Measurement method for optical retardation based on the phase difference effect of laser feedback fringes Peng Zhang, Ning Liu, Shijie Zhao, Yidong Tan, and Shulian Zhang* State Key Laboratory of Precision Measurement Technology and Instruments, Department of Precision Instruments, Tsinghua University, Beijing 100084, China *Corresponding author: zsl‑[email protected] Received 15 September 2014; revised 21 November 2014; accepted 26 November 2014; posted 26 November 2014 (Doc. ID 222930); published 7 January 2015

We present a measurement method for optical phase retardation, which utilizes a phase-difference phenomenon of the feedback fringes in orthogonally polarized directions of a laser with anisotropic weak feedback. This phase difference is dominated by the measured retardation. The measurement principles are given based on the 3-mirror cavity model, and experiments are conducted with quartz waveplates (WPs). A measurement range 0°–180° is achieved, the uncertainty is theoretically better than 0.5°, and the measurement precision can be improved further. This method does not require researchers to know beforehand the principal axes directions of the WPs, and has better anti-disturbance ability than the previous method based on laser feedback. © 2015 Optical Society of America OCIS codes: (120.0120) Instrumentation, measurement, and metrology; (140.0140) Lasers and laser optics; (260.1440) Birefringence. http://dx.doi.org/10.1364/AO.54.000204

1. Introduction

Waveplates (WPs) are widely used in many fields involving polarized light. To accurately determine the phase retardation of a WP is of great importance to its manufacturing process and application effect [1,2]. Nowadays, various measurement methods for the retardation of WPs have been developed [3–13]. Most of them are based on the traditional principles; for instance, the rotating extinction or the optical interferometry. However, the traditional principles are still not sufficient for actual requirements [14]. For example, the measurement range of some traditional methods (such as the rotating extinction method, and so on) is limited to the quarter-wave plate or other specific-retardation WPs, and a standard quarter-wave plate is commonly needed. Meanwhile, the exact principal axes direction needs to be 1559-128X/15/020204-06$15.00/0 © 2015 Optical Society of America 204

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known in advance for many traditional methods. The measurement accuracy of the ellipsometry method (which utilizes a white-light source) is largely influenced by the wavelength shift, especially for thick WPs. For instance, a 0.1 nm shift can cause a 1.3° error for a 1.5 mm thick WP. For the high precision measurement methods based on an interferometric system, the complex arrangement and operation is usually required. Besides those traditional ones, some novel methods have been proposed and investigated over the past decades, such as the nonlinear optical method [15], the time-domain method [16], and the laser frequency-splitting (LFS) method [17]. A measurement uncertainty (or sensitivity) of λ∕104 [15,17] even λ∕106 [16] has been reported. Nevertheless, rigorous operation requirement or limited measurement range may restrain their application. Moreover, most of the existing methods are not suitable for the online measurement during WP processing. That means the producer needs to remove the WP from the optical cement tray for

measuring the phase retardation, then attach the WP back, continue to polish, and repeat this procedure over and again. Recently, a laser feedback polarization-flipping (LFPF) method [18] has been reported, and it is proved to be feasible for the online measurement [19]. However, a locking phenomenon of polarization flipping is found when the phase retardation is small [20]; furthermore, high stability and a low noise level of the working environment is strictly required because of the sensitivity of the laser feedback to vibrations. When the feedback level is further weaken with respect to the LFPF method, the polarization-flipping phenomenon can not happen, but a phase-difference phenomenon of two feedback fringes in orthogonal polarization directions is found in Nd:YAG microchip lasers [21] and He–Ne lasers [22]. An approximately linear relationship between this phase difference and the retardation of WP is derived, and then this phase difference phenomenon is expected to be applied to the retardation measurement of WPs. By utilizing this phenomenon, we need not to know the principal axes direction of the measured WP in advance. Furthermore, the fast axis and slow axis azimuth can be differentiated based on this phenomenon [23]. In this paper, we demonstrate a measurement method for optical retardation based on this laser feedback fringe-phase-difference (LFFPD) phenomenon. The experimental setup and measurement principle are given in detail. The measurement results are compared with those obtained by the LFS method, a spectroscopic ellipsometer, and the LFPF method. Preliminary research shows that this method has good measurement repeatability, a large measurement range, and the potential to be improved further. The LFFPD method has similar advantages with the LFPF method, but also possesses better anti-disturbance ability compared with the latter. 2. Measurement setup and principle

