Desiging and modelling of a grating-based displacement micro

INSTITUTE OF PHYSICS PUBLISHING

NANOTECHNOLOGY

Nanotechnology 12 (2001) 308–315

PII: S0957-4484(01)19704-7

Designing and modelling of a grating-based displacement micro-transducer Bing Zhao1,2 , Ai-Qun Liu3 , Frank Chollet1 , JianMin Miao1 , HuiMin Xie1 and Anaud Asundi1 1 School of Mechanical and Production Engineering, Nanyang Technological University, 639798 Singapore 2 Key Laboratory for Solid Mechanics, Ministry of Education of China, Tongji University, Shanghai 200092, People’s Republic of China 3 Institute of Materials Research and Engineering, 117206 Singapore

Received 30 November 2000, in final form 23 July 2001 Published 28 August 2001 Online at stacks.iop.org/Nano/12/308 Abstract Fourier analysis is used to formulize and characterize grating-based displacement transducers. The analysis is carried out for linear and radial binary grating sensors, and for both single and grating-pair sensors. The influence of the grating (pair) structure and the receiving window on sensing sensitivity, linearity of response and dynamic range is investigated. Several types of photoresist grating are fabricated and are used to experimentally characterize the grating sensor. The tested result is found to be in general agreement with the analysis result. (Some figures in this article are in colour only in the electronic version)

1. Introduction

Light source

Grating-based displacement transducers are widely used in engineering for sensing and converting a displacement into a digital output, such as position sensing, velocity and acceleration sensing and motion encoding etc [1–7]. The basic structure (see figure 1) of optical motion sensors consists of an optical source, often a light-emitting diode (LED), one or several photodetectors, and one grating or grating pair. The principle of the sensor [1–6] relies on the measurement of intensity change due to the interaction between a light beam and grating(s). The output of the photodetector is controlled by the optical flux fed into the detector. The key factor that affects the sensitivity of motion sensor and dynamic response range is the relationship between the grating displacement and the optical flux. Previous works often used a simple linear relationship to estimate the system output response. The linear relationship cannot correctly and precisely represent the case of a grating-pair system where the response of optical flux to grating motion is often nonlinear. One of the objectives of this paper is to set up a correct and precise input–output relationship of a displacement transducer under certain conditions. Based on this relationship, some optimum designs of the optical structure of a grating-based displacement transducer can be carried out. 0957-4484/01/030308+08$30.00

© 2001 IOP Publishing Ltd

2. Moving grating 1. Fixed grating

Z

Light detector Transparent Output

Figure 1. Illustration of a traditional grating sensor transduction.

In the present work, a method for analysis of a gratingbased micro-displacement transducer is presented. We provide answers to questions such as the following: Why use a grating pair? What are the advantages and disadvantages of singlegrating and grating-pair systems? When will two gratings generate the moir´e phenomenon? What are the differences between sensors based on grating modulation and moir´e modulation? In section 2, a discussion on using the Fourier series method to analyse near-field diffraction is first given. The optical structure and its principle of linear and angular sensors, as well as the analysis method used in this paper, is introduced. In section 3, the optical fluxes with respect to

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308

Designing and modelling of a grating-based displacement micro-transducer

(a)

Z=Z2

Y

Plane wave

a

Fraunhofer diffraction a a 2 2

(b)

P

Fresnel diffraction

X

Z=0

Z=Z1

Figure 3. Illustration of diffraction and its transition region classification. In the region near the aperture, i.e. Z < Z1, the diffraction pattern can be seen as the projection of the aperture.

Y

The above expression can be expanded in form of a Fourier series as T (X) = A0 + 2

n=∞ 

An cos(2π nf X)

(2)

n=1

a + ∆X 2

with f = 1/P the spatial frequency of the grating. Coefficient An is

X

Figure 2. Illustration of binary linear grating structure. (a) Grating and its coordinate system; (b) the grating is shifted a distance X along the X axis.

grating displacement for single-grating and grating-pair linear sensor systems are established. The effect of grating structure on flux is analysed, and the optimum design of grating is discussed. A comparison between single-grating and gratingpair systems is carried out. In section 4, a similar analysis for an angular sensor is completed. In the appendix, the optical transmission functions (OTFs) of radial gratings are deduced.

