angers cedex, France*

**Abstract**

Thin polymer films are currently under intensive investigation owing to their promising optical and electrical properties. The roughness and refractive index variation in a film, and also presence of unwanted molecular inclusions in bulk created during film formation may lead to incoherent light scattering. Therefore the analysis of thin film optical spectra becomes more complicated. The aim of this work is presentation of light scattering phenomena occurring in polymer layer. They can be described by the Fresnel model for specular reflectance and as well Rayleigh, Debye and scalar theories of light scattering for non-directional reflection from a diffusive film. Furthermore, the optical methods based on combined spectro-goniometric measurements have been presented.

These investigations allow determining many important parameters as: optical constants, films thickness and their surface roughness and also optical scattering coefficients diffusive layers.

**Keywords**: elastic light scattering, spectroscopic ellipsometry diffusive reflectance and transmittance

**Introduction**

The description of optical phenomena in most thin film in general is commonly limited to presentation of interference effect occurring in layers [1,2]. Using Fresnel theory we are able to determine film thickness and as well their optical constants. In these studies one utilizes specular parts of reflected from or transmitted by film. The complex refractive index and film thickness of thin films may be effectively determined by the use of ellipsometric method [3]. Ellipsometry uses the light of known polarization incident on the studied film and detects the polarization state of the reflected light. Incident light is usually linearly polarized and the reflected light has elliptical polarization. Spectroscopic ellipsometry directly determines two angles *Ψ* and *Δ*, with:

(1)

where *Ψ* represents the angle determined from the amplitude ratio between *p-* and *s-* polarizations and *Δ* is the phase shift between the polarized waves. *r _{p}* and

*r*are the complex Fresnel reflection coefficients for

_{s}*p*- and

*s*- polarizations, respectively. Knowledge of ellipsometric angles allows us to determine dispersion of the refractive index

*n*and the extinction coefficient

*k*of the films.

The ellipsometry may be applied when most of reflected and transmitted radiation is coherent. Such condition is done in broad variety of homogenous flat layers, typical in inorganic coatings. In polymer films one may appear other phenomena related to non-coherent and non-specular light scattering. They result mostly from variation of refractive index in bulk material. It may result from non-regular distribution of polymer chains in films [4, 5]. As well in the layer may occur small scattering centers responsible for Rayleigh scattering. As representative samples in order to demonstrate light scattering phenomena the polyazomethine (PPI) thin films have been chosen. The PPI layers are important materials to obtain organic LED’s devices [6].

Thin PPI films on glass or Si substrate films exhibit partly amorphous and crystalline structure. The crystalline part in PPI compound can be high fraction of total layer volume As a result of crystallization, spherical structures (spherulites) with diameter @ 60 nm created in PPI films, being responsible for the Rayleigh scattering of light [6, 7]*. * Also one may to distinguish the second factor of bulk scattering. This scattering resulted on choice of deposition method. The variation of refractive index is responsible for volume scattering of light [8, 9].

The light scattering in thin organic films may be resulted from many different types of variations in surface material and in the bulk [10]. If variations are mild, then they can be characterized by weak single scatter events as in the case of smooth surface topography and where the scattering is caused by sparse particles embedded in a bulk of the film. Then one can to consider the total diffuse reflection as the sum three types of of scattering *I _{v}:*

- The surface scattering is caused by irregularities appearing on the film/substrate and film/air interfaces. Its arises from roughness appearing on the polymer-substrate and polymer-air interfaces. If roughness
*s*is much smaller than wavelength_{rms,}*,**l*the relation between the surface scattering reflectance*I*and root-mean square (rms) roughness_{ds}*s*according to scalar theory can be expressed as in the reference [11]:_{rms}

(2)

where *I _{0}* is the intensity of the specularly reflected light

- The volume scattering is produced by scattering centers causing the Rayleigh-type scattering of light. The Rayleigh approximation describe light scattering when size of the objects with different refractive index than surrounding medium is much smaller than light wavelength. In agreement with theory,
*s*at a given wavelength l equals [12]:_{v }

(3)

where *N* is concentration of scattering centers and *m * is constant specified for specified material

- The scattering may be result on refractive index fluctuations. The scattering by refractive index deviation is a part of elastic scattering observed in long spatial wavelengths. It is assumed there is no energy loss accompanying the scattering and the scattering medium is spherically symmetric. Debye and Bueche applied statistical theory to heterogeneous solid scattering and proposed that, in the range of small scattering angles, the scattering light is described by the following theoretical formula[13]:

(4)

where *s* and *r* are scattering and position vectors, respectively, and g(*r*) =exp(-*r/T*) is the correlation function corresponding to the fluctuation of polymer refractive index, where *T* is correlation distance and the scattering vector *s=2sin(**q*_{s}*/2), k=2**p/l**. *By substituting this correlation function into equation (4) and integrating it, one can obtain the following formula for scattering as follows [14]:

(5)

where *B* is the scattering factor. The formula (5) is valid only for *T* *> **l*.

