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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 16 — Jul. 30, 2012
  • pp: 18254–18267
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Analysis of reflective Mach-Zehnder interferometry for electro-optic characterization of poled polymer films in multilayer structures

Dong Hun Park and Warren N. Herman  »View Author Affiliations


Optics Express, Vol. 20, Issue 16, pp. 18254-18267 (2012)
http://dx.doi.org/10.1364/OE.20.018254


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Abstract

We consider the Mach-Zehnder interferometer (MZI) method that specifically uses a poled organic thin film as one of the reflective mirrors in order to characterize the two independent electro-optic tensor elements r 13 and r 33 . We discuss both a simple analysis based on a three-layer structure and a rigorous method including multiple reflection effects in a multilayer structure. In doing so, we find that the simple analysis of the reflective MZI method yields identical results to the reflection ellipsometric method (simple Teng-Man method), first introduced by Teng and Man as well as Shildkraut in 1990, when the ratio of r 13 to r 33 obtained from the MZI method is used in the analysis of the simple Teng-Man method. Error introduced by ignoring the multilayer nature of the sample structures in the MZI method is discussed and corrections are given for previous expressions in the literature for the simple analysis.

© 2012 OSA

1. Introduction

Nonlinear optical (NLO) polymers based on second-order nonlinearity have been widely studied because of their potential applications in electro-optic (EO) devices [1

1. W. N. Herman, S. R. Flom, and S. H. Foulger, eds., Organic Thin Films for Photonic Applications (ACS, Symposium Series 2010) Vol. 1039.

, 2

2. G. A. Lindsay and K. D. Singer, eds., Polymers for Second-Order Nonlinear Optics (ACS, Symposium Series 1995) Vol. 601.

]. Various techniques have been reported for characterization of EO coefficients of poled polymer thin films such as the reflection ellipsometric method [3

3. C. C. Teng and H. T. Man, “Simple reflection technique for measuring the electro‐optic coefficient of poled polymers,” Appl. Phys. Lett. 56(18), 1734–1736 (1990). [CrossRef]

7

7. D. H. Park, C. H. Lee, and W. N. Herman, “Analysis of multiple reflection effects in reflective measurements of electro-optic coefficients of poled polymers in multilayer structures,” Opt. Express 14(19), 8866–8884 (2006). [CrossRef] [PubMed]

], Fabry-Perot (FP) [8

8. C. A. Eldering, A. Knoesen, and S. T. Kowel, “Use of Fabry–Pérot devices for the characterization of polymeric electro‐optic films,” J. Appl. Phys. 69(6), 3676–3686 (1991). [CrossRef]

, 9

9. D. H. Park, J. Luo, A. K. Y. Jen, and W. N. Herman, “Simplified reflection Fabry-Perot method for determination of electro-optic coefficients of poled polymer thin films,” Polymers 3(3), 1310–1324 (2011). [CrossRef]

], attenuated total reflection (ATR) [10

10. D. Morichère, P.-A. Chollet, W. Fleming, M. Jurich, B. A. Smith, and J. D. Swalen, “Electro-optic effects in two tolane side-chain nonlinear-optical polymers: comparison between measured coefficients and second-harmonic generation,” J. Opt. Soc. Am. B 10(10), 1894–1900 (1993). [CrossRef]

12

12. D. H. Park, “Characterization of linear electro-optic effect of poled organic thin films”, Ph. D. Dissertation, University of Maryland, College Park, (2008).

], Michelson interferometer [13

13. H. Y. Zhang, X. H. He, Y. H. Shih, and S. H. Tang, “A new method for measuring the electro-optic coefficients with higher sensitivity and higher accuracy,” Opt. Commun. 86(6), 509–512 (1991). [CrossRef]

, 14

14. W. Shi, Y. J. Ding, X. Mu, X. Yin, and C. Fang, “Electro-optic and electromechanical properties of poled polymer thin films,” Appl. Phys. Lett. 79(23), 3749–3751 (2001). [CrossRef]

], and Mach-Zehnder interferometer (MZI) [14

14. W. Shi, Y. J. Ding, X. Mu, X. Yin, and C. Fang, “Electro-optic and electromechanical properties of poled polymer thin films,” Appl. Phys. Lett. 79(23), 3749–3751 (2001). [CrossRef]

19

19. C. Greenlee, A. Guilmo, A. Opadeyi, R. Himmelhuber, R. A. Norwood, M. Fallahi, J. Luo, S. Huang, X.-H. Zhou, A. K. Y. Jen, and N. Peyghambarian, “Mach–Zehnder interferometry method for decoupling electro-optic and piezoelectric effects in poled polymer films,” Appl. Phys. Lett. 97(4), 041109 (2010). [CrossRef]

]. Compared to the reflection ellipsometric method first introduced by Teng and Man [3

3. C. C. Teng and H. T. Man, “Simple reflection technique for measuring the electro‐optic coefficient of poled polymers,” Appl. Phys. Lett. 56(18), 1734–1736 (1990). [CrossRef]

] as well as Shildkraut [4

4. J. S. Schildkraut, “Determination of the electro optic coefficient of a poled polymer film,” Appl. Opt. 29(19), 2839–2841 (1990). [CrossRef] [PubMed]

] (so called simple Teng-Man method), other methods such as FP, ATR, and MZI have the advantage of separate measurement of the EO coefficients r13 and r33. In addition, ATR is less susceptible to properties of transparent conducting oxide (TCO) because the TCO layer doesn’t affect the waveguide modes very much [12

12. D. H. Park, “Characterization of linear electro-optic effect of poled organic thin films”, Ph. D. Dissertation, University of Maryland, College Park, (2008).

]. The reflection-type MZI (MZIR) method for characterization of the second-order nonlinearities in a poled polymer thin film was first proposed by Norwood et al. [14

14. W. Shi, Y. J. Ding, X. Mu, X. Yin, and C. Fang, “Electro-optic and electromechanical properties of poled polymer thin films,” Appl. Phys. Lett. 79(23), 3749–3751 (2001). [CrossRef]

] in 1994, later analyzed by Shin et al. [18

18. M. J. Shin, H. R. Cho, S. H. Han, and J. W. Wu, “Analysis of a Mach-Zehnder interferometry measurement of the Pockels coefficients in a poled polymer film with a reflection configuration,” J. Appl. Phys. 83(4), 1848–1853 (1998). [CrossRef]

], and recently modified by Greenlee, et al. [19

19. C. Greenlee, A. Guilmo, A. Opadeyi, R. Himmelhuber, R. A. Norwood, M. Fallahi, J. Luo, S. Huang, X.-H. Zhou, A. K. Y. Jen, and N. Peyghambarian, “Mach–Zehnder interferometry method for decoupling electro-optic and piezoelectric effects in poled polymer films,” Appl. Phys. Lett. 97(4), 041109 (2010). [CrossRef]

], who added the ability to independently determine the piezoelectric contribution by flipping the reflective sample. Singer et al. [16

