1. Introduction

The Michelson interferometer is an instrument of high precision in the displacement measurement, has been widely used in accuracy measuring, errors compensation and new product acceptance of CNC machine tools [1–3], as well as in the field of angle measurement [4]. Original Michelson laser interferometer which runs with a typical optical pathway based on plane mirrors is suitable for displacement measurements. However, this type is limited by effects from the pitch and yaw of the plane mirror during the measurement as soon as an plane mirror is considered as the retroreflector mounted at a moving target, which rigidly demands for a measured object (e.g. a guideway), otherwise, it will stop working because there isn’t a normal interference between measurement beam and reference beam. There is another type of Michelson laser interferometer which is using corner cube prisms to replace the plane mirrors. It can smoothly deal with the embarrassment of the original type, and it is immune to the measured object’s disturbances when a corner cube prism is adopted as the retroreflector. And it rapidly becomes an excellent metering instrument in one-dimensional measurement. But the second type also has a drawback that it doesn’t allow lateral movements of retroreflector because the measurement beam and the reference beam can’t interfere under the lateral moving [5–7]. So, the application of it is restricted in two-dimensional (2D) measurements for CNC machine tools.

In order to possess the immunity properties of the rotary disturbances and lateral moving of the retroreflector, a plane mirror interferometer with four times optical paths is presented later. This arrangement combines the advantages of the previous two methods, and has been developed for circular tests of CNC machine tools and positioning of a photomask measuring in 2D space ^[1^,2^]. Obviously, the errors analysis would be a critical issue that new arrangement should be performed to ensure high accuracy of the laser interferometer.

At present, measurement errors of laser interferometer are focused on misalignments, the Abbe error and rotary angles and so on [3, 5, 6, 8–12]. N. Bobroff, from IBM Thomas J Watson Research Center, has analyzed measurement errors of a plane mirror interferometer by using a vector method, and argued that the transducer axis for the interferometer is the normal direction of the target mirror under the misalignment analysis [8, 9]. W. Gao, et al, from Tohoku University, Japan, obtained a better straightness measurement accuracy through compensating for pitch and yaw angles of the target retroreflector in the measurement of a precision linear air-bearing stage [3]. H-J Büchner and G Jäger, from Technology University Ilmenau, Germany, presented a novel plane mirror interferometer without using corner cube reflectors, and pay efforts to reduce the influences from tilts of the plane mirrors, the Abbe error and quasi-orthogonality between the measurement beam and the measured mirror during the measurement [5]. Z. Zhang and C. H. Menq, from The Ohio State University, USA, have absorbed the coordinate transformation method to complete the errors modeling of a laser interferometer. Furthermore, they achieved six-axis motion measurement for real-time applications with this model [6]. S. Awtar and A. H. Slocum, from Massachusetts Institute of Technology (MIT), USA, discussed the various errors with the equations method systematically, and explained characteristics of the target block alignment error in XY stage metrology explicitly [12].

The above studies show that, target misalignments, rotary angles and the Abbe error are mainly reasons that measurement accuracy of an interferometer is weakened. Nevertheless, those methods are rather limited in the static state and separation of the disturbances, and can’t reveal the dynamic characteristics of the error and coupling effects of the factors. This paper firstly establishes the errors formula of the interferometer for a plane mirror interferometer according optical paths that the measurement optical beam propagates in a variety of components including a polarization beam splitter (PBS), a corner cube prism (C), and air, subsequently, deduces a dynamic error model by the partial differentiation theory, then studies each disturbed factor from the theoretical view systematically. Compared with the current proposed methods, the errors analysis for PBS and C are strengthened, and the entire measurement errors are classified into static and dynamic according to the characteristics of the disturbed factors, furthermore, the intercoupling effects of those two factors are particularly discussed to improve the measurement accuracy of the interferometer.