A schematic of the experimental setup is shown in Fig. 1. A half-intracavity 632.8 nm He–Ne laser is used. M 1 and M 2 compose the resonant cavity; the former is the rear output mirror (a concave mirror with reflectivity R1  99.5%) and the latter is the front output mirror (a plane mirror with R2  99%). The resonant-cavity length l ≈ 19 cm and the external-cavity length L ≈ 23 cm. The feedback mirror (M 3 ) is a plane mirror without a coating for weak feedback purposes (R3 ≈ 4%). The front output power is nearly 0.6 mW without optical feedback. Piezoelectric transducer PZT1 is attached to M 2 to maintain the polarization state of the laser output, and driven by an adjustable-voltage driver (PZTD1 ). PZT2 is attached to M 3 to make the latter move forth and back along the laser axis, and driven by a triangular wave voltage driver (PZTD2 ). A pair of collimation lenses (CLs) is used to expand and collimate the laser beam. The measured WP is put on the sample holder (S). A Wollaston prism (W) is used to separate two

Fig. 1. Schematic of the experimental setup; CL, collimation lens; D1;2 , photodetectors; DAQ, data acquisition card; G, gain tube; M 1;2 , cavity mirrors; M 3 , feedback mirror; PC, personal computer; PZT1;2 , piezoelectric transducers; PZTD1;2 , PZT drivers; S, sample holder; U 1 , detection unit; U 2 , laser unit; U 3 , feedback unit; U 4 , processing unit; W, Wollaston prism.

orthogonally polarized components of the laser rear output. Photodetectors D1 and D2 detect the two light intensities, respectively. The output signals of D1 and D2 are sent to the computer through a data acquisition card (DAQ, NI USB-6211), and are processed by a computer program. The program also outputs the control signal to PZTD2 through the DAQ. The laser works at single longitudinal mode, and the initial polarization state is linear polarization due to the residual anisotropy in the resonant cavity. We define the electric field without the measured WP as Ei. The experiments are prepared as the following steps. First, the measured WP is absent. The Wollaston prism is adjusted to make one of its optical axes parallel to Ei (i.e., only one beam outputs from the Wollaston prism), and afterward is rotated 45°. After this step, two identical modulated intensity signals (i.e., feedback fringes) are detected by D1 and D2 . Second, a measured WP is placed into the external cavity and oriented to make its fast or slow axis parallel to Ei ; at this time the intensity signals detected by D1 and D2 will return to be identical. Third, the WP is rotated around the axis parallel to its surface-normal, until 45° included-angle between its optical axes and Ei is achieved. During the experiments, the first step only needs to be performed once, and the latter two steps need to be done before each measurement. As the three steps are finished, the orientation relationship among Ei, the WP, and the Wollaston prism is as in Fig. 2(a). Ei can be resolved into two orthogonally polarized components after laser beam passing through the 10 January 2015 / Vol. 54, No. 2 / APPLIED OPTICS

205

Fig. 2. Schematics; (a) orientation relationship among the optical-axis directions of crystals and the polarization direction of Ei ; (b) laser feedback arrangement.

WP (Ef and Es , which are along the fast-axis and slow-axis directions, respectively). Both of them can be partially reflected by the feedback mirror and induce optical feedback. Based on the same consideration, the modulated laser output also contains two orthogonally polarized components corresponding to the two optical-axis directions of the WP. In Fig. 2(a), the optical-axis directions of the Wollaston prism are consistent to the ones of the WP, so the two output components can be just separated and detected respectively. The laser feedback arrangement is shown in Fig. 2(b), which is a 3-mirror cavity model [24]. Defining the electric field on the inner side of M 2 as E0, with the presence of optical feedback, the perturbation ΔE induced by a round-trip of the light in the external cavity is written as (assuming the resonant cavity is filled with active medium) 0

ΔE  t2 E0 e2iknL r3 ξt2 e2ikn l2 gl ;

0

0

(2)

(3)

where β  T 2 r3 ξ∕r2 , T 2  t22 is the transmittance of M 2 , and φ  2 knL is the phase of the roundtrip loop external to the laser. Without optical feedback, when the laser comes to equilibrium, the electric field should reappear after a roundtrip in the resonant cavity, so it can be derived from Eq. (2) that 0

jr1 r2 e2ikn l2 gl j  1: 206

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1 lnr1 r2 ; 2l

(5)

where g0 is the linear gain without optical feedback, and 0

je2ikn l j  1:

(6)

Similarly, with optical feedback, Eq. (7) should be satisfied according to the equilibrium condition ΔE  E00  E0 :

(7)

Substituting Eqs. (3) and (6) into Eq. (7), we get jr1 r2 e2 gl 1  βeiφ j  1:

(8)

