INTERNATIONAL JOURNAL OF PRECISION ENGINEERING AND MANUFACTURING Vol. 14, No. 3, pp. 381-386 MARCH 2013 / 381

© KSPE and Springer 2013

Optical Axis Alignment for Three-Aspherical MirrorsSystem Using Five Parallel Laser Beams and MoiréMethod

Der-Chin Chen1,#

1 Department of Electrical Engineering, Feng Chia University, 100, Wenhwa Road, Seatwen, Taichung, Taiwan, ROC, 40724# Corresponding Author / E-mail: kanatasan.tw@yahoo.com.tw, TEL: +886-4-2451-7250, Ext. 3845, FAX: +886-4-2451-6842

KEYWORDS: Alignment, Aspherical mirror, Effective focal length

In this paper, a new optical technique involving the five parallel laser beams alignment device and moiré method was developed to

rapidly and accurately align the optical axis of three-aspherical mirrors system (TAMS). In this method, the optical axis of a first

aspherical mirror is made parallel to the “five incident parallel laser beams” in the plane of incidence. The height and orientation

of the first aspherical mirror are fine tuned and the alignment is assured by examining the direction of these five reflected laser beams.

The optical axes of second and third aspherical mirror are aligned by five parallel laser beams in accordance with the configuration

parameter of optical system in sequence. The effective focal length (EFL) of the TAMS was measured by moiré method and image

formula. This novel method is used to align the optical axis of TAMS and the EFL deviation estimated has an accuracy of 1%.

Manuscript received: August 22, 2012 / Accepted: December 4, 2012

1. Introduction

Reflective optical systems provide superior thermal stability and

radiation resistance and offer lower image defects arising from

chromatic aberration. The reflective optical systems provide superior

performance over refractive optical systems as it can be made far more

compact and operated on a wider spectral range. Compared with the

refractive system, an all-reflective system increasing the number of

elements provides additional opportunities to eliminate or correct more

aberrations. It is widely used in evaluating the performance of optical

system and is applied to the reflecting telescope, and extreme

ultraviolet (EUV) lithographic projection. As three-aspherical mirrors

system (TAMS) becomes more complex, precise optical axis alignment

becomes more challenging as well. The manufacturing and alignment

of TAMS usually involves a sophisticated and expensive optical

instrument, such as laser unequal path interferometer, autocollimator or

knife edge method, which identify TAMS’s characteristics like effective

focal length, back focal length, and other optical parameter.1-4 Above

traditional optical instruments aligning the optical axis of TAMS cost

much of time. Because of the above concerns, a new optical technique

was designed to align the optical axis of the TAMS using the five

parallel laser beams and measure the EFL of TAMS by moir method.

There are some advantages in this method: (1) It is a rapid and simple

alignment method. (2) It simplifies the electro-optics system and lowers

the cost. (3) The different wavelengths of laser diodes are used to

enable getting wider spectral range of optical system alignment. (4)

The five parallel laser beams arrangement, it adjusts the optical axis of

TAMS with more freedoms of orientation. This is a rapid method for

measuring the EFL deviation up to 1% accuracy after aligning the

optical axis of the TAMS.