16. K. D. Singer, M. G. Kuzyk, W. R. Holland, J. E. Sohn, S. J. Lalama, R. B. Comizzoli, H. E. Katz, and M. L. Schilling, “Electro-optic phase modulation and optical second-harmonic generation in corona-poled polymer films,” Appl. Phys. Lett. 53(19), 1800–1802 (1988). [CrossRef]

] and Qiu, et al. [17

17. F. Qiu, K. Misawa, X. Cheng, A. Ueki, and T. Kobayashi, “Determination of complex tensor components of electro‐optic constants of dye‐doped polymer films with a Mach–Zehnder interferometer,” Appl. Phys. Lett. 65(13), 1605–1607 (1994). [CrossRef]

] demonstrated the method for the complex tensor components of EO coefficients with a MZI in a transmission scheme (MZIT). However, most analyses have used a simple analysis that ignores multiple reflections from a reflective layer, such as a TCO layer, and assumes a perfectly reflective metal electrode, in the MZIR case, or perfectly transmissive electrodes, in the MZIT case. Therefore, the simple analysis takes into account only optical properties of the EO materials, just as in the simple Teng-Man method, although the multilayer nature of the structures should be taken into account to obtain more reliable values for the EO coefficients [7

7. D. H. Park, C. H. Lee, and W. N. Herman, “Analysis of multiple reflection effects in reflective measurements of electro-optic coefficients of poled polymers in multilayer structures,” Opt. Express 14(19), 8866–8884 (2006). [CrossRef] [PubMed]

].

In this paper, we derive analytic expressions for the rigorous MZIR method to take into account multiple reflection effects and discuss relative errors that can be caused by using the simple MZIR analysis. We find that the results using the simple MZIR analysis are identical to the simple Teng-Man method provided the ratio of r13 to r33 obtained from the simple MZIR analysis is also used in the simple Teng-Man analysis, which requires an assumed value for this ratio (often 1/3). For the simple MZIR method, we also provide corrections to mathematical formulas for phase, effective EO coefficients, and the ratio of r13 to r33 given previously in the literature.

2. Theory

2.1 Rigorous Expressions for the rigorous MZIR model

The experimental setup for the MZIR method is shown in Fig. 1
Fig. 1 Schematic of a MZIR experimental setup. PD is a photo-detector. Laser is continuous-wave operating at a communication wavelength of 1.3 or 1.55 μm.
[14

14. W. Shi, Y. J. Ding, X. Mu, X. Yin, and C. Fang, “Electro-optic and electromechanical properties of poled polymer thin films,” Appl. Phys. Lett. 79(23), 3749–3751 (2001). [CrossRef]

, 18

18. M. J. Shin, H. R. Cho, S. H. Han, and J. W. Wu, “Analysis of a Mach-Zehnder interferometry measurement of the Pockels coefficients in a poled polymer film with a reflection configuration,” J. Appl. Phys. 83(4), 1848–1853 (1998). [CrossRef]

, 19

19. C. Greenlee, A. Guilmo, A. Opadeyi, R. Himmelhuber, R. A. Norwood, M. Fallahi, J. Luo, S. Huang, X.-H. Zhou, A. K. Y. Jen, and N. Peyghambarian, “Mach–Zehnder interferometry method for decoupling electro-optic and piezoelectric effects in poled polymer films,” Appl. Phys. Lett. 97(4), 041109 (2010). [CrossRef]

]. A thin glass plate in the reference arm is rotated to produce a phase shift relative to the laser light reflected from the multilayered sample (glass/TCO/NLO/gold). Interference intensity data without applying voltage to the sample can be collected from a photo-detector as a function of phase angle, Ωm, created by the rotating glass plate. The dc intensity is expressed as
Idc(m)=|E0|2|rmr1mt2mtgmt1mr2m|2,
(1)
where the subscript m represents s- or p- polarization and r1m(t1m) and r2m(t2m)are reflection (transmission) coefficients at the first and second beam splitters, respectively. The transmission coefficient of the thin rotating glass plate is defined by tgm. Assuming the two beam splitters are identical, that is, r1m=r2m and t1m=t2m, the dc intensity can be simplified to
Idc(m)=|r1mt1mE0|2|rmtgm|2=I04|rmtgm|2=Am+Bmsin2(ΨmΩm2),
(2)
where

Am=I04(|rm||tgm|)2,Bm=I0|rm||tgm|,I0=4|r1mt1mE0|2,rm=|rm|eiΨm,tgm=|tgm|eiΩm.
(3)

Application of a voltage to the sample changes the refractive indices and thickness of the sample, thus inducing a change in both the magnitude and the phase of the reflection coefficient rm. Similar to the dc intensity curve, the modulated intensity is collected from the lock-in amplifier under the application of AC voltage to the sample as a function of the phase Ωm created by the glass plate. To first order in V, the modulated intensity IM(m)is obtained by differentiating Idc(m) to get

IM(m)=δAm+δBmsin2(ΨmΩm2)+Bm2sin(ΨmΩm)δΨm.
(4)

From simple curve fitting of Idc(m) and IM(m), we can obtain A, B, δA, δB, and δΨ for s- and p- polarized light. We note that Eqs. (2) and (4) are quite similar to those in the rigorous Teng-Man method [7

7. D. H. Park, C. H. Lee, and W. N. Herman, “Analysis of multiple reflection effects in reflective measurements of electro-optic coefficients of poled polymers in multilayer structures,” Opt. Express 14(19), 8866–8884 (2006). [CrossRef] [PubMed]

]. We define B˜m=I0rmtgm*=Bmexp[i(ΨmΩm)] from which calculation of δB˜m/B˜mgives
δBmBm+iδΨm=δrmrm.
(5)
This formulation conveniently separates the modulation of the magnitude of the reflection coefficient from the modulation of the phase. In addition, the quantities on the left hand side are obtained from the experimental data as described above and the right hand side is calculated from theory as described below. The polymer material is usually spin-coated on a TCO-coated glass substrate. The first reflection off the air/glass interface and subsequent beams resulting from reflection of the first pass on its way out at the glass/air interface back into the polymer and out again are blocked. The reflection coefficients in the multilayered structure can be obtained by iteration of the Airy formula [20

20. M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th ed. (Cambridge University Press, 1999).

]. The s- and p- reflection coefficients for multilayered structures as shown in Fig. 2(a)
Fig. 2 Multilayered structures in (a) the rigorous model and (b) the simple model.
have the form
r=r23+r^34e2iβ3d31+r23r^34e2iβ3d3r^34=r34+r^45e2iβ4d41+r34r^45e2iβ4d4r^45=r45+r56e2iβ5d51+r45r56e2iβ5d5,
(6)
where we have omitted the s and p subscripts and the reflection coefficient at the single interface is given by
rjk=ZkZjZk+Zj,
(7)
with the s- and p- wave impedances (normalized to free space impedance) of each layer given by
Zs=1n˜o2N2,Zp=1n˜o1(Nn˜e)2,
(8)
where n˜o(=n˜s=no+iκo) and n˜e(=ne+iκe) are the complex ordinary and extraordinary indices of refraction and Nsinθ=n˜osinα˜s=n˜psinα˜p(Snell’s law), where α˜s and α˜p are the complex internal angles of incidence at s- and p- polarizations, respectively. The propagation constants are βs=k0n˜ocos(α˜s), βp=k0n˜pcos(α˜p) and n˜p is given by [20

