Nonlinearity error in homodyne interferometer ... - OSA Publishing

Vol. 25, No. 4 | 20 Feb 2017 | OPTICS EXPRESS 3605

Nonlinearity error in homodyne interferometer caused by multi-order Doppler frequency shift ghost reflections PENGCHENG HU, YUE WANG, HAIJIN FU,* JINGHAO ZHU, AND JIUBIN TAN Institute of Ultra-precision Optoelectronic Instrument Engineering, Harbin Institute of Technology, Harbin, 150080, China * [email protected]

Abstract: This paper reports a hitherto undiscovered cyclic error whose origin is different from that of conventional errors in homodyne interferometers. To explain this error, a model based on ghost reflections and the interference principle is developed. In general, in homodyne interferometers, multi-order Doppler frequency shift ghost beams participate in the final interference and generate multi-order cyclic errors. This “new” cyclic error is compared with conventional errors by means of Lissajous curves. And we establish a setup to validate our proposed model. We use a corner cube retroreflector to replace the mirror and we find the error is significantly reduced. We believe that our findings can contribute to the further development of highly accurate homodyne interferometers. © 2017 Optical Society of America OCIS codes: (120.0120) Instrumentation, measurement and metrology; (120.3180) Interferometry; (120.5700) Reflection.

References and links 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.

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1. Introduction #281787 Journal © 2017

https://doi.org/10.1364/OE.25.003605 Received 2 Dec 2016; revised 2 Feb 2017; accepted 7 Feb 2017; published 10 Feb 2017


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 frequencystabilized 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 quarterwave 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 I1 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 = arctan

Iy Ix

,

(1) (2)


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 = Ax cos (ϕ + δ ) + Bx , I y = Ay sin (ϕ ) + By ,

(3)

here, Ax and Ay represent the AC amplitudes and induced unequal AC amplitudes error if they

(

)

are not equal Ax ≠ Ay , Bx and By denote the DC offsets error, and δ is the quadrature difference. The periodic error can be expressed as

NL = arctan( I y / I x ) − ϕ + mπ .

(4)

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]. 4

CCR

PS DC AC PH

3

QWP1

2

v

1 0

Iy

Laser PBS1

-1

CCR

QWP2

-2

(a)

-3

(c) -4 -4

HWP

PBS2

8

I1

BS

I3

PBS3

I2 I4

Nonlinear error (nm)

6

QWP3

(b)

10

-2

0 Ix

2

4

6

4

PS DC AC PH

4 2 0 -2 -4 -6 -8 -10 0

(d) 2

8

φ(Rad)

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.

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. Second-order GRP

First-order GRP PBS

QWP

M

PBS QWP v

f

f f + fd (e)

v

PBS QWP v

PBS QWP CCR f

v

(d)

f

v

f +2fd

(f)

v f + 3fd

PBS QWP CCR

f +2 fd

M

f

f + 2fd (c)

(b) v

Second-order GB

M

f

f

(a)

First-order GB

PBS QWP

f

f + fd PBS QWP CCR

Main beam M

PBS QWP CCR f

v

f +3fd

(g)

(h)

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.

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 firstorder DFS. We can express the electric field of the main beam without ghost beams as

{

}

Em = Am exp i ω t + φ ( t ) + Ψ m  ,

(5)

where ω = 2π f represents the angular frequency, Am 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 n

Egh =  Aghk e 

i ω t + kφ ( t ) +Ψ k 

,

(6)

k =2

here, Aghk 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 secondorder 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

Er = Ar exp i (ωt + Ψ r )  ,

(7)

where Ar is the amplitude of reference beams, and Ψr 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 ( Em + E gh + Er ) ⋅ ( Em* + E gh + Er* )  ,

(8)

* here, Em* , Egh and Er* represent the complex conjugates of Em , E gh and Er , respectively.

Equation (8) can be expressed more compactly as

 n ikφ t  I s = Re  Γk e ( )  ,  k =1 

(9)

here, Γ0 - 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). (b)

MMS

MMS PIS2

Amplitude

Amplitude

(a)

0

fd

PIS3 PIS4 0

fd

2fd

3fd

4fd

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.

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 dφk = arctan

Γ1 sin φ ( t ) + Γ k sin kφ ( t )

Γ1 cos φ ( t ) + Γ k cos kφ ( t )

− φ (t ) ,

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

(10)

( 4π ) . The amplitude of

the kth-order ghost beam is significantly smaller than that of the main beam ( Γ k