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Collimation testing of the wavefront of interferometer by scanning cube corner retro-reflector method

Zhaodong Liu, Lei Chen*, Zhigang Han, Wulan Tuya

School of Electrical Engineering and Photoelectric Technology, Nanjing University of Science & Technology, 200 Xiao Ling Wei, Nanjing 210094, China

* Corresponding author:[email protected]

ABSTRACT

The wavefront collimation of the optical interferometric systems must be controlled strictly. A scanning cube corner retro-reflector is introduced to divide the wavefront of the interferometer into a series of sub-wavefronts. The tilt angles of all the sub-wavefronts are obtained to reconstruct the tested wavefront thus the collimation test of the wavefront can be realized. The wavefront in the horizontal direction of a 100mm aperture interferometer is tested and reconstructed. The PV and RMS value of the collimated wavefront are 0.629λ and 0.130λ. The effect of system error on the experimental results is analyzed. The experimental results show that this method is of low cost and high accuracy, and is particularly applicable to the collimation test of a large aperture wavefront.

Keywords: Optical testing, interferometer, collimation test, scanning cube corner retro-reflector.

1. INTRODUCTION In many applications, laser beam need to be expanded to a larger diameter and collimated. The collimation of the beam usually determines the performance of optical systems, especially in optical interferometric systems. Many methods have been developed to test the collimation of wavefronts, which includes techniques mainly based on shearing interferometry[1-3], Talbot interferometry[4-7] and so on. The characteristic of the shearing interferometry[8] is that the test wavefront interfere with a shifted or rotated version of itself, producing fringes which contain the information about the collimation of the interfering beams. Various techniques have been put forward to increase accuracy and collimation detection sensitivity based on Murty’s wedge-plate shearing interferometer[9]. In the Talbot interferometry[10], a grating is placed in the test beam, which in turn produces its self-images at several planes perpendicular to the direction of propagation. If another grating is placed in one of these planes of self-images, moiré fringes can be observed. The variation of the fringes gives the amount of collimation. Many types of gratings were used in the test, including dual field gratings[11], spiral gratings[7], evolute gratings[6], triangular gratings[12], circular gratings[5] and so on. Among them, the circular gratings have been demonstrated that they have the best performance in collimation test. Recently, several other collimation tests have been suggested, which includes testing with optically active mediums[13], using the wavefront analysis method of circular aperture sampling[14, 15], detecting optical phase singularity[15], etc.

In all the techniques reported above, reference optic with aperture as large as the wavefront to be tested are necessary to determine the collimation. So it is not suitable to use any of these techniques to test a large aperture wavefront because it will cost a lot to provide the reference. We had used a scanning pentaprism method to divide the wavefront of the interferometer into a series of sub-wavefronts, and the relative positions of the spot centroid according to every sub-wavefront are recorded on the CCD camera[16]. The normal directions of all the sub-wavefronts are obtained to reconstruct the tested wavefront. It can be used to test a large aperture but it takes a long time and its accuracy depends mainly on the straightness of the slideway. A cube corner retro-reflector (CCR) method[17] was proposed to test the degree of collimation caused by defocusing in the form of photographic records of the interferomgrams. But it could not test the collimation wavefront. In this paper, we propose a scanning CCR method to test the collimation wavefront of an interferometer, which is particularly applicable to the collimation test of a large aperture wavefront.

2. THEORY The configuration of a scanning CCR method testing the wavefront collimation of an interferometer is presented in Fig.1. A small aperture CCR is placed behind the transmission flat (TF) in a Fizeau interferometer. The reference beam

2011 International Conference on Optical Instruments and Technology: Optoelectronic Measurement Technology and Systems, edited by Xinyong Dong, Xiaoyi Bao, Perry Ping Shum, Tiegen Liu, Proc. of SPIE Vol. 8201, 82010K

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reflected from the TF interferes with the test beam reflected from the CCR and the interferogram is detected by the imaging CCD camera in the interferometer. The number and orientation of fringes will change during the CCR scanning along the horizontal direction of the wavefront if the wavefront is not a plane wave. So it can be very simple and fast to estimate whether the wavefront is collimated or not. The whole wavefront can be retrieved by polynomial fitting algorithm if each sub-aperture wavefront is calculated.