20. M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th ed. (Cambridge University Press, 1999).

]

1n˜p2=cos2α˜pn˜o2+sin2α˜pn˜e2.
(9)

Performing a Taylor expansion to first order, the right hand side of Eq. (5) can be written in the form
δrsrs=1rsrsn˜oδn˜o+1rsrsdδd,δrprp=1rprpn˜oδn˜o+1rprpn˜eδn˜e+1rprpdδd.
(10)
The variation of the complex refractive index can be expressed as
δn˜j=12n˜j3(rj3+isj3)E3,
(11)
where E3=V/d, j = 1 or 3 (n˜1=n˜oandn˜3=n˜e), and sj3 is the imaginary part of the complex EO coefficient leading to the electro-chromic effect [7

7. D. H. Park, C. H. Lee, and W. N. Herman, “Analysis of multiple reflection effects in reflective measurements of electro-optic coefficients of poled polymers in multilayer structures,” Opt. Express 14(19), 8866–8884 (2006). [CrossRef] [PubMed]

]. The strain resulting from the converse piezoelectric effect is the same for both s- and p- polarization and is given by [21

21. J. F. Nye, Physical Properties of Crystals (Oxford University Press, 1957).

]
δdd=d33E3.
(12)
Note that Ref. 19

19. C. Greenlee, A. Guilmo, A. Opadeyi, R. Himmelhuber, R. A. Norwood, M. Fallahi, J. Luo, S. Huang, X.-H. Zhou, A. K. Y. Jen, and N. Peyghambarian, “Mach–Zehnder interferometry method for decoupling electro-optic and piezoelectric effects in poled polymer films,” Appl. Phys. Lett. 97(4), 041109 (2010). [CrossRef]

uses a different convention with a factor of 1/2 here.

Equation (10) can thus be written as a linear function of the real and imaginary parts of the EO coefficients in the form
δrsrs=H1(s)(r13+is13)+Hd(s)d33,δrprp=H1(p)(r13+is13)+H3(p)(r33+is33)+Hd(p)d33,
(13)
where
Hj(m)n˜j3V2d1rmrmn˜j,Hd(m)=V1rmrmd.
(14)
Explicit forms for the derivatives with respect to refractive indices can be found in Appendix C of Ref. 7

7. D. H. Park, C. H. Lee, and W. N. Herman, “Analysis of multiple reflection effects in reflective measurements of electro-optic coefficients of poled polymers in multilayer structures,” Opt. Express 14(19), 8866–8884 (2006). [CrossRef] [PubMed]

(Note that H3(s)=0.) and the derivatives with respect to thickness are given by
Hd(m)=V1rrd=2iVβ4r^45e2iβ4de2iβ3d3(1+r23r^34e2iβ3d3)(1+r34r^45e2iβ4d)[1rr23][1r^34r34],
(15)
where we have omitted m (s or p) in reflection coefficients and propagation constants.

As described in Ref. 19

19. C. Greenlee, A. Guilmo, A. Opadeyi, R. Himmelhuber, R. A. Norwood, M. Fallahi, J. Luo, S. Huang, X.-H. Zhou, A. K. Y. Jen, and N. Peyghambarian, “Mach–Zehnder interferometry method for decoupling electro-optic and piezoelectric effects in poled polymer films,” Appl. Phys. Lett. 97(4), 041109 (2010). [CrossRef]

, the converse piezoelectric coefficient d33 is determined by a separate measurement with the sample flipped and the back electrode acting as a mirror while the modulating voltage is applied. In this case, there is only modulation of the phase of the reflected beam given by
δψ=2k0cosθδd=2k0Vd33cosθ,
(16)
where k0=2π/λand θ is the angle of incidence. Measurement of this phase change then determines d33 to be used in all subsequent calculations. The details are given in Appendix A.

Equation (5) together with Eq. (13) and d33 from Eq. (16) can be written in the matrix forms
(δΨsIm[Hd(s)]d33δBsBsRe[Hd(s)]d33)=(Im[H1(s)]Re[H1(s)]Re[H1(s)]Im[H1(s)])(r13s13),
(17)
for s-polarization, and
(δΨpIm[Hd(p)]d33δBpBpRe[Hd(p)]d33)=(Im[H1(p)]Re[H1(p)]Re[H1(p)]Im[H1(p)])(r13s13)+(Im[H3(p)]Re[H3(p)]Re[H3(p)]Im[H3(p)])(r33s33).
(18)
for p-polarization. The left hand sides are obtained from the experimental data by fitting Eqs.  (2) and (4) and using d33 from Eq. (16). The H functions in general depend on the refractive index and thickness of the layers in the sample structure. First, Eq. (17) is solved for r13 and s13. Then these values are used in Eq. (18) to determine r33 and s33.

A single angle of incidence measurement at 45° is usually performed in MZ interferometry, but it is possible, though cumbersome, to take data at multiple angles as demonstrated in Ref. 18

18. M. J. Shin, H. R. Cho, S. H. Han, and J. W. Wu, “Analysis of a Mach-Zehnder interferometry measurement of the Pockels coefficients in a poled polymer film with a reflection configuration,” J. Appl. Phys. 83(4), 1848–1853 (1998). [CrossRef]

. This analysis can also be applied to the MZIT method that uses a poled organic thin film as a rotating/fixed plate in a transmission scheme, in which case a multi-angle data set can be obtained by rotating the sample [17

17. F. Qiu, K. Misawa, X. Cheng, A. Ueki, and T. Kobayashi, “Determination of complex tensor components of electro‐optic constants of dye‐doped polymer films with a Mach–Zehnder interferometer,” Appl. Phys. Lett. 65(13), 1605–1607 (1994). [CrossRef]

]. Multiple angle data sets can be analyzed by implementing least square fitting as described in Ref. 7

7. D. H. Park, C. H. Lee, and W. N. Herman, “Analysis of multiple reflection effects in reflective measurements of electro-optic coefficients of poled polymers in multilayer structures,” Opt. Express 14(19), 8866–8884 (2006). [CrossRef] [PubMed]

. For the rigorous analysis, it is necessary to know detailed sample information such as complex refractive indices and thicknesses for each layer, as illustrated in Fig. 2(a).

2.2 Simple MZIR Model

Typically, in the simple model, the properties of the TCO layer and multilayered structure of the sample are ignored as shown in Fig. 2(b). The reflectance at the air/film interface is ignored and the metal electrode is considered as a perfect electric conductor (PEC). Assuming the anisotropic refractive indices are complex numbers and ignoring reflection at the air/NLO film interface, the reflection coefficients from the sample (air/NLO/PEC) are given by [20

20. M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th ed. (Cambridge University Press, 1999).

]
rs=ei2βsd,rp=ei2βpd,
(19)
where d is the NLO film thickness.

