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Nonlinear Errors Resulting from Ghost Reflection and Its Coupling with Optical Mixing in Heterodyne Laser Interferometers Haijin Fu 1,2,† , Yue Wang 1,† , Pengcheng Hu 1, * 1 2

* †

ID

, Jiubin Tan 1 and Zhigang Fan 2

Institute of Ultra-Precision Optoelectronic Instrument Engineering, Harbin Institute of Technology, Harbin 150001, China; [email protected] (H.F.); [email protected] (Y.W.); [email protected] (J.T.) Postdoctoral Research Station of Optical Engineering, Harbin Institute of Technology, Harbin 150001, China; [email protected] Correspondence: [email protected]; Tel.: +86-451-8641-2041 (ext. 803) These authors contribute equally to this study.

Received: 9 February 2018; Accepted: 28 February 2018; Published: 2 March 2018

Abstract: Even after the Heydemann correction, residual nonlinear errors, ranging from hundreds of picometers to several nanometers, are still found in heterodyne laser interferometers. This is a crucial factor impeding the realization of picometer level metrology, but its source and mechanism have barely been investigated. To study this problem, a novel nonlinear model based on optical mixing and coupling with ghost reflection is proposed and then verified by experiments. After intense investigation of this new model’s influence, results indicate that new additional high-order and negative-order nonlinear harmonics, arising from ghost reflection and its coupling with optical mixing, have only a negligible contribution to the overall nonlinear error. In real applications, any effect on the Lissajous trajectory might be invisible due to the small ghost reflectance. However, even a tiny ghost reflection can significantly worsen the effectiveness of the Heydemann correction, or even make this correction completely ineffective, i.e., compensation makes the error larger rather than smaller. Moreover, the residual nonlinear error after correction is dominated only by ghost reflectance. Keywords: laser sensor; heterodyne interferometer; optical nonlinearity; ghost reflection; optical mixing

1. Introduction Picometer level metrology faces increasing demand in numerous key fields, such as gravitationalwave detection, semiconductor industry, and nanotechnology [1–5]. In the case of Laser Interferometer Space Antenna (LISA) for detecting gravitational waves, pm-sensitivity is required to measure the transition of the inertial proof mass [1]. For precision positioning in semiconductor manufacturing, an uncertainty of about 25 pm should be available in the next decade according to the International Technology Roadmap for Semiconductors (ITRS) [2,3]. To deal with the challenges raised by the rapid progress of nanotechnology, the NANOTRACE project, funded by the European Metrology Research Programme (EMRP), has spearheaded the development of the next generation of optical interferometers with a target uncertainty of 10 pm [4]. Heterodyne laser interferometry is considered a promising candidate for the next generation of optical interferometer due to its high accuracy and robust capabilities. However, it is often subject to periodic nonlinear errors induced by optical mixing, with an amplitude averaging from several to tens of nanometers. Over the years, various solutions to this problem have been proposed, which fall into two overarching categories. The first adopts optical configurations with two spatially separated beams [6–10] and thereby avoids the optical mixing, but at the cost of complicated structures that are sensitive to air disturbance and thermal drift. Additionally, recent research indicates that these configurations still

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of 12 disturbance and thermal drift. Additionally, recent research indicates that 2these configurations still suffer from nonlinearities caused by multi-order Doppler frequency shift (DFS) arisingfrom fromnonlinearities ghost reflection [11]. The second solution keeps the common employs suffer caused by multi-order Doppler frequency shiftoptical (DFS) paths arisingand from ghost compensation algorithms, mainly the Heydemann correction [12–17]. This correction relies on the reflection [11]. The second solution keeps the common optical paths and employs compensation traditional model that first- and second-order nonlinearities are generated from optical mixing [18–20]. In algorithms, mainly the Heydemann correction [12–17]. This correction relies on the traditional model theory, the correction is capable of eliminating nonlinearities through elliptical fitting. Nevertheless, that first- and second-order nonlinearities are generated from optical mixing [18–20]. In theory, residual nonlinear errors from hundreds of picometers to several nanometers are still found even after the correction is capable of eliminating nonlinearities through elliptical fitting. Nevertheless, residual compensation [12–17], the cause and mechanism of which remain largely uninvestigated. nonlinear errors from hundreds of picometers to several nanometers are still found even after In this paper, to study the residual nonlinear errors impeding the realization of picometer level compensation [12–17], the cause and mechanism of which remain largely uninvestigated. metrology, a novel nonlinearity model based on ghost reflection and its coupling with optical mixing In this paper, to study the residual nonlinear errors impeding the realization of picometer level is proposed and then verified by experiments. Further, the influence of the developed model on the metrology, a novel nonlinearity model based on ghost reflection and its coupling with optical mixing Heydemann correction is analyzed to reveal the source and mechanism of the residual nonlinear is proposed and then verified by experiments. Further, the influence of the developed model on the errors after compensation. Heydemann correction is analyzed to reveal the source and mechanism of the residual nonlinear errors after compensation. 2. Nonlinearity Model Based on Ghost Reflection and Its Coupling with Optical Mixing

2. Nonlinearity Model onheterodyne Ghost Reflection and Its Coupling Optical Mixing The schematic of aBased typical interferometer is shownwith in Figure 1. The laser output, consisting of two orthogonal polarized beams with slightly different frequencies of f 1 and f2,output, is split The schematic of a typical heterodyne interferometer is shown in Figure 1. The laser in energy by a beam splitter (BS) into two parts: one is detected by the photodetector 1 (PD1) and consisting of two orthogonal polarized beams with slightly different frequencies of f 1 and f 2 , is split serves as the signal (BS) Ir; theinto other, an ideal is completely in polarization by in energy by reference a beam splitter twoinparts: onecase, is detected by theseparated photodetector 1 (PD1) and a polarizing beam splitter (PBS) into measurement and reference beams with frequencies of f 1 and f2, serves as the reference signal Ir ; the other, in an ideal case, is completely separated in polarization by The two beams areinto directed to the movable and fixed mirrors (TM and RM) the arespectively. polarizing beam splitter (PBS) measurement and reference beams with frequencies of fin 1 and measurement and reference arms, respectively. The reflected beams are recombined and then f 2 , respectively. The two beams are directed to the movable and fixed mirrors (TM and RM) in the detected by PD2 as the measurement signalThe Imreflected . The quarter-wave-plates (QWP1 and detected QWP2) measurement and reference arms, respectively. beams are recombined and then positioned in measurement the measurement and paths are used (QWP1 to rotate beampositioned polarization by PD2 as the signal Im . reference The quarter-wave-plates andthe QWP2) in orientation by 90°, and thereby a convenient layout with the input and output beams located on two the measurement and reference paths are used to rotate the beam polarization orientation by 90◦ , adjacent sides of the PBS can with be realized. measurement of phase difference and thereby a convenient layout the inputThe and output beams located on two adjacentbetween sides of the the measurement and reference signals, which is proportional to the displacement of the target mirror, PBS can be realized. The measurement of phase difference between the measurement and reference is carried out electronically. signals, which is proportional to the displacement of the target mirror, is carried out electronically.

