Design of beam scanners based on Talbot-encodedphase plates

Xin Zhao, Changhe Zhou, Liren Liu *

Shanghai Institute of Optics and Fine Mechanics, Chinese Academy of Sciences, P.O. Box 800-211, Shanghai 201800, China

Abstract

Beam scanners have wide applications in many fields. But conventional scanners have many shortcomings such as

limited scanning frequency, high cost and low reliability. At the same time, phase gratings are efficient devices for

deflecting light in prespecified direction. We propose a new kind of optical beam scanner by using a Talbot-encoded

phase plate. Our architecture is practical and easily fabricated and has stable capability. It includes a monochromatic

light source, a programmed controlled shifter, a detector and a phase plate. This phase plate consists of a pair of

complementary phase plates with Talbot phase coding structure. Talbot phase coding structure is a new kind of binary

phase coding structure. There are periodically repeated gratings on the surface of the phase plates. The complementary

plates are placed paralleled to each other and they can be shifted along the direction perpendicular to their central axes.

When the light passes through the pair of phase plates, it will be deflected. And the relative shift between the two plates

will determine the deflected angle. By changing the shift between the plates with the shifter, we can get accurately

deflected angles. When there is no shift between the plates, the light will go straight without any deflection. We have

designed the required phase plates according to the number of scanning points N , maximum deflected angle h and

the diffraction efficiency g. And satisfactory experimental results have been obtained. The phase plates can be mass-

produced by employing the binary-optical manufacture technology so that the costs of the scanners can be reduced.

� 2003 Elsevier Science B.V. All rights reserved.

Keywords: Beam scanner; Talbot-encoded phase plate; Deflected angle; Diffraction efficiency

1. Introduction

Beam scanners have very wide applications in

many aspects, for example, in laser printers and

TV sets. The conventional scanners have manydisadvantages, such as limited scanning frequency,

high cost and low reliability. No matter for com-

mercial or martial purpose, beam scanners which

can conquer the above shortcomings are in great

need [1]. Motamedi et al. proposed an optical

scanner employing micro-lens array, which was

manufactured by binary-optical techniques [2].

But it is very difficult to manufacture the micro-

lens array of high precision. The ideal lens array iscontinuous which requires infinite phase levels and

infinite phase values. But the binary-optical tech-

niques can only manufacture a limited number of

phase levels and a limited line width. Actually, the

micro-lens array, which is manufactured by the

binary-optical technique, is not an ideal binary-

phase function, particularly when the fine line

width on the outer part of the lens is concerned.

*Corresponding author. Fax: +86-21-59528885.

E-mail address: lirenliu@mail.shcnc.ac.cn (L. Liu).

0925-3467/03/$ - see front matter � 2003 Elsevier Science B.V. All rights reserved.

doi:10.1016/S0925-3467(02)00309-9

Optical Materials 23 (2003) 313–318

www.elsevier.com/locate/optmat

Because of the above reasons, the scanner will

have low capability and low diffraction efficiency

in practice. We put forward a new kind of beam

scanner which can be easily fabricated and has

reliable capability. The key component of our

scanner is a pair of Talbot phase coding phaseplates. This kind of phase plates with binary-phase

coding structure can remarkably improve the

capability of the scanner and decrease the cost.

2. Design of beam scanners based on Talbot-

encoded phase plates

2.1. Characteristic of Talbot-encoded phase plates

The Talbot coding structure is a new kind of

phase coding structure which will improve the

properties of the scanners dramatically and reduce

the production cost. As shown in Fig. 1, each pe-

riod is divided into M parts equally. In one period

the width of each phase level is the same, that is tosay, d1 ¼ d2 ¼ � � � ¼ dM ¼ d=M . The thickness of

the adjacent phase levels is unequal but the thick-

ness distribution is symmetric within one period.

The thickness difference of two adjacent phase

levels can be expressed as

hk � hk�1 ¼rð2k þ 1Þ

Mk

ðn� 1Þ ; ð1Þ

where k is the wavelength, n is the refractive indexof the phase plate, hk is the height of the kth phase

level and r is a positive integer which is the pa-

rameter of binary phase function.

The division number in each period d can be

calculated by the phase coding function of the

Talbot array illuminator [3]. When M is an evennumber, the phase distribution function is

uðkÞ ¼ r � k2

Mp; ð2Þ

and when M is an odd number, the phase distri-

bution function is

uðkÞ ¼ r � kðk � 1ÞM

p; ð3Þ

where k ¼ 1; 2; . . . ;M .