The modulated phase and reflectivity in the simple MZIR method can be derived based on Eqs. (17) and (18). We use a lowercase h, ψ, and b for the simple model instead of H, Ψ, and B, respectively, which include the effect of all layers. From Eq. (14), we obtain
h1(s)=ik0Vn˜o4n˜o2N2,h1(p)=ik0Vn˜o31(Nn˜e)2,h3(p)=ik0Vn˜on˜eN2n˜e2N2
(20)
and
hd(s)=2ik0Vn˜o2N2,hd(p)=2ik0Vn˜o1(Nn˜e)2
(21)
using Snell’s law, Nsinθ=n˜osinα˜s=n˜psinα˜p. Then, the modulated phase and reflectivity, δψm and δb/b can be obtained using Eqs. (17) and (18).

Outside the absorption band where κo,e1, we can approximate Eqs. (20) and (21) to obtain expressions for δψm and δb/b in the form
δψs2k0Vno2N2d33=k0Vno4no2N2r13,δbsbs=k0Vno4no2N2s13,
(22)
and
δψp2k0Vno1(Nne)2d33=k0Vno31(Nne)2r13k0VnoneN2ne2N2r33,δbpbp=k0Vno31(Nne)2s13+k0VnoneN2ne2N2s33.
(23)
Thus, in the simple model, reflectivity modulation depends only on sj3 and phase modulation only on rj3. In general, this is not true when all layers of the sample structure are taken into account. δψs and δψp are obtained from the experimental data, then Eq. (22) is first solved for r13 using the separately determined value of d33 as described above, and then the result used in Eq. (23) to obtain r33.

Subtracting δψp from δψs to compare with δψsp of the simple Teng-Man method, we get
δψsp2k0Vno(1(Nno)21(Nne)2)d33=k0V[(no4no2N2no3nene2N2)r13noneN2ne2N2r33].
(24)
For the isotropic case (nonen), Eq. (24) reduces to
δψsp=n2N2n2N2(r33r13)k0V,
(25)
which is independent of d33. We note that Eq. (24) (with d33 = 0) and Eq. (25) agree with those in Refs. 4

4. J. S. Schildkraut, “Determination of the electro optic coefficient of a poled polymer film,” Appl. Opt. 29(19), 2839–2841 (1990). [CrossRef] [PubMed]

7

7. D. H. Park, C. H. Lee, and W. N. Herman, “Analysis of multiple reflection effects in reflective measurements of electro-optic coefficients of poled polymers in multilayer structures,” Opt. Express 14(19), 8866–8884 (2006). [CrossRef] [PubMed]

. It is necessary to assume a value for the ratio of r13 to r33 in the simple Teng-Man method and it is apparent that when the ratio is obtained from the simple MZIR method using Eqs. (22) and (23), r33 by the simple Teng-Man method will be same as that obtained from simple analysis of MZIR data.

2.3 Comparison to previous MZIR simple model equations

For the case that sj3 = 0, we derive the expressions in Eqs. (22) and (23) for the voltage induced phase change by differentiation of βm=k0nmcos(αm) to get
δψm=δ(2βmd)=2k0[dcosαmδnm+nmcosαmδd],
(26)
where we have made use of δαm=tanαm(δnm/nm), which follows from Snell’s law. These are equivalent to the corresponding equations above.

The first term in Eq. (26) agrees with that given in Norwood, et al. [14

14. W. Shi, Y. J. Ding, X. Mu, X. Yin, and C. Fang, “Electro-optic and electromechanical properties of poled polymer thin films,” Appl. Phys. Lett. 79(23), 3749–3751 (2001). [CrossRef]

], but the corresponding term in Ref. 19

19. C. Greenlee, A. Guilmo, A. Opadeyi, R. Himmelhuber, R. A. Norwood, M. Fallahi, J. Luo, S. Huang, X.-H. Zhou, A. K. Y. Jen, and N. Peyghambarian, “Mach–Zehnder interferometry method for decoupling electro-optic and piezoelectric effects in poled polymer films,” Appl. Phys. Lett. 97(4), 041109 (2010). [CrossRef]

, which was drawn from Ref. 18

18. M. J. Shin, H. R. Cho, S. H. Han, and J. W. Wu, “Analysis of a Mach-Zehnder interferometry measurement of the Pockels coefficients in a poled polymer film with a reflection configuration,” J. Appl. Phys. 83(4), 1848–1853 (1998). [CrossRef]

, is not correct. The origin of the error in Ref. 18

18. M. J. Shin, H. R. Cho, S. H. Han, and J. W. Wu, “Analysis of a Mach-Zehnder interferometry measurement of the Pockels coefficients in a poled polymer film with a reflection configuration,” J. Appl. Phys. 83(4), 1848–1853 (1998). [CrossRef]

is speculative, but might be traced to Eq. (10) in Han and Wu [22

22. S. H. Han and J. W. Wu, “Single-beam polarization interferometry measurement of the linear electro-optic effect in poled polymer films with a reflection configuration,” J. Opt. Soc. Am. B 14(5), 1131–1137 (1997). [CrossRef]

]. The path length in Eq. (10) of Han and Wu corresponding to s in Ref. 18

18. M. J. Shin, H. R. Cho, S. H. Han, and J. W. Wu, “Analysis of a Mach-Zehnder interferometry measurement of the Pockels coefficients in a poled polymer film with a reflection configuration,” J. Appl. Phys. 83(4), 1848–1853 (1998). [CrossRef]

and sEO in Ref. 19

19. C. Greenlee, A. Guilmo, A. Opadeyi, R. Himmelhuber, R. A. Norwood, M. Fallahi, J. Luo, S. Huang, X.-H. Zhou, A. K. Y. Jen, and N. Peyghambarian, “Mach–Zehnder interferometry method for decoupling electro-optic and piezoelectric effects in poled polymer films,” Appl. Phys. Lett. 97(4), 041109 (2010). [CrossRef]

is not useful for phase determination because the separate parts of the expression correspond to different refractive indices; therefore the full expression cannot be multiplied by a single refractive index to provide a valid optical path length nds. However, the final expression [Eq. (20) in Ref. 22

22. S. H. Han and J. W. Wu, “Single-beam polarization interferometry measurement of the linear electro-optic effect in poled polymer films with a reflection configuration,” J. Opt. Soc. Am. B 14(5), 1131–1137 (1997). [CrossRef]

] for the modulated s-p phase difference is correct. The details of Eq. (26) from the point of view of diagrammatic phase delays in the thin film are given in Appendix A.

3. Results: numerical estimation of errors from use of the simple analysis

We used anisotropic refractive indices (no, ne) = (1.73, 1.79) at the wavelength of 1320 nm and a 150 nm thick Abrisa® ITO/glass substrate (the ITO refractive index is about 1 + 0.2i at 1320 nm). The EO coefficients are assumed to be 100 pm/V and 300 pm/V for r13 and r33, respectively. First, modulated phases of s- and p- waves, δψs and δψp, are calculated from the modulated reflection coefficients using Eqs. (5) and (10). From the modulated phases, the EO coefficients by the simple model are obtained from Eqs. (22) and (23) ignoring the converse piezoelectric effects. Figure 3(a)
Fig. 3 Plot of the error in r13 (black) and r33 (red) by the simple MZIR method for the case of thick ITO. Error in r33 (blue dot) by the simple Teng-Man method is plotted for comparison. Anisotropic index of refraction (1.73, 1.79) was used for the NLO thin film. For a TCO layer, (a) a 150 nm thick Abrisa ITO substrate and (b) a 45 nm thick TFD ITO substrate were used for this simulation.
shows errors in the EO coefficients calculated by the simple model analysis based on Eqs. (22) and (23). It should be emphasized that the r33 by the simple model analysis of MZIR will be exactly same as that by the simple Teng-Man method when the ratio of r13 to r33 obtained from the MZIR method is used in the simple Teng-Man analysis. Figure 3(b) shows errors when a 45 nm thick ITO (Thin Film Devices, Inc., with refractive index about 1.2 + 0.15i at 1320 nm) was used to compare with a thicker Abrisa ITO. It shows much smaller error in r33 compared with the Abrisa ITO because of much lower reflectivity from the ITO layer that reduces overall multiple reflection effects.

