The deviation of wave plates’ optical axes from their intended angles, which may result from either instability or assembly error, is the main cause of quadrature phase error in homodyne quadrature laser interferometers (HQLIs). The quadrature phase error sensitivity to wave plate angle deviations, which is an effective measure of HQLI robustness, is further amplified by beam splitter imperfections. In this paper, a new HQLI design involving non-polarization beam splitting is presented, and a method of making this HQLI robust by yawing the wave plates in the measurement and reference arms is proposed. The theoretical analysis results indicate that ultra-low quadrature phase error sensitivities to wave plate angle deviations can be realized and that non-polarizing beam splitter imperfections can be adequately compensated for. The experimental results demonstrate that the proposed method can reduce the quadrature phase error sensitivity by more than 1 order of magnitude, from a theoretical value of 1.4°/1° to 0.05°/1°.
© 2016 Optical Society of America
Homodyne quadrature laser interferometers (HQLIs) can currently achieve deep sub-nanometer resolutions. However, their measurement accuracies are significantly limited by cyclic nonlinear errors ranging from several sub-nanometers to several nanometers [1, 2]. Due to the high accuracy requirements of advanced scientific research projects such as the Laser Interferometer Gravitational Wave Observatory (LIGO), ultralow-frequency vibration calibration methods , and nanometrological atomic force microscopes , HQLI measurements must be robust against slight optical component misalignments .
Although DC offset and unequal AC amplitude errors can easily be corrected with simple gain and offset correction methods [6, 7], quadrature phase errors can only be corrected by employing complicated and time-consuming algorithms [8–11]. Thus, quadrature phase errors are the main causes of cyclic nonlinear errors and significantly limit the real-time performance of HQLIs, and quadrature phase error correction is currently a primary research focus.
HQLI measurements must be highly accurate and robust against slight deviations of critical optical elements, such as wave plates, from their intended angles, which could be caused by instability or assembly errors . Therefore, error sensitivity measurements can be employed to assess the robustness of different HQLI configurations. Error analyses have revealed that quadrature phase error sensitivity to wave plate angle deviations, i.e., deviations of wave plates’ optical axes from their intended angles, is a key problem in HQLIs. Optical component imperfections, especially polarizing beam splitter (PBS) polarization crosstalk and non-polarizing beam splitter (NBS) phase shift errors, further amplify the quadrature phase error sensitivity to wave plate angle deviations. For instance, in a conventional four-detector HQLI , in which a quarter-wave plate (QWP) is located before the NBS in the detection part, a 1° angular deviation of the half-wave plate (HWP) theoretically causes a quadrature phase error of ~4°, regardless of whether or not imperfections in the NBS are considered, yielding a quadrature phase error sensitivity of ~4°/1°. In a similar configuration in which a HWP is located before the NBS in the detection part , the quadrature phase error sensitivity to HWP angular deviations is zero if NBS and PBS imperfections are not considered. However, the sensitivity can reach 1.5°/1° if NBS imperfections are accounted for. In 2009, Gregorčič et al. proposed an HQLI design in which two detectors are used . The quadrature phase error sensitivity to angular deviations of the octadic-wave plate can be as large as 3°/1° if the NBS phase shift error is considered. In 2015, Hu et al. proposed a DC-offset-free HQLI in which Wollaston prisms (WPs) with extinction ratios of up to 100,000:1 are employed rather than PBSs (extinction ratios ~1000:1) in the detection part . A theoretical quadrature phase error sensitivity to QWP angular deviations of 2.7°/1° results from PBS polarization crosstalk in the interferometer part and NBS phase shift error in the detection part. Similar conclusions can be drawn for other HQLI configurations.
In this paper, a new HQLI configuration is proposed that uses non-polarization beam splitting to avoid the polarization crosstalk that occurs during beam splitting. Then, a method of making this configuration robust by yawing the wave plates in the measurement and reference arms is presented. The quadrature phase error sensitivity to wave plate angle deviations can thereby be reduced to zero, due to the NBS phase shift error compensation provided by yawing the wave plates.