Figure 1. Schematic of a typical heterodyne interferometer. DFL: DFL: dual-frequency dual-frequency laser. laser. BS: BS: beam beam splitter. splitter. PBS: polarizing polarizing beam splitter. splitter. RM: reference mirror. TM: target mirror. mirror. QWP: quarter quarter wave-plate. wave-plate. P: Polarizer. PD1: photodetector photodetectorfor forthe thereference referencesignal. signal.PD2: PD2:photodetector photodetectorfor forthe themeasurement measurementsignal. signal.

Optical mixing mixing is is inevitable inevitable in in aa real case due due to to the of the the laser laser Optical real case the ellipticity ellipticity and and nonorthogonality nonorthogonality of source, optical defects, and alignment errors. Therefore, each arm of the heterodyne interferometer source, optical defects, and alignment errors. Therefore, each arm of the heterodyne interferometer contains aa small small component component of of the the frequency frequency intended intended for for the the other other arm. arm. In In this this case, contains case, the the laser laser beams beams back from the measurement and reference arms are given by back from the measurement and reference arms are given by E m  A exp  i 1t       exp  i  2 t     , Em = A exp[i (ω1 t + φ)] + β exp[i (ω2 t + φ)],

(1) (1)

E  B exp  i t    exp  i t  ,

(2) (2)

r Er = B exp(iω22t) + α exp(iω1 1 t),

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where A and B are amplitudes of the intended measurement and reference beams, respectively; β and where A and B are of amplitudes of the intended measurement beams, respectively; and α are amplitudes the unwanted leakage laser beams inand the reference measurement and reference β arms, α are amplitudes of the unwanted leakage laser beams in the measurement and reference arms, respectively. ω1 = 2πf1 and ω2 = 2πf2 are angular frequencies of the laser beams, ϕ = 4πL/λ = 4πvt/ respectively. ω1 = 2πf ω2 = 2πf angular of the lasercaused beams,by ϕ target = 4πL/λ = 4πvt/λ denotes the phase shift resulting from DFS whichfrequencies is equivalent to that displacement. 1 and 2 are denotes the phase shift resulting from DFS which equivalent For simplicity, initial phases of the laser beams areisset at zero. to that caused by target displacement. For simplicity, initial phases the laser beams that are set at zero. Besides optical mixing,ofghost reflection occurs at the interface of optical components in Besides optical mixing, thaterrors occurs the interface of optical practical applications can alsoghost lead reflection to nonlinear byatinducing multi-order DFScomponents [11]. This isina practical applications can also lead to nonlinear errors by inducing multi-order DFS [11]. Thison is a potential obstacle of picometer level metrology as the existing compensation algorithms rely potential of picometer level metrology as the existing compensation algorithms on a traditionalobstacle nonlinearity model based on optical mixing. As illustrated in Figure 2a–c, the rely intended traditional nonlinearity modeltobased on opticalDFS mixing. Asby illustrated in Figure 2a–c, thereflections intended measurement beam is subject the first-order caused target motion, while ghost measurement beam is surfaces subject toofthe DFS by target motion, whileDFS, ghost reflections in the inner and outer thefirst-order QWP result in caused the zerothand second-order respectively. in inner outer surfaces of the QWP the surface zeroth- of and DFS, In the Figure 2d,and ghost reflection happens twice result in the in outer thesecond-order QWP, leading to arespectively. third-order In Figure 2d, ghost happens twice in the outer of the QWP, arm leading a third-order DFS. Similarly, thereflection unwanted leakage laser beam in surface the measurement is to also subject to DFS. Similarly, the unwanted leakage laser beam in the measurement arm is also subject to multi-order multi-order DFS, and it will participate in the inference that produces the measurement signal. In this DFS, andbeam it will participate the inference that producesas the measurement signal. In this case, case, the back from the in measurement arm is rewritten the beam back from the measurement arm is rewritten as n  n  Em (An  k exp i 1t  k  )  exp i 2t  k   ，) (3)    (  nk  k 0   k 0  Em = A ∑ γk exp[i (ω1 t + kφ)] + β ∑ γk exp[i (ω2 t + kφ)] , (3) k =0 k =0 where where

(  r k 0  k  r k = ,0 2 k 1 γk = 1  r  2r k−1 k  1 , k≥1 (1 − r ) r

(4) (4)

represents the the order order of of DFS DFS and and rr denotes denotes the the ghost ghost reflectance. reflectance. kk represents PBS

QWP

M

M v

f f

f + fd

(b)

(a) PBS

QWP

PBS v

f

QWP

M

PBS v

f

f + 2fd

QWP

M v

f

f + 3fd

(c)

(d)

Figure 2. 2. Presentation of ghost beams in the measurement Figure measurement arm arm of of aa heterodyne heterodyne laser laser interferometer. interferometer. (a) Intended beam with first-order Doppler frequency shift (DFS); (b) Ghost beam with zeroth-order second-order DFS; DFS; (d) (d) Ghost Ghost beam beam with with third-order third-order DFS. DFS. DFS; (c) Ghost beam with second-order

The optical optical components components in laser interferometer interferometer are are generally generally coated coated with with anti-reflection anti-reflection films films The in aa laser which make the ghost reflectance quite small. Therefore, when evaluating the measurement signal, a which make the ghost reflectance quite small. Therefore, when evaluating the measurement signal, are athird-order third-orderapproximation approximationisissufficient, sufficient,i.e., i.e.,only onlythe thezero-, zero-,first-, first-, secondsecond- and and third-order third-order DFS DFS are necessary for computation, with any other high-order DFS omitted. Regarding any beams back from necessary for computation, with any other high-order DFS omitted. Regarding any beams back the reference arm, arm, they they are are freefree of DFS and can still the from the reference of DFS and can stillbebeexpressed expressedas as Equation Equation (2). (2). Under Under the approximation, the the measurement measurement signal signal of of the the interferometer interferometer can can be be calculated calculated as as follows: follows: approximation, Im