Figure 4
Fig. 4 Ratio of r33 to r13 for the case of 150 nm thick Abrisa ITO and 45 nm thick TFD ITO at the wavelength of 1320 nm resulting from simple analysis of MZIR when the correct value is 3.
shows the simulated ratio of r33 to r13 as a function of thickness of the nonlinear polymer for the case of thick ITO (Abrisa) and thin ITO (TFD). When the thickness is less than ~0.2 μm, the ratios for both cases are larger than 10, which is much higher than the correct value of 3 that was assumed in the simulation. For the thickness larger than ~0.2 μm, the simulated ratio for the case of Abrisa ITO shows a large cyclic variation from approximately 0 to 8, as expected in the r33 and r13 from Fig. 3(a), whereas the ratio for the TFD ITO is much smaller than that for the Abrisa ITO because of lower absorption of the TFD ITO at the operating wavelength of 1320 nm. Nevertheless, even with the less reflective TFD ITO, values of the ratio r33/r13extracted from experimental results using the simple models can vary ± 33% from the correct value. These considerations point out the difficulty in obtaining reliable r-coefficient ratios if the multilayer nature of the sample is neglected.

4. Conclusions

We have also presented mathematical expressions for phase modulation including converse piezoelectric effects as well as for the effective EO coefficient reff(p) in the simple model. These provide corrections to earlier expressions in the literature. If the light reflected once through the sample is detected and other reflections are blocked, thick films/crystals such as LiNbO3 can be analyzed using the simple MZIR method because the thickness is large enough to avoid multiple reflections. However, it is not generally possible to avoid multiple reflection effects in a thin NLO film, because reflections from interfaces in multilayer structures overlap. We have also discussed the relative error caused by ignoring multiple reflection effects. Depending on the optical properties of NLO and TCO such as refractive index and thickness, the error in using the simple model can be large (>100%) in both r13 and r33 and oscillates from positive and negative, as can be seen in the simple Teng-Man method as discussed in Ref. 7

7. D. H. Park, C. H. Lee, and W. N. Herman, “Analysis of multiple reflection effects in reflective measurements of electro-optic coefficients of poled polymers in multilayer structures,” Opt. Express 14(19), 8866–8884 (2006). [CrossRef] [PubMed]

.

Appendix A: calculation of the modulated phase term for reflection from a thin slab

To further justify the simple model equations discussed above compared to Refs. 18

18. M. J. Shin, H. R. Cho, S. H. Han, and J. W. Wu, “Analysis of a Mach-Zehnder interferometry measurement of the Pockels coefficients in a poled polymer film with a reflection configuration,” J. Appl. Phys. 83(4), 1848–1853 (1998). [CrossRef]

and 19

19. C. Greenlee, A. Guilmo, A. Opadeyi, R. Himmelhuber, R. A. Norwood, M. Fallahi, J. Luo, S. Huang, X.-H. Zhou, A. K. Y. Jen, and N. Peyghambarian, “Mach–Zehnder interferometry method for decoupling electro-optic and piezoelectric effects in poled polymer films,” Appl. Phys. Lett. 97(4), 041109 (2010). [CrossRef]

, here we follow the diagrammatic approach in Ref. 18

18. M. J. Shin, H. R. Cho, S. H. Han, and J. W. Wu, “Analysis of a Mach-Zehnder interferometry measurement of the Pockels coefficients in a poled polymer film with a reflection configuration,” J. Appl. Phys. 83(4), 1848–1853 (1998). [CrossRef]

and derive the modulated phase term for reflection in the three different cases illustrated in Fig. 5
Fig. 5 Reflection with a voltage bias across the film. (a) Back-side reflection from metal (gold) surface. Horizontal dashed line represents a voltage-induced thickness change. (b) Front-side reflection from a glass-film-gold with only an EO effect. (c) Front-side reflection with only a converse piezoelectric effect.
: (a) back-side reflection from gold for the converse piezoelectric effect, (b) front-side reflection through the NLO film with only an EO effect, and (c) front-side reflection through the NLO film with only a converse piezoelectric effect.

First, the phase change for reflection from the back side of the EO sample with a voltage modulation is considered. In the ray traces shown in Fig. 5(a), beams b1 and b2 are the reflections at the air-gold interfaces with and without a voltage-induced thickness change, respectively. Then, since the optical paths are always in air, the path length difference is simply 2AB¯AD¯ and the phase change is given by
δψ=k0nair(2AB¯AD¯)=2k0cosθδd,
(29)
where AB¯=δd/cosθand AD¯=2AB¯sin2θ. This agrees with our Eq. (16) above.

Second, consider the phase change from reflection in a glass-film-gold structure with only an EO effect. In Fig. 5(b), beam b1 is the reflection at the glass-film interface, beam b2 is the single pass trace through the film reflected off the gold, and beam b2 is the single pass trace after application of a voltage. The phase difference between beam b2 and the voltage-deviated beam b2 is given by
δψm=k0[nm(AB¯+BC¯)nm(AB¯+BC¯)ngDD¯]=k0{[nm(AB¯+BC¯)ngAD¯][nm(AB¯+BC¯)ngAD¯]}=2k0d(nmcosαmnmcosαm)=2k0dδ(nmcosαm),
(30)
where ngsinθ1=nmsinαm=nmsinαm (Snell’s law) has been used, along with the relations AB¯=BC¯=d/cosαmand AD¯=2dtanαmsinθ1. Here nm and αm are the refractive index and the internal angle of incidence with a voltage bias. This then leads directly to the first term on the right hand side in Eq. (26) upon use ofδαm=tanαm(δnm/nm).

Referring to Fig. 5(c), the induced phase change when the applied voltage only induces a change in thickness with no EO effect is similarly calculated to be
δψm=ψm(b2)ψm(b2)=k0[nm(AB¯+BC¯)nm(AB¯+BC¯)ngDD¯]=k0[nm(AB¯+BC¯)nm(AB¯+BC¯)ng(AD¯AD¯)]=k0[nm(2dcosαm)nm(2dcosαm)ng(2dtanαmsinθ12dtanαmsinθ1)]=2k0nmcosαmδd,
(31)
which agrees with the second term on the right in Eq. (26). One can derive the same result assuming the thickness change δdis all at the top of the film rather than at the bottom.

