Measurement of phase retardation of optical multilayer films based on laser feedback system

Jiyang Li; Yanxiong Niu; Haisha Niu

doi:10.1364/OE.24.000409

1.Introduction

With the rapid development of optoelectronics industry, optical films are showing great application potential in fields as biomedical diagnostics [1], chemical analysis [2], femtosecond laser technology [3] and high-speed optical fiber communication [4]. The photometric properties like transmittance and reflectance used to be the main concern of optical films application. Recently, the phase property of optical films is becoming a novel research focus. The phase property of optical films can be utilized to make phase retarders, phase shift masks and dispersion compensators in optical systems instead of costly crystal wave plates such as an azo-polymer film in phase-shifting interferometer [5]. On the other hand, the inhomogeneous coating and the internal stress during the coating process will also cause phase property change in optical films. It will bring harmful effects upon systems demanding for polarization -preserving property such as influencing the pulse shape in femtosecond pulse laser [6]. Hence the precise measurement of the phase property in optical films is significant for optical systems design.

When a beam is reflected on the interface of an optical medium coated with optical films, because of different penetration depth [7], retardance will be induced between s- and p-polarization and the phase shift is caused. This retardance is called reflection-induced retardance and can be used to make reflection-induced phase retarders. In order to measure the reflection-induced retardance, Fresnel formulae are not suitable for calculating due to the characteristics of multilayer optical films. Babinet compensator can be used to compensate the phase shift directly, but resolution is ensured at the cost of extremely precise manufacturing process. Spectroscopic ellipsometry can measure the retardance over a wide wavelength with high precision and accuracy [8], but the equipment is somehow expensive and complicated. Zhao and Wang [9] reported a method to measure the retardance by employing two linear polarizers. The error is 2.86 degrees and depends mostly on the adjustment of polarizers which cannot fulfill the demand for high precision measurement. Xue and Shen [10] reported a method utilizing a white-light interferometer and Fourier transformation, but the precision was limited because of the uncertainty and disturbance brought by Fourier transformation algorithm.

Since observed by King in 1963, massive amount of both theoretical and experimental work has been done in the field of laser feedback [11]. According to the laser feedback theory, the laser intensity, the polarization state and the phase position of a laser could be modified by introducing coherent optical feedback from an external surface. Laser feedback system has especial sensitivity to the change in feedback signal. As described in citation, the author put forward the scheme of laser frequency-shifted feedback, which gave rise to great potential in non-cooperative measurement on various of targets, such as piezoelectric transducers, soft materials and the liquids [12]. With the laser intensity being modified and detected, the laser feedback system is widely used in precise measurement fields such as measurement of displacement [13], optical axis azimuth [14], thermal expansion coefficient [12], liquid evaporation rate [15]. Benefiting from the high sensitivity and resolution of laser feedback, the authors also presented a promising application on biological imaging and inner structure test of materials, on which the pins inserted at ~1 mm beneath the onion surface were high-sensitively detected and positioned in two dimentions. It was a great breakthrough for laser feedback techique in bio-imaging detection and laser feedback tomography [16,17]. The polarization state modulation in laser feedback can be applied in generating ultrahigh frequency optical modulation [18]. The phase position modulation as phase difference in external feedback birefringence cavity [19] can be used as displacement measurement interferometer and phase difference detector [20]. In citation, the cause of the phase position modulation was explained clearly, which greatly expanded the application prospects [21]. Though the laser feedback system for phase retardance measurement has been built [22], the system can only measure the transparent, symmetrical samples with optical axis known in advance as wave plates. The samples must have flat surface and are not allowed to rotate in the system. The application of laser feedback technology on the measurement of reflection-induced retardance on optical films is not yet reported. To achieve accurate measurement of phase shift brought by reflection on optical films, we propose a method based on both the laser intensity and polarization state modulation in laser feedback theory in this paper. The system is suitable for phase property measurement of both single-layer and multi-layers in a convenient way at a high certainty of measurement which is less than 0.5 degree. The system has no limit on the samples and can measure the reflection-induced retadance at different angles of incidence.

2.Experimental apparatus and theory analysis

The apparatus setup is as shown in Fig. 1. Experiments are carried out on a single mode, linearly polarized semi-external cavity He-Ne laser with the output wavelength of 632.8 nm. The ration of gaseous pressure in laser is He: Ne = 9:1 and Ne²⁰:Ne²² = 1:1. The directions of three reference axises are along x, y and z. The initial polarization direction of the laser is along the y axis.