2. HQLI using non-polarization beam splitting
2.1 Optical configuration
The proposed HQLI is illustrated in Fig. 1. In the interferometer part, a frequency-stabilized and linearly polarized He-Ne laser (Melles Griot 25 STP 912, wavelength: 632.8 nm, output power: 1 mW, noise < 0.05% rms) is used. The laser beam is polarized along the x-axis by rotating the cylindrical laser head around its axis. An optical Faraday isolator (OFI) is employed to prevent any light from re-entering the laser. After the beam exits the OFI, its polarization direction is changed to 45° with respect to the x-axis, in the xy plane. The linearly polarized laser beam is then split into the reference and measurement arms by the first non-polarizing beam splitter (NBS1), whose nominal split ratio is 50%–50%. In the reference arm, the laser beam first passes through the first QWP (QWP1), whose fast axis is parallel to the y-axis, causing a 90° phase difference between the two orthogonal polarization elements. Then, the beam is reflected by a high-reflectivity reference mirror (RM). The phase of the reflected beam is changed by an additional 90° when it returns through QWP1, and half of the beam then passes through NBS1. In the measurement arm, a HWP whose fast axis is parallel to the y-axis is employed, and its role in achieving robustness will be discussed in Section 3. The beam is reflected by a high-reflectivity target mirror (TM) and is half-reflected by NBS1. The TM is driven by a piezoelectric transducer (PZT), which is employed to generate standard displacements during experiments. The measurement beam experiences a phase shift δ due to the movement u(t) of the TM. The reference and measurement beams output by the interferometer part are orthogonally polarized and together serve as the incident light for the detection part.
In the detection part, the incident light is first evenly split into two beams by the second non-polarizing beam splitter (NBS2), whose split ratio is 50%–50%, like that of NBS1. The reflected beam passes through the second QWP (QWP2), whose fast axis has an angle of 45° with respect to the y-axis, before reaching the WPs. The ordinary and extraordinary lights of the reference and measurement beams are then respectively separated by WP1 and WP2, which are located in the xz plane, and are detected by four photodiodes (PDs), PD1–PD4. Half the laser power is lost due to the back reflection towards the laser.
The widely used cube NBSs with anti-reflective coating on the surfaces are adopted. The anti-reflective coating which is clearly indicated in Fig. 2(a), mostly reduces the stray light beams reflected from surfaces under zero incident angles. Such reflections which are often called ghost images, are known to be a critical performance issue if they enter one or more of the relevant photodiodes. They are further reduced by the proposed robust method of yawing wave plates which is discussed in Section 3.
Ideally, the laser beams incident on the PDs should produce four sinusoidal signals with the same AC amplitudes, the same DC offsets, and 90° phase shifts with respect to one another. The four signals, I1–I4, can be expressed as
The two signals Ix and Iy, obtained by subtracting the signals with 180° and 270° phases from those with 0° and 90° phases, respectively, can be written as
Then, u(t) can be calculated using
In the proposed HQLI configuration, NBSs and WPs rather than PBSs are used for beam splitting in both the interferometer and detection parts, and the use of polaroids is deliberately avoided. Thus, polarization crosstalk during beam splitting can be prevented.
2.2 Jones matrices of a real NBS
To determine the beam splitting mechanism and characteristics of NBSs and to provide a theoretical basis for the method of achieving robustness that is proposed in Section 3, the beam splitting model of a real NBS is discussed theoretically and experimentally in this section, and revised Jones matrices are provided.
A cube NBS consists of two triangular glass prisms with coatings and cement in between, as shown in Fig. 2(a). The Jones matrices of an ideal NBS with a 50%–50% split ratio can be written as
As shown in Fig. 2, there are four beam-splitting cases, in which the beam enters the cement through face A, B, C, or D. In each case, the transmitted beam travels through the cement once. When the beam enters through face A or B, the reflected beam travels through the cement twice and is reflected by side I of the coatings. When the beam enters through face C or D, the reflected beam does not pass through the cement at all and the beam is reflected by side II of the coatings.