Appendix B: Effective EO coefficient for p-polarization

Here we derive the effective electro-optic coefficient appropriate for p-polarized light propagating at non-normal incidence through a poled polymer film. Electro-optic coefficients are typically defined in connection with the electric impermeability tensorB=ε01 [23

23. I. P. Kaminow, An Introduction to Electrooptic Devices (Academic Press Inc., 1974).

]. For the case of a poled polymer, ε=ε0diag(no2no2ne2), so
B=diag[1no21no21ne2].
(32)
To first order in the applied field, Ek, electro-optically induced changes in the refractive index are represented by δBpq=rpq,kEk, where (k, p, q = 1, 2, 3) [23

23. I. P. Kaminow, An Introduction to Electrooptic Devices (Academic Press Inc., 1974).

]. The linear electro-optic coefficients rpq,k are symmetric in the first two indices and usually written using reduced 6 × 3 tensor notation rsk, where pqsaccording to (11→1, 22→2, 33→3, 23→4, 31→5, 12→6). Thus, corresponding to Eq. (32), we have the usual δ(1/no2)=δB11=r13E3 and δ(1/ne2)=δB33=r33E3 when the applied field is in the z-direction only.

Expressions for effective electro-optic coefficients for s-and p- polarizations are desired in the form
δ(1nm2)=reff(m)E3,
(33)
recalling that m = s or p. Because ns=no, the case of s-polarization is straightforward and reff(s)=r13. For the case of p-polarization, we recall the well-known expression [24

24. A. Yariv and P. Yeh, Optical Waves in Crystals: Propagation and Control of Laser Radiation (John Wiley & Sons, 2003).

] resulting from Maxwell’s equations for a plane electric wave Eωei(krωt)propagating in the direction k^with constant Eω through an anisotropic material,
n2[Eωk^(Eωk^)]1ε0Dω=0,
(34)
where Dω is the electric displacement vector and n is the refractive index appropriate to the propagation direction and polarization. In the absence of nonlinear effects, Dω=εEω. As is well-known, Dωk^=0, but in general Eωk^0. If we insertEω=ε1Dω=(1/ε0)BDω, define u^ to be a unit vector in the direction of Dω, and then take the dot product of Eq. (34) with u^, we get
u^Bu^=1n2,
(35)
having used u^k^=0. Note that the substitution u^=(1/n)(xyz)gives the well-known equation for the index ellipsoid. For p-polarization, we have k^p=(sinαp0cosαp) and u^p=(cosαp0sinαp), which when inserted in Eq. (35) together with Eq. (32) gives Eq. (9).

For the case of a bulk material when the direction of propagation is fixed during the application of an electric field so that αp is constant, we have
δ(1np2)=u^pδBu^p=cos2αpδB11+sin2αpδB33(αpfixed).
(36)
Thus,
[reff(p)]αp=constant=r13cos2αp+r33sin2αp.
(37)
This is the expression used in Refs. 8

8. C. A. Eldering, A. Knoesen, and S. T. Kowel, “Use of Fabry–Pérot devices for the characterization of polymeric electro‐optic films,” J. Appl. Phys. 69(6), 3676–3686 (1991). [CrossRef]

and 17

17. F. Qiu, K. Misawa, X. Cheng, A. Ueki, and T. Kobayashi, “Determination of complex tensor components of electro‐optic constants of dye‐doped polymer films with a Mach–Zehnder interferometer,” Appl. Phys. Lett. 65(13), 1605–1607 (1994). [CrossRef]

, but is not appropriate for a finite thickness film when the external angle of incidence is fixed under the application of an applied field because a change in the refractive index, by Snell’s law, changes the internal angle αp and thus u^p. Consequently, for both Teng-Man and MZIR we have
δ(1np2)=δu^pBu^p+u^pδBu^p+u^pBδu^p=no2np2[r13cos2αp+r33sin2αp]E3,
(38)
having used δαp=(1/2)np2tanαpδ(1/np2), which follows from Snell’s law, together with (1/ne21/no2)np2sin2αp=1np2/no2, which follows from Eq. (9). From this result, we get the appropriate expression for reff(p)given in Eq. (27).

The MZIR analysis used in Ref. 19

19. C. Greenlee, A. Guilmo, A. Opadeyi, R. Himmelhuber, R. A. Norwood, M. Fallahi, J. Luo, S. Huang, X.-H. Zhou, A. K. Y. Jen, and N. Peyghambarian, “Mach–Zehnder interferometry method for decoupling electro-optic and piezoelectric effects in poled polymer films,” Appl. Phys. Lett. 97(4), 041109 (2010). [CrossRef]

follows that of Ref. 14

14. W. Shi, Y. J. Ding, X. Mu, X. Yin, and C. Fang, “Electro-optic and electromechanical properties of poled polymer thin films,” Appl. Phys. Lett. 79(23), 3749–3751 (2001). [CrossRef]

and relies on yet a different expression for reff(p)that was derived from the nonlinear polarization, PωNL, where PωjNL=ε0nj4rj3EωjE3, (j=1,3) for p-polarization [25

25. For discussion of the relationship between EO coefficients and nonlinear polarization, see, e.gK. D. Singer, M. G. Kuzyk, and J. E. Sohn, “Second-order nonlinear-optical processes in orientationally ordered materials: relationship between molecular and macroscopic properties,” J. Opt. Soc. Am. B 4(6), 968–976 (1987).

]. To examine this approach, we substitute Dω=εEω+PωNL in Eq. (34) and take the dot product of the resulting equation with e^p, where Eω=Eωe^pto get
np2[e^pEωe^pk^(Eωk^)]1ε0e^pεEω=1ε0e^pPωNL,
(39)
where e^p=[cos(αpγ)0sin(αpγ)] and γ is the walk-off angle [26

26. W. N. Herman and L. M. Hayden, “Maker fringes revisited: second-harmonic generation from birefringent or absorbing materials,” J. Opt. Soc. Am. B 12(3), 416–427 (1995). [CrossRef]

] between Dω and Eω, i.e., DωEω=DωEωcosγ. Separating out the zero-applied field contribution in Eq. (39), we find
δ(np2)=1ε0e^pPωNLEωcos2γ=[no4r13cos2(αpγ)cos2γ+ne4r33sin2(αpγ)cos2γ]E3,
(40)
which, after comparison with δnp=12np3reff(p)E3, implies
[reff(p)]αp=constant=1np4[no4r13cos2(αpγ)cos2γ+ne4r33sin2(αpγ)cos2γ],
(41)
noting that αp is constant in this derivation. If one ignores the walk-off angle, as done in Ref. 14

14. W. Shi, Y. J. Ding, X. Mu, X. Yin, and C. Fang, “Electro-optic and electromechanical properties of poled polymer thin films,” Appl. Phys. Lett. 79(23), 3749–3751 (2001). [CrossRef]

and puts γ=0in this expression, the result agrees with that given in Ref. 14

14. W. Shi, Y. J. Ding, X. Mu, X. Yin, and C. Fang, “Electro-optic and electromechanical properties of poled polymer thin films,” Appl. Phys. Lett. 79(23), 3749–3751 (2001). [CrossRef]

. However, ignoring the walk-off angle implies nonenp. Finally we note that in Appendix A of Ref. 26

26. W. N. Herman and L. M. Hayden, “Maker fringes revisited: second-harmonic generation from birefringent or absorbing materials,” J. Opt. Soc. Am. B 12(3), 416–427 (1995). [CrossRef]

it was discussed that the relations
cos(αpγ)=[npno]2cosγcosαpandsin(αpγ)=[npne]2cosγsinαp,
(42)
follow directly from np2[e^pk^(e^pk^)](1/ε0)εe^p=0. These relations turn Eq. (40) into Eq. (37), so the approach using the nonlinear polarization is consistent with using the electric impermeability tensor. But, again, this is for the case that the applied field does not have the effect of also changing the angle αp. The appropriate reff(p) for both the MZIR and Teng-Man sample structures is given by Eq. (27), which can also be derived starting from Eq. (38) with considerably more complication in algebra.