The parameter r is determined by

p � r ¼ kr �M þ 1; ð4Þwhere p, r and M are positive integers and kr is aninteger not less than zero.

For an arbitrary positive integer M and another

positive integer p which is smaller than M , the

phase distribution of the Talbot array illuminatorcan be deduced from the above Eqs. (2)–(4). It can

be known from Eq. (4) that p ¼ 1, r ¼ 1 and kr ¼ 0

is always one set of solutions of Eq. (4). This set of

solutions is also the most common situation that

we consider. As the phase of the Talbot array il-

luminator is the quadratic function of the para-

meter k, the scanning function can be realized by

employing just a pair of Talbot array illuminatorphase plates. The number L is defined as the

amount of phase levels that are different from each

other in one period. Some numerical solutions of

the relationship among the phase distribution

function uðkÞ, M and L are listed in Table 1. The

phase distributions of the positive phase plate and

the negative one are uðkÞ and �uðkÞ, respectively.

Fig. 1. The schematic phase distribution of our Talbot-encoded

phase plates within one period: (a) phase distribution of a phase

plate whenM is an even integer; (b) phase distribution of a plate

when M is an odd integer.

314 X. Zhao et al. / Optical Materials 23 (2003) 313–318

The symmetry of the phase distribution will be

different considering the division number M is odd

or even in one period d. When M is an even inte-

ger, from Eq. (2), the symmetry of the phase

distribution can be expressed as (as shown in

Fig. 1(a))

uðM � kÞ ¼ uðkÞ: ð5ÞWhen M is an odd integer, from Eq. (3), the

symmetry of the phase distribution can be ex-

pressed as (as shown in Fig. 1(b))

uðM � k þ 1Þ ¼ uðkÞ; ð6Þ

where k ¼ 1; 2; . . . ;M .

A pair of complementary phase plates with

Talbot phase coding is used. And one is called the

positive phase plate, the other is called the negative

one. As shown in Fig. 2(a), in the positive phase

plate, the height and the width of the first phaselevel in each period are h11 and d11, respectively. h12and d12 for the second phase level and h1M and

d1M for the Mth phase level. All the phase levels

have thesamewidth, that is,d11 ¼ d12 ¼ � � � ¼ d1M ¼d=M . In one period, the height of each phase level

differs from that of the adjacent one. But the

height distribution is symmetric in one period, as

shown in Fig. 2(b). The phase changes accordingto the height. The height difference of the two

adjacent phase levels can be expressed by the ad-

jacent phase difference

D/ðkÞ ¼ uðk þ 1Þ � uðkÞ; ð7Þ

where k ¼ 1; 2; . . . ;M � 1.

When M is an even integer, the adjacent phasedifference is

DuðkÞ ¼ r � ð2k þ 1ÞM

p: ð8Þ

When M is an odd integer, the adjacent phase

difference is

DuðkÞ ¼ 2r � kM

p; ð9Þ

where k ¼ 1; 2; . . . ;M � 1.According to Eq. (1), we can get the height

difference between the first and the second phase

levels of the positive phase plate

Table 1

The relationship among uðkÞ (the Talbot phase function), M(the division number in one period) and L (the number of phaselevels)