For a He–Ne laser, β  T 2 r3 ξ∕r2 ≪ 1, so

Hence, the combined field E is E  ΔE  E00  r1 r2 E0 e2ikn l2 gl 1  βeiφ ;

g0  −

(1)

where l and L are defined as previously mentioned, n is the refractive index of the medium in the external cavity, n0 is the corresponding parameter in the resonant cavity, t2 is the transmission coefficient of M 2, r3 is the reflection coefficient of M 3, ξ is the effective feedback ratio, and g is the linear gain per unit length. The electric field after a round-trip in the resonant cavity is expressed as E00  r1 r2 e2ikn l2 gl E0 :

Then we solve the Eq. (4) as

(4)

g−

1 lnr1 r2   β cos φ: 2l

(9)

Hence the excess required gain Δg with optical feedback is Δg  g − g0  −

β cos φ: 2l

(10)

The light intensity with optical feedback can be expressed as [25] I  I 0 1  κΔg;

(11)

where κ is a constant, and I 0 is the light intensity without optical feedback. When a measured WP is placed into the external cavity and adjusted as in the previous steps, the light intensities in the directions of the fast-axis and slow-axis can be written as   κβ I f  I 0f 1  κΔgf   I 0f 1 − cosφf  αf  ; (12a) 2l   κβ cosφf  2δ  αs  ; I s  I 0 s 1  κΔgs   I 0 s 1 − 2l (12b)

Fig. 3. Typical experimental phenomena; (a) about 84° WP; (b) about 28° WP.

where δ is the retardation of the measured WP, and αf and αs are the additional phases caused by other factors. Because κ, β, and l are all constants for a certain laser, it can be seen clearly from Eq. (12) that the two modulated intensities have a phase difference ϕ to each other ϕ  2δ  α;

(13)

where α  αs − αf . A functional relationship between the phase difference of the modulated intensity signals and the retardation of the measured WP is now obtained. In ideal conditions, α  0, ϕ is just twice that of δ. Thus, when the phase difference of the modulated intensity signals is obtained, we can calculate the retardation of the measured WP directly. Two typical experimental phenomena are shown in Fig. 3. In Fig. 3, the upper two curves are the intensity signals detected by D1 and D2 , and the lower curve is the signal of the PZT driving voltage output by the computer program. It is clear to see that a phase difference exists between the two modulated intensity signals and alters with the optical retardation of the WP. By the laser feedback effect, the phase retardation in the electric field is transferred to the phase difference of two feedback fringes which can be observed and detected directly. This is an advantage of the LFFPD method. 3. Experimental results

The discrete Fourier transform (DFT) is a universal operation for the data processing of finite-length Table 1.

discrete signal sequences. In practice, the fast Fourier transform (FFT), which is the fast algorithm of the DFT, is commonly used [26]. In the current experimental system, the phase information of feedback fringes signals is extracted by using the FFT module of LabVIEW software (Version 8.6). The results of FFT contain the frequency, amplitude, and phase of each component of the original signal. The phase information of the fundamental-frequency component is extracted by judging the maximum amplitude. With this processing, the phase of the intensity signal I 1 is defined as ϕ1 ; correspondingly, the phase of the intensity signal I 2 is defined as ϕ2, and then the phase difference is obtained by jϕ1 − ϕ2 j. The calculations repeat within five periods of the triangular-wave driving voltage, so the phase difference is obtained in each rising edge and falling edge, respectively, and the result is calculated by averaging the 10 data. The experimental results of a series of 9-order quartz WPs are listed in Table 1. The Reference Values 1 are calibrated by the LFS method [17], and they are regarded as the standard values due to the traceability of this method. The Reference Values 2 are measured by a spectroscopic ellipsometer (J. A. Woollam Company, M-2000UI). The measured values are obtained after fitting all experimental data refer to the standard values. We can see that the errors of the measured values to the standard values are less than 2° for most WPs, and similar to the magnitude of the Reference Values 2. The repeatability is better than 0.1°, which is obtained by 5 times repeated measurements for

Experimental Resultsa

Code Reference Value 1b Reference Value 2c Measurement Value Code Reference Value 1b Reference Value 2c Measurement Value 1# 2# 3# 4# 5# 6#

16.42 27.89 44.87 69.48 78.95 83.81

17.53 28.94 45.8 70.58 80.35 85.12

16.52 28.22 45.44 71.2 78.17 85.15

7# 8# 9# 10# 11# 12#

97.19 106.82 128.97 138.19 157.09 —

99.39 109.29 131.34 140.48 159.35 176.38

97.15 104.46 131.22 137.77 156.75 177.69

a

Temperature, 28°C; humidity, 60%. All reported values are in degrees. Measured by the LFS method at 30°C, except for 12#. c Measured by a spectroscopic ellipsometer at 22°C. b

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Fig. 4. Measurement comparisons of the LFFPD method with the LFPF method.

one WP; the reproducibility is better than 1°, which is obtained by the measurements of different operators at different dates. The theoretical measurement uncertainty uC ≈ 0.31°, which is estimated with considering the rotation angle error, the temperature, alignment error of the measured WP, and so on. The experimental results indicate that the LFFPD method has the feasibility of measuring wide range of phase retardations. It should be noted that the nonlinear factors existing in the system is a major error source for now, e.g., the residual anisotropy in laser cavity and the nonlinearity of PZT may influence the waveform and phase of the intensity signals. The deviation between the optical axes of the WP and the Wollaston prism is another major error source according to the measurement principle. Moreover, the environmental temperature difference is a major error source for the data comparison in Table 1.

