Wave plates are important optical elements that can change the polarization states of incident lights,^{[1, 2]} and are widely used in polarization related systems such as optical isolators,^{[3, 4]} optical attenuators,^[5] and polarization rotators.^[6] The retardation accuracy of the wave plate has a direct effect on the performance of the system.^[7] Therefore, accurate measurement of retardation in wave plates plays an important role in processing high quality optical elements and improving the performances of optical systems.

Wave plate measurement based on the laser self-mixing interference (SMI) effect,^{[8, 9]} is a burgeoning technique for its advantages such as simplicity, low-cost, and high precision.^[10] However, for non-coated wave plates, the repeatability of the method is too poor to meet the measurement application. In order to reduce the influence of the reflected light on the surface of sample, Chen^[11] has proposed a refractive index matching method and achieved good results under laboratory conditions. The method requires the upper and lower surfaces of the wave plate being coated with a refractive index-matching fluid, and sandwiched between a glass substrate and a cover sheet that coated with the antireflective film. The need to align the instrument for each piece of wave plate and the need to clean off the index-matching liquid from the sample make these measurements complex and inefficient, so refractive index matching method is not suitable for large-scale industrial production applications.

We calculate the laser output characteristics with double external cavities and propose a synchronous tuning laser SMI system in this paper. The system eliminates the sub-external cavity effect by tuning the external and sub-external cavity at the same frequency. The measurement errors of non-coated wave plates have been proved to be reduced at the same time. The research plays an important role in improving the performance and enlarging application range of the SMI system.

As shown in Fig. 1, M₁ and M₂ constitute the resonator of the laser, whose length is L. WP is the non-coated wave plate, and its surface is perpendicular to the laser axis. M₃ is the reflective mirror with reflectivity of 10%. M₂ and M₃ constitute the laser external cavity, whose length is l₂. WP and M₂ constitute the sub-external cavity, whose length is l₁. As the reflectivities of M₃ and WP are low, only one round-trip of the beam is considered.

Fig. 1.

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	Fig. 1. (color online) Sub-external cavity effect.

After one round-trip in the laser cavity, the laser needs to have the same amplitude and phase to be excited, the condition is still tenable for SMI system. Assuming that the starting point of the light wave is M₂, and the initial electric field can be expressed as 1 where ω is the angular frequency of the laser, and φ is the amplitude and initial phase of the light-wave, respectively. After a round-trip in the laser cavity, the electric field of light is denoted as E₂. After passing through M₂, the light is reflected by non-coated wave plate and coupled into the inner cavity, and is denoted as E₃. The light coupled into inner cavity by M₃ is denoted as E₄. According to the condition of laser excitation, when the system is stable, superposition of the three electric fields should have the same amplitude and phase with the initial electric field, which can be expressed as 2 E₂, E₃, and E₄ are expressed as follows:3 4 5 where r₁, r₂, r₃, and are the reflectivities of M₁, M₂, M₃, and the wave plate, respectively; t₁, t₂, t₃, and are the corresponding transmittances. Substituting E₂, E₃, and E₄ into Eq. (2), the expression can be written as

Fig. 2.

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	Fig. 2. (color online) Theoretical output intensity of laser with external and sub-external cavity.

A laser SMI experiment with double external cavities is carried out to verify the theoretical analysis. As shown in Fig. 3(a), M₁ and M₂ is the laser cavity, M₃ and M₄ are reflecting mirrors. PZT₁ and PZT₂ are connected to M₃ and M₄, respectively. Triangular voltages with different frequency are applied to PZT₁ and PZT₂ to tune the external and sub-external cavity at the same time, the output of the laser is shown in Fig. 3(b).

Fig. 3.

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	Fig. 3. (color online) Experimental tuning curve.

The envelope of the curve is generated by the tuning of the external cavity, and the movement of the sub-external cavity mirror is superimposed on the external cavity tuning curve. The experimental result is consistent with theoretical calculations.

In the laser SMI wave plate measurement system, when the length of the sub-external cavity is changed by the environmental disturbance, the generated harmonic would cause the distortion of the measuring signal, which directly leads to unreliable precision of measurement results.