Any real NBS is designed so that the incident light is split with a certain power ratio, such as 50%–50%. However, the phase shift of the transmitted beam is not guaranteed to be zero if the incident beam enters through a different surface, and that of the reflected beam is not guaranteed to be 180°. When an NBS is used at an oblique incident angle (45° in Fig. 2), the transmissivity and reflectivity of its coatings introduce additional phase shifts . Therefore, the Jones matrices of a real NBS must be rewritten asEq. (4).
The Jones vector of a polarized laser beam can be expressed as E = , where a and b are the electric field vector amplitudes in the horizontal and vertical directions, respectively, and δ0 is the difference between the phase retardations of the orthogonal polarization elements. This vector can be obtained with the DM using the following procedure. When the WP is in the xz plane, the intensities of the horizontal and vertical polarized beams split by the WP, Io and Ie, respectively, are detected. Then, the WP is rotated until the difference between the intensities of the split beams is maximized, and the intensities of the horizontal and vertical polarized beams, Imax and Imin, respectively, are measured. Finally, the Jones vector parameters can be determined by using
The Jones matrices of a real NBS can be determined by using the following procedure. First, the laser and OFI depicted in Fig. 1 are employed to generate an incident beam that is linearly polarized at 45°, and the DM is used to determine its Jones vector. Second, the NBS is placed in the incident beam so that the incident beam is perpendicular to face A, and the DM is used to determine the Jones vectors of the transmitted and reflected beams. Finally, the Jones matrices of the NBS can be determined by usingFig. 3. The Jones matrices of the NBS when the beam enters through faces B, C, and D can be obtained following the same procedure.
2.3 Quadrature phase error sensitivity of proposed HQLI
The quadrature phase error sensitivity to angular deviations of the wave plates in the proposed HQLI was theoretically analyzed using the Jones calculus method [4, 9], which is a convenient means of analyzing laser beam propagation in an HQLI. The Jones matrices listed in Table 1 were employed to examine a situation in which a beam enters NBS1 through face A and NBS2 through face C. When the angular deviation of one wave plate was analyzed, the deviations of the other two wave plates were set to zero. The intended angles of the optical axes of QWP1 and QWP2 were 0° and 45°, respectively, relative to the y-axis. Theoretically, the optical axis of a HWP can be at any angle relative to the y-axis, and in this study, 0° was used.
The analysis results are shown in Fig. 4. In the legend, “Eq. (4)” indicates that the Jones matrices of an ideal NBS, which are shown in Eq. (4), were used, while “Eq. (5)” signifies that the Jones matrices of a real NBS, which are provided in Eq. (5) and account for NBS imperfections, were used.
From the results obtained using Eq. (4), the following conclusions can be made: (1) no quadrature phase error is introduced when the angles of the optical axes of the HWP, QWP1, and QWP2 equal to their designed values, i.e., 0°, 0°, and 45°, respectively, relative to the y-axis; (2) deviations of the angles of the optical axes of the HWP and QWP1 from their ideal values do not cause quadrature phase errors, while those of the optical axis of QWP2 introduce obvious quadrature phase errors; (3) the quadrature phase error sensitivity to angular deviations of QWP2 is near zero when the optical axis of QWP2 has an angle of approximately 45°. However, when Eq. (5) is used, (1) a quadrature phase error of 2.7° is introduced even when the optical axes of the HWP, QWP1, and QWP2 are oriented at their intended angles of 0°, 0°, and 45°, respectively; this error is mostly caused by imperfection of a real NBS in the HQLI; (2) deviations of the optical axis of the HWP do not cause additional quadrature phase errors, while those of the optical axes of QWP1 and QWP2 do introduce obvious additional quadrature phase errors; (3) the quadrature phase error sensitivity to angular deviations of QWP1 and QWP2 when their optical axes are approximately equal to 0° and 45°, respectively, are significantly amplified. The quadrature phase error sensitivity to angular deviations of QWP2 is 0.15°/1° when the angle of its optical axis is near 45°. The quadrature phase error sensitivity to angular deviations of QWP1 can be up to 1.4°/1° when the angle of its optical axis is approximately 0°.