 I m  Re  Em  Er    Em  Er∗  = Re ( Em + Er ) · ( Em + Er )  cos(∆ωt t  3+) 3φ  0 (cos 11cos t +  22φ  1) cos tφ)+ Γ−  )+ Γ2 −cos( t (∆ωt) = Γ0cos ∆ωt (∆ωt + Γ1cos  + 2 cos {z  }  |  {z } |  {z  } |  |  {z   } 

basebase

1st harmonic −1st harmonic

2nd harmonic

−2nd harmonic

+Γ2 cos −φt )+ Γ4 cos cos( ∆ωt cos   2)+   2(∆ωt  Γ3cos  t −33φ (∆ωt  −t 2φ  ,), 3 cos     } | {z  } |  {z }  | {z 2nd harmonic 2nd harmonic

harmonic 3rd 3rd harmonic

4th harmonic 4th harmonic

1st harmonic

1st harmonic

(5) (5)

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where ∆ω = ω1 − ω2 = 2π∆f and

A 0 1  A 1 2  A 2 3  AB 1 where ∆ω = ω 1 − ω 2 = 2π∆f 0 and   1  A 0 2  A 1 3  AB 2 ,  2  A 0 3  AB 3  Γ0  = Aβγ0 γ12 + Aβγ2 1 γ2 + Aβγ  2 γ32 + ABγ1  2   (6)  = 1  A 0  A 1  A 2  A 3  AB 0   0  B  Γ Aβγ γ + Aβγ γ + ABγ 0 2 2 , Γ−2 = Aβγ0 γ3 + ABγ3  −1 1 3 2 0 1  A2 1 2  A 2 2 3   2  2Aβγ  A 1 (6) Γ1 = 0 + Aβγ1 + Aβγ2 + Aβγ3 + ABγ0 + αβγ0 + Bα             A A , A .       Γ = Aβγ γ + Aβγ γ + Aβγ γ + αβγ 2 2 34 3  2 3 0 10 2 11 23 10 3   Γ3 = Aβγ0 γ2 + Aβγ1 γ3 + αβγ2 , Γ4 = Aβγ0 γ3 + αβγ3 . In Equation (5), the first, fourth, and fifth terms are the base signal, the first- and second-order nonlinear harmonics, respectively. They to those of the conventional model based on In Equation (5), the first, fourth, and are fifthsimilar terms are the base signal, the first- and second-order optical mixing, but with different They magnitudes duetotothose ghost The second third terms, nonlinear harmonics, respectively. are similar of reflection. the conventional modeland based on optical which fromdifferent the secondand third-order DFS, are referred to as negative and negative mixing,arise but with magnitudes due to ghost reflection. The second and thirdfirstterms, which arise second-order harmonics, respectively, for their phases are opposite in sign to that of the traditional from the second- and third-order DFS, are referred to as negative first- and negative second-order firstand second-order harmonics. sixthare and seventh in terms, areofthe and fourth-order harmonics, respectively, for their The phases opposite signwhich to that thethirdtraditional first- and harmonics, are induced by coupling of ghost reflection andthe optical Figure 3 harmonics, shows the second-order harmonics. Thethe sixth and seventh terms, which are third- mixing. and fourth-order simulation the measurement signal with different degrees of optical mixing and ghost are inducedspectra by the of coupling of ghost reflection and optical mixing. Figure 3 shows the simulation reflectance. Formeasurement convenience, in this paper A and B are set atof 1, optical whereasmixing α and βand represent the degree spectra of the signal with different degrees ghost reflectance. of mixing. in When α = β =A0.03, 0.01, byα Figure 3a, besidesthe thedegree base signal, the Foroptical convenience, this paper andrB= are setasatillustrated 1, whereas and β represent of optical traditional first-α and second-order harmonics, the 3a, negative firstsecond-order mixing. When =β= 0.03, r = 0.01,nonlinear as illustrated by Figure besides the and basenegative signal, the traditional nonlinear harmonics coming fromharmonics, multi-order can also beand observed, with magnitudes smaller first- and second-order nonlinear theDFS negative firstnegative second-order nonlinear relative to those of the traditional firstand second-order nonlinear harmonics, respectively; the harmonics coming from multi-order DFS can also be observed, with magnitudes smaller relative thirdand harmonics, by coupling of ghost reflectionrespectively; and optical mixing, areand too to those offourth-order the traditional first- andcaused second-order nonlinear harmonics, the thirdsmall to be seen. If r increases or α and β are reflection enlarged to 0.1, as shown in Figure fourth-order harmonics, causedtoby0.05 coupling of ghost and optical mixing, are too 3b,c, smallthe to third-order harmonic observed to the enhancement coupling of ghost be seen. If r nonlinear increases to 0.05 or α can and be β are enlargeddue to 0.1, as shown in Figureof3b,c, the third-order reflection and optical mixing; however, thethe fourth-order nonlinear harmonic still cannot and be seen. In nonlinear harmonic can be observed due to enhancement of coupling of ghost reflection optical addition, compared to Figure 3a, the apparent growth of the first-, negative first-, secondand mixing; however, the fourth-order nonlinear harmonic still cannot be seen. In addition, compared negative harmonics Figure 3b can be ascribed an increase the ghost reflectance; to Figuresecond-order 3a, the apparent growthinof the first-, negative first-,tosecondandofnegative second-order in Figure in 3c,Figure the increase the leakage beams of leads to growth in the firstand harmonics 3b can beofascribed to an increase the ghost reflectance; in traditional Figure 3c, the increase second-order harmonics. of the leakage beams leads to growth in the traditional first- and second-order harmonics.

-1st

-40 -60 -80

0

(a)

Base

2nd -2nd 1.5 2.0 2.5 3.0 3.5 Frequency (MHz)

 = 0.03  = 0.03 1st -20 r = 0.05

-40

2nd

-1st

-2nd

-60 -80

0

Base (b)

3rd 1.5 2.0 2.5 3.0 3.5 Frequency (MHz)

Magnitude (dB)

Magnitude (dB)

 = 0.03  = 0.03 -20 r = 0.01 1st

Magnitude (dB)

0

 = 0.1 1st  = 0.1 -20 r = 0.01

2nd

-40

Base (c)

-1st

-60 -80

3rd

-2nd

1.5 2.0 2.5 3.0 3.5 Frequency (MHz)

Figure 3. Simulation Figure 3. Simulation spectra spectra of of the the measurement measurement signal signal with with different different degrees degrees of of optical optical mixing mixing and and ghost reflectance, where whereAA==BB== 1, 1, ∆f Δf = = 2.33 2.33 MHz, MHz, vv == 105 105 mm/s, mm/s, (a) = ββ == 0.03, ghost reflectance, (a) with with α α= 0.03, rr == 0.01; 0.01; (b) (b) with with α = ββ ==0.03, α= 0.03,rr==0.05; 0.05;(c) (c)with withαα==ββ==0.1, 0.1,r r= =0.01. 0.01.

