Relations between nonlinearity and PBS in heterodyne Michelson interferometer with different optical structures

Optik 126 (2015) 5061–5066

Contents lists available at ScienceDirect

Optik journal homepage: www.elsevier.de/ijleo

Relations between nonlinearity and PBS in heterodyne Michelson interferometer with different optical structures Haijiang Hu ∗ , Juju Hu College of Physics and Communication Electronics, Jiangxi Normal University, Nanchang 330022, Jiangxi, China

a r t i c l e

i n f o

Article history: Received 10 November 2014 Accepted 11 September 2015 Keywords: Nonlinearity PBS Heterodyne interferometer Nanometric measurement

a b s t r a c t Heterodyne interferometer is an important optical instrument which is widely used in the precision measurement and positioning. As the important optical part, the polarized beam splitter (PBS) divides a polarized laser beam with the random direction into two orthogonal linear polarized beams. This paper studies the relation between the nonlinearity errors and PBS in two typical heterodyne Michelson interferometer structures and proves that the nonlinearity errors from the nonideal PBS are related with the optical structures of interferometer. Moreover, the details on the mathematical analysis and the simulations for the relations are demonstrated in this paper. © 2015 Elsevier GmbH. All rights reserved.

1. Introduction Heterodyne interferometer is an important optical instrument which is widely used in the precision measurement and positioning. In contrast with the homodyne interferometer, the heterodyne interferometer is thought to be less affected by the environmental impacts. According to the difference in the structures of the optical systems, the heterodyne interferometer can efficiently measure many physical or geometric quantities, such as length, velocity, and so on. However, the various errors reduce the measurement accuracy, especially in the nanometric or picometric measurement. Among these error sources, the nonlinear error is the most important error source in the heterodyne interferometer, which is initially proposed in [1] and firstly described experimentally in [2]. The periodic nonlinear error usually ranges from several nanometers to tens of nanometers. [3,4] analyze the relation between the nonlinearity error and the behavior of the optical parts. The main error sources include the elliptic polarization from the imperfect dual-frequency laser and the incomplete separation of the orthogonal polarization light from the imperfect polarized beam splitter (PBS). Moreover, the nonlinearity error does not exist independently and usually mixes with the Abbe error, the deadpath error, and the environmental error. Many scientists study the property of nonliearity errors [3–5] and the methods for error detection and elimination [6–12].

PBS is an ordinary optical part in the optical instruments, which can divide a polarized laser beam with the random direction into two orthogonal linear polarized beams. As the one of the core components in the heterodyne interferometer, PBS separates the two orthogonal beams with the different frequencies from an incident mixed beam. Although the former studies [4,5] analyze the nonlinear error is affected by imperfect PBS, the influence contrast from different optical structure is not studied. This paper studies the relation between nonlinearity and PBS in two typical heterodyne Michelson interferometers. Moreover, the paper also studies the relation between nonlinear error and optical subdivision from imperfect PBS. This study is useful for the elimination of nonlinear error and the PBS design.

2. Laser beam propagation in nonideal PBS in structure A 2.1. Laser beam propagation in ideal PBS At first the ideal scheme is analyzed. Fig. 1 shows a typical heterodyne interferometer. We assume both the laser beam and the PBS are ideal in Fig. 1. The ideal laser beam often includes two orthogonal components, which is described as follows:



E1 = E1a sin (2f1 t + ϕ1a ) E2 = E2a sin (2f2 t + ϕ2a )

∗ Corresponding author. Tel.: +86 18170030428. E-mail address: [email protected] (H. Hu). http://dx.doi.org/10.1016/j.ijleo.2015.09.134 0030-4026/© 2015 Elsevier GmbH. All rights reserved.

(1)

where E1a and E2a are amplitudes, f1 and f2 are frequencies, ϕ1a and ϕ2a are initial phase, respectively.