Fig. 1 Experimental apparatus of laser feedback system. M₁, M₂, high-reflectivity mirrors; W, output window; PZT₁, PZT₂, piezoelectric transducers; M_E, laser feedback mirror; P, polarizer; PD₁, PD₂, photo detectors; S, sample coated with optical films; ERT, electric rotary table.

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As is shown in Fig. 1, the laser cavity is made up of high-reflectivity mirrors M₁ and M₂ with reflectivity of 99.8% and 98.8%, respectively. The length of the laser cavity is 150 mm. W is the antireflective layer film coated output window. M_E is a 10% reflectivity mirror which reflects laser beam back into the laser cavity to produce laser feedback. M_E and M₂ form the external cavity. PZT₁ and PZT₂ are piezoelectric transducers (PZT) whose length is linear to the controlling voltage. PZT₁, controlled by an external DC voltage, drives M₂ to modulate the length of laser cavity to ensure stable output and single longitudinal mode of laser. PZT₂ is controlled by a periodical triangular-wave voltage and drives M_E to do the reciprocating movement in order to modulate the length of external cavity along z axis. P is the polarizer to distinguish the polarization state of the laser beam. PD₁ and PD₂ are photo detectors. The intensity of the modulated laser is detected by PD₁ while the polarization state is detected by PD₂. Only when the laser is in single longitudinal mode can the intensity of laser detected by PD₂ be extinguished by rotating the polarizer P. S is the sample to be measured with optical films on the surface. ERT is an electric rotary table controlled by a precise controller.

To analysis the modulation function of laser feedback, a model of transferring three cavity mirrors into two mirrors is employed. The model was first set by P.J.Groot [24]. In the model, instead of external cavity, an equivalent model of Fabry-Perot cavity is utilized to analyze the effect of feedback on the mode structure of the laser. The laser intensity $E$ can be expressed as:

\begin{array}{l} E = R_{1}^{1 / 2} R_{2}^{1 / 2} E_{0} e^{i 2 K L} e^{2 g L} + (1 - R_{2}) R_{E}^{1 / 2} R_{1}^{1 / 2} E_{0} e^{i 2 K (L + l)} e^{2 g L} \\ = R_{1}^{1 / 2} e^{i 2 K L} e^{2 g L} E_{0} (R_{2}^{1 / 2} + (1 - R_{2}) R_{E}^{1 / 2} e^{i 2 K l}) \end{array}

Where L is the length of laser cavity, l is the length of external cavity; E₀ is the initial intensity of laser; R₁, R₂ and R_E denote the reflectivity of mirrors M₁, M₂ and M_E; k is the wave number under vacuum and g is the gain coefficient of the laser. Suppose the equivalent reflectivity of output mirror is R_eff , it can be derived as:

\begin{array}{l} R_{e f f} = {(R_{2}^{1 / 2} + (1 - R_{2}) R_{E}^{1 / 2} e^{i 2 K l})}^{2} \\ = R_{2} + 2 R_{2}^{1 / 2} (1 - R_{2}) {(1 - R_{E})}^{1 / 2} \cos (2 K l) + {(1 - R_{2})}^{4} R_{E} \end{array}

With R₂ ≈1, ignore the high order term,

R_{e f f} = R_{2} + 2 R_{2}^{1 / 2} (1 - R_{2}) {(1 - R_{E})}^{1 / 2} \cos (2 K l)

Then the laser intensity $E$ can be expressed as:

\begin{array}{l} E = R_{1}^{1 / 2} e^{i 2 K L} e^{2 g L} E_{0} R_{e f f} = R_{1}^{1 / 2} e^{i 2 K L} e^{2 g L} E_{0} R_{2} + \\ 2 R_{1}^{1 / 2} e^{i 2 K L} e^{2 g L} E_{0} R_{2}^{1 / 2} (1 - R_{2}) {(1 - R_{E})}^{1 / 2} \cos (2 K l) \end{array}

From Eq. (4), as the intensity of the laser is proportional to the equivalent reflectivity, when the length of external cavity is changed by the reciprocating movement of the feedback mirror, the intensity of the laser is modulated cosinoidally.