2. Optical paths modeling for components

Figure 1(a) is a typical arrangement of a plane mirror interferometer with a PBS, and is composed of two λ/4 wave plates (W), two plane reflecting mirrors (M), one PBS and one polarizer (P) and one photodetector (PD), etc. This type is similar to the original Michelson laser interferometer, and is used for homodyne interferometer and also heterodyne interferometer with the similar work principles. Taking heterodyne interferometer for an example, a dual-frequency beam hits the PBS, and is split into two beams which vibration directions are orthogonal and we named them as S-beam and P-beam respectively. S-beam is reflected to the reference mirror (M) by the PBS and is rebounded by the M, through the W. Light polarization direction rotates 90 degrees because of passing through a λ/4 wave plate two times. It transmits the PBS, reaches the PD. S-beam is used for reference beam. Accordingly, P-beam transmits the PBS and W, reaches the target mirror (M), and returns through the W again. This time it is reflected by the PBS and also reaches the PD. This beam is considered as measurement beam. The interference can be found at the PD. It is also influenced by misalignments, rotary angles and the Abbe error of the target plane mirror.

 

Fig. 1 (a) Typical arrangement of a plane mirror interferometer. (b) Optical paths of the plane mirror interferometer with four times paths.

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Figure 1(b) shows that a plane mirror interferometer with four times optical paths is composed of the PBS, C, M, W, P, PD.

The principle is following: a beam with dual-frequency from a laser is split into S-beam and P-beam by a PBS. The S-beam is reflected to C, and rebounded which is parallel with original direction and again reflected by the PBS. Here S-beam works as reference beam. The P-beam is transmitting through the PBS and a W, hitting M as the measured target, and reflected by the M, which transmits through the W again, arrives at the PBS, because light polarization direction rotates 90 degrees through a λ/4 wave plate twice, and reflected to another C by the PBS, rebounded which is parallel with original direction, and reflected again by the PBS, subsequently transmits through the W, reaches the M secondly and reflected, goes through the W and arrives at the PBS, then it is transmitting over the PBS due to changed the polarization direction. This is considered as measurement beam. At last, the reference and measurement beams both through a P, and interference occurs. Interference signal of displacement is detected by a PD. At the PD, the optical interference signal is transformed to electrical signal which subtracts the reference signal from the laser measuring head, then can be used for solving of the measured displacement value by counting the signal frequency. So, the displacement measurement is achieved by this method [1].

Due to the measurement beam reaches the target mirror twice, the interferometer achieves four times optical paths and features the higher resolution than the common. The method has been described and implemented in detail in Ref [1]. The optical path difference (OPD) changes due to measured displacement, can be obtained by counting the interference signal frequency to calculate the target value of the displacement. But the OPD is also changed by some disturbed factors, which would bring out errors during the measurement. Therefore, the disturbed factors should be considered for the error analysis.

Comparing with the arrangement of the plane mirror interferometer with PBS, optical paths of the plane mirror interferometer with four times paths is more complex for error analysis, it is also a typical and usual interferometer, e. g. applied to nano-metrology of the photomask and the micro-semiconductor industry, etc [2, 13–15]. Moreover, there are common actions actually induced by the misalignments and rotations. So, it is applicable to improve the measurement accuracy though error analysis of a plane mirror interferometer is mainly conducted on the optical paths of the plane mirror interferometer with four times paths.

2.1 Optical path modeling for the PBS

Figure 2(a) shows the measurement schematic diagram of the interferometer. Figure 2(b) and 2(c) illustrate that this arrangement is immune to the disturbed tilted angle of the retroreflector. However, the plane mirror has a pitch or a yaw angle, and will bring out the errors for the measurement optical paths. It is necessary to focus on the errors caused by the disturbed angles.

 

Fig. 2 (a) The measurement schematic diagram of the interferometer. (b) Optical paths with the pitch angle. (c) Optical paths with the yaw angle.

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Suppose the PBS is perfect and doesn’t have manufacturing defects. According to the geometric paths in Fig. 2(a) and 2(b), the optical paths of measurement beam (OPMB) for the PBS could be divided into two parts that one part will change with the angles and the other part is constant. So, OPMB that only contains the deviation of the pitch angle could be showed in the Fig. 2(b), it can be expressed as:

Similarly, OPMB for the PBS that only contains the deviation of the yaw angle could be showed in the Fig. 2(c), it can also be written as:

Apparently, for an arbitrary angle θ in space, OPMB for the PBS could be expressed as:

An arbitrary angle can be decomposed into the roll angle around the X axis, the pitch angle around Y axis, and the yaw angle around the Z axis in the Cartesian. Because the roll angle of the plane mirror doesn’t affect the optical paths, so it can be ignored. Therefore, OPMB for the PBS will change due to components of the pitch and yaw in the case that the plane mirror only has an arbitrary rotary angle, and will be proportional to the pitch and yaw angle respectively.