TransmissionFlat

Collimator

ScanningCCR

CCD

Lens

BeamSplitter

PointLight

Source

Fig.1 Diagram of the scanning cube corner retro-reflector test system

The emergent wavefront of the interferometer is divided into a series of sub-apertures in the process of CCR scanning. The interferogram of each sub-aperture will keep the same when the wavefront is a perfect plane wave. And the emergent wavefront can be regarded as a parabolic[18] when the collimator is axially defocused as shown in Fig.2.

CCR

W(x,y)Wtest(¦ Ñ0,¦ È0)

Reference Surface

d

aD

¦ Èa

h

Y'

Z X

Wref(¦ Ñ0,¦ È0) Y

Z' X'

Fig.2 Schematic of the scanning cube corner retro-reflector collimation testing system

Establish a Cartesian coordinates XOY, whose coordinate origin is in the center line of the wavefront. The emergent wavefront can be denoted as (x, y)W . Another Cartesian coordinates ' ' 'X OY and Polar coordinates 0 0( , )ρ θ are

established in the XOY plane, whose coordinate origin named ' ( , )O m n is arbitrary point in the XOY coordinate. Then the relationship between the two Cartesian coordinates is given by

'

'

x x ny y m

⎧ = −⎨

= −⎩ (1)



And the center of the CCR front surface is coincide with ' ( , )O m n , the CCR front surface is in the ' ' 'X OY plane. Then

the arbitrary point ' '( , )x y in the ' ' 'X OY plane can be expressed as

'0 0

'0 0

sincos

xy

ρ θρ θ

⎧ =⎨

= −⎩ (2)

In Cartesian coordinate ' ' 'X OY the reference wavefront 0 0( , )refW ρ θ reflected by reference surface can be regarded as the mirror of the interferometer wavefront, which is given by

0 0 0 0( , ) ( , )refW Wρ θ ρ θ= − (3)

According to the reflective properties of CCR, the test beam 0 0( , )testW ρ θ from the CCR should be mirrored and rotated compared to the incident beam[19], and is written as

0 0 0 0( , ) ( , )testW Wρ θ ρ θ π= − + (4)

So the optical path difference (OPD) function between the reference beam and the test beam can be derived as

0 0 0 0 0 0

0 0 0 0

( , ) ( , ) ( , )

( , ) ( , )ref testOPD W W

W Wρ θ ρ θ ρ θ

ρ θ π ρ θ

= −

= + − (5)

Assuming the aperture of CCR is much smaller than the tested wavefront, the sub-aperture wavefront can be approximated as a tilted plane wavefront, which can be written as

0 0 0 0 0 0( , ) sin costW sρ θ ρ θ ρ θ+= , (6)

where s is the tilt coefficient in X direction and t is the tilt coefficient in Y direction. Then substituting Eq.(6) into Eq.(5), the OPD function is given by

0 0 0 0 0 0

0 0

cos2 ( , )

( , ) 2 sin 2W

OPD s tρ θ

ρ θ ρ θ ρ θ= −= − −

(7)

Transformed to Cartesian coordinate ' ' 'X OY , Eq.(7) can be written as

' ' ' ', 2 ( , )( )x y W x yOPD −= (8)

Substituting Eq.(1) into Eq.(8), Eq.(8) can be written as

, 2 ( ) 2 ( )2 ( , )

( )x n y m s x n t y mW x n y m

OPD − − = − − + −= − − −

(9)

If only Y direction is considered, the one-dimensional wavefront function is given by

1( )2

( ) ( )W y m OPD y m t y m− = − − = − − (10)

In the process of CCR scanning the slope ' ( )nW y in the center of each sub-aperture wavefront is given by



' '0 1 0

' '1 2 1

' '

1( ) ( )21( ) ( )2

1( ) ( )2n n n

W y t OPD y

W y t OPD y

W y t OPD y

⎧ = − = −⎪⎪⎪ = − = −⎪⎨⎪⎪⎪ = − = −⎪⎩

M

(11)

where ( 0,1,2,...)ny n = are the relative positions of scanning CCR. Since the sub-aperture is very small, the OPD

function of each sub-aperture can be approximated as a straight line whose slope is nt . So the slope of each sub-aperture

nt can be calculated by the polynomial fitting algorithm. Assuming the wavefront W(y) presented by a polynomial as

( ) nnW y a y=∑ (12)

Deriving Eq.(12) and substituting Eq.(11) into it can obtain: ' 1( ) ( )n

n nW y a ny t y−= ⋅ =∑ (13)

Eq.(13) shows that each scanning position ny corresponds to a slope nt . So ' ( )W y can be obtained by the polynomial

fitting algorithm. Then the wavefront of the interferometer can be retrieved by integrating the ' ( )W y .