2. Basic principle

TAMS that consists of three mirrors and two thicknesses is

indicated in the Fig. 1. Three mirrors, named first aspherical mirror,

second aspherical mirror and third aspherical mirror, are arranged in

TAMS configuration that reflects light from the object to the image

through three mirrors in order. There is at least one intermediate image

formed between the first and third mirror and at least one of the three

aspherical mirrors is non-rotationally symmetric. The radiuses of three

mirrors are labeled r1, r2 and r3, the axial distance between first mirror

to second mirror is d1, the axial distance between the second mirror to

third mirror is d2. The conic constants of three surfaces are k1, k2 and

DOI: 10.1007/s12541-013-0053-7

382 / MARCH 2013 INTERNATIONAL JOURNAL OF PRECISION ENGINEERING AND MANUFACTURING Vol. 14, No. 3

k3 and the aperture sizes of three mirrors are labeled h1, h2 and h3. The

surface profile parameters of optical system can be described as three

parts: (1) the first obstacle ratio is given by: α1 = l2/f1 = h2/h1, where α1

is the obstacle ratio of the second mirror relative to the first mirror, l2

is the axial distance of vertex point of second mirror to the focal point

of first mirror and f1 is the focal length of first mirror. (2) similarly, the

second obstacle ratio is given by: α2 = l3/l’2 = h3/h2, where α2 is the

obstacle ratio of third mirror relative to second mirror, l’2 is axial

distance of the vertex point of second mirror to the focal point

combining the first mirror with the second mirror, l3 is axial distance

of the vertex point of third mirror to the focal point combining the first

mirror with the second mirror. (3) eventually, the magnification

equations are β1 = l’2/l2 ≒u2/u’2 and β2 = l’3/l3≒ u3/u’3, where β1 is

the magnification of second mirror, β2 is the magnification of third

mirror; l’3 is the vertex point of third mirror to the focal point of TAMS

axial distance. The structure parameters of optical system are derived

by the ynu paraxial ray tracing in Gaussian optics.5,6 These structure

parameters are given by the expression:

(1)

(2)

(3)

(4)

(5)

where f is the focal length of TAMS. When the following aberration

parameters, the spherical aberration, coma, astigmatism given by

system, the surface profile parameters α1, α2, β1, β2 and conic constant

of three surfaces could be designed. Then the solution of system

structure parameter will be calculated by the above five equations and

configuration parameters. When α1 > 1, α2 < 0, β1 < 1, and β2 > 0,

TAMS with at least one intermediate image located between the first

and third mirrors is one type of three TAMS solution. Besides above

eight alignment configuration parameters of TAMS, the concerns about

the depth of focus (DOF) in the process of alignment to the optical axis

should be deliberated.

The concept of depth of focus rests on the assumption that for a

given optical system, there exist a blur (due to defocusing) of small

enough size such that it will not adversely affect the performance of the

system. The depth of focus, shown in Fig. 2, is the amount by which the

image may be shifted longitudinally (δ) with respect to the reference

plane and which will introduce no more than the acceptable blur. The

size of the DOF of lens using geometry optics could be obtained.

In Fig. 2, D is the distance from exit pupil to the reference plane,

and B is the diameter of the blur spot or linear blur. Then it shows

(6)

where ψ is called angular blur.

Apparently, if reference plane is sit on the image plane, then B = 0,

and ψ = 0.

According to the analysis of DOF, this paper selects the smallest

blur that serves as the focal point to implement the alignment to the

optical axis and measurement of focus length.7

3. The optical alignment system

We develop a specific technique to align the optical axis of the

TAMS by five parallel laser beams and then measure its EFL by moiré

method. This alignment system can applies to UV ~ IR regions with

the different wavelengths of alignment laser diode. The optical

alignment system in Fig. 3 is composed of as followings: (1) the

TAMS as shown in Fig. 4(a), (2) the alignment device as shown in Fig.

r1

2

β1β2

-----------f=

r2

2α1

1 β+1

( )β2

-----------------------f=

r3

2α1α2

1 β+2

( )----------------- f=

d1

r1

2---- 1 α

1–( )=

d2

r1

2----α

1β1

1 α2

–( )=

ψB

D----=

Fig. 1 Three-aspherical mirrors system

Fig. 2 Depth of focus δ of optical system

Fig. 3 The configuration of the optical alignment system

INTERNATIONAL JOURNAL OF PRECISION ENGINEERING AND MANUFACTURING Vol. 14, No. 3 MARCH 2013 / 383

4(b) and (c), (3) pinhole and the line grating and (4) optical table and

mount. Functions of four parts are simply described as follows.

3.1 Three-aspherical mirrors system

The objective of alignment to TAMS, the main part of the optical

alignment system and consisting of three aspherical mirrors, is to adjust

three dimensional optical alignments. Fig. 4(a) shows the photograph

of TMAS under alignment and use for the reflective collimator of a

dynamic infrared scene projector system.