5062

H. Hu, J. Hu / Optik 126 (2015) 5061–5066

Obviously, E21 contains E2 and the part of E1 . After E11 and E21 reach this PBS again, E12 and E22 can be described as follows:



E11 = (1 − a) E1a sin (2f1 t + ϕ1a + ϕ1r )

(8)

E21 = (1 − a) E2a sin [2 (f2 + f ) t + ϕ2a + ϕ2r ]

The reason why E22 does not contain the part of E1 is that this part of E1 in E21 transmits through this PBS completely. Therefore, Im1A = Fig. 1. Typical heterodyne interferometer, structure A: BS, beam splitter; PBS, polarizing beam splitter; QWP, quarter wave plate; P, polarization analyzer.

Because of the Doppler effects, the laser beam in photosensor D2 is described as follows when the plane mirror P2 moves:



E1 = E1a sin (2f1 t + ϕ1a + ϕ1r ) E2 = E2a sin [2 (f2 + f ) t + ϕ2a + ϕ2r ]

(2)

where f is the frequency shift because of the movement of plane mirror P2 , ϕ1r and ϕ2r are additional phase shift for the optical path, respectively. The photosensors D1 and D2 convert laser beam into electrical signal. The electrical signal is shown as follows: I=

1 (E1 + E2 )2 . 2

Im = I1m cos [2 (f1 − f2 − f ) t + ϕm ] = I1m cos [2 (f1 − f2 ) t + ϕm + ϕ]

Im1A =

For this PBS, we assume the fraction b of E2 reaches plane mirror P1 . Similarly, in this scheme, E11 and E21 can be described as follows:



E21 = (1 − b) E2a sin (2f2 t + ϕ2a + ϕ2r )

.

(11)

And E12 and E22 can be expressed as follows: (5)



E11 = (1 − b) E1a sin (2f1 t + ϕ1a + ϕ1r ) E21 = (1 − b) E2a sin [2 (f2 + f ) t + ϕ2a + ϕ2r ]

(6)

In this scheme we assume the optical parts in the heterodyne interferometer are ideal except the PBS that are shown as Fig. 1. This PBS is an imperfect PBS with the perfect transmission of P polarized beam. In this PBS, we assume the fraction a of E1 reaches the plane mirror P2 . In Fig. 2, E11 and E21 can be expressed as follows: E11 = (1 − a) E1a sin (2f1 t + ϕ1a + ϕ1r )

E11 = E1a sin (2f1 t + ϕ1a + ϕ1r ) + bE2a sin (2f2 t + ϕ2a + ϕ1r )

(4)

2.2. Imperfect PBS with the perfect transmission of P polarized beam

E21 = aE1a sin (2f1 t + ϕ1a + ϕ2r ) + E2a sin (2f2 t + ϕ2a + ϕ2r )

(10)

2.3. Imperfect PBS with the perfect reflection of S polarized beam

(3)

where  are the nonlinearity errors which is the function of ϕ.



1 (1 − a)2 E1a E2a cos [2 (f1 − f2 ) t + ϕm + ϕ] 2

From Eq. (10), we can see  (ϕ) = 0, Therefore, an imperfect PBS with the perfect transmission of P polarized beam can not cause the nonlinearity errors in structure A.

If the nonlinearity errors exist, Eq. (5) can be expressed as: Im = I1m cos [2 (f1 − f2 ) t + ϕm + ϕ +  (ϕ)]

(9)

and

In ideal conditions, reference signal from D1 and measurement signal from D2 are: Ir = I1r cos [2 (f1 − f2 ) t + ϕr ]

1 (1 − a)2 E1a E2a cos [2 (f1 − f2 − f ) t + ϕm ] 2

(7)

.

(12)

From Eq. (12), we can obtain: Im2A =

1 2 (1 − b) E1a E2a cos [2 (f1 − f2 ) t + ϕm + ϕ] . 2

(13)

In Eq. (13),  (ϕ) = 0. So, an imperfect PBS with the perfect reflection of S polarized beam also can not cause the nonlinearity errors in structure A. 2.4. Imperfect PBS with nonideal transmission and nonideal reflection In this scheme, we assume the fraction a of E1 reaches plane mirror P2 and the fraction b of E2 reaches plane mirror P1 . E11 and E21 can be expressed as follows:



E11 = (1 − a) E1a sin (2f1 t + ϕ1a + ϕ1r ) + bE2a sin (2f2 t + ϕ2a + ϕ1r ) E21 = aE1a sin (2f1 t + ϕ1a + ϕ2r ) + (1 − b) E2a sin (2f2 t + ϕ2a + ϕ2r )

(14)

.