Similar to the traditional nonlinear harmonics, the new harmonics arising from ghost reflection Similar to the traditional nonlinear harmonics, the new harmonics arising from ghost reflection and its coupling with optical mixing will also contribute to the nonlinearity of an interferometer. and its coupling with optical mixing will also contribute to the nonlinearity of an interferometer. According to Equation (5), the overall nonlinear phase error can be calculated by using the phasor According to Equation (5), the overall nonlinear phase error can be calculated by using the phasor analysis method [21] which is given as follows: analysis method [21] which is given as follows:

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    nn= 44    n sin(n )  2, n  0 Γ n sin( nφ )    n ∑ .  =arctan  n=−n2,  4 n 6 =0   δϕ = arctan . =4  n cos  n      n Γ cos nφ ( ) 2 n  n∑ 

(7) (7)

n=−2

Inreal realapplications, applications,the the magnitudes of the leakage beams are significantly smaller than In magnitudes of the leakage laserlaser beams are significantly smaller than those those of the intended beams and the ghost reflection is comparably slight, which makes Γ n ≪ Γ0, of the intended beams and the ghost reflection is comparably slight, which makes Γn  Γ0 , n 6= 0. n ≠ 0. Accordingly, Equation be further simplified as follows: Accordingly, Equation (7) can (7) be can further simplified as follows:  1   -1  2   -2 4 Γ Γ−2 sin(2 )  3Γsin Γ4  ).  Γ−  sin( ) Γ2 − sin(4 3  3   1− 1 δϕ ≈ sin(2φ) + sin(4φ).  0sin(φ) +   0 sin(3φ)+ 0 Γ Γ0 Γ00 Γ0 0

(8) (8)

can be be seen seen from fromEquation Equation (8) (8)that thatthe thenegative negativefirstfirst-and andnegative negativesecond-order second-order harmonics harmonics ItIt can arising from ghost reflection result in the negative firstand negative second-order nonlinear errors, arising from ghost reflection result in the negative first- and negative second-order nonlinear errors, opposite in sign to the traditional firstand second-order nonlinearities. Additionally, the thirdand opposite in sign to the traditional first- and second-order nonlinearities. Additionally, the third- and fourth-order harmonics, which are induced by the coupling of ghost reflection and optical mixing, fourth-order harmonics, which are induced by the coupling of ghost reflection and optical mixing, lead to completely new nonlinearities. lead to completely new nonlinearities. 3. Experimental Experimental Validation Validation 3. To verify verify the the developed developed model, model, as as shown shown in in Figure Figure 4,4, an an experimental experimental setup setup of of aa typical typical To interferometer was was established, established, where where the the output output laser laser of of aa commercial commercial laser laser source source (Agilent (Agilent 5517B, 5517B, interferometer Agilent Technologies, Santa Clara, CA, USA) consisted of two orthogonal linear polarized Agilent Technologies, Santa Clara, CA, USA) consisted of two orthogonal linear polarized beamsbeams with a frequency difference 2.33atMHz at the central wavelength 632.8 nm.wave-plate A half wave-plate awith frequency difference of 2.33of MHz the central wavelength of 632.8ofnm. A half (HWP) (HWP) was employed to beam rotate polarization the beam polarization orientation and thereby to adjust the was employed to rotate the orientation and thereby to adjust the amplitudes of amplitudes of leakage beams. To simulate variable ghost reflectance, a reflective neutral density filter leakage beams. To simulate variable ghost reflectance, a reflective neutral density filter (NDF) with with a continuously tunablewas reflectance wasbetween positioned between a target mirror and a(NDF) continuously tunable reflectance positioned a target mirror and QWP1. TheQWP1. target The target mirror was mounted on a linear motorized translation stage. A PIN photodetector mirror was mounted on a linear motorized translation stage. A PIN photodetector (HCA-S-200M-SI, (HCA-S-200M-SI, FEMTO GmbH, Berlin, Germany) was adoptedconversion to conduct FEMTO Messtechnik GmbH, Messtechnik Berlin, Germany) was adopted to conduct photoelectric of photoelectric conversion of the was measurement signalbywhich was then analyzed by a spectrum the measurement signal which then analyzed a spectrum analyzer (Agilent N9010A,analyzer Agilent (Agilent N9010A, Technologies, Santa Clara, CA, USA). Technologies, SantaAgilent Clara, CA, USA). y x

QWP2 Fast axis: 45°

RM

HWP

NDF

TM

DFL PBS P PD

y QWP1 x Fast axis: 45° Spectrum analyzer

Figure4. 4. Experimental Experimental interferometer interferometersetup setupused usedfor forvalidation. validation. DFL: DFL: dual-frequency dual-frequencylaser. laser.BS: BS:beam beam Figure splitter. PBS: polarizing beam splitter. RM: reference mirror. TM: target mirror. QWP: quarter splitter. PBS: polarizing beam splitter. RM: reference mirror. TM: target mirror. QWP: quarter wave-plate. wave-plate. P: Polarizer. PD: photodetector. HWP: halfNDF: wave-plate. neutral P: Polarizer. PD: photodetector. HWP: half wave-plate. neutral NDF: density filter. density filter.

Figure 5 shows the experimental spectra of the measurement signal with different degrees of Figure 5 shows the experimental spectra of the measurement signal with different degrees of optical mixing and ghost reflectance. The red horizontal line in each panel denotes a background optical mixing and ghost reflectance. The red horizontal line in each panel denotes a background noise noise level of −70 dB. Nonlinear harmonics below this line will not appear in real spectra. In level of −70 dB. Nonlinear harmonics below this line will not appear in real spectra. In Figure 5a Figure 5a (without the HWP and NDF), the leakage beams and the ghost reflectance are relatively (without the HWP and NDF), the leakage beams and the ghost reflectance are relatively small. The thirdsmall. The third- and fourth-order nonlinear harmonics, which are theoretically generated by the coupling of ghost reflection and optical mixing, could not be identified. When enlarging the leakage

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and fourth-order nonlinear harmonics, which are theoretically generated by the coupling of ghost Sensors 2018, 18, x 6 of 12 reflection and optical mixing, could not be identified. When enlarging the leakage beams by rotating the HWP or increasing the ghost reflectance by adjusting the NDF, as shown in Figure 5b, the third-order beams by rotating the HWP or increasing the ghost reflectance by adjusting the NDF, as shown in nonlinear harmonic broke through the background andthe presented in the spectra, as illustrated Figure 5b, the third-order nonlinear harmonic brokenoise through background noise and presented in in the Figure 5b,c, but the fourth-order nonlinear harmonic was still submerged below the red line. spectra, as illustrated in Figure 5b,c, but the fourth-order nonlinear harmonic was still submerged In below addition, Figure 5a, the enlarged ghost reflectance Figure 5b also led to evident the compared red line. Into addition, compared to Figure 5a, the enlargedinghost reflectance in Figure 5b growth of the first-, negative first-, secondand negative second-order harmonics, which is largely also led to evident growth of the first-, negative first-, second- and negative second-order harmonics, consistent the variation Figures 3a and 3b. InFigure Figure3a5c, theFigure increased leakage which iswith largely consistenttrend with between the variation trend between and 3b. In Figurebeams 5c, also resulted in distinct of the traditional first-growth and second-order harmonics, butsecond-order their influence the increased leakage growth beams also resulted in distinct of the traditional first- and on harmonics, the negative firstnegative harmonics was much lower, which corresponds with but theirand influence on second-order the negative firstand negative second-order harmonics was much which corresponds the3a,c. variation trend between Figure 3a,c. thelower, variation trend between with Figure