3.2 Alignment device

The alignment device includes five parallel semiconductor lasers,

the rotation mechanism, precise adjustable holder and the fixture. The

semiconductor laser beams on the fixture are made parallel each other

with precise adjustable holder and perpendicular to the surface fixture.

An adjustable holder for mounting and orienting the semiconductor

laser is set by a removable retaining laser within a mount of ring.

The rotation mechanism adjusts the initially horizontal laser beams

to any direction as shown Fig. 4(b). For example, the three laser beams

lines horizontally, i.e., #1, #2, and #3 from right to left and the three

laser beams lines vertically, i.e., #5, #2, and #4 from up to down as

shown in Fig. 4(b). In the rotation mechanism, it can rotate any degree

to adjust the three-dimensioned optical axis of TAMS at any tilted and

yawed angle.

3.3 Pinhole and grating

The pinhole has a size of 30ìm. When the diffraction pattern of

concentric circular rings was appeared, the laser beam passes through

the pinhole accurately. The diffraction patterns formed by a pinhole

consist of a central bright spot surrounded by a series of bright and dark

rings. We can describe the pattern in terms of the angle θ, representing

the radius angle of each ring. If the aperture diameter is W (mm) and

the wavelength is λ (mm), the radius angle θ of the first dark ring is

given by sin θ = 1.22 (λ/W). With this diffraction technique, the

pinhole is used to precise align laser beam parallel to the TAMS of the

alignment system. The two gratings are used generated Moiré pattern

for measuring the EFL of TAMS. Transverse mode in which a Moiré

pattern, formed by two sets of parallel lines (Ronchi grating), one set

inclined at an angle to the other is shown in Fig. 5(a). Let us consider

two patterns made of parallel and equidistant lines, e.g., vertical lines.

The pitch of the first pattern is p, the step of the second is p + δp, with

0 < δ < 1. The distance d between the middle of a pale zone and a dark

zone of Moiré of parallel patterns is d = p2/(2δp) or

1/d = 1/p-1/(p + δp) (7)

as shown in Fig. 5(b), that is longitudinal mode. From this formula, we

can see that: the bigger the step, the bigger the distance between the

pale and dark zones; the bigger the discrepancy δp, the closer the dark

and pale zones; a great spacing between dark and pale zones mean that

the patterns have very close steps.

Fig. 4 (a) Photograph of TMAS under alignment (b) configuration of

alignment device, (c) Photograph of the alignment device

Fig. 5 Moiré pattern

384 / MARCH 2013 INTERNATIONAL JOURNAL OF PRECISION ENGINEERING AND MANUFACTURING Vol. 14, No. 3

4. Experiments and Results

The conditions and specifications used for the experiment are listed

in Table 1. These specifications consist of three parts, TAMS that has

75 mm EFL, two linear gratings and a CCD camera. Due to

sophisticated experiments, experimental procedures are divided into

two parts as following:

4.1 Pre-Experiment

Pre-Experiment of five semiconductor laser beams fine adjustment:

(1) The fixture of alignment device is vertically (perpendicular) put on

the optical table such that laser beam #2 is nearly parallel to the

surface of the optical table.

(2) The precise adjustable holder is fine adjusted to make the laser

beam #2 parallel to the surface of the optical table. To check the

parallelism, the pinhole is moved on the optical table from

position 1 to position 2 (about 50 cm distance) with a fixed height

and the pinhole diffraction pattern that the laser beam passes

through the pinhole accurately was observed. Above method is

repeated until the laser beam #4, #5, #3 and beam #1 are parallel

to the surface of the optical table, respectively.

(3) The plane mirror is put on the optical table at 3 meter distance

from the alignment device. To adjust the plane mirror, the

reflected beam #2 is made parallel to the optical table. To check

the plane mirror perpendicular to the optical table, a piece of paper

was used to chop reflected beam #2. The reflected beam #2 must

be coaxial to the incident laser beam #2 by adjusting the precise

adjustable holder.

(4) Step (3) is repeated until the reflected laser beam #4, #5, #3 and

#1 are coaxial to the incident laser beam #4, #5, #3 and #1,

respectively.

4.2 Experiment with TAMS

The TAMS alignment experiment can be divided into two main parts.