3.Experiments and results

The sample to be measured is a high reflection mirror coated with dielectric films made by Daheng Optics, Inc. The base of the sample is a piece of K9 glass and the coating is made of two different materials, represented as H and L. Materials H and L have refractive index as 2.308 and 1.469, respectively. The coating has sixty layers with different thickness of odd and even layers in every twenty layers. When the sample is inserted into the external cavity, the beam is reflected on the interface of the sample coated with multilayer optical films. The polarization state is divided into two polarization states after reflection, namely s- and p- polarization. The two polarization states are orthogonal to each other. Because of the equivalent penetration depth difference of the laser beam into optical films between s- and p- polarization, the length of the laser cavity is split into two physical lengths along s- and p- polarization. The curve of laser intensity and the polarization state is as shown in Fig. 2. Laser intensity curve is detected by PD₁ and polarization state square wave is detected by PD₂. When the sample is inserted into the system, the intensity curve of laser does not show the ordinary cosinoidal trend as in Fig. 2(a). Instead, the polarization flipping phenomenon can be observed from the laser feedback system as in Fig. 2(b). The theory analysis is as followed.

Fig. 2 Experimental curves of laser feedback and polarization flipping in laser feedback system. (a) The sample is not inserted in the external cavity. (b) The sample is inserted in the external cavity.

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Suppose the phase shift caused by reflection on optical films is $δ$ , the equivalent reflectivity of the two polarizations in the laser feedback system can be expressed as:

\begin{array}{l} R^{s}_{e f f} = R_{2} + 2 R_{2}^{1 / 2} (1 - R_{2}) {(1 - R_{E})}^{1 / 2} \cos (2 K l) \\ R^{p}_{e f f} = R_{2} + 2 R_{2}^{1 / 2} (1 - R_{2}) {(1 - R_{E})}^{1 / 2} \cos (2 K l + 2 δ) \end{array}

According to Eq. (5), since there are two polarizations in laser, when the periodical triangular-wave voltage drives the movement of PZT₂ and M_E, the length of the external cavity l and the equivalent reflectivity of the two polarizations are tuned. Mode competition between the two polarizations will occur. The different modes correspond to two different equivalent reflectivity. When one equivalent reflectivity is higher, the cavity loss is less. Consequently, the corresponding mode will have more superiority in mode competition. The polarization state of the beam is induced by the result of mode competition. As a result, there will be a mutation in laser intensity and a flipping in the polarization state of the beam. In one period of laser feedback intensity curve, the duty ratio of the width between the s- and p- polarization is proportional to the phase shift. Consequently, the position of polarization flipping in the curve is determined by the phase shift $δ$ . By this method, the phase shift induced by reflection on optical films can be measured.

In the experiment, the room temperature is 18.4°C. The laser intensity and polarization state together with the PZT₂ driving triangular voltage curves are as shown in Fig. 2. In Fig. 2 (a), the sample is not inserted into the external cavity and the laser intensity curve shows the cosinoidal trend as described in Eq. (4). In Fig. 2(b), the polarization flipping phenomenon can be observed from the laser intensity curve and there is mutation in polarization state. In the increasing region of PZT driving voltage, the polarization flipping point in the laser intensity curve is A and its corresponding isocandela point is A₁. The tuning period is from C to C₁. The same goes to the decreasing region with B, B₁, D and D₁. From Eq. (1) and Eq. (5), the phase shift $δ$ can be deduced as:

δ = \frac{l_{A A_{1}}}{l_{C C_{1}}} \times 180^{\circ} = \frac{l_{B B_{1}}}{l_{D D_{1}}} \times 180^{\circ}

Where l_XX1 is the interval between point X and X₁. However, in the actual measurement, the two results in Eq. (6) are not so identical. According to the voltage-displacement characteristic of PZT, under the same driving voltage, the displacement during increasing and decreasing period is not the same. To eliminate the effect of nonlinear factors in the movement of PZT on the system, measurement results are averaged from both increasing and decreasing period. Then, Eq. (6) can be rewritten as:

δ = \frac{1}{2} \times (\frac{l_{A A_{1}}}{l_{C C_{1}}} + \frac{l_{B B_{1}}}{l_{D D_{1}}}) \times 180^{\circ}

When the sample is inserted into the system at the incident angle of 45°, the reflection-induced phase shift measured in laser feedback system is 135.32°. To evaluate the change in phase shift at the different angles of incidence, an electric rotary table is employed to drive the sample to rotate to ensure the precise control of incident angle. The rotary table is made by Daheng Optics, Inc. The table is controlled by a precise controller and its positioning accuracy is less than 0.0020°. The experimental curves of polarization flipping in laser feedback system at different angle of incidence are as shown in Fig. 3.

Fig. 3 Experimental curves of laser feedback at different incident angles. (a) 35°. (b) 36°. (c) 37°. (d) 38°. (e) 44°. (f) 45°.