-1st

-40

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2nd -2nd

-60 3rd

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(c)

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(b)

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-20

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0

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(a)

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-60 3rd -80

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Figure 5. Experimentalspectra spectraofofthe themeasurement measurement signal. signal. (a) Adopting Figure 5. Experimental (a) Without WithoutHWP HWPand andNDF; NDF;(b) (b) Adopting NDF to enhance the ghost reflectance; (c) Adopting HWP to increase the leakage beams. The red line NDF to enhance the ghost reflectance; (c) Adopting HWP to increase the leakage beams. The red line in each panel denotes a background noise level of −70 dB. in each panel denotes a background noise level of −70 dB.

4. Influence of the Proposed Nonlinearity Model on the Heydemann Correction 4. Influence of the Proposed Nonlinearity Model on the Heydemann Correction In real applications, the leakage beams and the ghost reflectance cannot be very large. This In real applications, theharmonics leakage beams and ghost reflectance cannot be very This means means that the nonlinear induced bythe ghost reflection and its coupling withlarge. optical mixing that nonlinear harmonics induced by traditional ghost reflection andharmonics its coupling optical might mixing arethe generally relatively smaller than the nonlinear and with accordingly arecontribute generallylittle relatively smaller than the traditional nonlinear harmonics and accordingly to the overall nonlinear error. However, if we want to achieve sub-nanometermight or contribute little to theand overall nonlineartake error. However, if interferometers we want to achieve sub-nanometer picometer accuracy simultaneously advantage of the with common paths, or such picometer accuracy and simultaneously take advantage of thethe interferometers common as relatively simple configuration and higher thermal stability, periodic error inwith this kind of paths, such as relatively configuration and higher thermal the periodic error in this interferometers must simple be addressed. One well-known method stability, is the Heydemann correction kind of interferometers must addressed. One well-known method is one the needs Heydemann correction algorithm [12]. To apply thisbe method to heterodyne laser interferometers, to transform the measurement and reference signals to into orthogonallaser forminterferometers, [13]. More specifically, theto measurement algorithm [12]. To apply this method heterodyne one needs transform the signal needsand to bereference mixed with two reference signals 90° out[13]. of phase, Im ⨂ Ir (0°) the and measurement Im ⨂ Ir (90°). measurement signals into orthogonal form Morei.e., specifically, N N ◦ ◦ After a low-pass filter, two slow varying direct current (DC) terms can be obtained. For in ◦ ). signal needs to be mixed with two reference signals 90 out of phase, i.e., Im Ir (0 ) andthe Im setup Ir (90 Figure 1, the measurement signal I m can be calculated by Equation (5) and the reference signal I r can After a low-pass filter, two slow varying direct current (DC) terms can be obtained. For the setup in be simplified as cos (∆ωt).signal Therefore, thebe slowly varying terms can derived as follows: Figure 1, the measurement Im can calculated byDC Equation (5) be and the reference signal Ir can be simplified as cos (∆ωt). Therefore, slowly varying   3 DC terms 1  0   the  can   4 be derived as follows:  2 cos    1 cos  2   2 cos  3  I x  2   2 2 2    Γ + Γ Γ + Γ4 3 term −2 TF term   Γ1 + Γ0 +Γ2 cos(φ) + −1 DF  Ix = cos(2φ) + cos(.3φ)  (9) 2 2   2   4 }  |2 2 {z  }  I   2   0 sin     3 |  1 sin{z  2   sin  3   y DF term TF term  2 2 2 . (9)   Γ3 − Γ−1DF term Γ4 −TFΓterm −2  Γ2 − Γ0    I = sin φ + sin 2φ + sin 3φ ( ) ( ) ( )  y 2    | 2 {z } | 2 {z }  The high-order terms in Equation (9),DFincluding the DF (double frequency) and TF (triple term TF term frequency) terms, come from the new nonlinear harmonics in the proposed nonlinearity model, The are high-order terms inreflection Equationand (9),itsincluding the optical DF (double frequency) and TF (triple which induced by ghost coupling with mixing. Figure 6a illustrates the Lissajous terms, trajectory between andnew Ix, with α = β = 0.03, r = 0.05.in For comparison, graph based on frequency) come fromIythe nonlinear harmonics the proposed the nonlinearity model, the conventional model is also presented. For the conventional model, those high-order terms in Equation (9) are zero, thus the trajectory of Iy versus Ix is a perfect ellipse. As for the proposed model,

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which are induced by ghost reflection and its coupling with optical mixing. Figure 6a illustrates the Lissajous trajectory between Iy and Ix , with α = β = 0.03, r = 0.05. For comparison, the graph based on the conventional model is also presented. For the conventional model, those high-order terms Sensors 2018, 18, x of 12 in Equation (9) are zero, thus the trajectory of Iy versus Ix is a perfect ellipse. As for the 7proposed model, the Lissajous trajectory still looks like an ellipse and the major visible changes are at the the Lissajous trajectory still looks like an ellipse and the major visible changes are at the center as well centerasas well as and the major axes. However, if the difference between the in two trajectories the major minor and axes.minor However, if the difference between the two trajectories Figure 6a is in Figure 6a is amplified by a factor of 5, as shown in Figure 6b, for the proposed model, the Lissajous amplified by a factor of 5, as shown in Figure 6b, for the proposed model, the Lissajous trajectory trajectory like a cardioid. can be interpreted by(9). Equation (9).theRegarding high-order looks looks like a cardioid. This canThis be interpreted by Equation Regarding high-orderthe terms, the of the of DFthe terms are significantly larger than the TF terms, thus distortion is terms,amplitudes the amplitudes DF terms are significantly largerthose thanofthose of the TF terms, thus distortion mainly causedby bythe theDF DFterms. terms. If Equation (9) (9) becomes a cardioid is mainly caused If the the TF TF terms termsare areneglected, neglected, Equation becomes a cardioid equation. In real applications, as the ghost reflectance generallyquite quitesmall, small, the the distortion equation. In real applications, as the ghost reflectance is isgenerally distortionmight might not not be identified easily. be identified easily. 0.8

0.8

(a)

0.6

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0.2

0.2 Iy

Iy

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CM NM

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(b)

CM NM

-0.8 -0.8 -0.6 -0.4 -0.2 0.0

0.8

0.2

0.4

0.6

0.8

Ix

Ix

Figure 6. The Lissajous trajectoriesfor forthe theconventional conventional model and thethe proposed newnew model Figure 6. The Lissajous trajectories model(CM) (CM) and proposed model (NM), with α = β = 0.03, r = 0.05. (a) The difference between the two trajectories is amplified by a factor (NM), with α = β = 0.03, r = 0.05. (a) The difference between the two trajectories is amplified by a factor of 1. (b) The difference between the two trajectories is amplified by a factor of 5. of 1. (b) The difference between the two trajectories is amplified by a factor of 5.