3. EXPERIMENT We used the scanning CCR method to test and obtain the wavefront (Wa) of a 100mm aperture wavelength phase-shifting interferometer, whose working wavelength is 632.8nm. The setup is shown in Fig.3. The TF and a 200mm long slideway are placed on the right side of the interferometer. A 13mm aperture CCR is placed on the top of the slideway with its center height same as that of the interferometer optical axis. Move the CCR to scan the interferometer wavefront with steps of 5mm, calculating each sub-aperture wavefronts and recording the corresponding CCR positions. The typical interferograms of sub-apertures in the process of CCR scanning are shown in Fig.4. The amount and direction of fringes change obviously during the scanning process. As shown in Fig.5, the peak-to-valley (PV) error is 0.629λ and the root-mean-square (RMS) error is 0.130λ.

Interferometer

TF

Slideway

CCR

Fig.3 Scanning CCR testing the interferometer wavefront



Fig.4 Interferograms of scanning CCR

Fig.5 Interferometer wavefront tested by scanning CCR

4. DISCUSSION 4.1. Quality of CCR

Quality of CCR will have an effect on the test result of the wavefront. The single pass wavefront distortion of CCR can be tested by ZYGO interferometer and can be stored as the system error to be eliminated during the CCR scanning. Thereby the accuracy of the collimation test can be improved. The test result of CCR in Fig.6 shows that the PV value is 0.168λ and RMS value is 0.024λ.



Fig.6 Test result of CCR by ZYGO

4.2. The straightness requirement of slideway

Due to the straightness of the slideway, the CCR will have changes in yaw, roll and pitch angle in the scanning process. The important characteristics of CCR is that, no matter which direction light incident in from the surface, after three right-angle surface reflections, the emergent ray is always parallel to the incident light. So when the CCR rotates, the quantity and direction of fringes will not change with the CCR movement during the scanning CCR test. Therefore the error caused by the straightness of the slideway in the process of CCR scanning can be ignored.

4.3. The accuracy of the test

It is assumed that the interferometer wavefront is a parabolic, as shown in Fig.2. Adjust the transmission flat in order that the emergent ray is perpendicular to the reference surface of TF at the center (the intensity of the interferogram is nearly uniform) when CCR is placed at the center of TF. In another word the normal of the wavefront center is perpendicular to the reference surface of TF. The sub-aperture wavefront tilt aθ is given by

28 /a ah Dθ = , (14)

where D is the diameter of the tested wavefront, a is the distance between the center of the wavefront and that of the sub-aperture, and h is vector height of the defocused wavefront. And number of the fringe aN is given by

216 / ( )aN a h d D λ= ⋅ ⋅ ⋅ , (15)

where d is the sub-aperture diameter of CCR. When the wavefront is defocused, the tilt of sub-aperture at the edge of wavefront compared to the ideal plane (parallel to the tangent plane at the center of wavefront) goes to a maximum, which is defined as maxaθ , and is given by

max 2a dδθ = , (16)

where δ is the measurement accuracy of the interferometer. From Eq.(14) and (16), we can get

2

min 16Dh

a dδ ⋅

=⋅

, (17)

In our test we used a 13mm aperture CCR to test a 100mm aperture interferometer whose accuracy is /10λ , so the precision of the wavefront vector height test can achieve as 0.1λ using this method in this interferometer.

5. CONCLUSION The Scanning CCR method tests the collimation wavefront of the interferometer using the characteristics of reflecting the incident ray in a parallel direction, which can be very effective to guide the alignment of the wavefront of the interferometer. A simple mechanical structure is used in this method instead of a high-precision slideway. So the test is



much easier to implement than other techniques such as shearing interferometry, scanning pentaprism method and the test cost can be reduced greatly. It can be very simple and fast to estimate whether the wavefront is collimated or not through the changes of the amount and orientation of fringes in the process of CCR scanning. Each sub-aperture wavefront tilt can be calculated using the phase-shifting interferometry so as to retrieve the wavefront by polynomial fitting algorithm. The wavefront collimation of a Φ100mm interferometer was tested using a Φ13mm CCR, and test accuracy of the wavefront vector height can be achieved as 0.1λ in this interferometer.

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