M p uðkÞ ðk ¼ 1; 2; . . . ;M) L

2 1 p=2, 0 2

3 0, p=2 2

3 1 0, 2p=3, 0 2

2 0, 2p=3, 2p=3 2

4 2p=3, 0, 0 2

5 2p=3, 0, 2p=3 2

4 1 p=4, 4p=4, p=4, 0 3

3 0, p=4, 0, 5p=4 3

5 p=4, 0, p=4, 4p=4 3

7 0, 5p=4, 0, p=4 3

5 1 0, 2p=5, 6p=5, 2p=5, 0 3

2 2p=5, 0, 4p=5, 4p=5, 0 3

3 0, 4p=5, 2p=5, 4p=5, 0 3

4 6p=5, 0, 2p=5, 2p=5, 0 3

6 6p=5, 2p=5, 0, 0, 2p=5 3

7 4p=5, 0, 2p=5, 0, 4p=5 3

8 2p=5, 4p=5, 0, 0, 4p=5 3

9 2p=5, 0, 6p=5, 0, 2p=5 3

6 1 p=6, 4p=6, 9p=6, 4p=6, p=6, 0 4

5 0, 3p=6, 4p=6, 3p=6, 0, 7p=6 4

7 4p=6, p=6, 0, p=6, 4p=6, 9p=6 4

11 3p=6, 0, 7p=6, 0, 3p=6, 4p=6 4

7 1 2p=7, 4p=7, 8p=7, 0, 8p=7, 4p=7, 2p=7 4

2 0, 4p=7, 2p=7, 8p=7, 8p=7, 2p=7, 4p=7 4

3 4p=7, 0, 6p=7, 8p=7, 6p=7, 0, 4p=7 4

4 0, 2p=7, 8p=7, 4p=7, 4p=7, 8p=7, 2p=7 4

5 0, 6p=7, 4p=7, 8p=7, 4p=7, 6p=7, 0 4

6 8p=7, 0, 4p=7, 6p=7, 6p=7, 4p=7, 0 4

8 0; 8p=7, 4p=7, 2p=7, 2p=7, 4p=7, 8p=7 4

9 8p=7, 2p=7, 4p=7, 0, 4p=7, 2p=7, 8p=7 4

10 8p=7, 6p=7, 0, 4p=7, 4p=7, 0, 6p=7 4

11 4p=7, 8p=7, 2p=7, 0, 2p=7, 8p=7, 4p=7 4

12 8p=7, 4p=7, 6p=7, 0, 0, 6p=7, 4p=7 4

13 6p=7, 4p=7, 0, 8p=7, 0, 4p=7, 6p=7 4

8 1 p=8, 4p=8, 9p=8, 0, 9p=8, 4p=8, p=8, 0 4

3 0, p=8, 8p=8, 5p=8, 8p=8, p=8, 0, 5p=8 4

5 0, 7p=8, 8p=8, 3p=8, 8p=8, 7p=8, 0,3p=8

4

7 11p=8, 0, 3p=8, 4p=8, 3p=8, 0, 11p=8,4p=8

4

9 9p=8, 4p=8, p=8, 0, p=8, 8p=8, 5p=8 4

11 8p=8, p=8, 0, 5p=8, 0, p=8, 8p=8, 5p=8 4

13 8p=8, 7p=8, 0, 3p=8, 0, 7p=8, 8p=8,3p=8

4

15 3p=8, 0, 11p=8, 4p=8, 11p=8, 0, 3p=8,4p=8

4

X. Zhao et al. / Optical Materials 23 (2003) 313–318 315

hpo2 � hpo1 ¼r � ð2k þ 1Þ

M

� �k=ðn� 1Þ; ð10Þ

where k ¼ 1 and the subscript �po� stands for thepositive phase plate.

In the case of the negative phase plate, each

phase level also has the same width which is equal

to that of the positive one. And the phase differ-

ence related to the height of each phase is com-

plementary to that of the positive phase plate.

2.2. Our design of the complementary plates

Here we discuss the process of beam scanning in

detail. When there is no shift between the two

phase plates, the light beam can pass through the

scanner without any deflexion.

When the shifter gives the negative plate a dis-placement of d=M , the phase difference will be

upoðk þ 1Þ � uneðkÞ ¼r � ð2k þ 1Þ

Mp; ð11Þ

where k ¼ 1; 2; . . . ;M . The subscript �ne� stands forthe negative plate. Hence the light beam will be

deflected. And it can be seen that the phase dif-

ference is a linear phase factor. The deflected angle

h1 (where r ¼ 1) is

h1 ¼kd: ð12Þ

The deflected angle will increase when the shift

increases by d=m. If the shift is D, for an even

number M , the phase function of the positive

phase plate relative to the negative one will be

upoðk þ DÞ ¼ r � ðk þ DÞ2

Mp: ð13Þ

And the phase difference compared with the

negative phase plate is

upoðk þ DÞ � uðkÞne ¼r � ð2k þ DÞD

Mp; ð14Þ

where k ¼ 1; 2; . . . ;M .

The above formula is the simple function of D.In the situation when there is no shift between the

plates, D ¼ 0 and Eq. (14) equals to zero. The light

beam can pass through the plates without being

defleted for there is no phase difference. When

there is a shift between the plates, D 6¼ 0 and the

slope of the term in Eq. (14) is a linear function of

D. The more D increases, the more the deflected

angle becomes. The similar results can be obtainedwhen M is an odd number.

The relationship between D and the diffraction

efficiency is

g ¼ sincDM

� �2

: ð15Þ

Fig. 2. The surface profile of the Talbot encoded phase plate (a) and its corresponding phase distribution in gray-scale illustration (b).