Figure 7a shows the Lissajous trajectories after Heydemann correction for the conventional

Figure shows the Lissajouserrors trajectories Heydemann correction for the conventional model. model. 7a In theory, the nonlinear can beafter completely corrected through elliptical fitting, making In theory, the nonlinear cancircle be completely through ellipticalFor fitting, making the the corrected trajectoryerrors a perfect centering at corrected the origin of the coordinates. the proposed model, the corrected trajectory overlaps thatofofthe thecoordinates. conventional For model. shown in corrected trajectory a perfect circlealmost centering at thewith origin the As proposed model, Figure 7a,trajectory there is no distinguishable deviation. theconventional difference between two corrected the corrected almost overlaps with that ofIfthe model.these As shown in Figure 7a, is amplified bydeviation. a factor of 7,Ifasthe illustrated in Figure 7b, the deviation be identified. For theretrajectories is no distinguishable difference between these two can corrected trajectories is the proposed model, the corrected graph is still a cardioid. However, in a real case, this deviation amplified by a factor of 7, as illustrated in Figure 7b, the deviation can be identified. For the proposed might be invisible due to the generally quite small ghost reflection. model, the corrected graph is still a cardioid. However, in a real case, this deviation might be invisible due to the generally quite small ghost reflection. 1.6 1.6 1.2 1.6 0.8 1.2

CM NM

(a)

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(b)

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Figure 7. The corrected LissajousIxtrajectories for the conventional model I(CM) and the proposed new x model (NM), with α = β = 0.03, r = 0.05. (a) The difference between the two trajectories is amplified by Figure 7. Theofcorrected Lissajous between trajectories for the conventional model and the proposed new a factor 1. (b) The difference the two trajectories is amplified by(CM) a factor of 7.

model (NM), with α = β = 0.03, r = 0.05. (a) The difference between the two trajectories is amplified by In order to The quantify the influence the high-order nonlinear harmonics, the of nonlinear phases a factor of 1. (b) difference betweenof the two trajectories is amplified by a factor 7. before and after the Heydemann correction were calculated. Results are provided in Figure 8. It can

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In2018, order Sensors 18, xto quantify the influence of the high-order nonlinear harmonics, the nonlinear phases 8 of 12 before and after the Heydemann correction were calculated. Results are provided in Figure 8. It can be be seen Figure 8a,the forseveral the several giventhat cases, the periodic are dominated by the seen fromfrom Figure 8a, for given cases, thethat periodic errors areerrors dominated by the first-order first-order nonlinearity. When theofamplitudes of leakage at αghost = β =reflectance 0.03 and ghost nonlinearity. When the amplitudes leakage beams are keptbeams at α = are β = kept 0.03 and is set reflectance is set 0, 0.01, 0.03, andthe 0.05, respectively, nonlinearities for overlapped, these cases are almost at 0, 0.01, 0.03, andat0.05, respectively, nonlinearities forthe these cases are almost indicating overlapped, indicating that ghost reflection generally contributes to the if overall However, that ghost reflection generally contributes little to the overall error.little However, α anderror. β increase from if α to and β increase to 0.05 and 0.1, there significant increases of thethe nonlinearities. 0.03 0.05 and 0.1, from there 0.03 are significant increases of are the nonlinearities. Therefore, magnitudes Therefore, the magnitudes of the nonlinearities are determined predominantly by the amplitudes of of the nonlinearities are determined predominantly by the amplitudes of leakage beams. Figure 8b leakage beams. Figure 8b shows the nonlinear phases after the Heydemann correction, when the shows the nonlinear phases after the Heydemann correction, when the ghost reflection is zero, such as ghost reflection zero, asαthe 0.03, r = 0 anderrors α = βcan = 0.1, = 0; corrected the nonlinear errors the cases α = β = is 0.03, r =such 0 and = βcases = 0.1,αr ==β0;=the nonlinear be rfully regardless can be fully corrected regardless of the magnitudes of leakage beams. When r = 0, the proposed of the magnitudes of leakage beams. When r = 0, the proposed nonlinearity model is identical to the nonlinearity modelOtherwise, is identicalthese to theerrors traditional Otherwise, these errors cannot traditional model. cannotmodel. be effectively compensated even withbea effectively tiny ghost compensated with a tiny ghost reflectance r = 0.01.sharply Moreover, residual errorreflectance. increases reflectance of reven = 0.01. Moreover, the residual errorofincreases withthe increased ghost sharply with increased ghost reflectance. For the case α = β = 0.03, r = 0.01, the residual error is smaller For the case α = β = 0.03, r = 0.01, the residual error is smaller than that before correction. For the two than that correction. For the two cases same reflectance, α =r = β =0.03, 0.03,the r = residual 0.03 and cases withbefore the same ghost reflectance, i.e., αwith = β =the 0.03, r =ghost 0.03 and α = β = i.e., 0.05, α = β =are 0.05, r = 0.03, residual errors arethat almost same.errors This suggests that the residual errors are errors almost the the same. This suggests the the residual are determined only by the ghost determined only by the ghost reflectance and are irrelevant to the leakage beams. Additionally, the reflectance and are irrelevant to the leakage beams. Additionally, the magnitude of the residual error magnitude foristhe casetoαthat = β =before 0.03, r =correction. 0.03 is closeIntocontrast, that before for the caseof α the = βresidual = 0.03, rerror = 0.03 close forcorrection. the case α In = contrast, β = 0.03, for the case α = β = 0.03, r = 0.05, the residual error is even significantly larger than that before correction. r = 0.05, the residual error is even significantly larger than that before correction. Therefore, based Therefore, based on thethough above analysis, though ghost reflection and its coupling with optical has on the above analysis, ghost reflection and its coupling with optical mixing has a mixing negligible a negligible influence on the overall nonlinearity, it can significantly reduce the effectiveness of the influence on the overall nonlinearity, it can significantly reduce the effectiveness of the Heydemann Heydemann correction or even make this compensation algorithm completely ineffective. correction or even make this compensation algorithm completely ineffective. == 0.03, r = 0 == 0.1, r = 0 == 0.03, r = 0.01 == 0.03, r = 0.03 == 0.03, r = 0.05 == 0.05, r = 0.03

8 4

(a)

0 -4 -8 -12

== 0.03, r = 0 == 0.1, r = 0 == 0.03, r = 0.01 == 0.03, r = 0.03 == 0.03, r = 0.05 == 0.05, r = 0.03

12

Nonlinear phase (deg.)