The contribution of scattering due to refractive index variation to the total non-specular reflectance is larger for small scattered angles. This is mainly significant in translucent materials with a small extinction coefficient [9].

The Rayleigh scattering is strongly dependent on wavelength. It also concerns Debye scattering but in a smaller extent. If concentration of scattering centers is small as in our case we may apply Fresnel equations for description of polarization state of specular reflectance. Therefore ellipsometry is good idea for slightly scattered samples. On the other hand only spectral reflectance measurements are not sufficient to obtain reliable values of optical constants and film thickness. For this reason the envelope method based on determination maxima and minima from transmitted and reflected light may gives wrong optical constants [2].

However wavelength positions for maximum reflection do not change due to surface and bulk scattering. Therefore if refractive index *n* is known then from elementary optics we can use well known formula for thickness d determination:

(6)

where *l _{1}* and

*l*denote adjacent maxima position in reflectance spectrum.

_{2}The aim of this work is deeper analysis of optical properties of diffusive polymer PPI films described earlier in [15]

**Experimental method**

The PPI films were obtained by polycondensation of two components (TPA) and (PPDA) in vacuum system using the CVD method described in [16]. The samples exhibited slight volume scattering. It caused gentle yellow color of PPI samples observed in diffuse reflectance.

The spectral dependence for optical indices and thickness of films were determined by the use Woollam M2000 spectroscopic ellipsometer [17]. For PPI samples the ellipsometric measurements were performed in wavelength range 400-1000 nm. The samples were measured for incidence angles - 70º and 75º. For data analysis, all angular spectra were fitted simultaneously.

The Debye scattering in PPI films were studied by the using small angles light scattering method (SALS). The SALS represents an angular distribution of scattered intensity as a function of scattering angle *q** _{s}* [18].

SALS measurements were carried out with an automatic scatterometer. It consisted of the 635 nm laser diode as the light source. The CCD ruler with 512 diode elements was applied to scattered light detection. The resolution angle per pixel was equal 0.06 what was suitable value for that type of measurements. The scattering angles *q** _{s}* ranged from 1

^{o}to 11°. The measurements were done in the plain of light incidence.

**Result and discussion**

In goal to estimate of share of scattering caused by the surface roughness s of the PPI films the formula (2) was applied. The roughness s were determined by atomic force microscope Topometrix 2000 yielding roughness values of the PPI films lower than 7.5 nm. Thus, the contribution of the surface scattering to the total diffuse reflectance, calculated according to formula (2), is equal to 2% and 0.4% for 600 nm and 1000 nm, respectively. Thus the surface scattering was neglected in the further consideration.

As well from AFM study the thickness of PPI films was found. In this work the thickness *d* was determined. For two PPI samples the values thickness and roughness of films are presented in column 2 and 3of Table 1 respectively.

Optical studies comprised were divided on investigations using specular part of reflectance (ellipsometry) and non-directional scattering (integrating sphere and SALS methods).

In goal to determine optical constants and films thickness the ellipsometric measurements by the use M2000 ellipsometer were performed. In Fig. 1 the spectral dependence of ellipsometric angles *Y* and *D* for sample 2 from Table 1 is presented. As well in Fig. 1 the theoretical dependences of angles *Y* and *D* determined from the Kramers- Krönig relations have been shown [19] . The Kramers–Krönig relations are mathematical relations, connecting the real *n *and imaginary *k *parts of complex refractive index regarded as analytic function in studied reflectance and ellipsometric angles spectra.

In Fig. 2 dispersive relations for *n* and *k* for sample no. 2 determined from ellipsometric measurements in spectral range 400 to 1000 nm have been shown. Values of *n* for light wavelength = 633 nm and as well thickness of films are presented in column 5 and 4 in Table 1 respectively. The values of refractive index of PPI layers are substantially higher than for others polymers used in optoelectronics [20,21]. As may be seen in Table 1, thickness values of PPI layer determined from ellipsometric study are slight higher but similar to values obtained from AFM measurements.