And E12 and E22 can be expressed as follows:

⎧ E11 = (1 − a) (1 − b) E1a sin (2f1 t + ϕ1a + ϕ1r ) ⎪ ⎪ ⎪ ⎨ + abE2a sin (2f2 t + ϕ2a + ϕ1r )

E21 = abE1a sin [2 (f1 + f ) t + ϕ1a + ϕ2r ] ⎪ ⎪ ⎪ ⎩

+ (1 − a) (1 − b) E2a sin [2 (f2 + f ) t + ϕ2a + ϕ2r ]

Fig. 2. Propagation of polarized light in PBS in structure A.

Therefore we can get:

.

(15)

H. Hu, J. Hu / Optik 126 (2015) 5061–5066 0.5

0.1 ideal P-ideal S-ideal PS-imperfect

0.4 0.3

5063

0.5

ideal P-ideal S-ideal PS-imperfect

0.08 0.06

0.1 ideal a=0.1 a=0.2 a=0.3

0.4 0.3

0.06

0.2

0.04

0.2

0.04

0.1

0.02

0.1

0.02

0

0

0

0

-0.1

-0.02

-0.1

-0.02

-0.2

-0.04

-0.2

-0.04

-0.3

-0.06

-0.3

-0.06

-0.4

-0.08

-0.4

-0.08

-0.5

-0.1

0

1

2

3

4

-0.5

2.45

2.5

2.55

-6

x 10

0

1

2

3

4 x 10

-6

x 10

0.5

1 2 = (1 − a)2 (1 − b) E1a E2a cos [2 (f1 − f2 − f ) t + ϕ1a + ϕ1r − ϕ2a − ϕ2r ] 2

+ +

0.3

.

1 2 cos (2ft + ϕ2r − ϕ1r ) (1 − a) (1 − b) abE1a 2 1 2 cos (2ft + ϕ2r − ϕ1r ) (1 − a) (1 − b) abE2a 2

2.45

2.5

2.55 x 10

-6

0.1 ideal a=0.1 a=0.2 a=0.3

0.4

1 + a2 b2 E1a E2a cos [2 (f1 − f2 + f ) t + ϕ1a + ϕ2r − ϕ2a − ϕ1r ] 2 + (1 − a) (1 − b) abE1a E2a cos [2 (f1 − f2 ) t + ϕ1a − ϕ2a ]

-0.1

-6

Fig. 4. Relation between a and Im in structure A with the imperfect PBS (b = 0).

Fig. 3. Im in the structure A.

Im3A

ideal a=0.1 a=0.2 a=0.3

0.08

(16)

In Eq. (16), if a = 0 or b = 0, Eq. (16) equals to Eq. (10) or Eq. (13). But if a and b don’t equal to 0, the nonlinearity errors exists. Consequently, the nonlinearity errors are caused by the imperfect PBS with the nonideal transmission and the nonideal reflection in structure A. 2.5. Simulations In order to testify our analysis, we use Matlab 6.5 to simulate the above scheme. Fig. 3 shows the Im signal in four different schemes. The first scheme is that the PBS is ideal. The second scheme uses an imperfect PBS with the perfect transmission of P polarized beam. The third scheme uses an imperfect PBS with the prefect reflection of S polarized beam. The fourth scheme uses Imperfect PBS with nonideal transmission and nonideal reflection. All four schemes use the structure A that is shown as Fig. 1. From Fig. 3, we can see the imperfect PBS with the perfect transmission of P polarized beam or the imperfect PBS with the perfect reflection of S polarized beam only leads to reduce the signal amplitude, and does not produce the nonlinear error. However, the imperfect PBS with nonideal transmission and nonideal reflection not only leads to the signal amplitude decrease, but also generates the nonlinear error. Fig. 4 shows the relation between fraction a and signal amplitude in structure A with the imperfect PBS with the perfect transmission of P polarized beam. We can see the nonlinear error is not produced in this scheme no matter how the fraction a changes. However, if the fraction a is larger, the signal amplitude is smaller. Fig. 5 shows the relation between fraction a and signal amplitude in structure A with the imperfect PBS with the imperfect transmission of P polarized beam and the imperfect reflection of S polarized beam. In Fig. 5, when b is not zero, the nonlinear error is produced. Furthermore, if the fraction a is larger, the signal amplitude is smaller. The relation between fraction b and signal amplitude in structure A is similar to the result of the relation between fraction a and signal amplitude. If the fraction b is larger, the signal amplitude is