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0 -4 -8 -12

0

30 60 90 120 150 180 210 240 270 300 330 360

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Phase (deg.)

Figure 8. 8. Nonlinear phases with different degrees of optical mixing and ghost reflectance, (a) before before Figure Heydemann correction; correction; (b) (b) after after Heydemann Heydemann correction. correction. Heydemann

5. Mechanism Mechanism of of the the Residual Residual Nonlinear Nonlinear Errors Errors after after the the Heydemann Heydemann Correction Correction 5. As described described above, above, an an “abnormality” “abnormality” is is aa case case in in which which the the Heydemann Heydemann correction correction makes makes As nonlinear errors larger. For instance, when α = β = 0.03 and r = 0.05, as in Figure 7a, there is no visible nonlinear errors larger. For instance, when α = β = 0.03 and r = 0.05, as in Figure 7a, there is no visible deviation in in the the corrected corrected Lissajous Lissajous trajectory trajectory when when compared compared to to the the conventional conventional model. model. In deviation In contrast, contrast, in Figure 8b, the compensated phase is higher than that before correction. The Heydemann correction in Figure 8b, the compensated phase is higher than that before correction. The Heydemann correction is an an ellipse-fitting it it fits the Lissajous trajectory to is ellipse-fitting algorithm algorithm based basedon onthe theleast leastsquare squaremethod. method.First, First, fits the Lissajous trajectory obtain thethe ellipse semimajor and and to obtain ellipseparameters, parameters,including includingthe thecenter centerofofthe the ellipse ellipse xx00 and and yy00,, the the semimajor semiminor axes of the ellipse a and b, and the non-orthogonal angle φ 0 between the signals Ix and Iy. semiminor axes of the ellipse a and b, and the non-orthogonal angle φ0 between the signals Ix and these parameters are used to correct the original signals and thereby remove the nonlinear INext, y . Next, these parameters are used to correct the original signals and thereby remove the nonlinear errors. Finally, Finally, the the real real measurement measurement phase phase can can be be obtained obtained with with the the corrected corrected signals signals by by arctangent arctangent errors. operation. In order to analyze the above-mentioned abnormality for the Lissajous trajectory based on on operation. In order to analyze the above-mentioned abnormality for the Lissajous trajectory based the proposed proposed model, model, aa series series of of ellipse-fittings ellipse-fittings were were carried carried out out and and the the fitted fitted parameters parameters are are listed listed the in Table 1. The amplitudes of the nonlinear harmonics Γn (n = −2, −1, 0, 1, 2) are also presented in the table.

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in Table 1. The amplitudes of the nonlinear harmonics Γn (n = −2, −1, 0, 1, 2) are also presented in the table. Table 1. Ellipse-fitting parameters with different degrees of optical mixing and ghost reflectance. α

0.03

0.1

0.03

0.05

0.03

0

0

β

0.03

0.1

0.03

0.05

0.03

0

0

r

0

0

0.03

0.03

0.05

0.05

0.03

Γ1

0.06

0.2

0.0866

0.1244

0.1046

0.05

0.03

Γ −1

0

0

0.0283

0.0283

0.0453

0.0451

0.0282

0.01

0.0025

0.0051

0.0034

0

0

0.0023

0.0023

8.5 × 10−4

Γ2

9×

10−4

10−4

10−4

Γ −2

0

0

Γ0

1

1

0.9425

0.9436

0.9051

0.9025

0.9409

x0

0.03

0.1

0.0575

0.0765

0.0751

0.0476

0.0291

y0

0

0

0

0

0

0

0

a

0.5005

0.5050

0.4729

0.4748

0.4554

0.4524

0.4709

b

0.4996

0.4950

0.4705

0.4697

0.4520

0.4524

0.4709

φ0

0

0

0

0

0

0

0

8.5 ×

8.5 ×

According to Table 1, it can be found that the ellipse-fitting parameters and the amplitudes of the nonlinear harmonics meet the relation as follows: x0 ≈

Γ 1 + Γ −1 Γ 0 + Γ −2 + Γ 2 Γ 0 + Γ −2 − Γ 2 , y0 = 0, a ≈ , b≈ . 2 2 2

(10)

Generally, Γ2 , Γ−2  Γ0 , thus Equation (10) can be further rewritten as x0 =

Γ0 Γ 1 + Γ −1 , y0 = 0, a ≈ b ≈ . 2 2

(11)

In this approximation, the “corrected signals” of Equation (9) can be obtained as        Ix0 =     

 Ix − x0 a

 Γ −1 = 1a  − 2 +

 Γ0 + Γ2 2

cos(φ) +

 Γ −2 + Γ 4 Γ −1 + Γ 3 cos(2φ) + cos(3φ)  | 2 {z } | 2 {z } DF term

TF term



        Iy0 =    

Iy −y0 b

(12)

 Γ −Γ  Γ 3 − Γ −1 Γ − Γ −2 2 0 ≈ 1a  sin(2φ) + 4 sin(3φ)  2 sin(φ) + . | 2 {z } | 2 {z } DF term

TF term

As Equation (12) is an approximate formula, to verify its reliability, when α = β = 0.03, r = 0.05, the Lissajous trajectory calculated by using Equation (12) is compared with the corrected Lissajous trajectory of Equation (9), which is obtained by directly applying the Heydemann correction to Equation (9). As illustrated in Figure 9a, these two trajectories almost overlap, indicating a good approximation. If the TF terms in Equation (12) are omitted, Equation (12) will be a cardioid equation, which is the source of the heart-shaped distorted trajectory in Figure 7b. With reference to the relationship between Equations (8) and (9), the residual nonlinear phase error in Equation (12) can be backward derived as δϕ0 ≈ −

2Γ−1 Γ 2 − Γ −2 Γ3 Γ sin(φ) + sin(2φ) + sin(3φ) + 4 sin(4φ). Γ0 Γ0 Γ0 Γ0

(13)

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0.4 Iy

Error calculated by using Equation (13) Error after Heydemann correction Difference

4

0.8

0.0 -0.4 -0.8

6

Lissajous trajectory of Equation (12) Corrected Lissajous trajectory of Equation (9)

1.2

(a)

-1.2 -1.6 -1.2 -0.8 -0.4 0.0 0.4 0.8 1.2 1.6 Ix

2 0 -2 -4 -6

(b) 0

50

100

150

200

250

300

350

Phase (deg.)