Part 1: the optical axis of first aspherical mirror alignment

(1) The first (1st) aspherical mirror is adjusted at first so that its optical

axis is in the incident plane made by three horizontal parallel

incident laser beams.

(2) The1st aspherical mirror is adjusted again so that the three

horizontal parallel reflected beams come to a focus at the same

point, i.e., the saggital optical plane of the 1st aspherical mirror

was found.

(3) Three vertical parallel incident beams are parallel to optical axis by

adjusting the 1st aspherical mirror and the three vertical reflected

beams also come to a focus at the same point obtained in (1), i.e.,

the tangential optical plane of the 1st aspherical mirror is found.

(4) The horizontal focal point and vertical focal point are gone to one

point together.

(5) Capture the focus pattern at the reference plane (ref. Fig.2) with

the CCD camera. The reference plane is moved from left to right

of the image plane to see if it fits Eq. (6).

Part 2: the alignment of the optical axis of the second (2nd) and third

(3rd) aspherical mirror in sequence

(1) These tangent lines at Q2 and Q3 point calculated by differentiation

method are on the optical axis of 2nd aspherical mirror and 3rd

aspherical mirror, respectively. The point Q2 ,Q3 and slope angle

θ are shown in Fig. 6.

(2) D21 is the horizontal distance between 2nd aspherical mirror and 3rd

mirror and D3 is the horizontal distance between 3rd aspherical

mirror and the focal point.

(3) The 1st plane mirror and 2nd plane mirror are seated at position Q2

and Q3, respectively according to the calculation results from step

(1) and (2).

(4) These reflection angle α and γ, calculated by optical ray tracing as

shown in Fig. 6, can be used to align the 1st plane mirror and 2nd

plane mirror.

(5) The purpose of the step for finding approximate focal plane of this

system is solved based on D3 obtainable from the optical ray tracing.

(6) According to above adjusting result of the plane mirror, we can get

optimizing tangent line (at Q2 and Q3), afterwards the 2nd

aspherical mirror and 3rd aspherical mirror should replaced 1st

plane mirror and 2nd plane mirror.

(7) All the parameters including D21, D3 and reflection angle α, γ as

shown in Fig. 6, are calculated by ray tracing.

(8) After ascertaining D21, D3 and reflection angle α and γ, we can

adjust the 2nd aspherical mirror and 3rd aspherical mirror, and then

find the focal plane of system.

(9) The normal vector of two gratings separately at the position P1 and

position P2 were aligned to parallel the optical axis of aspherical

mirror, as shown in Fig. 7.

(10) We adjust the first grating at P1 position to optimize the image

distance X2 and to get the Morie pattern, the sharpness of which

can be observed by a CCD camera that monitor and measure it’s

the value of Moiré pitch d, as shown in Fig. 7.

(11) The effective focal length (EFL) of the three aspherical mirror

system was calculated by the image formula (m = h2/h1= EFL/

x1 = -x2/EFL, where h1 and h2 represent the object height and the

image height respectively.)

The focus pattern curves were measured by a CCD camera to

illustrate the relationship between size of focus spot and the position of

the reference plane. Focus sizes of reference plane shift are captured

Table 1 The experiment conditions

Three-aspherical mirrors system

System EFL 75 mm

First-aspherical mirror 88.6 mm

Second-aspherical mirror 29.4 mm

Third-aspherical mirror 17.5 mm

total length 110 mm

Grating

The pitch of 1st grating 0.75 mm

The pitch of 2nd grating 0.015 mm

Semiconductor laser of alignment device

Wavelength (λp) 635 nm

Output power 3 mW

Beam Divergence < 2 mrad

Beam Dimeter 3.3 mm

CCD camera

Resolution 752 × 582

Spectral range 350 ~ 1100 nm

INTERNATIONAL JOURNAL OF PRECISION ENGINEERING AND MANUFACTURING Vol. 14, No. 3 MARCH 2013 / 385

with CCD camera as shown in Fig. 8. The eleven focus patterns from

left of the image plane to right of it are measured. The 30% of the

center of the maximum intensity is defined as the focus pattern of the

TAMS. The alignment of the aspherical mirror is finished when getting

relative minimum focus size. The influence of Moiré pitches on the

measurement of EFL will be discussed before the measuring EFL of

TAMS by Moiré method. Under the assumption that the pitch of 1st and

2nd grating are 0.75 mm and 0.015 mm, the change of Moiré pitch d

induced by the variation of the image pitch of 1st grating (h2) on the

image plane is shown as Table 2.

