An 11-frame phase shifting algorithm in lateral shearing interferometry

Chao Fang; Yang Xiang; Keqi Qi; Chunlei Zhang; Changsong Yu

doi:10.1364/OE.21.028325

1. Introduction

With the development of projection lithography technology [1], more attentions to lateral shearing interferometry (LSI) are paid due to its advantages of no need of standard reference wavefront and insensitivity to environmental factor [2–4]. It has been used in surface measurement of optical element and in measurement of optical system’s wavefront [5–7]. Phase Shifting Interferometry (PSI) is widely used to calculate the phase of shearing wavefront in LSI, and the Bruning 4-frame algorithm, Schwider–Hariharan 5-frame algorithm, Schmit 6-frame algorithm and de Groot 7-frame algorithm are the common algorithms [8–11], which are not insensitive to phase shifting errors. For the double grating lateral shearing interferometer [12,13], with the order-selection mask for spatial filtering the zeroth order diffracted beam, the ± 1st order diffracted beams interfere with each other. By suppressing the undesired 0th order beam with the spatial filter, there is still a certain percentage of the beam leaking through the 1st order window. The common algorithms can compensate the phase shifting errors well, but they cannot compensate the errors caused by zeroth order effect. Therefore it is significant to design a special phase shifting algorithm to suppress the zeroth order effect and also to compensate the phase shifting error. In 2004, Zhu et al proposed a 9-frame phase shifting algorithm to suppress the 0th order effect [14]. However, as the phase shifting error increases, the accuracy of the algorithm drops considerably. In practice, the grating with small grating constant should be chosen because to set up the micro displacement stage and detector needs enough space. In this case, there is a large amount of phase shifting error caused by the limits of manufactures and installations of the displacement stage and grating. So it is necessary to design an algorithm of phase calculation, which is much more insensitive to phase shifting error and also can suppress zeroth order effect.

An 11-frame phase shifting algorithm to eliminate zeroth order effect and compensate the phase shifting error is proposed in this paper. The analytical expression of phase-restoration error function is derived. The principle of the phase shifting error compensation and the capability of suppressing zeroth order effect are explained. Finally the simulations are given.

2. Lateral shearing interferometer and phase shifting algorithm

The typical double grating lateral shearing interferometer is shown in Fig. 1. The beam diffracted by grating goes through the projection lens. The ± 1st order diffracted beams form the interference pattern, and the 0th and higher order diffracted beams are eliminated by the spatial filter. Although the spatial filter is used, there is still a certain percentage of the 0th order beam reaching CCD detector, which generates interference pattern with ± 1st order diffracted beams.

Fig. 1 Block diagram of double grating lateral shearing interferometer.

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The phase shifting amount of ϕ can be written as

ϕ = \frac{2 π m x}{p},

where m is diffracted order, x is grating displacement and p is grating constant. By controlling the grating displacement of x, the phase shifting amount between ± 1st order beams is as

ϕ_{j} = j \frac{4 π x}{p} = j \frac{π}{2} .

The ± 1st order beams are signals and the 0th order beam is noise. The complex amplitudes of them can be expressed as

E_{- 1} = a_{- 1} \exp (k W_{- 1}),

E_{1} = a_{1} \exp (k W_{1}),

E_{0} = a_{0} \exp (k W_{0}),

where E_-1,E₁ and E₀ are the complex amplitudes of −1st,1st and 0th order beams respectively, a_-1, a₁, a₀ are the amplitudes, W_-1, W₁, W₀ are the wavefronts, and k is the wave number. The fringe intensity distribution of I_j for j-th frame is

\begin{array}{l} I_{j} = a_{0}^{2} + a_{1}^{2} + a_{- 1}^{2} + 2 a_{0} a_{1} \cos [k (W_{1} - W_{0}) + ϕ'_{j}] \\ + 2 a_{0} a_{- 1} \cos [k (W_{0} - W_{- 1}) + ϕ "_{j}] + 2 a_{1} a_{- 1} \cos [k (W_{1} - W_{- 1}) + ϕ_{j}], \end{array}

where

ϕ'_{j} = ϕ_{j} / 2

is the phase shifting amount between 1st order and 0th order,

ϕ''_{j} = ϕ_{j} / 2

is that between 0th order and −1st order,

k (W_{1} - W_{- 1})

is unknown phase needing to be restored,

k (W_{1} - W_{0})

and

k (W_{0} - W_{- 1})

are undesired phases distorted by 0th order beam. We suppose:

Q = a_{0}^{2} + a_{1}^{2} + a_{- 1}^{2}

,

V_{1} = 2 a_{1} a_{- 1}

,

V_{2} = 2 a_{0} a_{1}

,

V_{3} = 2 a_{0} a_{- 1}

,

Equation (6) can be written as following:

I_{j} = Q + V_{1} \cos (θ_{1} + ϕ_{j}) + V_{2} \cos (θ_{2} + ϕ_{j} / 2) + V_{3} \cos (θ_{3} + ϕ_{j} / 2) .