Acknowledgments

The authors gratefully acknowledge discussions with Professor Robert A. Norwood at the Optical Sciences Center of the University of Arizona.

References and links

1.

W. N. Herman, S. R. Flom, and S. H. Foulger, eds., Organic Thin Films for Photonic Applications (ACS, Symposium Series 2010) Vol. 1039.

2.

G. A. Lindsay and K. D. Singer, eds., Polymers for Second-Order Nonlinear Optics (ACS, Symposium Series 1995) Vol. 601.

3.

C. C. Teng and H. T. Man, “Simple reflection technique for measuring the electro‐optic coefficient of poled polymers,” Appl. Phys. Lett. 56(18), 1734–1736 (1990). [CrossRef]

4.

J. S. Schildkraut, “Determination of the electro optic coefficient of a poled polymer film,” Appl. Opt. 29(19), 2839–2841 (1990). [CrossRef] [PubMed]

5.

Y. Shuto and M. Amano, “Reflection measurement technique of electro‐optic coefficients in lithium niobate crystals and poled polymer films,” J. Appl. Phys. 77(9), 4632–4638 (1995). [CrossRef]

6.

F. Michelotti, G. Nicolao, F. Tesi, and M. Bertolotti, “On the measurement of the electro-optic properties of poled side-chain copolymer films with a modified Teng–Man technique,” Chem. Phys. 245(1-3), 311–326 (1999). [CrossRef]

7.

D. H. Park, C. H. Lee, and W. N. Herman, “Analysis of multiple reflection effects in reflective measurements of electro-optic coefficients of poled polymers in multilayer structures,” Opt. Express 14(19), 8866–8884 (2006). [CrossRef] [PubMed]

8.

C. A. Eldering, A. Knoesen, and S. T. Kowel, “Use of Fabry–Pérot devices for the characterization of polymeric electro‐optic films,” J. Appl. Phys. 69(6), 3676–3686 (1991). [CrossRef]

9.

D. H. Park, J. Luo, A. K. Y. Jen, and W. N. Herman, “Simplified reflection Fabry-Perot method for determination of electro-optic coefficients of poled polymer thin films,” Polymers 3(3), 1310–1324 (2011). [CrossRef]

10.

D. Morichère, P.-A. Chollet, W. Fleming, M. Jurich, B. A. Smith, and J. D. Swalen, “Electro-optic effects in two tolane side-chain nonlinear-optical polymers: comparison between measured coefficients and second-harmonic generation,” J. Opt. Soc. Am. B 10(10), 1894–1900 (1993). [CrossRef]

11.

A. Chen, V. Chuyanov, S. Garner, W. H. Steier, and L. R. Dalton, “Modified attenuated total reflection for the fast and routine electro-optic measurement of nonlinear optical polymer thin films,” Technical Digest of the Organic Thin Films for Photonics Applications, OSA Technical Digest Series 14, 158–160 (1997).

12.

D. H. Park, “Characterization of linear electro-optic effect of poled organic thin films”, Ph. D. Dissertation, University of Maryland, College Park, (2008).

13.

H. Y. Zhang, X. H. He, Y. H. Shih, and S. H. Tang, “A new method for measuring the electro-optic coefficients with higher sensitivity and higher accuracy,” Opt. Commun. 86(6), 509–512 (1991). [CrossRef]

14.

W. Shi, Y. J. Ding, X. Mu, X. Yin, and C. Fang, “Electro-optic and electromechanical properties of poled polymer thin films,” Appl. Phys. Lett. 79(23), 3749–3751 (2001). [CrossRef]

15.

R. A. Norwood, M. G. Kuzyk, and R. A. Keosian, “Electro‐optic tensor ratio determination of side‐chain copolymers with electro‐optic interferometry,” J. Appl. Phys. 75(4), 1869–1874 (1994). [CrossRef]

16.

K. D. Singer, M. G. Kuzyk, W. R. Holland, J. E. Sohn, S. J. Lalama, R. B. Comizzoli, H. E. Katz, and M. L. Schilling, “Electro-optic phase modulation and optical second-harmonic generation in corona-poled polymer films,” Appl. Phys. Lett. 53(19), 1800–1802 (1988). [CrossRef]

17.

F. Qiu, K. Misawa, X. Cheng, A. Ueki, and T. Kobayashi, “Determination of complex tensor components of electro‐optic constants of dye‐doped polymer films with a Mach–Zehnder interferometer,” Appl. Phys. Lett. 65(13), 1605–1607 (1994). [CrossRef]

18.

M. J. Shin, H. R. Cho, S. H. Han, and J. W. Wu, “Analysis of a Mach-Zehnder interferometry measurement of the Pockels coefficients in a poled polymer film with a reflection configuration,” J. Appl. Phys. 83(4), 1848–1853 (1998). [CrossRef]

19.

C. Greenlee, A. Guilmo, A. Opadeyi, R. Himmelhuber, R. A. Norwood, M. Fallahi, J. Luo, S. Huang, X.-H. Zhou, A. K. Y. Jen, and N. Peyghambarian, “Mach–Zehnder interferometry method for decoupling electro-optic and piezoelectric effects in poled polymer films,” Appl. Phys. Lett. 97(4), 041109 (2010). [CrossRef]

20.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th ed. (Cambridge University Press, 1999).

21.

J. F. Nye, Physical Properties of Crystals (Oxford University Press, 1957).

22.

S. H. Han and J. W. Wu, “Single-beam polarization interferometry measurement of the linear electro-optic effect in poled polymer films with a reflection configuration,” J. Opt. Soc. Am. B 14(5), 1131–1137 (1997). [CrossRef]

23.

I. P. Kaminow, An Introduction to Electrooptic Devices (Academic Press Inc., 1974).

24.

A. Yariv and P. Yeh, Optical Waves in Crystals: Propagation and Control of Laser Radiation (John Wiley & Sons, 2003).

25.

For discussion of the relationship between EO coefficients and nonlinear polarization, see, e.gK. D. Singer, M. G. Kuzyk, and J. E. Sohn, “Second-order nonlinear-optical processes in orientationally ordered materials: relationship between molecular and macroscopic properties,” J. Opt. Soc. Am. B 4(6), 968–976 (1987).