2. Formulation of grating-based transducers 2.1. Single-linear-grating system Single-grating structures are becoming more popular especially in MEMS sensors/transducers, because of their simple structure. In figure 1, if the fixed grating is removed or replaced by a receiving window, the system becomes a single-grating transducer. In this case, no moir´e pattern appears. The illumination beam is modulated by the grating and detected directly by the photodetector, without using the fixed grating to demodulate the optical signal. Figure 2(a) shows a binary linear transmitting grating, which consists of a series of bars and spaces. This type of grating is frequently used in a MEMS structure, as it is easy to design and manufacture it. With reference to the coordinate system in figure 2(a), the OTF of the grating is defined as   1, nP − a
X
nP + a , n = 0, ±1, ±2 . . . 2 2 T (X) =  0, other region. (1)

An =

 π na  1 sin = Rsinc(nR), πn P a A0 = R = P

n = 1, 2, 3, . . .

(3) where R = a/P is the ratio of slot to pitch, called the open ratio or fill factor or duty circle of the grating. Equation (1) shows the amplitude modulation of the binary grating. Equations (2) and (3) demonstrate the signal composition of the transmitted light from the grating, such as the harmonics, direct current component, noise etc. Furthermore, it shows the relationship between these harmonics and the grating structure such as the fill factor and grating pitch. If the grating is shifted a distance X along axis X (see figure 2(b)), the OTF of the grating becomes T (X) = A0 + 2

n=∞ 

An cos[2πnf (X − X)].

(4)

n=1

Following the Fourier optics [7, 8], the intensity distribution in the near field is also periodical and can be approximated as [7, 8] I (X, Z) = I0 + 2

n=∞ 

In (Z) cos[2πnf (X − X)]

(5)

n=1

where I0 is the intensity of incident light. In is the amplitude of the nth harmonic; Z is the distance shown in figures 1 and 3. Figure 3 illustrates a grating diffraction under plane wave illumination and its classification. When Z > Z2, Fraunhofer diffraction (or far field) occurs. When Z1 < Z < Z2, it is the near-field Fresnel diffraction region. When Z < Z1, this area is defined as the geometric projection area, and the diffraction intensity can be approximated by I0 T (X). In this 309

Bing Zhao et al

case, In (Z) = I0 = constant. Z2 can be generally estimated as [8] Z2 = a 2 /λ. (6)

window width, and RX the relative displacement of the movable grating. The above expression can be simplified, for example,

a is defined in figures 1 and 3. Following the present MUMPS design rule for MEMS, a must be larger than 2 µm. The frequently used a is between 2–20 µm. Thus we have Z2 ≈ 6–600 µm. Generally Z is several micrometres, for example, 2 µm in [6]. Therefore, there is no doubt that near-field diffraction must be considered for a MEMS product design. Equation (5) can be used for MEMS grating design in the near-field case. However, In (Z) in this equation cannot be easily derived, and it is strongly associated with the grating geometry. Therefore, it is difficult to use this formula for theoretical analysis. Fortunately, Z, the distance between grating and photodetector, is very short in some applications, so it is considered as Z < Z1. So far, we have not found the exact expression for Z1. In this paper, we suggest using the following expression as criterion:

F ∗ = R + 0.5[R + RW − 2RX − 2RRW ]  0 < 2RX < RW < 0.5 RW − 2RX < R < RW + 2RX .

Z1 =

1 a2 Z2 = . 5 5λ

(7)

It should be emphasized that all the contribution from adjacent facets has already been considered although only one facet of the grating is shown. For Z < Z1, there is no coupling effect of adjacent facets. When Z > Z1, the contribution from adjacent facets will overlap and modify the distribution as the light progresses. At distance Z = 2a 2 /λ, the self-image (or Talbot image) [8] appears. Some grating transducers use this effect for design. For most MEMS grating displacement transducer designing [3, 4], the condition given in equation (7) can be satisfied. In this case, equation (5) can be rewritten as I (X, Z) = I0 A0 + 2I0

n=∞ 

An cos[2πnf (X − X)].