Using 11 step phase shifting (

j = - 5, - 4, \dots 0, \dots 4, 5

), we have

(I_{5} - I_{- 5}) - 7 (I_{3} - I_{- 3}) + 8 (I_{1} - I_{- 1}) = - 32 V_{1} \sin θ_{1},

- 4 (I_{4} + I_{- 4}) + 8 (I_{2} + I_{- 2}) - 8 I_{0} = - 32 V_{1} \cos θ_{1} .

The phase to be restored is calculated as following:

θ_{1} = \tan^{- 1} [\frac{(I_{5} - I_{- 5}) - 7 (I_{3} - I_{- 3}) + 8 (I_{1} - I_{- 1})}{- 4 (I_{4} + I_{- 4}) + 8 (I_{2} + I_{- 2}) - 8 I_{0}}] .

From Eq. (10), we conclude that the phase of θ₁ to be restored is not affected by 0th order diffraction under the condition of error free of phase shifting, because θ₁ is unrelated with θ₂ and θ₃.

3. Error analyses and comparisons

Due to the manufacture error and alignment error of micro displacement stage and grating, a mount of phase shifting error always exists in the phase shifting interferometry. So the phase shifting amount is different from Eq. (2), which is as following:

ϕ_{j} = j \cdot \frac{π}{2} + j \cdot ε,

where ε is the phase shifting error. It is convenient to divide the numerator part of Eq. (10) into 3 parts, named as A₁, A₂ and A₃. Part A₁ contains terms with θ₁ and constant, and can be expressed as:

A_{1} = [16 \cos (ε) - 16 \cos {(ε)}^{3} - 32 \cos {(ε)}^{5}] \sin θ_{1} .

We expand Eq. (12) as Taylor expansions for the phase shifting error of ε, and omit the small quantity of high order

A_{1} = (- 32 + 96 ε^{2} - 100 ε^{4}) \sin θ_{1} .

Part A₂ contains term with θ₂, denoted as:

A_{2} = - \sqrt{2} [8 \cos (ε) + 8 \sin (ε) - 7 \cos (\frac{3}{2} ε) + 7 \sin (\frac{3}{2} ε) - \cos (\frac{5}{2} ε) - \sin (\frac{5}{2} ε)] \frac{V_{2}}{V_{1}} \cos θ_{2} .

Equation (14) can be expressed as Taylor expansion form as:

A_{2} = [- 12 \sqrt{2} ε + O (ε^{2})] \frac{V_{2}}{V_{1}} \cos θ_{2} .

For the grating in the interferometer, the designed ± 1st order diffraction efficiencies are 40%. Actually they are 35%, so the 0th order diffraction efficiency is less than 30%. Calculation shows that the amplitude of 0th order beam going through spatial filter is less than 2% of those of ± 1st order beams. So V₂ and V₃ are much smaller than V₁. And phase shifting error of ε is a small value. A₂ is approximately zero, because A₂ is products of two small quantities (polynomial of ε and V₂/V₁). Similarly, Part A₃ contains terms with θ₃, and it is also approximately zero.

It is convenient to divide the denominator of Eq. (10) into 3 parts, named as B₁, B₂ and B₃. Part B₁ contains terms with θ₁ and constant, and can be expressed as

B_{1} = [32 \cos {(ε)}^{2} - 64 \cos {(ε)}^{4}] \cos θ_{1} .

We expand Eq. (16) as Taylor expansions for the phase-shift error of ε, and omit the small quantity of high order. Then we have:

B_{1} = (- 32 + 96 ε^{2} - 96 ε^{4}) \cos θ_{1} .

Part B₂ contains term with θ₂, denoted as:

B_{2} = [- 16 \sin (ε) - 16 + 16 \cos {(ε)}^{2}] \frac{V_{2}}{V_{1}} \cos θ_{2} .