2.2 Optical path modeling for the C

Figure 3 shows optical paths of the light beam in the C. The H means the height of the C, and α means the angle constant. The β and β’ mean incident and refraction angles respectively. OPMB for the C will be influenced with the changes of the incident angle when the light beam transmits from the air to the C [16]. In addition, the defects of the C will cause a phase error [17]. OPMB for the C is a constant value with a certain incident angle on the assumption that the C is zero defect. Here, it is focused on deviations that extra optical paths are brought by the changes of the incident angle.

 

Fig. 3 Optical paths of the light beam in the C with a certain incident angle.

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Figure 3(c) shows that the geometric path JMNB is equivalent to the straight line JB’. So, the OPMB for the C can be expressed as:

2.3 Optical path modeling for the air

Figure 4 shows misalignments of the plane mirror. To further analyze the changes of the measurement vectors, the ideal coordinate system is introduced to error demonstrations for the measurement. Assume that the ideal coordinate index direction is the nominal motional direction.

 

Fig. 4 (a) Misalignments of the plane mirror in the ideal coordinate system. (b) Misalignments of the plane mirror during the measurement without θ ₀. (c) Optical paths of the measurement beam in the air.

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In the Fig. 4(a), $\overrightarrow{I_x}$ and $\overrightarrow{I_x^{\text{'}}}$mean the vectors of the incident and reflective light respectively during the X direction measurement; $\overrightarrow{N_x}$ means a normal vector to the plane mirror of the X direction; $\overrightarrow{D_x}$ means the vector of the measured displacement in the X direction practically; θ₀ means an angle between ideal coordinate system and the normal direction to the plane mirror; θ is an arbitrary tilted angle in space in the previous section, here it also means the incident angle of $\overrightarrow{I_x}$; ϕ means the angle between $\overrightarrow{N_x}$ and $\overrightarrow{D_x}$.

To simplify error modeling, the θ ₀ can be eliminated with the refined operation, making the normal direction of the plane mirror similarity to the measured nominal motional direction. Figure 4(c) shows the optical paths of the measurement beam in the air. Suppose $\overrightarrow{I_x}$ is perpendicular to the side of the PBS (it may be qusi-perpendicular in practice with few effects produced for the geometric modeling). In the Fig. 4(c), the OPMB for the air and other geometric relationships can be expressed as:

The OPMB for the air can be obtained by Eqs. (9), (10) and (11) as:

L_air, reflected twice by the M in the Fig. 2, and means the OPMB for the air; L_AI means sum of the geometric dimension of $\overrightarrow{I_x}$ and $\overrightarrow{I_x^{\text{'}}}$ in the Fig. 4(c) (is equal to $EF+FG$); L_IX means geometric dimension between the incident point of the plane mirror and the PBS in the $\overrightarrow{I_x}$ direction (see EF); x means geometric dimension between the incident point of the plane mirror (F point) and the PBS in the $\overrightarrow{D_x}$ direction. Besides, the OPMB for the component W is considered as a constant value because the W is so thin that the OPMB almost doesn’t shift with the changes of the incident angle in a small range. It can be signed as B.

So, the sum of the OPMBs for the components can be calculated in the initial state where the tilted angle of the plane mirror is θ. For the OPMB for the C, the β is equal to 2θ in the previous section. And Eq. (8) can be written as:

The sums of OPMBs can be expressed as:

Here, f_M means sum of OPMBs for the components; B means OPMB for the W, a constant value. According to Eqs. (5), (7), (12) and (14), the sums of the OPMBs can be also expressed as:

3 Optical path difference (OPD) modeling for the interferometer

Generally, the reference arm is static and its optical path keeps constant during the measurement. Its optical path can be signed as a constant value R. Then, the OPD of the interferometer can be expressed as:

Here, f means the OPD of the interferometer; f_R means the optical paths of the reference arm, which is equal to the constant value R.