Therefore, deviations of the wave plates’ optical axes from their ideal angles, resulting from either instability or assembly error, are the main causes of quadrature phase errors in HQLIs. In terms of error sensitivity, NBS imperfections significantly amplify the quadrature phase error sensitivity to wave plate angle deviations, further increasing the error.
In terms of error sensitivity, NBS imperfections significantly amplify the quadrature phase error sensitivity to wave plate angle deviations.
Subsequent analysis revealed that, since the differences between the real transmissivity and reflectivity (around 0.65–0.69) and their ideal values (0.707), as well as the difference between the real transmissivity and real reflectivity themselves, were very small, as shown in Table 1, and since transmissivity and reflectivity errors mostly cause only DC offset and AC amplitude errors, the quadrature phase errors caused by transmissivity and reflectivity errors are negligible. Meanwhile, the differences between the real phase shifts of the transmitted and reflected beams and their ideal values (0° and 180°, respectively) are considerable. Phase shift errors are the main NBS imperfections that introduce quadrature phase errors in the proposed non-polarization beam splitting HQLI.
3. Method of achieving robustness
3.1 Basic concept
According to the analysis presented in Section 2.3, NBS imperfections, especially phase shift errors of the transmitted and reflected beams, increase the quadrature phase error sensitivity to wave plate angle deviations. Thus, even slight deviations of the wave plates’ optical axes from their intended angles, caused by either instability or assembly errors, result in significant measurement errors and decrease the robustness of the proposed non-polarization beam splitting HQLI.
NBS imperfections, especially phase shift errors, have two primary effects in the proposed HQLI: (1) the electric field vectors output from the interferometer part, Em and Er in Fig. 1, are not orthogonal, unlike in an ideal HQLI, because the sums of the phase shifts of the measured and reference beams produced by NBS1 in the interferometer part are unequal, i.e., τA1 + γC1 ≠ τB1 + γA1 in this research, where τij and γij are the phase shifts of the transmitted and reflected beams at face i of the jth NBS; (2) the measured and reference beams are not linearly polarized before passing through QWP2.
The basic concept of the proposed method of achieving robustness is to eliminate the two abovementioned effects, by compensating for NBS imperfections by yawing the HWP and QWP1 to tune their phase retardations. If these effects can be avoided, the polarization characteristics of the interferometer can be made close to those of an ideal HQLI, and the interferometer’s robustness can be improved.
According to crystal optics, when an incident laser beam is not normal to the wave plate due to the wave plate yawing away from its ideal angle, the effective thickness of wave plate increases, and the phase retardation of the wave plate can be changed.
In the subsequently presented theoretical analysis of the proposed method of achieving robustness, two assumptions have been made:
- (1) The influences of the transmissivity and reflectivity errors of NBS1 and NBS2 can be ignored, and their split ratios are ideal, i.e., 50%–50%.
- (2) All assembly errors can be ignored, except for those introduced by the yawing of the HWP and QWP1. The optical axes of the HWP, QWP1, and QWP2 are oriented at 0°, 0°, and 45°, respectively, relative to the y-axis.
The phase retardations of the HWP and QWP1 are denoted as ΓH and ΓQ, respectively. The intensities I1–I4 reaching PD1–PD4 depend on δ, the phase shifts of the NBSs, ΓH, and ΓQ:
When M - R = π, M = 4π, and R = 3π, four quadrature signals can be obtained. The expression M - R = π means that the electric field vectors output by the interferometer part are strictly orthogonal, while M = 4π and R = 3π indicate that the measured and reference beams, respectively, that are obtained before passing through QWP2 are strictly linearly polarized. Therefore, the two aforementioned effects have been eliminated.
Using the detected NBS phase-shift parameters listed in Table 1, the required phase retardations of the HWP and QWP1 can be calculated to be
Therefore, if the required values of ΓH and ΓQ can be realized by yawing the HWP and QWP1, the two aforementioned effects can be effectively avoided, and the NBS imperfections can be adequately compensated for. Thus, by following this procedure, the primary remaining causes of quadrature phase error in the proposed non-polarization beam splitting HQLI can be eliminated.