Two series of quartz WPs are measured to compare the LFFPD method with the LFPF method, and the results are shown in Fig. 4. It can be seen that the tendencies of the two measurement series are nearly consistent, which means the accuracy of the LFFPD method is similar to that of the LFPF method. On the other hand, the LFFPD method has better anti-disturbance ability due to its measurement principle. Because that the two intensity signals are separated from the same beam, and their modulations come from the same PZT, the influence of the external disturbances on the two feedback fringes is simultaneous and nearly same. As the phase difference is obtained via subtraction operation, the influence becomes very small as long as the disturbances are not large enough to transform the cycle characteristics of the modulated signals. However, for the LFPF method, because its measurement principle completely depends on the feedback fringes’ quality, any disturbance on the fringes can cause a large error. That is why the LFPF method must have the strict vibration isolation measures and a quiet working environment. Figure 5 shows some cases of the disturbed feedback fringes, which are caused by people speaking or a slight vibration of the experimental bench. The experimental results of the new method without and with the disturbance are shown in Table 2. It can be seen from Table 2 that for the new method, the measurements with disturbance can obtain similar results as the normal case. However, for the LFPF method, the measurement can not even be operated in those disturbance cases. Due to the high sensitivity of the laser feedback to environmental vibration, those disturbance phenomena are very common for practical applications. Thus, this characteristic of the new method indicates that it has a lower stability requirement than the other methods based on the laser feedback, and is suitable for an ordinary working environment.

Fig. 5. Disturbance phenomena on laser feedback fringes. 208

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Table 2.

Experimental Results without/with Disturbancea

With Disturbance Without Disturbance 84.85

1

2

3

4

5

85.05

85.07

84.85

84.7

84.7

a

Temperature, 25°C; humidity, 10%. All reported values are in units of degrees.

4. Conclusions

A measurement method for the phase retardation of WPs is investigated based on the phase difference effect of the feedback fringes in anisotropic weak optical feedback. The measurement principle is described theoretically, and the experimental results have shown the feasibility for practical applications. Comparing with the previous measurement method based on the laser feedback, this method has better anti-disturbance ability due to its measurement principle and also shares the advantages of the laser feedback technique. Moreover, no remarkable locking phenomenon is observed, which means the new method has a larger measurement range. This method also has the potential to measure the inner stress in transparent materials. At present, the main problem of the new method is that the theoretical model is idealized, and both higher measurement precision and repeatability need to be studied in the future. The authors acknowledge the support of the Key Program of the National Natural Science Foundation of China (Grant No. 61036016), and are indebted to Haisha Niu for the comparison experiments. References 1. M. P. Kothiyal and C. Delisle, “Polarization component phase shifters in phase shifting interferometry: error analysis,” Opt. Acta 33, 787–793 (1986). 2. P. D. Hale and G. W. Day, “Stability of birefringent linear retarders (waveplates),” Appl. Opt. 27, 5146–5153 (1988). 3. B. R. Grunstra and H. B. Perkins, “A method for the measurement of optical retardation angles near 90 degrees,” Appl. Opt. 5, 585–587 (1966). 4. L. H. Shyu, C. L. Chen, and D. C. Su, “Method for measuring the retardation of a wave plate,” Appl. Opt. 32, 4228–4230 (1993). 5. X. J. Chen, L. S. Yan, and X. S. Yao, “Waveplate analyzer using binary magneto-optic rotators,” Opt. Express 15, 12989–12994 (2007). 6. B. L. Wang and W. Hellman, “Accuracy assessment of a linear birefringence measurement system using a Soleil–Babinet compensator,” Rev. Sci. Instrum. 72, 4066–4070 (2001). 7. P. A. Williams, A. H. Rose, and C. M. Wang, “Rotatingpolarizer polarimeter for accurate retardance measurement,” Appl. Opt. 36, 6466–6472 (1997).

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Measurement method for optical retardation based on the phase difference effect of laser feedback fringes.

We present a measurement method for optical phase retardation, which utilizes a phase-difference phenomenon of the feedback fringes in orthogonally po...
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