Figure Figure 9. 9. (a) (a) Lissajous Lissajous trajectories trajectories calculated calculated by by using using Equation Equation (12) (12) and and corrected corrected Lissajous Lissajous trajectory trajectory of Equation (9); (b) Nonlinear errors calculated by using Equation (13) and residual of Equation (9); (b) Nonlinear errors calculated by using Equation (13) and residual error error obtained obtained by by directly the Heydemann Heydemann correction correction to to Equation Equation (9). (9). A A= =B directly applying applying the B == 1, 1, αα ==ββ==0.03, 0.03,rr==0.05. 0.05.

As illustrated in Figure 8, for the proposed model, if the ghost reflectance is not zero, whether When α = β = 0.03, r = 0.05, the nonlinear error calculated by using Equation (13) is shown in the Heydemann correction is applied or not, the first-order nonlinearity dominates the overall Figure 9b. For comparison, the residual error obtained by directly applying the Heydemann correction nonlinear error. Generally, the ghost reflectance is a quite small value, according to to Equation (9) is also illustrated in the same figure. The two curves are basically consistent and the Equations (4)–(6) and (8). Before correction, the amplitude of the first-order nonlinearity can be small difference can be attributed largely to the approximation of a ≈ b. For the proposed model, evaluated by Equation (13) can be used to estimate the residual phase after the Heydemann correction. A  Bif  the ghost reflectance is not zero, whether the As illustrated in Figure 8, for the  proposed 1   1  model,     . (14) AB nonlinearity dominates the overall nonlinear Heydemann correction is applied or not,the first-order 0 error. Generally, the ghost reflectance is a quite small value, according to Equations (4)–(6) and (8). For simplicity, in this paper, the values A and B are always set at 1. Accordingly, before the Before correction, the amplitude of the first-order nonlinearity can be evaluated by Heydemann correction, the first-order nonlinearity is determined mainly by the optical mixing degree. Similarly, according to Equation after correction, the amplitude of the first-order Aβ + Bα (Γ1 − Γ−1(13), ) = α + β. (14) ≈ nonlinearity can be evaluated by Γ0 AB 1 For simplicity, in this paper, the values 2Aand the  2Br , are always set at 1. Accordingly, before(15)  0 is determined mainly by the optical mixing degree. Heydemann correction, the first-order nonlinearity Similarly, according to Equation (13), after correction, the amplitude of the first-order nonlinearity can which is only determined by the ghost reflectance. With the help of Equations (14) and (15), the be evaluated by above-mentioned abnormality in Figure 8 can be reasonably interpreted. For case α = β = 0.03, r = 0 2Γ−1 ≈ 2r, and α = β = 0.1, r = 0, as the ghost reflectance is zero, there is no residual error after correction. (15) For Γ0 case α = β = 0.03, r = 0.01, 2r < α + β, the residual error decreases after correction. For case α = β = 0.03, which only determined byclear the ghost reflectance. With the help ofthe Equations and (15), r = 0.03,isbecause 2r = α + β, no difference can be observed between nonlinear(14) errors before the above-mentioned abnormality case α = β = after 0.03, and after correction. For case α = βin= Figure 0.05, r =8 can 0.03,be asreasonably 2r < α + β, interpreted. the residual For error decreases rcorrection, = 0 and α but = β is = very 0.1, rclose = 0, as the ghost reflectance is zero, there is no residual error after correction. to case α = β = 0.03, r = 0.03, due to the same ghost reflectance. By contrast, Forcase case = 0.03, r = 0.01, the residual leads error to decreases after instead. correction. For case in α =α β==β0.03, r = 0.05, then 2r 2r >< αα++β,β, compensation a larger error α = β = 0.03, r = 0.03, because 2r = α + β, no clear difference can be observed between the nonlinear errors before and after correction. For case α = β = 0.05, r = 0.03, as 2r < α + β, the residual error 6. Conclusions decreases after correction, but is very close to case α = β = 0.03, r = 0.03, due to the same ghost Even after the Heydemann correction, nonlinear ranging from hundreds of reflectance. By contrast, in case α = β = 0.03, rresidual = 0.05, then 2r > α +errors, β, compensation leads to a larger picometers to several nanometers, are still found in heterodyne laser interferometers. This is a crucial error instead. factor impeding the realization of picometer level metrology, but its source and mechanism have barely been investigated. To study this problem, a novel nonlinear model based on optical mixing 6. Conclusions and coupling with ghost reflection is proposed and then verified by experiments. After intense Even after the Heydemann correction, residual nonlinear errors, ranging from hundreds of investigation of this new model's influence, results indicate that the new additional high-order and picometers to several nanometers, are still found in heterodyne laser interferometers. This is a crucial negative-order nonlinear harmonics, arising from ghost reflection and its coupling with optical factor impeding the realization of picometer level metrology, but its source and mechanism have mixing, have only a negligible contribution to the overall nonlinear error. In real applications, any barely been investigated. To study this problem, a novel nonlinear model based on optical mixing effect on the Lissajous trajectory might be invisible due to the small ghost reflectance. However, even and coupling with ghost reflection is proposed and then verified by experiments. After intense a tiny ghost reflection can significantly worsen the effectiveness of the Heydemann correction, or even make this correction completely ineffective, i.e., compensation makes the error larger rather than smaller. Moreover, the residual nonlinear error after correction is dominated only by the ghost

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investigation of this new model’s influence, results indicate that the new additional high-order and negative-order nonlinear harmonics, arising from ghost reflection and its coupling with optical mixing, have only a negligible contribution to the overall nonlinear error. In real applications, any effect on the Lissajous trajectory might be invisible due to the small ghost reflectance. However, even a tiny ghost reflection can significantly worsen the effectiveness of the Heydemann correction, or even make this correction completely ineffective, i.e., compensation makes the error larger rather than smaller. Moreover, the residual nonlinear error after correction is dominated only by the ghost reflectance. Therefore, for real applications that intend to achieve sub-nanometer or picometer level accuracy, ghost reflection and its coupling with optical mixing should be taken into serious consideration, and the design must be elaborated to restrict ghost reflection. This study is expected to promote the development of ultra-high precision laser interferometry. Acknowledgments: This research was financially supported by National Natural Science Foundation of China (NSFC) (51675138, 91536224, 61675058 and 51605120). Author Contributions: Haijin Fu contributed to developing the ideas of this research. Haijin Fu and Yue Wang were involved in the mathematical development, experiment setting as well as drafting of the paper. Pengcheng Hu carried out the data analysis. Jiubin Tan and Zhigang Fan supervised the work and proofread the manuscript. All of the authors approved the final version of the manuscript. Conflicts of Interest: The authors declare no conflict of interest.

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