Equation (18) can be expressed as Taylor expansion form as:

B_{2} = [- 16 ε + O (ε^{2})] \frac{V_{2}}{V_{1}} \cos θ_{2} .

By the method similar to analysis of A₂, B₂ is approximately zero. Part B₃ containing terms of θ₃ is also approximately zero.

From Eqs. (15) and (19), it is known that the part containing θ₂ and θ₃ in numerator and denominator is zero. So we can ignore the shearing phase generated by zeroth order beam even in the presence of phase shifting error. The algorithm can eliminate the undesired zeroth order effect.

Substituting Eqs. (13), (15), (17) and (19) into Eq. (10), the phase of β₁ with phase shifting error of ε is restored as:

\tan β_{1} = (1 + \frac{ε^{4}}{8 - 24 ε^{2} + 24 ε^{4}}) \tan θ_{1} \approx (1 + \frac{ε^{4}}{8}) \tan θ_{1} .

Considering the phase-restoration error of ∆θ₁ is small amount, the phase-restoration error function with 11-frame algorithm can be expressed as

Δ θ_{1} = β_{1} - θ_{1} \approx \tan (β_{1} - θ_{1}) = \frac{ε^{4}}{16} \sin (2 θ_{1}) .

With 9-frame phase shifting algorithm, the phase can be restored as following expression [14]:

θ'_{1} = \tan^{- 1} [\frac{2 (I_{3} - I_{- 3}) - 2 (I_{1} - I_{- 1})}{(I_{4} + I_{- 4}) - 2 (I_{2} + I_{- 2}) + 2 I_{0}}] .

With above method Eq. (22) is analyzed, and the phase of β’₁ with phase shifting error of ε, is restored:

\tan β'_{1} = (1 + \frac{ε^{2}}{2 - 6 ε^{2}}) \tan θ'_{1} \approx (1 + \frac{ε^{2}}{2}) \tan θ'_{1} .

The phase-restoration error function of Δθ’₁ with 9-frame algorithm is expressed as

Δ θ'_{1} = β'_{1} - θ'_{1} \approx \tan (β'_{1} - θ'_{1}) = \frac{ε^{2}}{4} \sin (2 θ'_{1}) .

From Eqs. (21) and (24), in the case of constant phase shifting error such as ε = π/6, the relationships between phase-restoration error and shearing phase for 11-frame and 9-frame algorithm are shown in Fig. 2. The two phase-restoration errors vary as sine of double shifting phases, and the worst condition is at π/4. With the same shearing phase, the restoration error of 11-frame algorithm is smaller than that of 9-frame algorithm. So the 11-frame algorithm is less sensitive to phase shifting error and has more extensive adaptability.

Fig. 2 The relationship between phase-restoration error and shearing phase.

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Figure 3 shows the phase-restoration error by 11-frame and 9-frame algorithm in the case of same shifting phase, such as shearing phase of π/4, according to Eqs. (21) and (24). The restoration error of 11-frame algorithm is proportional to biquadratic of phase shifting error, and that of 9-frame algorithm is proportional to square of phase shifting error. So with the same phase shifting error, restoration error of 11-frame algorithm is less than that of 9-frame algorithm. Especially, with large phase shifting error, the accuracy of 11-frame algorithm is much higher than that of 9-frame algorithm.

Fig. 3 The relationship between phase-restoration error and phase shearing error.

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4. Simulations

In this section, we simulate the procedures of lateral shearing interference and shearing phase restoration with zeroth order effect to verify the analyses in Section 3. In the simulations a wavefront of spherical aberration is chosen, which is a common aberration in the measurement of wavefront. The wavefront of 0.5λ spherical aberration located in unit circle is shown in Fig. 4. The wavefront is sheared in x direction, shearing ratio is assumed as 0.1, and the noise is V₂ = V₃ = 2%V₁. When the phase shifting amount is zero, the shearing phase and its interference pattern are shown in Fig. 5.

Fig. 4 The wavefront of 0.5λ spherical aberration.

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Fig. 5 The shearing phase (a) and interference pattern (b) when shearing ratio is 0.1, V₂ = V₃ = 0.02V₁ and phase shearing amount is 0.

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11 interference patterns are obtained with 11 step phase shifting. The phase shifting amount is π/2 for one step, and the phase shifting error is π/6. The shearing phase is restored with 11-frame phase shifting algorithm. The restoration phase and its error are shown in Fig. 6.

Fig. 6 The shearing phase (a) and phase-restoration error (b) restored with 11-frame algorithm.