From Eq. (15) and Eq. (16), and make K = A + B − R, the f can be written as:

Suppose that the PBS and the C is made of the same material and their refractive indices are the same and constant. It is signed as,

According Eqs. (6), (13) and (18), and the following equation is obtained as:

Equation (17) can be abbreviated by the Eq. (19) as:

Equation (6) with Eq. (18) can be transformed as:

The OPD can be solved ultimately by Eq. (20) and Eq. (21), as:

Equation (22) indicates that the OPD of the interferometer mainly determined by the x, θ, ϕ and n, and it is a function of four variables x, θ, ϕ and n in the certain static state. Then, each variable has a variable micro-quantity such as displacement dx, rotary angle dθ, rotary angle dϕ and air refractive index fluctuation dn, which will cause the changes of the OPD. In principle, the contribution of the dx does is expected for the changes of the OPD, therefore, dθ, dϕ and dn are the sources of the measurement errors.

4 Dynamic error modeling and coupling effect analysis

In the last section, the OPD can be obtained by Eq. (22), and this formula reveals the static error characteristics of the measurement optical paths. Especially, variables θ and ϕ depend on misalignments of the initial state during the measurement. Sometimes, it is determined by the manual factors.

4.1 General modeling for dynamic error and coupling effect analysis

In order to further revealing the error characteristics, the OPD variable micro-quantity df (x, θ, ϕ, n) is disturbed by the dθ, dϕ and dn, which can be studied with the partial differentiation. Equation (22) can be transformed as:

Then,

Equation (24) shows kinetic relationships between the OPD and the variables. Suppose the nominal displacement of the plane mirror is dx, and the disturbed factors have a few variable quantities dθ, dϕ and dn during the measurement. The dx, dθ, dϕ and dn are differential terms of the df (x, θ, ϕ, n). So, the measurement value of the displacement can be calculated by which df (x, θ, ϕ, n) is divided by the optical subdivision time (here it is 4) and the refractive index of the air n. According to the error principle, the measurement error is the difference between measurement value and the nominal value. Its variable micro-quantity can be expressed as:

Substitution of Eq. (24) into Eq. (25), and given: $\frac{{\cos }^2\theta \cos (\theta +\phi )}{\cos 2\theta }-1=M$, $\frac{{\eta }^2(L_0+H)n\sin 4\theta }{2\sqrt{{({\eta }^2-n^2{\sin }^{2}2\theta )}^3}}+\frac{x(\sin 2\theta \cos (\theta +\phi )-\cos 2\theta \sin (\theta +\phi ){\cos }^2\theta )}{{\cos }^{2}2\theta }=N$,$\frac{x{\cos }^2\theta \sin (\theta +\phi )}{\cos 2\theta }$ $=P$, and $\frac{{\eta }^2(L_0+H){\sin }^{2}2\theta }{2\sqrt{{({\eta }^2-n^2{\sin }^{2}2\theta )}^3}}+\frac{x{\cos }^2\theta \cos (\theta +\phi )}{n\cos 2\theta }=Q$.

And then,

The Eq. (26) describes the dynamic error characteristics for the measurement optical paths, may be named error dynamic equation. Obviously, during the measurement, the error is kinetic with the change of the variables.

To simplify Eq. (26) for the following analysis, the four variables are written as a vector, and given as $\overrightarrow{u}={[x,\theta ,\phi ,n]}^T$ and $\overrightarrow{T}=[M,N,-P,Q]$.

From Eq. (26),

Suppose, the plane mirror moves from the initial position to another one, accordingly, the vector $\overrightarrow{u}$, which describes the state of the plane mirror, will change from $\overrightarrow{u_1}$ to $\overrightarrow{u_2}$ during the measurement. And the error of this process can be expressed as the following integration:

So, the errors can be ultimately revealed by Eq. (26) and Eq. (28). In the displacement measurement, df_e is influenced together by the M, N, P, Q and dx, dθ, dϕ and dn.

The M, N, P and Q, mean misalignments and the air refractive index of the initial state for the plane mirror, can be defined as the static error factors (SEF). They depend on the adjustments of the optical paths and represent the static characteristics for the interferometer. And they are also the coefficients of the errors in Eq. (28) and constant for the certain state of the plane mirror.