(8)

n=1

Suppose that only the light beam passing through the receiving window can be detected by the photodetector. Considering a receiving window of height H (in the Y direction in figure 2) and width W , the optical flux sensed by the photodetector can be obtained by integrating the light intensity over the receiving window area S:  F = I0 T (X) dX dY s

= RI0 S + 2I0 S ×

n=∞  n=1

1 πf W

1 An sin(π nf W ) cos(2πnf X). n

(9)

Denoting RW = W/P , RX = X/P , the above expression can be rewritten as 2 F ∗ = F /(I0 S) = R + 2 π RW n=∞  1 × sin(π nR) sin(π nRW ) cos(2π nRX ) (10) n2 n=1 where F0 = I0 S is the initial light flux impinging on the grating. F ∗ is the relative light flux or dimensionless light flux impinging on the photodetector. RW is the relative receiving 310

(11)

The calculation of equation (11) is simple and fast. The series in the equation will be convergent when n is taken as 100. Equation (11) implies that the optical flux is a sectional linear function of grating displacement X. From equations (9)– (11), some conclusive remarks can be drawn. (i) Light flux depends not on the absolute amount of grating displacement X, but on its relative displacement RX . Similarly, flux changes as the relative geometric size of the grating and receiving window width (R and RW ) varies. (ii) The term sin(nπR W )/RW in equation (10) shows the multiple influences of the receiving window. Divide RW , into two parts as RW = RWI + RWF , where RWI is the integer part of RW , and RWF the fractional part. Case 1. RWF = 0, RWI = 0. In this case, the receiving window width W is an integer multiple of the grating pitch P . We have sin(nπR W )/RW = 0; i.e. the optical flux will be constant, no matter how the grating moves. This is a very important result, and is useful for design specifications. Case 2. RWF = 0.1 ∼ 1.0, RWI = 0. This case shows the influences of RWF (see figure 4). From the curves in figure 4, we know that (a) system sensitivity F /RX could be improved by using a receiving window with small RWF and (b) there are some zero-incremental zones in most of the curves except the curve of RWF = 0.5. In these zones, sensitivity F /RX = 0; i.e. the optical flux will be constant or zero increment regardless of how the grating moves. Here, RX is an increment of the relative grating displacement RX , and F is the increment of optical flux due to RX . In order to avoid this case, RWF = 0.5 is the only choice. Case 3. RWF = 0, RWI = 0. Relative light flux is approximately inversely proportional to RW (see equation (9)), or approximately to RWI . It is clear that the maximum flux would be obtained at RWI = 0, or RW = RWF ; i.e. the receiving window width W is less than the grating pitch P . (iii) The term sin(nπ R) represents the influence of the open ratio of grating R that ranges from zero to unity. The numerical result following equation (6) demonstrates that relative optical flux F ∗ is symmetrical about R = 0.5, and the maximum dynamic range of measurement can be obtained at R = 0.5. Furthermore, there is no zeroincremental zone in this case. From the above discussion, we conclude that RW = 0.5, R = 0.5 is a good choice for grating sensor design. In this

Designing and modelling of a grating-based displacement micro-transducer

F* 1 .5

.4

.3

.2

RW

= 0 .1

.6

0 .8

.7 .8

0 .6

.9

1 .0

0 .4

0 .2 RX 0 0 .0 0

0 .2 0

0 .4 0

0 .6 0

0 .8 0

1 .0 0

Figure 4. Curves of relative optical flux F ∗ versus the relative grating displacement RX , with respect to different relative lengths of filter window RW , which varies from 0.1 to 1.0.

design, the grating is simplified with only one slot. The optical flux of equation (10) can then be further simplified as  1 1 n=∞ F ∗ = 0.5 + 4 2 cos [2(2n + 1)π RX ] π n=0 (2n + 1)2 = 1 − 2|RX | = 1 − 2|X|/P

− 0.5
RX
0.5 − 0.5P
X
0.5P .