For light wavelengths smaller than 500 nm the abnormal dispersion of *n* and *k * occurs. In this range the Tauc-Lorentz derived model may be applied [22]. However optical properties in shorter wavelength range are out of scope of this work.

Ellipsometric study also allows determining the depolarization coefficient. Its defines relative contribution of incoherent light intensity in specular reflection from a sample. The depolarization of reflected light is caused by two factors: first – by thickness inhomogeneity of thin polymer films and second - due to variation of bulk refractive index [3, 5]. The spectral dependence of depolarization coefficient. for two angles: 70^{0} and 75^{0} for sample no. 2 are presented in Fig. 3. As may be noticed depolarization coefficients determined for PPI samples are smaller for lower incidence angles.

The light intensity coming from Rayleigh scattering was collected by integrating sphere [23, 24]. The scattering coefficients *s _{v}(*

*l)*were determined from total and specular transmittance. The light intensity transmitted trough thin film sample is given by:

(7)

(8)

Where *I _{0}*, is the incident light intensity,

*I*are diffuse and specular light intensities transmitted trough sample,

_{diff}, I_{spec}*R*– is the reflection coefficient,

*a=4*

*p*

*k*

*/l*

*is the absorption coefficient. From formulas (6) and (7) we obtain:*

(9)

According to formula (9) scattering coefficient *s _{v}* should be linear function of

*1/*

*l*. This dependence is fulfilled for all samples presented in Table 1. It has been presented in Fig 4. It shows the ratio of

^{4}*ln(I*divided by the sample thickness versus (1/wavelength)^4 within wavelength range 500 to 800 nm.

_{diff}/I_{spec})As may be noticed the absorption in studied PPI layers is small for wavelengths longer than 600 nm but Rayleigh scattering is still large so it mainly influences on effective loss function in PPI films. For l > 600 nm the absorption practically disappears and light losses come mainly from elastic scatterings. Applying formula (9) the values of scattering coefficient *s _{v}* at 633 nm have been determined. The values of

*s*are presented in column 6 of Table 1.

_{v}As results from Table 1 for thinner PPI films the scattering coefficients *s _{v }* is smaller. It suggests increase of scattering centers numbers when time of deposition is longer.

In goal to characterizing light scattering produced by refractive index variation the SALS study was done. In Fig. 5 relative light intensities obtained from SALS measurements have been shown. The light scattering in longer spatial wavelengths originated from refractive index variation of in PPI films. The Debye model was adopted to the experimental data. Fitted values of correlation lengths *T *and scattering factors *B* are presented in column 7 and 8 in Table 1 respectively.

**Conclusion**

The PPI diffusive layers demonstrate two types of scattering Rayleigh and Debeye-Bueche scattering as many other polymer films. As may be noticed, the PPI films presented in this work exhibit weak optical scattering within visible spectal range. For better clearity optical properties of PPI layers have been discussed above absorption edge i.e in spectral wavelength range 500-1000 nm. This limitation allowed to simplify substantially the description of optical phenomena occurring in presented samples.

The Rayleigh scattering were found from the integrating sphere measurements. This technique is particularly recommended when light beam passing by sample is scattered in broad solid angle. If the refractive index variation of bulk of a film occurs the scattered light propagates near specular direction around small solid angle. The light scattering due to fluctuation of refractive index is spread near specular direction so well it can’t be measured by integrating sphere. Therefore the results obtained from sphere are only related to Rayleigh scattering.

If autocorrelation length describing refractive index variation is larger than light wavelength then scattering can be describe by the Debye-Bueche model and in that case SALS measurements give reliable results for this kind of scattering. Combined techniques such: ellipsometry, total and diffuse transmittance/reflectance measurements and SALS study allowed to determine optical constants and film thickness as well many others parameters describing surface and volume light scattering.

** **

**References**

[1] C. Bosshard,M. Canva,L. Dalton, Kwang-Sup Lee, Polymers for photonics applications I, Springer – Verlag 2002.

[2] J.C. Martinez-Anton., *Appl. Opt.* 1 September (2000), Vol.39, No. 25

[3] R.M.A Azzam, N.M. Bashara, Ellipsometry and polarized light, North-Holland, Amsterdam 1987.