The dx is expected and reasonable, means the nominal displacement. It makes contribution to the error due to the coupling with the M of the SEF. And the dθ, dϕ and dn, mean the changes of the disturbed factors, can be defined as the dynamic error factors (DEF). The dθ, dϕ, implying the rotary angles of the plane mirror during the measurement, are associated with the accuracy of the measured target (e.g. a guideway or motion platform). The dn of the DEF yield the error independently because it still operates even if other error factors are cleared away. And it has a coupling action with the Q of the SEF if other errors exist.

Thus, the coupling effects between SEF and DEF can be revealed by the error dynamic equations for the measurement error.

4.2 Error analysis for a certain initial state

The errors can be calculated during the measurement that the plane mirror displaces from the one position to another position by Eq. (28). To uncover the errors without the mistake of the manual adjustment, assuming the misalignment of the plane mirror can be removed in the initial state, and the effect of the fluctuation of the air can be compensated. Given as, $\overrightarrow{\Delta u}=\overrightarrow{u_2}-\overrightarrow{u_1}=[\Delta x,\Delta \theta ,\Delta \phi ,0]$, $\overrightarrow{u_1}=[x,0,0,n]$ and $\overrightarrow{u_2}=[x+\Delta x,\Delta \theta ,\Delta \phi ,n]$.

From Eq. (25),

Then, from Eq. (22) and Eq. (29),

Signed $J=\eta (L_0+H)$, a constant value, it is a basic parameter of the PBS and the C. So, Eq. (30) can be abbreviated as:

This is the error formula during the measurement. It proves that the dθ and dϕ of the DEF are the sources for the errors, and still have a coupling action for the error with the x and n of the SEF. The error is proportional to the x + Δx which are decided by the initial and the range of the measured displacement. For the certain range, to diminish the error, an appropriate initial position should be chosen to ensure absolute value of the x + Δx as small as possible.

4.3 Sensitivity and quantitative analysis

In order to analyze sensitivity of the errors, the error curves can be obtained by the Eq. (31) for the specific conditions. Generally, value of the n is 1 for the air and value of the η is 1.516 for the K9 material of the PBS and C. Values of the L₀ and H are 25.4 mm and 19.1 mm respectively.

When the measuring range is 1000 mm, Δx is +/−500 mm, and as the certain initial position is determined, e. g. value of x is 1000 mm, the error of the variable Δθ or Δϕ (The Δθ and Δϕ all mean the rotation, which are given as equal.) can be shown in the Fig. 5(a) . In the Fig. 5(a), the curve illustrates that the error has a parabolic growth with the increase of the Δθ or Δϕ, and the error of the A point is equal to 1.5 μm when the Δθ or Δϕ is equal to 10⁻³ rad.

 

Fig. 5 (a) Errors of the variable Δθ or Δϕ; (b) Errors of the variable x.

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In conditions of the certain Δθ or Δϕ, e. g. 10⁻³ rad and the Δx is +/−500 mm, the error of the variable x is obtained as shown as the Fig. 5(b). Obviously, the error is proportional to the variable x and the A point in Fig. 5(a) and Fig. 5(b) occupy the same meaning. If the value of x is zero, the error is the smallest and equal to 0.53 μm. This is the B point of the Fig. 5(b). In practice, x is all positive value because the incident point of the plane mirror can’t position the other side of the PBS in the $\overrightarrow{D_x}$ direction, but considering the influence of the PBS and C, its error is 0.029 μm if the value of x is −500 mm (see the C point of the Fig. 5(b)).

5 Conclusions

The error modeling for the plane mirror interferometer with four times optical paths has been completed by the OPMB of each component. And with the model, static error and dynamic error have been discussed by the partial differential theory.

The traditional methods have disadvantages which mainly analyze the errors based on the static state and separations of the disturbances, and they ignore the dynamic errors during the measurement. In order to deal with the problem, the SEF and DEF have been presented to exactly explain the intercoupling effects between the misalignment of the initial state and the disturbed rotary angle during the displacement measurement in this paper. Then, their characteristics have been expounded from the aspect of the error contributions. The analysis indicates that the error of manual adjustments of the initial and the rotary error of the measured target during the displacement are the sources of the error.

In the condition of the ideal initial state and no fluctuations of the air, the measurement error has been obtained during the measurement from initial state to end state. And a tactic is proposed to improve the measurement accuracy, which an appropriate initial position should be chosen to ensure the absolute value of the x + Δ x as small as possible.