(12)

Such a design possesses some important advantages: • system response is linear, i.e. the relationship between F and X is linear; • a simple system, and easy manufacture, as the grating is reduced to a slot; • high dynamic range of measurement; • no zero-increment zone exists. 2.2. Linear grating-pair system

T1 (X) = A10 + 2

A1m cos(2πmf1 X)

m=1

T2 (X) = A20 + 2

n=∞ 

(13)

A2n cos[2πnf2 (X − X)].

n=1

The combined OTF of the grating-pair sensor is the production of the two OTFs of the two gratings [9, 10], i.e. T (X) = T1 (X) × T2 (X): T (X) = T1 (X)T2 (X) n=∞  = A10 A20 + 2A20 A1m cos(2πmf1 X) +2A20

n=∞  m=1

m=1

A2n cos[2πnf2 (X − X)]

+2

m=∞  n=∞  m=1 n=1 m=∞  n=∞ 

A1m A2n cos[2πmf1 X − 2πnf2 (X − X)] A1m A2n cos[2πmf1 X + 2πnf2 (X − X)].

m=1 n=1

(14) The relative optical flux is therefore  

 ∗ I0 T (X) dX dY F = (I0 S) s

= R1 R2 + + +

The condition for using the OTF method for a high-density grating-pair system is that the gap of the grating pair is very small. Under this condition, the diffraction effect and light source effect can be neglected [7]. Supposing grating 1 is fixed and grating 2 is movable, the OTF of the fixed and moving gratings for a grating-pair sensor (see figure 1) can be expressed by [9, 10] n=∞ 

+2

 1 2R2 m=∞ sin(πmR1 ) sin(π mRW1 ) 2 π RW1 m=1 m2

 1 2R1 n=∞ sin(π nR2 ) sin(π nRW2 ) cos(2π mRX2 ) 2 π RW2 n=1 n2  n=∞  1 sin (π mRW1 − π nRW2 ) 2 m=∞ π 3 m=1 n=1 mn mRW1 − nRW2

× sin(π mR1 ) sin(πnR2 ) cos(2π nRX2 )  n=∞  1 sin (π mRW1 + π nRW2 ) 2 m=∞ + 3 π m=1 n=1 mn mRW1 + nRW2 × sin(π mR1 ) sin(πnR2 ) cos(2π nRX2 )

(15)

where RX2 = X/P2 is the relative displacement of the moving grating and RW1 = W/P1 and RW2 = W/P2 are the dimensionless receiving window widths relative to gratings 1 and 2. Therefore, RW1 = RW2 means that the two gratings have the same pitch. Terms 1 and 2 in the above equation are constants during the movement of the grating. They represent the direct current component of an output signal impinging on the photodetector. Terms 2 and 3 indicate the influences of the fixed grating and moving grating respectively; terms 4 and 5 reflect the coupled effect of two gratings. Term 4 represents the moir´e effect, of which the spatial fringe frequency is the difference of the frequencies of the two gratings. 1/(RW1 − RW2 ) = W/WF is actually the ratio of receiving window width to moir´e fringe width, where fringe width is denoted as WF . This term plays an important role in the contribution to the optical flux. If a moir´e fringe is formed under condition of P1 ≈ kP2 (k is 311

Bing Zhao et al

F* RW1 = 10.3, RW2 = 10.6

0.4

Term 10 Term 21

0.2

Term 32 Term 43 Term 54 Sum

0

R

X

-0.2 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure 5. Illustration of the nonlinear relationship of relative optical flux F ∗ and relative grating displacement RX , and a comparison of contributions of the terms in equation (11), where R1 = R2 = 0.5.

an integer constant), all the terms except terms 1 and 4 could be treated as high-frequency noise and therefore neglected. Generally, the last term, i.e. term 5, is relatively small, and could be neglected, as sin(π mRW1 +πnRW2 )/(mR W1 +nR W2 ) is usually a small quantity. The term represents the influence from the so-called additive moir´e. The spatial frequency of this additive moir´e, which is the sum of the frequencies of two gratings, is frequently so high that cannot be distinguished. Therefore, this term is considered as noise and is neglected. In the following paragraphs, some discussions on the grating-pair transducer system are carried out based on equation (15).