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The result of reflection-induced retardance is averaged from both the increase and decrease process of incident angle to eliminate the effect of error in the electric rotary table. The measurement results of reflection-induced retardance at different angle of incidence are as shown in Fig. 4(a).

Fig. 4 Measured reflection-induced retardance curve at different incident angle. (a) The sample from Daheng, Inc. (b) The sample from Zolix, Inc.

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From the Fig. 4(a) above, the retardance varies with the incident angle. The curve of reflection-induced retardance changes in semi-periodical trends. In this sample, the maximum value of the phase shift caused by reflection is 165.24° and the minimum value is 12.78°. In the angle of incident sections from 30° to 35°and 50° to 55°, the retardance changes in slow ascendant trends. When the angle of incidence is around the maximum point, there is a rapid change in the retardance curve. The result shows that the reflection-induced retardance is sensitive to the change of incident angles and changes under certain rules. To testify the reflection- induced retardance measurement performance of the system, another piece of sample is also measured in the system. The sample to be measured is a high reflection mirror coated with dielectric films made by Zolix, Inc. The measurement result is in Fig. 4(b) and the curve shows the similar trend and change regulation as Fig. 4(a).

The laser feedback system has a simple structure so as to bring least disturbance on the measurement results. Unlike the other methods mentioned above, because the phase shift is calculated by the duty ratio between the two polarization states, the laser feedback system is insensitive to the drift in laser intensity. The measuring time for the retardance is quite short that the impact of laser frequency shift can hardly affect the accuracy. The error is mainly caused by the misalignment of the polarization direction between the polarization state and the laser beam anisotropy. The error analysis is as followed. The misalignment of the polarization direction can cause the distortion in the polarization state square wave. And because the position of the flipping point and the corresponding isocandela point is determined by the square wave, the error can be caused by the false internal. According to the experiment results, there is no more than 0.001 error in the result of the ratio in $l_{A A_{1}} / l_{C C_{1}}$ and the same in $l_{B B_{1}} / l_{D D_{1}}$ in Eq. (7), which results in 0.18 degree error in results. Because of the rapidity in measurement process and the averaging in results, the error caused by the laser beam anisotropy and the variation of laser intensity is less than 0.1 degree. The error caused by the temperature drift and the nonlinear factors in the movement of PZT is about 0.05 degree. Above all, the total error is as followed:

δ Δ = \sqrt{\sum_{i} {(δ Δ_{i})}^{2}} = {0.212}^{°}

Since the reflection- induced retardance of the samples is sensitive to the angle of incidence, the samples are hard to be measured in other systems. To testify the accuracy of the laser feedback system. a comparative experiment is conducted with a set of standard waveplates. The retardance of the waveplates is measured both by the laser feedback system and the V- VASE spectroscopic ellipsometry from J.A. Woollam Co. whose accuracy is 0.01 degree. The measured results are shown in Table 1.

Table 1. Results of Retardance of Standard Waveplates

View Table

Considering all the factors mentioned above, the accuracy of the laser feedback system is less than 0.5 °, which greatly increase the precision in the measurement of the reflection-induced retardance. Due to the fact that the light path is reflective one in the system, the system is especially suitable for phase property measurement of incident angle sensitive optical films like the samples measured above.

4.Conclusion

In this paper, a measurement system for reflection-induced retardance of optical films based on laser feedback theory is built. Compared with the conventional measurement methods, laser feedback system is more compact, more insensitive to environment disturbance and is easier to collimate in both laboratories and factories while still remains a high accuracy of less than 0.5 °. The retardance of two samples made of K9 glass covered with multilayer optical films are measured in this system. The results show that the reflection-induced retardance is sensitive to the change of incident angles. The measurement of reflection-induced retardance affords great significance for the phase property study of optical films. With the reflective light path, the samples to be measured are allowed to rotate and it is of great importance for measuring samples with films sensitive to the angle of incidence.

Acknowledgments

This work is supported by the Key Program of the National Natural Science Foundation of China (NSFC) (No. 61036016) .

References and links

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Waveplate Number	1	2	3	4	5	6
Spectroscopic Ellipsometry	69.48	83.81	97.19	106.82	128.97	138.19
Laser Feedback System	69.32	84.00	97.37	106.49	128.74	138.10
Result Deviation	-0.16	0.21	0.18	0.33	-0.23	0.09

Measurement of phase retardation of optical multilayer films based on laser feedback system

Abstract

1.Introduction

2.Experimental apparatus and theory analysis

3.Experiments and results

4.Conclusion

Acknowledgments

References and links

Cited By

Figures (4)

Tables (1)

Equations (8)

Optics Express