Nonlinearity error in homodyne interferometer caused by multi-order Doppler frequency shift ghost reflections

Pengcheng Hu; Yue Wang; Haijin Fu; Jinghao Zhu; Jiubin Tan

doi:10.1364/OE.25.003605

1. Introduction

Rapid progress in the fields of nanotechnology, ultrasonic signal detection, and semiconductor industry has created an increasing demand for highly accurate dimensional measurements [1–4]. In this context, homodyne interferometry is an attractive measurement technique that offers noncontact measurement with high resolution [3]. Temperature variation, vibration, and air turbulence are the main error sources of homodyne interferometers. Even when these parameters lie within specified limits, the device accuracy is still limited by the periodic error, which lies in the range of sub-nanometers to tens of nanometers [5–7].

A homodyne interferometer consists of beam splitters, polarizing beam splitters (PBSs), waveplates, and reflectors. Imperfections in the PBS in a homodyne interferometer lead to “leakage” of the resulting s- and p-polarized beams. For example, an undesired s-polarized beam can be transmitted along the p-polarized beam direction and induce measurement errors. Further, imperfections in the waveplates yield a retardation error. For example, an imperfect quarter-waveplate can yield a retardation of 45° along with an additional unwanted error. In this context, Wu has reported phase mixing caused by these two factors [8]. Further, the axes of the optics systems may not be aligned perfectly. In addition to alignment errors, Heydemann has reported that unequal gains of the detectors in the detection system also induce a periodic error, which does not arise from the interferometer itself [5]. In addition, the power of the single-frequency laser used in the homodyne interferometer is often unstable. All these factors induce periodic errors such as DC offsets error, unequal AC amplitudes error, and quadrature phase delay error [7]. Many methods have been proposed to compensate for periodic errors [5–13], which have been found to efficiently offset nonlinearity in the errors. Nevertheless, a nonlinearity whose origin is different from that of these “conventional errors” still exists in the measurement signal.

In this study, we report a hitherto undiscovered periodic error caused by ghost beams, and we propose a new model to explain the error. We compare the Lissajous trajectories of the “new” periodic error and conventional periodic errors. In a homodyne interferometer, multiple reflections at the interfaces of the target mirror induce multi-order Doppler frequency shift (DFS) ghost beams. The multi-order DFS ghost beams participate in the final interference and lead to multi-order periodic errors. We use a corner cube retroreflector (CCR) to substitute the mirror as a reflector and find that the periodic error resulting from ghost beams significantly decreases.

2. Conventional nonlinearity error in homodyne interferometer

Figure 1 shows the schematic of the homodyne interferometer used in the study along with Lissajous curves of the periodic errors. In this setup [Figs. 1(a) and 1(b)], a frequency-stabilized 633-nm He–Ne laser beam with 45° linear polarization is incident on a polarizing beam splitter (PBS₁). The main beam is divided into two beams; one passes through a quarter-wave plate (QWP₁) and is directed to a fixed reference reflector while the other is incident on a moving target reflector after passing through QWP₂. Both the beams from the mirrors travel back and interfere at PBS₁. After passing through the quadrature detection system, the interference signals detected by four detectors (corresponding to signals $I_{1}$ to $I_{4}$ in the figure) have phases of 0°, 90°, 180°, and 270° in that order. We can obtain two orthogonal signals via subtraction of each of the two signals with a phase difference of 180°. In the ideal case, the intensity of the orthogonal signals is expressed as

I_{x} = A \cos (φ), I_{y} = A \sin (φ),

φ = arc \tan \frac{I_{y}}{I_{x}},

here, A and φ represent the amplitude and phase difference of the two signals, respectively. The Lissajous trajectory of the two orthogonal signals is a perfect circle, as shown in Fig. 1(c). With conventional periodic errors, the intensity of the two signals is expressed as

I_{x} = A_{x} \cos (φ + δ) + B_{x}, I_{y} = A_{y} \sin (φ) + B_{y},

here,

A_{x}

and

A_{y}

represent the AC amplitudes and induced unequal AC amplitudes error if they are not equal

(A_{x} \neq A_{y}),

B_{x}

and

B_{y}

denote the DC offsets error, and δ is the quadrature difference. The periodic error can be expressed as

Fig. 1 Schematic of homodyne interferometer and illustration of conventional periodic errors. (a) Schematic of homodyne interferometer with polarizing beam splitter (PBS), quarter-wave plate (QWP), and reflectors, (b) Schematic of quadrature detection system with PBS, QWP, half-wave plate (HWP), beam splitter (BS). (c) Depiction of Lissajous distortion in homodyne interferometer. (d) Depiction of nonlinear errors in homodyne interferometer. PS: perfect signal, DC: DC offsets error, AC: unequal AC amplitudes error, PH: quadrature phase delay error.