316 X. Zhao et al. / Optical Materials 23 (2003) 313–318

Because the light can be deflected along the

direction of the positive order or negative order,

the amount of the total scanning points are

N ¼ 2D þ 1: ð16ÞWhen D ¼ 0, the theoretical diffraction effi-

ciency is 100%. If D ¼ 1, the efficiency will be

smaller. It will decrease because of the increase of

D. And when D is given, the efficiency will change

along with M .

After the required diffraction efficiency is given,

D can be calculated according to the requiredscanning points N . Then from Table 2, we can get

the division number M . At last, the corresponding

L can be obtained from Table 3 or Table 4.2.3. Structure of our optical scanner

The schematic structure of our scanner is shown

in Fig. 3. It includes a monochromatic light

source, a pair of Talbot coding phase plates, a

programmed controlled shifter and a light detec-

tor. The phase plates with binary-phase function

distribution consist of a positive phase plate and anegative one. O1O1 and O2O2 are the axes of the

two plates, respectively. The surfaces of the two

plates with grating lines are opposite to each other

and there is a tiny distance between them.

3. Experimental results

We have fabricated a pair of phase plates with

their period d ¼ 90 lm. And there are 20 grating

lines on each plate. The monochromatic light with

the central wavelength k of 0.6328 lm is used as

the light source. When O2O2 (the axis of the neg-

ative plate) departs every 10 lm from O1O1 (the

Table 4

The relationship between M (the division number in one period) and L (the number of phase levels in one period)

M 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

L 2 3 3 4 4 4 4 6 6 6 7 8 6 7 9

18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33

8 10 9 8 12 12 8 11 14 11 12 15 12 16 12 12

34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49

18 12 12 19 20 14 12 21 16 22 18 18 24 24 14 22

50 51 52 53 54 55 56 57 58 59 60 61 62 63 64

22 18 21 27 22 18 16 20 30 30 18 31 32 16 23

Table 2

The simplified relationship between M (the division number in

one period) and L (the number of phase levels in one period)

M L

T ðt þ 1Þ=22t t þ 1

t2 tðt � 1Þ=2þ 1

t1t2ðt1 6¼ t2Þ ðt1 þ 1Þðt2 þ 1Þ=4t3 ðt2ðt � 1Þ þ t þ 1Þ=2t1t2t3ðt1 6¼ t2; t1 6¼ t3; t2 6¼ t3Þ ðt1 þ 1Þðt2 þ 1Þðt3 þ 1Þ=82n �MðoddÞ Lð2nÞ � L½MðoddÞMð1Þ � � �MðnÞ L½Mð1Þ � � �L½MðnÞ

Table 3

The relationship among uðkÞ (the phase distribution function),

g (the diffraction efficiency) and N (the number of scanning

points)

g 81 88 91 93 95

M 4D 5D 6D 7D 8D

Fig. 3. Schematic structure of Talbot-encoded optical scanner.

X. Zhao et al. / Optical Materials 23 (2003) 313–318 317

axis of the positive one), the light beam will be

deflected by 0.4� (k=d ¼ 0:4�). The total deflectedangle which the scanner can offer is 3.2�. The de-flected angle will become larger if d is smaller.

4. Conclusion

The technology for manufacturing the phase

plates in our approach is similar to the micro-

electronic technology. As mentioned above, we

can design the template of the required phases

according to scanning points N , deflected angle hand diffraction efficiency g. And this kind of binaryphase plate can be fabricated by means of micro-

electronic-lithography technology. Then the mask

pattern on the template can be transferred onto

pieces of glass with photoresist via wet chemical

etching method or reactive ion etching method,

etc. Finally, multi-level phase plates can be fabri-

cated by repeating the above procedure.

Our approach possesses the following advan-tages: (1) It can make better use of the light energy

so as to enhance the diffraction efficiency. (2) Dur-

ing the course of designing, the phase coding dis-

tribution can be derived from L which can be

fabricated in practice. Hence the theoretical errors

will be reduced and the signal-to-noise (SNR) of

the scanner will be enhanced. (3) The deflected

angle of the light beam changes along with the

shift between the plates and precise shift distance is

easy to be realized with the technologies in exis-

tence. The scanner will be easily used and con-

trolled. (4) The phase plates can be mass-producedvia the binary-optical manufacture technology so

that the cost of the scanners can be reduced.

Acknowledgements

The authors acknowledge the support from Na-

tional Natural Science Foundation of China

(60125512, 60177016) and Shanghai Science and

Technology Committee (011661032, 012261011).

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