Deprived from the Moiré theory and the assumption that the pitch

of 1st and 2nd grating are 0.75 mm and 0.015 mm, Table 3 shows that

the greater the Moiré pitch d, the value of EFL approaches closely the

designed value. The greater Moiré pitch d serves to better measurement

accuracy in the practice as well. Above the calculation results also

show that the EFL accuracy of TAMS falls within 1% when the Moiré

pitch d locates from 1.2 mm to 2.4 mm.

This alignment experiment repeatedly does twenty five times. The

test results show that average EFL is 74.2 mm and the standard

deviation is 0.388 mm as shown in Fig. 9(a). Moiré fringe pattern is

Fig. 6 The configuration of optical axis alignment of TAMS in process

by two plane mirror

Fig. 7 The configuration of EFL measurement of TAMS

Fig. 8 The focus patterns when reference plane shift

Table 2 The change of Moiré pitch d induced by the variation of the

image pitch p of 1st grating

h2 x2 d δp

0.017 mm 1.7 mm 0.127 mm 0.002 mm

0.016 mm 1.6 mm 0.24 mm 0.001 mm

0.0155 mm 1.55 mm 0.465 mm 0.0005 mm

0.0154 mm 1.54 mm 0.577 mm 0.0004 mm

0.0153 mm 1.53 mm 0.765 mm 0.0003 mm

0.0152 mm 1.52 mm 1.14 mm 0.0002 mm

0.0151 mm 1.51 mm 2.265 mm 0.0001 mm

Table 3 The error of EFL induced by the variation of the Moiré pitch d

h2 efl d δp*

0.015 mm 75 mm ~~~~~~ 0

0.0155 mm 72.58 mm 0.465 mm 0.0005 mm

0.0154 mm 73.05 mm 0.577 mm 0.0004 mm

0.0153 mm 73.53 mm 0.765 mm 0.0003 mm

0.0152 mm 74 mm 1.14 mm 0.0002 mm

0.0151 mm 74.5 mm 2.265 mm 0.0001 mm

*δ is between 0 to 1

Fig. 9 (a) EFL measurement result of the three-aspherical mirror

system by Moiré method (b) Moiré fringe pattern is captured using the

CCD camera and optimized by image processing

386 / MARCH 2013 INTERNATIONAL JOURNAL OF PRECISION ENGINEERING AND MANUFACTURING Vol. 14, No. 3

captured using the CCD camera and its phase information is extracted.

The Moiré fringe pattern that optimized by image processing are

showed in Fig. 9(b). The experimental results indicate that the new

alignment device and moiré method is a rapid method for aligning and

measuring the EFL deviation up to 1% accuracy after aligning the

TAMS.

5. Conclusion

This new technique has effectively developed by using the five

parallel laser beams alignment device and moiré method to align the

optical axis of the TAMS. The EFL of the TAMS was measured by

moiré method and image formula. The measuring accuracy of EFL is

sensitive to the environmental conditions and Moiré pattern profile.

Also, its measuring accuracy is greatly affected by the algorithm

choices for Moiré pattern image processing. By utilizing the merging

geometry center approximation for moiré image processing, it indeed

succeeds in increasing accuracy of measuring EFL. This novel method

is used to align the optical axis of the TAMS and the EFL deviation has

an accuracy of 1%.The advantages of the alignment techniques are: (1)

it is simple to operate; (2) it simplifies the optical test system and

lowers the cost and (3) it is less expensive to maintain the equipments.

ACKNOWLEDGEMENTS

The authors would like to give a great thanks to National Science

Council (NSC101-2623-E-035-003-D) in Taiwan which offers funds

for this research.

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