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N L = arc \tan (I_{y} / I_{x}) - φ + m π .

Figure 1(c) depicts the Lissajous trajectory with nonlinear errors, i.e., DC offsets error, unequal AC amplitudes error, and quadrature phase delay error. The corresponding nonlinearity error is shown in Fig. 1(d). The conventional nonlinearity error usually ranges from sub-nanometers to tens of nanometers [7].

3. Nonlinearity model based on multi-order DFS ghost beams

In a homodyne interferometer, ghost reflections occur at each interface through which the laser beams have travelled. In order to explain our nonlinearity model, in the measurement arm of our homodyne interferometer, we include a target reflector, PBS, and QWP, as shown in Fig. 2. A plane mirror [14] is used as the reflector in Figs. 2(a)-2(d), while a CCR is used in Figs. 2(e)-2(h). As shown in Fig. 2(e), when the target reflector moves, a first-order Doppler frequency shift (DFS) beam is generated. In addition, second-order and third-order DFS ghost beams are generated, as shown in Fig. 2. Parasitic interference signals (PIS) are generated along with the main measurement signal (MMS) when multi-order DFS ghost beams participate in the final interference. Note that the kth-order PIS is induced not only by the interference between the kth-order ghost beam and the reference beam but also by the interference between the nth-order ghost beam and the (n-k)th-order ghost beam in a homodyne interferometer.

Fig. 2 Ghost beams in interferometer with mirror and CCR. (a) Main beam with first-order Doppler frequency shift (DFS). (b) Ghost beams with zero-order DFS. (c) Ghost beams with second-order DFS. (d) Ghost beam with third-order DFS. (e) Main beam with first-order DFS. (f) and (g) Ghost beams with second-order DFS. (h) Ghost beam with third-order DFS. M: mirror, GRP: ghost reflect point, GB: ghost beam.

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In order to investigate the periodic error resulting from the multi-order DFS ghost beams, we establish a model based on the interference principle. As shown in Figs. 2(a) and 2(e), when the target reflector moves, the main beam in the measurement arm generates a first-order DFS. We can express the electric field of the main beam without ghost beams as

E_{m} = A_{m} \exp {i [ω t + ϕ (t) + Ψ_{m}]},

where

ω = 2 π f

represents the angular frequency,

A_{m}

denotes the amplitude,

Ψ_{m}

is the initial phase of the main measurement beam, and

ϕ (t)

represents the phase shift caused by the DFS.

Simultaneously, the multi-order DFS ghost beams generated by the ghost reflections trace the same path and participate in the final interference along with the main beam as shown in Fig. 2. The multi-order DFS ghost beams from second-order to nth-order can be expressed as

E_{g h} = \sum_{k = 2}^{n} A_{g h k} e^{i [ω t + k ϕ (t) + Ψ_{k}]},

here,

A_{g h k}

is the amplitude of kth-order ghost beam and

Ψ_{k}

denotes the corresponding initial phases of the kth-order ghost beam. In mirror model [Fig. 2(c)], the second-order ghost beam is generated after main beam reflects at the surface of QWP and mirror. In CCR model [Figs. 2(f) and 2(g)], comparing to that in the mirror model, the generation of the second-order ghost beam needs an additional reflection at the surface of CCR, whose reflectivity is significantly smaller than 1. Main beam reflects twice at the surface of QWP to generate a third-order ghost beam in both model in Figs. 2(d) and 2(h). This means the amplitude of the second-order ghost beam in CCR model is much smaller, while the third-order ghost beam is almost equal to that in mirror model. After several times reflections, the amplitude of nth-order (n > 3) ghost beam is extremely smaller than the second- and third-order ghost beams.