26.

W. N. Herman and L. M. Hayden, “Maker fringes revisited: second-harmonic generation from birefringent or absorbing materials,” J. Opt. Soc. Am. B 12(3), 416–427 (1995). [CrossRef]

OCIS Codes
(120.3180) Instrumentation, measurement, and metrology : Interferometry
(190.4710) Nonlinear optics : Optical nonlinearities in organic materials
(310.6860) Thin films : Thin films, optical properties

ToC Category:
Instrumentation, Measurement, and Metrology

History
Original Manuscript: May 14, 2012
Revised Manuscript: July 19, 2012
Manuscript Accepted: July 20, 2012
Published: July 25, 2012

Citation
Dong Hun Park and Warren N. Herman, "Analysis of reflective Mach-Zehnder interferometry for electro-optic characterization of poled polymer films in multilayer structures," Opt. Express 20, 18254-18267 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-16-18254


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References

  1. W. N. Herman, S. R. Flom, and S. H. Foulger, eds., Organic Thin Films for Photonic Applications (ACS, Symposium Series 2010) Vol. 1039.
  2. G. A. Lindsay and K. D. Singer, eds., Polymers for Second-Order Nonlinear Optics (ACS, Symposium Series 1995) Vol. 601.
  3. C. C. Teng and H. T. Man, “Simple reflection technique for measuring the electro‐optic coefficient of poled polymers,” Appl. Phys. Lett.56(18), 1734–1736 (1990). [CrossRef]
  4. J. S. Schildkraut, “Determination of the electro optic coefficient of a poled polymer film,” Appl. Opt.29(19), 2839–2841 (1990). [CrossRef] [PubMed]
  5. Y. Shuto and M. Amano, “Reflection measurement technique of electro‐optic coefficients in lithium niobate crystals and poled polymer films,” J. Appl. Phys.77(9), 4632–4638 (1995). [CrossRef]
  6. F. Michelotti, G. Nicolao, F. Tesi, and M. Bertolotti, “On the measurement of the electro-optic properties of poled side-chain copolymer films with a modified Teng–Man technique,” Chem. Phys.245(1-3), 311–326 (1999). [CrossRef]
  7. D. H. Park, C. H. Lee, and W. N. Herman, “Analysis of multiple reflection effects in reflective measurements of electro-optic coefficients of poled polymers in multilayer structures,” Opt. Express14(19), 8866–8884 (2006). [CrossRef] [PubMed]
  8. C. A. Eldering, A. Knoesen, and S. T. Kowel, “Use of Fabry–Pérot devices for the characterization of polymeric electro‐optic films,” J. Appl. Phys.69(6), 3676–3686 (1991). [CrossRef]
  9. D. H. Park, J. Luo, A. K. Y. Jen, and W. N. Herman, “Simplified reflection Fabry-Perot method for determination of electro-optic coefficients of poled polymer thin films,” Polymers3(3), 1310–1324 (2011). [CrossRef]
  10. D. Morichère, P.-A. Chollet, W. Fleming, M. Jurich, B. A. Smith, and J. D. Swalen, “Electro-optic effects in two tolane side-chain nonlinear-optical polymers: comparison between measured coefficients and second-harmonic generation,” J. Opt. Soc. Am. B10(10), 1894–1900 (1993). [CrossRef]
  11. A. Chen, V. Chuyanov, S. Garner, W. H. Steier, and L. R. Dalton, “Modified attenuated total reflection for the fast and routine electro-optic measurement of nonlinear optical polymer thin films,” Technical Digest of the Organic Thin Films for Photonics Applications, OSA Technical Digest Series14, 158–160 (1997).
  12. D. H. Park, “Characterization of linear electro-optic effect of poled organic thin films”, Ph. D. Dissertation, University of Maryland, College Park, (2008).
  13. H. Y. Zhang, X. H. He, Y. H. Shih, and S. H. Tang, “A new method for measuring the electro-optic coefficients with higher sensitivity and higher accuracy,” Opt. Commun.86(6), 509–512 (1991). [CrossRef]
  14. W. Shi, Y. J. Ding, X. Mu, X. Yin, and C. Fang, “Electro-optic and electromechanical properties of poled polymer thin films,” Appl. Phys. Lett.79(23), 3749–3751 (2001). [CrossRef]
  15. R. A. Norwood, M. G. Kuzyk, and R. A. Keosian, “Electro‐optic tensor ratio determination of side‐chain copolymers with electro‐optic interferometry,” J. Appl. Phys.75(4), 1869–1874 (1994). [CrossRef]
  16. K. D. Singer, M. G. Kuzyk, W. R. Holland, J. E. Sohn, S. J. Lalama, R. B. Comizzoli, H. E. Katz, and M. L. Schilling, “Electro-optic phase modulation and optical second-harmonic generation in corona-poled polymer films,” Appl. Phys. Lett.53(19), 1800–1802 (1988). [CrossRef]
  17. F. Qiu, K. Misawa, X. Cheng, A. Ueki, and T. Kobayashi, “Determination of complex tensor components of electro‐optic constants of dye‐doped polymer films with a Mach–Zehnder interferometer,” Appl. Phys. Lett.65(13), 1605–1607 (1994). [CrossRef]
  18. M. J. Shin, H. R. Cho, S. H. Han, and J. W. Wu, “Analysis of a Mach-Zehnder interferometry measurement of the Pockels coefficients in a poled polymer film with a reflection configuration,” J. Appl. Phys.83(4), 1848–1853 (1998). [CrossRef]
  19. C. Greenlee, A. Guilmo, A. Opadeyi, R. Himmelhuber, R. A. Norwood, M. Fallahi, J. Luo, S. Huang, X.-H. Zhou, A. K. Y. Jen, and N. Peyghambarian, “Mach–Zehnder interferometry method for decoupling electro-optic and piezoelectric effects in poled polymer films,” Appl. Phys. Lett.97(4), 041109 (2010). [CrossRef]
  20. M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th ed. (Cambridge University Press, 1999).
  21. J. F. Nye, Physical Properties of Crystals (Oxford University Press, 1957).
  22. S. H. Han and J. W. Wu, “Single-beam polarization interferometry measurement of the linear electro-optic effect in poled polymer films with a reflection configuration,” J. Opt. Soc. Am. B14(5), 1131–1137 (1997). [CrossRef]
  23. I. P. Kaminow, An Introduction to Electrooptic Devices (Academic Press Inc., 1974).
  24. A. Yariv and P. Yeh, Optical Waves in Crystals: Propagation and Control of Laser Radiation (John Wiley & Sons, 2003).
  25. For discussion of the relationship between EO coefficients and nonlinear polarization, see, e.gK. D. Singer, M. G. Kuzyk, and J. E. Sohn, “Second-order nonlinear-optical processes in orientationally ordered materials: relationship between molecular and macroscopic properties,” J. Opt. Soc. Am. B4(6), 968–976 (1987).
  26. W. N. Herman and L. M. Hayden, “Maker fringes revisited: second-harmonic generation from birefringent or absorbing materials,” J. Opt. Soc. Am. B12(3), 416–427 (1995). [CrossRef]

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