It can be found from Eq. (13) that, if the external and sub-external cavity can be tuned at the same frequency, the sub-external cavity is modulated to the amplitude and phase of the measurement signal rather than distortion.

In order to achieve synchronous tuning, a cavity tuning device is designed to be placed in the sub-external cavity. The synchronous tuning system is shown in Fig. 4. The light source is a half-external cavity, single-mode, linearly polarized He–Ne laser, and the working wavelength is 632.8 nm. The ratio of gaseous pressure in the laser is He:Ne = 9:1 and Ne²⁰:Ne²² = 1:1 to eliminate lamb dip.^[12] The laser cavity is made up of mirrors M₁ and M₂ with reflectivity of 99.8% and 98.8% respectively and the length is 150 mm. The initial polarization state of the laser is parallel to the Y axis. PZT₁ is connected to M₂ to make the longitudinal mode work in the center of the bandwidth. P is the polarizer whose polarization direction is in the X axis. WP is the non-coated wave plate, D₁ and D₂ are photodetectors. D₁ is used to detect the light intensity, and D₂ is used to detect the laser intensity behind the polarizer. Because the polarization direction of P is perpendicular to the initial polarization state of laser, the light cannot reach D₂ at the beginning. The lengths of external and sub-external cavity are l₂ and l₁, respectively.

Fig. 4.

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	Fig. 4. (color online) Structure of the improved system.

The cavity tuning device is composed of a pair of glass wedges and a piezoelectric ceramics PZT₂, and the surface of the glass wedges where light passes through are coated with antireflection film for 632.8-nm wavelength. As shown in Fig. 5, θ, , , and α represent the wedge angle, the refraction index of the optical wedge, the refractive index of air, and the refraction angle, respectively. When triangular voltage is applied on PZT₂ and optical wedge 2 is pushed at displacement of S, the change of optical length of the laser in the two optical wedges and the air gap between them can be expressed as follows: 14 The displacement of the light spot on optical wedge 2 is S₂, and S₂ can be expressed as follows: 15

Fig. 5.

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	Fig. 5. (color online) Operational principle of cavity tuning device.

The material of the optical wedge is K9 glass, the refractive index of which is 1.514 at the wavelength of 632.8 nm. The refractive index of air is between and under the temperature of 25 °C.^[13] We take 1.000 in the calculation because the fractional part has little effect on the result. The wedge angle θ is 30°. Substituting the above parameter into Eqs. (14) and (15), we obtain the following relationship 16 It can be seen from Eq. (16) that S₁ and S₂ are proportional to the displacement of PZT₂. When triangular voltage is applied on PZT₂, the length of both external and sub-external cavities are linearly tuned by S₁ synchronously. If S is set to – , and S₂ are 0 nm–987 nm, 0 nm–756 nm, respectively. The light spot of the laser is about 1.1 mm, S₂ is quite small in comparison with the size of the light spot and it has no influence on the measurement.

According to Luo’s research,^{[14, 15]} the photonic spin-dependent splitting occurs when the refractive index gradient exists for spin Hall effect, so the transmitted beam splits by a fraction of wavelength into its two left- and right-handed circularly polarized components ( and ) after passing through the cavity tuning device. The spin-dependent splitting is restricted to a few tens of nanometers,^{[16, 17]} and its scale is much smaller than the radius of laser spot. The circularly polarized components have opposite rotating direction and the superposition of them is linearly polarized, but the polarization direction is deflected by 2° compared with the initial state through our experimental test.

As shown in Fig. 6(a), assuming that the incident light is beam 1, the light passing through the cavity tuning device is beam 2, the light reflected by the feedback mirror is beam 3, and the feedback light emitted from the cavity tuning device is beam 4, the polarization directions of them can be indicated in Fig. 6(b). Above all, the light re-injected into the laser cavity has the same polarization direction with the intra-cavity light field. Therefore, the impact of photonic spin-dependent splitting can be ignored in the SMI system.

Fig. 6.

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	Fig. 6. (color online) Spin-dependent splitting phenomena and polarization states.