As for the reference arm in Fig. 1(a), the beam split by the PBS is incident upon the reference mirror. After several reflections at different surfaces, ghost beams are generated. However, the reflector in the reference arm is fixed, and thus, no DFS is generated. The electric field of the reference beam can be expressed as

E_{r} = A_{r} \exp [i (ω t + Ψ_{r})],

where

A_{r}

is the amplitude of reference beams, and Ψ_r denotes the initial phase of the reference beam.

After passing through the homodyne interferometer in Fig. 1(a), according to the interference principle, we can express the intensity of the final interference signal as

I_{s} = Re [(E_{m} + E_{g h} + E_{r}) \cdot (E_{m}^{*} + E_{g h}^{*} + E_{r}^{*})],

here,

E_{m}^{*}

,

E_{g h}^{*}

and

E_{r}^{*}

represent the complex conjugates of

E_{m}

,

E_{g h}

and

E_{r},

respectively. Equation (8) can be expressed more compactly as

I_{s} = Re [\sum_{k = 1}^{n} Γ_{k} e^{i k ϕ (t)}],

here,

Γ_{0} - Γ_{n}

denote the relative intensities of various orders of DFSs. Here, we remark that there is more than one frequency component in the final interference signal expressed in Eq. (9).

As shown in Fig. 3, MMS is the normal beat signal, while the PISs contribute to the generation of periodic errors. Via analyzing each PIS component, the phase error resulting from a kth-order parasitic interference signal can be expressed as

d ϕ_{k} = arc \tan \frac{Γ_{1} \sin ϕ (t) + Γ_{k} \sin k ϕ (t)}{Γ_{1} \cos ϕ (t) + Γ_{k} \cos k ϕ (t)} - ϕ (t),

where k = 2, 3,…, n. The corresponding periodic error is

ε_{k} = λ d ϕ_{k} / (4 π) .

The amplitude of the kth-order ghost beam is significantly smaller than that of the main beam (

Γ_{k}

<<

Γ_{1}

). So the periodic error can be further expressed as

ε_{k} = \frac{Γ_{k}}{Γ_{1}} \sin [(k - 1) ϕ (t)] \times \frac{λ}{4 π},

and further, the peak-to-peak value of the periodic error induced by the kth-order PIS is

ε_{k \max} = λ Γ_{k} / (2 π Γ_{1}) .

It can be observed from Eq. (11) that the periodic error is determined by the phase

ϕ (t),

order k, and relative intensities of the ghost beam and main beam. As shown in Fig. 3(b), there are several frequency components in the final interference signal, namely, MMS, PIS2, PIS3, etc, in that order. All of these frequency components contribute to the nonlinearity error. For the model without considering ghost beams, there is only one frequency component in the measurement signal, as shown in Fig. 3(a).

Fig. 3 Comparison of periodic error source between interferometers without and with consideration of multi-order ghost beams. (a) and (b) are spectrums of interference signal without and with multi-order DFS ghost beams, respectively.

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As for the interferometer with a quadrature detection system shown in Fig. 1(b), the measurement beam, reference beam, and ghost beam are classified into four beams each: $E_{m i}, E_{g h i}, E_{r i} (i = 1 - 4) .$ Consequently, the intensity signal from each detector in the detection system in Fig. 1(a) can be expressed as

I_{i} = Re [(E_{m i} + E_{g h i} + E_{r i}) \cdot (E_{m i}^{*} + E_{g h i}^{*} + E_{r i}^{*})] .

In the detection system, the intensity signals at the detectors are ideally two sinusoids with a 90° phase difference in the absence of conventional errors. In consideration of the ghost reflection, we express the intensity relations of phase quadrature signals as:

I_{x} = I_{1} - I_{3} = \sum_{k = 1}^{n} Ω_{k} \cos k ϕ (t),

I_{y} = I_{2} - I_{4} = \sum_{k = 1}^{n} Ω_{k} \sin k ϕ (t),

where

Ω_{1} - Ω_{n}

represents the relative intensities of various orders of the DFS ghost beams in the x and y channels. The nonlinearity of the phase measurements can then be expressed as

N L = arc \tan \frac{I_{y} (t)}{I_{x} (t)} - ϕ (t) + m π .