3.2 Wave plate phase retardation change by yawing
In this section, a method of changing a wave plate’s phase retardation by yawing is proposed. The relationship between the phase retardation and the yaw angle can be theoretically modeled according to the law of refraction.
Zero-order crystalline quartz wave plates are widely used in interferometers. A zero-order wave plate is designed so that the phase shift it creates is exactly one-quarter or one-half of a wavelength. This type of wave plate consists of two plates stacked together, where the fast axis of one plate is aligned with the slow axis of the other, as shown in Fig. 5. When the incident ray is perpendicular to the wave plate surface, the phase retardation of the wave plate satisfies
However, when the incident ray is not perpendicular to the wave plate surface and the yaw angle is β, as shown in Fig. 5(b), then the phase retardation of the wave plate can be calculated as follows by using the law of refraction:
In this research, according to the material and dimensional parameters of the HWP and QWP, no = 1.542605; ne = 1.551650; d1 = 1 mm; d2 = 0.965 mm and 0.982 mm for the HWP and QWP, respectively; and β ∈ [0°, 5°]. The phase retardations after yawing are shown in Fig. 6, which reveals that the phase retardations of the HWP and QWP are 199° and 119.2° when the yaw angles are 3.5° and 4.4°, respectively.
Therefore, the phase retardations of the wave plates can be changed to the values required in Section 3.1 by yawing.
3.3 Quadrature phase error sensitivity after wave plate yawing
Using the same analysis method as in Section 2.3, the quadrature phase error sensitivity to wave plate angle deviations after applying the method used to achieve robustness was theoretically analyzed, and the results are shown in Fig. 7.
Based on Fig. 7, it is evident that applying the proposed method has the following effects: (1) the quadrature phase error is zero when the optical axes of the HWP, QWP1, and QWP2 are at their ideal angles of 0°, 0°, and 45, respectively; (2) the angular deviation of QWP1 does not cause quadrature phase error, the quadrature phase error resulting from the angular deviation of the HWP is negligible, and the quadrature phase error due to the angular deviation of QWP2 is greatly suppressed; (3) the error sensitivities to wave plate angle deviations are obviously reduced and are all zero when the wave plates’ optical axes are oriented at their intended angles; most notably, the quadrature phase error sensitivity to angular deviation of QWP1 is reduced from 1.4°/1° to zero; (4) the three curves obtained after yawing are nearly identical to those in Fig. 4 that correspond to the ideal case and were determined using Eq. (4).
Therefore, by applying the proposed method of yawing the wave plates, ultra-low quadrature phase error sensitivity to wave plate angle deviations can be obtained, and the HQLI can be made significantly more robust than the existing configurations.
3.3 Procedure used to improve HQLI robustness
Figure 8 lists the steps used to improve the HQLI’s robustness by yawing the HWP and QWP1. In this procedure, the HQLI is first set up with the optical axes of the HWP, QWP1, and QWP2 oriented at 0°, 0°, and 45°, respectively. The PZT is moved uniformly, and the signals are sampled to enable calculation of the quadrature phase error using the method described in . Second, QWP1 is yawed to cause the Lissajous graph of I1 versus I2 to become a straight line with a negative slope, indicating that I1 and I2 have opposite phases. Third, the HWP is yawed slowly to minimize the quadrature error. Then, Step 2 is repeated. Finally, once the HWP and QWP1 have been yawed to the required angles, the HQLI will exhibit ultra-low error sensitivity and outstanding robustness.
4.1 Experimental setup
Experiments were conducted to verify the proposed method of achieving HQLI robustness. The wave plates were set on rotatable mounts to adjust their optical axes. A single-axis PZT (PI P753.1CD, resolution: 0.05 nm) was used to generate standard displacements during the experiments. A triangle-wave voltage signal was generated by a signal generator (Keysight 33522B, 16Bits) and was amplified by a piezo amplifier and controller (PI E665). The PZT was driven to move uniformly. The photocurrent signals of PD1–PD4 were converted into voltage signals by a signal-processing board made by our research group and were then equidistantly sampled by a four-channel data acquisition card (NI PXI 4462, 24 bits, sampling rate: 200 kHz). A heat-shielding box was used to enclose the experimental setup, thus reducing the effects of environmental temperature variations. The experimental setup was set on a self-leveling isolation system (Thorlabs PFA51508). The calculations and data analysis were performed offline on a National Instrument (NI) computer. The Lissajous graphs and interference signals were displayed using the LabVIEW platform.