Figures 7(a)–7(c) show the tuning curves when triangular voltage is applied on PZT₂. Curves 2 are laser output intensities detected by D₁, while curves 3 are polarization states curves detected by D₂. Figure 7(a) shows that when there is no wave plate, the output intensity of the laser is modulated in the form of cosine, and each modulation period corresponds to 1/2 wavelength change of the external cavity length; when the wave plate is inserted in the external cavity, the polarization state of the laser flips from the initial direction to its orthogonal direction, as shown in Figs. 7(b) and 7(c). When the retardation changes, the flipping point changes accordingly, as shown in curves 3. AC and ac are modulation periods; B and b are polarization flipping points for different wave plates.

Fig. 7.

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	Fig. 7. (color online) Intensity tuning curves of the laser SMI system.

According to the equivalent cavity of a Fabry–Perot interferometer, reflectivities of laser in X and Y directions can be expressed as 17 As effective gain of the laser is proportional to the equivalent reflectivity, the intensity of the laser is sinusoidally modulated. The phase retardation of the wave plate is half the phase difference of the two polarization states. The phase retardation σ can be obtained by the equation 18 where points b and are a pair of isocandela points and denotes the time interval between the isocandela points, so as ac.

S₁ displays nonlinearity for the nonlinearity of PZT₂ and non-parallelism of the glass wedges, which is the major error source. We measure the nonlinearity of S₁ by a heterodyne interferometer with accuracy of sub-nanometer.^[18] The measuring principle diagram is shown in Fig. 8. The output of the laser is an orthogonal polarized laser with the frequency of f₁ and f₂, light is split into two parts by a beam splitter (BS). One part passes through the polarizer (P) and generates reference signal , which is detected by detector D₁. The other part is split into f₁ and f₂ by polarizing beam splitter (PBS). f₁ is reflected by measuring corner prism(CCR₂), f₂ is reflected by reference corner prism(CCR₁) and then merges in PBS with f₁. When PZT₂ is moving with the speed of v, Doppler frequency shift is generated in the measuring optical path. Measuring signal is detected by detector D₂. By the comparison of reference signal and measuring signal, the change of optical path can be obtained.

Fig. 8.

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	Fig. 8. (color online) Measurement diagram.

The measurement result is shown in Fig. 9. It can be found that the displacement nonlinearity of differs from the rising edge and falling edge of the driving voltage. If the average of the displacement–voltage curve is taken for the rising and falling edges, the nonlinearity is reduced.

Fig. 9.

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	Fig. 9. (color online) Displacement nonlinearity of S1.

Referring to Fig. 7, equation (18) can thus be written as follows: 19 The phase retardation of a quartz wave plate is measured repeatedly and the results are shown in Table 1. Table 1 shows that the repeatability of the system is 0.16°.

Table 1.

Table 1.

Table 1. Repeatability of improved system (deg). . Ordinal 1 2 3 4 5 Result/(°) 36.202 36.141 36.193 36.097 36.042

Table 1. Repeatability of improved system (deg). .

Five non-coated wave plates with different orders and phase retardations are measured with the SMI system and ellipsometer for comparison, the measurement accuracy of the ellipsometer was better than 0.1°, and the error of laser SMI system is in the range of −0.435°∼0.387°, as shown in Table 2.

Table 2.

Table 2.

Table 2. Comparison result (deg). . Number Order SMI/(°) Ellipsometer/(°) Error/(°) 1 16 36.135 36.397 0.262 2 14 48.952 49.321 0.369 3 5 62.481 62.868 0.387 4 16 105.071 104.823 −0.248 5 13 110.541 110.106 −0.435

Table 2. Comparison result (deg). .

In this paper, a self-consistent model for the laser operation with double external cavities is established for a self-mixing interference laser wave plate measurement system. According to the conclusion, a synchronous tuning method of external cavity and sub-external cavity is proposed to eliminate the disturbance of the sub-external cavity. Experiments are carried out on the synchronously tuning feedback system. Compared with the ellipsometer, the error of the system is in the range of −0.435°∼0.387° with repeatability of 0.16°.