(v)

(i) Generally, terms 4 and 5 are not linear functions of the (vi) relative displacement of the moving grating RX . However, the total flux is sometimes a sectional linear function of grating displacement or RX . (ii) Following equation (15) terms 2, 3 and 5 are approximately inversely proportional to RW1 and RW2 . Therefore, terms 1 and 4 will be two determinative (vii) parameters, even there is no visible moir´e pattern, only if RW1 and RW2 are two large amounts. In this case, the optical flux will be a nonlinear function of RX , as term 1 is a constant; term 4 is generally a nonlinear function of RX . Figure 5 shows these effects, where RW1 = 10.3, RW2 = 10.6. It is clear that the terms 2, 3 and 5 can be neglected, and the nonlinearity of the optical flux F ∗ versus relative displacement of the moving grating RX is due to the contribution of term 4. (iii) Influences of RW1 and RW2 , case 1: RW1 < 1, RW2 < 1. (viii) In this case, the grating-pair sensor has a linear response, i.e. F ∗ is a sectional linear function of RX . This is because the nonlinear parts due to terms 4 and 5 in equation (11) cancel each other. (iv) Influences of RW1 and RW2 , case 2: RW1 = RW2 + J , where J is an integer. Following equation (15), in order to obtain a maximum dynamic range, two measures can be used. One is designing the moving grating with small RW value. Another measure is to let the two grating structures 312

have parameters RW1 = RW2 + J . For these two cases, terms 3 or 4 will become larger. Influences of RW1 and RW2 , case 3: RW1 = RW2 . When RW 1 (or RW2 ) is less than 0.5, there are some zeroincremental areas, even if R1 = R2 is chosen as its optimum value 0.5. When RW1 = RW2 = 0.5, a maximum dynamic range of response is obtained, and the response is linear. When RW1 (RW2 ) is larger than 0.5, the response is still linear, but the dynamic range is decreased. The simulation results also provide an important point, that is, the maximum flux F ∗ fluctuates a little around 0.5 in case of I  1. Influences of RW1 and RW2 , case 4: RW1 = C1 + 0.5, RW2 = C2 + 0.5, where C1 and C2 are two positive integers. In this case, the result is similar to case 1; i.e. optical flux would be a linear function of RX . This is because terms 4 and 5 are no longer nonlinear functions of RX . Influences of RW1 and RW2 , case 5: RW1 = RW2 (RW1 > 1, RW2 > 1), (RW1 = C1 + 0.5, RW2 = C2 + 0.5). The discussion on the relationship between F ∗ ∼ RX will become complicated. The numeric analysis shows the following. (a) The dynamic range will seriously decrease when RW1 and RW2 are greater than unity. (b) Optical flux would be a nonlinear function of RX if RW1 and RW2 were larger than unity. This is very important for the gratingpair sensor design. (c) It is difficult to obtain a dynamic range larger than 0.5 when RW1 > 5, RW2 > 5. Figure 6 illustrates the influence of the grating structures. Several F ∼ RX curves are shown, with respect to different combinations of R1 and R2 . It is clear that there are two zero-incremental zones in the vicinity of RX = C, or C + 0.5, where C is an integer. Another point drawn from this figure is that the variation of R1 and R2 will not cause nonlinearity of the F ∼ RX response. Let us discuss two special cases of R1 = R2 and R1 + R2 = 1. If R1 = R2 , the zero-incremental zone in the vicinity of RX = C disappears. Furthermore, if R1 = R2 = 0.5, the

Designing and modelling of a grating-based displacement micro-transducer

F* RW1 = RW2 = 10.5

0.7

( R1 , R 2 )

(0.4, 0.8)

0.5

(0.4, 0.3) (0.5, 0.1) (0.7, 0.9)

0.3

(0.1, 0.3) (0.7, 0.6) 0.1 R

X

-0.1 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure 6. Curves of relative optical flux F ∗ versus relative grating displacement RX , with different combinations of R1 and R2 , to illustrate the influence of the grating structures.