0.06

0.2

0.04

0.1

0.02

0

0

-0.1

-0.02

-0.2

-0.04

-0.3

-0.06

-0.4

-0.08

-0.5

0

1

2

3

4 x 10

ideal a=0.1 a=0.2 a=0.3

0.08

-0.1

2.45

2.5

2.55

-6

x 10

-6

Fig. 5. Relation between a and Im in structure A with the imperfect PBS (b = 0.2).

smaller. When a is zero, the nonlinear error is not produced. When a is not zero, the structure A has the nonlinear error. Fig. 6 shows the relation between fraction b and signal amplitude in structure A with the imperfect PBS with the imperfect transmission of P polarized beam and the imperfect reflection of S polarized beam. 3. Laser beam propagation in nonideal PBS in structure B 3.1. Laser beam propagation in ideal PBS In this section we study another heterodyne interferometer structure that is shown as Fig. 7. In this scheme all optical parts are ideal. This structure is different from the structure in Fig. 1 and does not use the quarter wave plates. Fig. 8 shows the details on the propagation of polarized light in PBS and corner cube prisms.

0.5

0.1 ideal b=0.1 b=0.2 b=0.3

0.4 0.3

0.06

0.2

0.04

0.1

0.02

0

0

-0.1

-0.02

-0.2

-0.04

-0.3

-0.06

-0.4

-0.08

-0.5

0

1

2

3

4 x 10

ideal b=0.1 b=0.2 b=0.3

0.08

-6

-0.1

2.45

2.5

2.55 x 10

Fig. 6. Relation between b and Im in structure A with the imperfect PBS (a = 0.1).

-6

5064

H. Hu, J. Hu / Optik 126 (2015) 5061–5066

3.2. Imperfect PBS with the perfect transmission of P polarized beam Then the structure B with imperfect PBS is studied. In this scheme, we assume the fraction a of E1 reaches the corner cube prism M2 . If the PBS is imperfect with the perfect transmission of P polarized beam, E11 and E21 in Fig. 8 can be expressed as follows:



E11 = (1 − a) E1a sin (2f1 t + ϕ1a + ϕ1r ) E21 = aE1a sin (2f1 t + ϕ1a + ϕr2 ) + E2a sin (2f2 t + ϕ2a + ϕ2r ) (22)

Fig. 7. Typical heterodyne interferometer, structure B: BS, beam splitter; PBS, polarizing beam splitter; P, polarization analyzer.

The same to structure A, the ideal laser beam also includes two orthogonal components, which is described as follows:



E1 = E1a sin (2f1 t + ϕ1a ) E2 = E2a sin (2f2 t + ϕ2a )

(17)

where E1a and E2a are amplitudes, f1 and f2 are frequencies, ϕ1a and ϕ2a are initial phase, respectively. When the plane mirror P2 moves, the laser beam in photosensor D2 is described as follows:



E1 = E1a sin (2f1 t + ϕ1a + ϕ1r ) E2 = E2a sin [2 (f2 + f ) t + ϕ2a + ϕ2r ]

(18)

where f is the frequency shift based on Doppler effects when the plane mirror P2 moves, ϕ1r and ϕ2r are additional phase shift for the optical path, respectively. The photosensors D1 and D2 convert laser beam into electrical signal. In ideal conditions, reference signal from D1 and measurement signal from D2 are: Ir = I1r cos [2 (f1 − f2 ) t + ϕr ] Im = I1m cos [2 (f1 − f2 − f ) t + ϕm ] = I1m cos [2 (f1 − f2 ) t + ϕm + ϕ]

(19)

(20)