In homodyne interferometers, the existence of ghost beams distorts the Lissajous trajectory, which is originally a perfect circle. In Fig. 4(a), the space between the distorted circle and its least-square circle is amplified by a factor of $1 .5 \times 10^{5},$ so we could get a clear look of nonlinearity of the “new” error. The reflectance of surface of the CCR is 0.5%. Besides, the reflectance of QWP is 0.15%, which is usually the smallest reflectance of high-quality anti-reflection coating. As shown in Fig. 4(b), the nonlinearity error caused by ghost reflections is about 0.4 pm.

Fig. 4 Depictions of the nonlinear error in homodyne interferometer. (a) Lissajous distortion. (b) Nonlinearity error, DS: distorted signal, PS: perfect signal.

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4. Experimental validation

In order to verify the above model, we establish an experimental setup as shown in Fig. 5. In the setup, a beam from a single-mode frequency-stabilized laser is split by a PBS and divided into two beams that are polarized vertically and horizontally. These two beams propagate along their separate paths, i.e., the measurement arm and the reference arm. Subsequently, a phase difference is generated between the two beams. After the reflected beams pass through the QWPs for the second time, their polarization states are rotated by 90°, and both propagate again through the PBS. The final beam is received by the detector and transformed into an electrical signal. During the experiment, a neutral density filter (NDF) is positioned between QWP₂ and reflector [mirror in Fig. 5(a) or CCR in Fig. 5(b)], so we can vary of the reflectance via adjustment of the NDF transmittance. Besides, the reflectance at the surface of CCR is less than 0.5%.

Fig. 5 Experimental interferometer setups used for validation. (a) Setup of mirror model. (b) Setup of CCR model. P: polarizer.

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In Fig. 6, we observe that there is more than one peak in both sets of detected signals, that is, there is more than one frequency component in the final signal. These components correspond to MMS, PIS2, PIS3, etc. in that order in Fig. 3(b). All the PISs contribute to the periodic errors. Because the interference signal is detected in the absence of the quadrature detection system, the DC component is larger than MMS.

Fig. 6 Experimental results obtained in mirror model (a)-(c) and CCR model (d)-(f). (a) and (d) are obtained using NDF with a ghost reflectance of 3.77%; (b) and (e) are obtained using NDF with a ghost reflectance of 6.67%; (c) and (f) are obtained using NDF with a ghost reflectance of 18.53%.

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In Fig. 7, with the increase of reflectance, the error caused by ghost reflections increases, which is different from conventional periodic errors. Comparing to a plane mirror, using a CCR as the reflector leads to a much smaller second-order periodic error and an almost equal third-order periodic error. In CCR model, the incident beam and reflect beam travel in an unaxial way, while they travel in an axial way in mirror model. So there are less ghost beams interfering with other beams, This result matches the theory well, showing that using the CCR as a reflector would significantly reduce the periodic error caused by ghost reflection.

Fig. 7 Different order periodic errors caused by different reflectance in mirror and CCR model.

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5. Conclusion

In general, periodic errors in homodyne interferometers, such as DC offsets error, unequal amplitudes error, and quadrature phase delay error, are caused by imperfect optics, misalignment of the axes of the optics, unequal gains of the detectors, and so on. In this study, we demonstrate the existence of a hitherto undiscovered periodic error whose origin is different from those of the abovementioned errors. Further, we construct a model to explain this “new” error. This model considers the fact that in a homodyne interferometer, multi-order DFS ghost beams are generated when the target mirror moves. A parasitic interference signal is generated when the ghost beams participate in the final interference. And it proves that using CCR as the reflector is effective to reduce periodic error caused by ghost reflections. Our findings indicate that this model can be used in the further development of highly accurate optical interferometers.

Funding

National Natural Science Foundation of China (NSFC) (51675138, 51305105).

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Nonlinearity error in homodyne interferometer caused by multi-order Doppler frequency shift ghost reflections

Abstract

1. Introduction

2. Conventional nonlinearity error in homodyne interferometer

3. Nonlinearity model based on multi-order DFS ghost beams

4. Experimental validation

5. Conclusion

Funding

References and links

Cited By

Figures (7)

Equations (15)

Optics Express