[4]. D. Fu, W. Choi, Y. Sung, S. Oh, Z. Yaqoob, Y. Park, R. R. Dasari, and M. S. Feld, Ultraviolet refractometry using field-based light scattering spectroscopy, Opt. Express 17, (2009). 18878–18886

[5]. K. Koikea, T. Araki, Y. Koikea, Influence of dielectric fluctuation on light-scattering properties of random copolymers in bulk, Polymer 55 (2014) 2697–2703.

[6] H. R. Kricheldorf (Ed.) Polycondensation 2002, progress in step-growth polymerization and structure, Willey-VCH, 2003.

[7]. H. C. d. Hulst, Light Scattering by Small Particles, Structure of Matter Series Wiley, 1957.

[8] A.R. Stokes, The Theory of Optical Properties of Inhomogeneous Materials, E. &F.N. Spon Limited, London 1963.

[9] C. C. Han, , A. Z Akcasu, Plane Waves, Scattering, and Polymers Wiley & Sons 2011

[10] J.C. Stover., Optical Scattering. Measurements and Analysis –second edition. SPIE 1995

[11] M. Bennet, L Mattson., Introduction to surface roughness and scattering, Optical Society of __America__ Washington DC 1999.

[12] C.F. Bohren, D.R. Huffman, Absorption and Scattering of Light by Small Particles, Wiley, New York, 1983.

[13] P. Debye, , A. M Bueche, J. Appl. Phys., 120 (1949) 518.

[14] J. Zhou. , J. Sheng, J. Polymer Vol. 38 No. 15 (1997) 3727-3731.

[15] J. Jaglarz J., Cisowski J., Jurusik J., Domański M., Rev.Adv.Mater.Sci., 8 (2004) 82.

[16] B. Jarząbek, J. Weszka, M. Domański, J. Jurusik,J. Cisowski, J. Non-Cryst.Sol.*,* 354 (2008) 856-862.

[17] J.A. Woollam, P.G. Synder, Fundamentals and applications of variable angle spectroscopic ellipsometry, Mater Sci. Eng., B5 (1990) 279.

[18] F. Scheffold, R. Cerbino, Colloid & Interface Science, 12 (2007) 50–57.

[19] V. Lucarini, J. Jarkko, K. Saarinen, K. Peiponen, E. M. Vartiainen,: Kramers-Kronig relations in Optical Materials Research, Springer, Heidelberg, 2005.

[20] www.texloc.com/closet/cl_refractiveindex.html

[21] J. Jaglarz, T. Wagner, J. Cisowski, J. Sanetra, * *Opt. Mat*.*, 29 (2007) 908–912.

[22] J. Tauc in Fundamentals of Semiconductors by P. Y. Yu and M. Cardona (Springer, New York, 1996.

[23] A. Arecchi, K.A Carr, Guide to integrating sphere theory and application, Labsphere Technical Guide 1997.

[24] J. Jung, Yo. Park, Spectro-angular light scattering measurements of individual microscopic objects, Optics Express, 22 (2014) 4108-4114

**Descriptions of the table and figures **

** **

Table 1. Values of optical parameters of thin PPI films determined from optical studies described in text.

Fig. 1. Spectral dependence of ellipsometric angles *Y* and *D* for PPI for sample 2.

Fig. 2. Spectral dependence of *n* and *k* indices for PPI film (sample 2).

Fig. 3. Spectral dependence of depolarization coefficient in PPI film (sample 2).

Fig. 4. Log ratio I_{diff}/I_{spec} versus (1/wavelength)^4 for measured PPI samples.

Fig. 5. SALS normalized intensity vs. scattering vector. Experimental data (symbols) and fitted theoretical Debye’s model.

Tab-1

Sample |
n (l =nm) | thickness [mm] | roughness s [nm] | s[1/mm]
_{v }(l = 635 nm) |
a_{corr }[mm] |
B x 10^{2} [mm^{-3}] (l = 635 nm) |

1 | 2.84 | 327 | 7.5 | 0.246 | 2.71 | 8.96 |

2 | 2.63 | 223 | 5.7 | 0.215 | 2.89 | 7.19 |

3 | 2.69 | 189 | 6.4 | 0.183 | 2.97 | 5.43 |

4 | 2.81 | 55 | 6.3 | 0.139 | 3.29 | 4.57 |

Tab-1

Fig-1

Fig. 2

Fig-3

Fig. 4

Fig. 5