Obviously, E21 contains E2 and the part of E1 . After E11 and E21 reach this PBS again, E12 and E22 can be described as follows:

⎧ E = − a)2 E1a sin (2f1 t + ϕ1a + ϕ1r ) ⎪ ⎨ 12 (1 ⎪ ⎩

(23)

+ E2a sin [2 (f2 + f2 ) t + ϕ2a + ϕ2r ] Therefore we can get:

Im1B =

1 2 a E1a E2a cos [2 (f1 + f1 − f2 − f2 ) t + ϕ1a + ϕr2 − ϕ2a − ϕ2r ] 2

1 + a2 (1 − a)2 E1a 2 cos [2 (f1 ) t + ϕ1r − ϕr2 ] 2

(24)

1 + (1 − a)2 E1a E2a cos [2 (f1 − f2 − f2 ) t + ϕ1a + ϕ1r − ϕ2a − ϕ2r ] 2

Obviously, the nonlinearity errors are caused by the imperfect PBS with the nonideal reflection in structure B. This result is different from the former result in the same PBS in structure A. 3.3. Imperfect PBS with the perfect reflection of S polarized beam In the third scheme we assume the fraction b of E2 reaches the corner cube prism M1 . If the PBS is imperfect with the perfect transmission of S polarized beam, E11 and E21 in Fig. 8 can be expressed as follows:



E11 = E1a sin (2f1 t + ϕ1a + ϕ1r ) + bE2a sin (2f2 t + ϕ2a + ϕr1 ) E21 = (1 − b) E2a sin (2f2 t + ϕ2a + ϕ2r )

If the nonlinearity errors exist, Eq. (20) can be expressed as: Im = I1m cos [2 (f1 − f2 ) t + ϕm + ϕ +  (ϕ)]

E22 = a2 E1a sin [2 (f1 + f1 ) t + ϕ1a + ϕr2 ]

(25) (21)

where  are the nonlinearity errors which is the function of ϕ. Therefore, the principle of the structure B is the similar to the principle of structure A.



And E12 and E22 can be expressed as follows: E11 = E1a sin (2f1 t + ϕ1a + ϕ1r ) + b2 E2a sin (2f2 t + ϕ2a + ϕr1 ) 2

E21 = (1 − b) E2a sin [2 (f2 + f2 ) t + ϕ2a + ϕ2r ] (26)

So the signal amplitude can be obtained as follows: Im1B =

Fig. 8. Propagation of polarized light in PBS in structure B.

1 2 b E1a E2a cos [2 (f1 − f2 ) t + ϕ1a + ϕ1r − ϕ2a − ϕr1 ] 2

+

1 2 2 b (1 − b) E2a 2 cos [2 (f2 ) t + ϕr1 − ϕ2r ] 2

+

1 2 (1 − b) E1a E2a cos [2 (f1 − f2 − f2 ) t + ϕ1a + ϕ1r − ϕ2a − ϕ2r ] 2

(27)

Obviously, the nonlinearity errors are caused by the imperfect PBS with the nonideal transmission in structure B. This result is also different from the former result in the same PBS in structure A.

H. Hu, J. Hu / Optik 126 (2015) 5061–5066 0.5

0.1 ideal P-ideal S-ideal PS-imperfect

0.4 0.3

5065

0.5 ideal P-ideal S-ideal PS-imperfect

0.08 0.06

0.1 ideal a=0.1 a=0.2 a=0.3

0.4 0.3

0.06

0.2

0.04

0.2

0.04

0.1

0.02

0.1

0.02

0

0

0

0

-0.1

-0.02

-0.1

-0.02

-0.2

-0.04

-0.2

-0.04

-0.3

-0.06

-0.3

-0.06

-0.4

-0.08

-0.4

-0.08

-0.5

0

1

2

3

4

-0.1 2.5

2.55

2.6

2.65

-6

x 10

2.7 -6

(28)

⎧ E11 = (1 − a)2 E1a sin (2f1 t + ϕ1a + ϕ1r ) ⎪ ⎪ ⎪ ⎨ E21 ⎪ ⎪ ⎪ ⎩

3

4

-0.1 2.5

2.55

2.6

2.65

-6

2.7 -6

x 10

x 10

0.1 ideal a=0.1 a=0.2 a=0.3

0.4

And E12 and E22 can be expressed as follows:

+ b2 E2a sin (2f2 t + ϕ2a + ϕr1 ) = a2 E1a sin [2 (f1 + f1 ) t + ϕ1a + ϕr2 ]

2

0.5

In the fourth scheme we assume the fraction a of E1 reaches plane mirror M2 and the fraction b of E2 reaches plane mirror M1 . E11 and E21 in Fig. 8 can be expressed as follows: E21 = aE1a sin (2f1 t + ϕ1a + ϕr2 ) + (1 − b) E2a sin (2f2 t + ϕ2a + ϕ2r )

1

Fig. 10. Relation between a and Im in structure B with the imperfect PBS (b = 0).

3.4. Imperfect PBS with nonideal transmission and nonideal reflection

E11 = (1 − a) E1a sin (2f1 t + ϕ1a + ϕ1r ) + bE2a sin (2f2 t + ϕ2a + ϕr1 )

0

x 10

Fig. 9. Im in the structure B.



-0.5

ideal a=0.1 a=0.2 a=0.3

0.08

(29)

0.3 0.2

0.06 0.04

0.1

0.02

0

0

-0.1

-0.02

-0.2

-0.04

-0.3

-0.06

-0.4

-0.08

-0.5

ideal a=0.1 a=0.2 a=0.3

0.08

0

1

2

3

4 x 10

-0.1 2.5

2.55

2.6

2.65

-6

2.7 x 10

-6

2

+ (1 − b) E2a sin [2 (f2 + f2 ) t + ϕ2a + ϕ2r ]

Fig. 11. Relation between a and Im in structure B with the imperfect PBS (b = 0.2).

Therefore we can get: Im1B

1 = b2 (1 − a)2 E1a E2a cos [2 (f1 − f2 ) t + ϕ1a + ϕ1r − ϕ2a − ϕr1 ] 2 +

1 2 2 a (1 − b) E1a E2a cos [2 (f1 + f1 − f2 − f2 ) t + ϕ1a + ϕr2 − ϕ2a − ϕ2r ] 2

+

1 2 a (1 − a)2 E1a 2 cos [2 (f1 ) t + ϕr1 − ϕr2 ] 2

+

1 2 2 (1 − a) (1 − b) E1a E2a cos [2 (f1 − f2 − f2 ) t + ϕ1a + ϕ1r − ϕ2a − ϕ2r ] 2

+

1 2 2 a b E1a E2a cos [2 (f1 + f1 − f2 ) t + ϕ1a + ϕr2 − ϕ2a − ϕr1 ] 2

+

1 2 b (1 − a)2 E2a 2 cos [2 (f2 ) t + ϕ2r − ϕr1 ] 2

.

imperfect PBS with nonideal transmission and nonideal reflection not only leads to the signal amplitude decrease, but also generates the nonlinear error. It’s similar to the result in structure A. Fig. 10 shows the relation between fraction a and signal amplitude in structure B with the imperfect PBS with the perfect transmission of P polarized beam. We can see the nonlinear error is produced in this scheme when a is not zero. Moreover, if the fraction a is larger, the signal amplitude is smaller. Fig. 11 shows the relation between fraction a and signal amplitude in structure A with the imperfect PBS with the imperfect transmission of P polarized beam and the imperfect reflection of S polarized beam. In Fig. 11, when b is not zero, the nonlinear error is produced whether a is zero. Furthermore, if the fraction a is larger, the signal amplitude is smaller.