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4 interference patterns are obtained with 4 step phase shifting. The phase shifting amount is π/2 for one step, and the phase shifting error is π/6. The shearing phase is restored with Bruning 4-frame phase shifting algorithm. The restoration phase and its error are shown in Fig. 7.

Fig. 7 The shearing phase (a) and phase-restoration error (b) restored with 4-frame algorithm.

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From Figs. 6 and 7, for the same wavefront with the same phase shifting error, the maximum absolute value of 11-frame algorithm’s restoration error is smaller than 0.04 rad, and that of 4-frame algorithm is smaller than 0.30 rad. The restoration accuracy of 11-frame algorithm is higher than that of 4-frame algorithm.

9 interference patterns are generated with 9 step phase shifting. The phase shifting amount is π/2 for one step, and the phase shifting error is π/6. The shearing phase is restored with 9-frame phase shifting algorithm. The restoration phase and its error are shown in Fig. 8.

Fig. 8 The shearing phase (a) and phase-restoration error (b) restored with 9-frame algorithm.

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From Figs. 6 and 8, we know that the shearing phase can be restored with both 11-frame and 9-frame algorithm when zeroth order effect exists. For the same wavefront with the same phase shifting error, the maximum absolute value of 11-frame algorithm’s restoration error is smaller than 0.04 rad, and that of 9-frame algorithm is smaller than 0.09 rad. Therefore, the restoration accuracy of 11-frame algorithm is higher than that of 9-frame algorithm. This proves the analyses in Section 3. With different phase shifting errors the different interference patterns are generated respectively with 11step and 9 step phase shifting, and the restoration errors are obtained. The restoration rms errors of those algorithms are shown in Fig. 9. The phase-restoration errors increase as the absolute values of phase shifting errors. With the same phase shifting error, restoration accuracy of 11-frame algorithm is higher than that of 9-frame algorithm. Especially, under the condition of large phase shifting error, the accuracy difference is much more remarkable. The restoration rms error with phase shifting error of -π/6 for 11-frame algorithm is 0.011 rad, and it is 0.056 rad for 9-frame algorithm. 11-frame algorithm’s error is 1/5 of 9-frame algorithm’s error. The simulations are in good agreement with the tendency of Fig. 3, the analyses results in Section 3.

Fig. 9 The relationship between phase-restoration error and phase shearing error given by simulation with 11-frame and 9-frame algorithms under the condition of V₂ = V₃ = 2%V_1.

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With the simulation conditions of V₂ = V₃ = 0.5%V₁ and V₂ = V₃ = 1%V₁, the restoration rms errors of 9-frame algorithm and 11-frame algorithm are shown in Figs. 9 and 10 respectively. From Figs. 9–11, for different degrees of zeroth order effect with large amount of phase shifting error, phase-restoration error of 11-frame algorithm is much smaller than that of 9-frame algorithm.

Fig. 10 The relationship between phase-restoration error and phase shearing error given by simulation with 11-frame and 9-frame algorithms under the condition of V₂ = V₃ = 0.5%V_1.

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Fig. 11 The relationship between phase-restoration error and phase shearing error given by simulation with 11-frame and 9-frame algorithms under the condition of V₂ = V₃ = 1%V_1.

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5. Conclusions

The 11-frame phase shifting algorithm is proposed, and the analytical expression of phase-restoration error function is derived. The principle of phase shifting error compensation and the capability of suppressing zeroth order effect are explained. The algorithm is compared with existing algorithm. From analysis results, we conclude that it is better to use 11-frame algorithm to restore shearing phase with zeroth order effect and to compensate phase shifting error. This algorithm’s phase-restoration error is proportional to sine of double shearing phase and to biquadratic of phase shifting error. With the same phase shifting error, phase-restoration error of 11-frame algorithm is smaller than that of existing algorithm. Finally, we simulate the interference patterns of 11-frame algorithm and existing algorithm, restore the shearing phases and calculate the phase-restoration errors. The simulations verify our theoretical analyses.

Acknowledgment

This work was supported by National S&T Special Program (No. 2009ZX02005).

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An 11-frame phase shifting algorithm in lateral shearing interferometry

Abstract

1. Introduction

2. Lateral shearing interferometer and phase shifting algorithm

3. Error analyses and comparisons

4. Simulations

5. Conclusions

Acknowledgment

References and links

Cited By

Figures (11)

Equations (24)

Optics Express