4.2 Experimental quadrature phase error sensitivity results
The quadrature phase error sensitivities to wave plate angle deviations were investigated experimentally by measuring the quadrature phase errors corresponding to various angles of the optical axes of the HWP, QWP1, and QWP2 after assembly. A residual quadrature phase error of ~0.05° was achieved after the method of achieving robustness was applied.
The experimental results are presented in Fig. 9. It is evident that (1) the quadrature phase error is approximately zero when the optical axes of the HWP, QWP1, and QWP2 are oriented at 0°, 0°, and 45°, respectively; (2) angular deviations of the HWP and QWP1 introduce negligible quadrature phase errors, while those of QWP2 do introduce significant errors; (3) the quadrature phase error sensitivities to angular deviations of the HWP, QWP1, and QWP2 are 0, 0.05°/1°, and 0, respectively, when their optical axes are oriented at their intended angles. The differences between the experimental and theoretical results were mainly caused by errors in the yaw angles of the HWP and QWP1 as well as other assembly errors. Disturbances and noise also influenced the detected quadrature signal amplitudes, thus introducing quadrature phase calculation error.
4.3 Cyclic nonlinearity after performing method of achieving robustness
A nonlinearity experiment was conducted after applying the proposed method of achieving robustness. In this experiment, the PZT was driven to move linearly for a distance equal to one wavelength. The simple gain and offset correction method was applied to correct the DC offset and AC amplitude errors in the quadrature signals, and third-order polynomial fitting was used to remove the PZT nonlinearity [7, 8]. The results of this experiment are depicted in Fig. 10, which shows that the nonlinearity of the proposed HQLI is 0.1 nm (peak-to-valley). Theoretically, the nonlinearity resulting from residual quadrature phase error in the proposed HQLI is equal to two cycles per fringe. However, the periodicity of two cycles per fringe is not obvious in the experimental results. Therefore, the residual nonlinearity mainly arises from PZT position variations, electrical noise, local variations of the refractive index of air, and environmental vibrations.
In this report, a new HQLI configuration and a method of improving its robustness and achieving long-term accuracy were proposed and demonstrated. Since this HQLI is based on non-polarization beam splitting, polarization crosstalk is effectively avoided in both its interferometer and detection parts. The theoretical derivations and experimental applications of the Jones matrices of NBSs were also provided in this paper, and theoretical analyses of the influences of NBS phase shift errors were presented. Based on the quadrature phase error sensitivity to wave plate angle deviations, a method of compensating for NBS phase shift errors by yawing the wave plates, and thereby increasing the robustness of the proposed HQLI, was described. Our experimental results indicate that the HQLI has quadrature phase error sensitivities of 0, 0.05°/1°, and 0 to angular deviations of the HWP, QWP1, and QWP2, respectively, when they are assembled with their optical axes at the intended angles. A peak-to-valley nonlinearity error of 0.1 nm was achieved and could be further improved if the noise and environmental disturbances could be reduced.
If the proposed method of achieving robustness is applied, slight deviations of the wave plates in the designed HQLI, which could be caused by instability or assembly errors, will not result in considerable measurement errors, and long-term accuracy and outstanding robustness can be realized. The proposed HQLI and method of achieving robustness can be employed wherever long-term accuracy, ultra-low cyclic nonlinearity, and real-time performance are required. Our further research will focus on dynamic applications of this robust HQLI, such as high speed vibration measurement with high accuracy, low data sampling rate, and small memory assumption.
National Natural Science Foundation of China (NSFC) (51675139, 51105115).
The authors would like to thank Dr. Junying Li and Master Yuanwei Jiu for their useful suggestions and experimental assistances
† These authors contributed equally to this work.
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