(30) 0.5

From Eq. (30), we know the nonlinearity errors are caused by the imperfect PBS with the nonideal transmission and the nonideal reflection in structure B. This result is similar to the result in structure A. 3.5. Simulations The same to Section 2, we also simulate the four schemes to testify the relations between PBS and the nonlinear error. All four schemes use the structure B that is shown as Fig. 7. Fig. 9 shows the Im signal in four different schemes. From Fig. 9, we can see the imperfect PBS with the perfect transmission of P polarized beam or the imperfect PBS with the perfect reflection of S polarized beam not only leads to reduce the signal amplitude, but also produce the nonlinear error. It’s different from the result in structure A. The

0.1 ideal b=0.1 b=0.2 b=0.3

0.4 0.3

0.06

0.2

0.04

0.1

0.02

0

0

-0.1

-0.02

-0.2

-0.04

-0.3

-0.06

-0.4

-0.08

-0.5

0

1

2

3

4 x 10

ideal b=0.1 b=0.2 b=0.3

0.08

-6

-0.1 2.5

2.55

2.6

2.65

2.7 x 10

-6

Fig. 12. Relation between b and Im in structure B with the imperfect PBS (a = 0.1).

5066

H. Hu, J. Hu / Optik 126 (2015) 5061–5066

The relation between fraction b and signal amplitude in structure B is similar to the result of the relation between fraction a and signal amplitude. If the fraction b is larger, the signal amplitude is smaller. Fig. 12 shows the relation between fraction b and signal amplitude in structure A with the imperfect PBS with the imperfect transmission of P polarized beam and the imperfect reflection of S polarized beam. 4. Conclusions In this paper the relations between the nonlinearity errors and the PBS in the heterodyne interferometer are analyzed. From the analysis, in the typical heterodyne interferometer which is shown in Fig. 1, the nonlinearity errors don’t exist in the imperfect PBS with the nonideal transmission or the nonideal reflection, while the nonlinearity errors are caused by the imperfect PBS with the nonideal transmission and the nonideal reflection. However, in another typical heterodyne interferometer which is shown in Fig. 7, the nonlinearity errors exist not only in the imperfect PBS with the nonideal transmission or the nonideal reflection, but also in the imperfect PBS with the nonideal transmission and the nonideal reflection. It’s both an important property of PBS and a useful conclusion in the optical design of heterodyne interferometer. Acknowledgements The generous support of National Natural Science Foundation of China (11404150) and The Open Fund Project for Key Laboratory of

Photoelectronics and Telecommunication of Jiangxi are gratefully acknowledged. References [1] R.C. Quenelle, Nonlinearity in interferometric measurement, Hewlett Packard J. 34 (1983) 3–13. [2] C.M. Sutton, Nonlinearity in length measurement using heterodyne laser Michelson interferometry, J. Phys. E: Sci. Instrum. 20 (1987) 1290–1292. [3] C.M. Wu, C.S. Su, Nonlinearity in measurement of length by optical interferometry, Meas. Sci. Technol. 7 (1996) 62–68. [4] W. Hou, G. Wilkenling, Investigation and compensation of the nonlinearity of heterodyne interferometers, Precis. Eng. 14 (1992) 91–98. [5] W. Hou, Optical parts and the nonlinearity in heterodyne interferometers, Precis. Eng. 30 (2006) 337–346. [6] K. Joo, J.D. Ellis, E.S. Buice, J.W. Spronck, R.H.M. Schmidt, High resolution heterodyne interferometer without detectable periodic nonlinearity, Opt. Express 18 (2010) 1159–1165. [7] T.L. Schmitz, D. Chu, H.S. Kim, First and second order periodic error measurement for non-constant velocity motions, Precis. Eng. 33 (2009) 353–361. [8] S. Dubovitsky, O.P. Lay, D.J. Seidel, Elimination of heterodyne interferometer nonlinearity by carrier phase modulation, Opt. Lett. 27 (2002) 619–621. [9] W. Hou, Y. Zhang, H. Hu, A simple technique for eliminating the nonlinearity of a heterodyne interferometer, Meas. Sci. Technol. 20 (2009) 105303. [10] A. Yacoot, M.J. Downs, The use of X-ray interferometry to investigate the linearity of NPL differential plane mirror optical interferometer, Meas. Sci. Technol. 11 (2000) 1126–1130. [11] T.L. Schmitz, H.S. Kim, Monte Carlo evaluation of periodic error uncertainty, Precis. Eng. 31 (2007) 251–259. [12] J. Hu, H. Hu, Y. Ji, Detection method of nonlinearity errors by statistical signal analysis in heterodyne Michelson interferometer, Opt. Express 18 (2010) 5831–5839.