optimum case where there is no zero-incremental zone and maximum dynamic range, linear F ∼ RX response is obtained. If R1 + R2 = 1, the zero-incremental zone only occurs in the vicinity of RX = C. From the above discussion, an important conclusive remark could be drawn; that is, the optimum geometric structure parameters should be R1 = R2 = 0.5, RW1 = RW2 = integer k + 0.5. When k = 0, this is a best choice. 2.3. Radial grating system A radial grating is generally used for angular sensing and transduction. In this section, only single-grating sensor analysis is demonstrated. There is no substantial difficulty in analysing a grating-pair sensor, and this is left for interested readers. Considering a filter window illustrated in figure 7 by the shaded area, the optical flux sensed by the photodetector can be obtained by integrating the light intensity over the receiving window area S:   !0 /2 1 1 F∗ = I0 T (θ) dS = T (θ)rr dθ. (16) I0 S I0 S −!0 /2 s Upon substituting the expression of T (θ ) giving by equation (A.6) (see the appendix), the above expression can be rewritten as  1 2 n=∞ sin(π nR ) sin(π nR! ) cos(2πnRθ ) π 2 R! n=1 n2 (17) where S = rr! is the receiving window area; R = α/β is the angle open ratio of the grating; R! = !/β is the relative angle of the receiving window; Rθ = θ/β is the relative rotation angle of the grating. Comparing equation (17) with (10), we find the two equations share the same form. This means that the performance of the angular sensor and the linear sensor could be analysed with the same equation if the relative parameters are used. Some conclusions similar to the analysis in section 2.3 could be drawn as follows. F ∗ = R +

(i) R should be 0.5; i.e. the transmitting slot should the same size as the non-transmitting part. (ii) Decreasing the relative receiving window size will increase the system sensitivity and dynamic range. R! = 0.5, i.e. the filter angle is half of the grating period, will be an optimum choice. (iii) The relative filter window size R! determines the period of the output signal. For example, if R! = 8.3, i.e. the filter window covers about eight slots of the grating, the optical flux will change eight times while the grating rotates through one revolution.

3. Experimental characterization In order to validate the above analysis, a series of gratings were fabricated and tested. The photoresist gratings were produced on transparent plastic films using photolithography, and the gratings were tested without removing the substrate. The grating pitches (P ) vary from 25 µm to 1 mm; the open ratio (R = a/P ) is either 0.3 or 0.5. The procedure for fabricating the grating is as follows. The surface of the specimen was coated with a layer of positive photoresist (submicrometre thick). The resist is exposed under a photomask in a vacuum exposure machine to UV lamp radiation. The exposed part of the photoresist is removed on development. A layer of chrome (sub-micrometre thick) is then deposited on the specimen surface. After removing the residual photoresist between the deposited layer and specimen surface using photoresist remover, an amplitude grating is formed on the plastic substrate. Figure 8 shows the laboratory setup to characterize the grating sensor. When performing single-grating testing, grating 2 is removed. For grating-pair testing, the two grating films should be placed face to face. The light source is four GaAs LEDs or a structured light projector (Lasiris Inc, < 1 mW). The gap between two gratings, and the movement of the movable grating, is controlled by a three-dimensional translation stage with a resolution of 5 micrometres. A 313

Bing Zhao et al

(a)

(b)

Y

α/2+∆θ

β α/2

Y α/2-∆θ

α/2

X

(c)

X

Y Λ

∆r

r

Figure 7. Radial grating and filter structure. (a) Radial grating and its coordinate system. (b) Grating is rotated an angle of θ . (c) Filter window and grating. 1 2 3 4 5 6 Light source

those in figure 9(a). The two curves of F ∼ RX in figure 9(b) show the comparable tendency between the experimental and simulated results.

4. Discussion and conclusion The comparison of two types of linear grating sensor is as follows. 3-D translation stage

Figure 8. Schematic diagram of experimental system: 1, 4, transparent grating substrate; 2, 3, tested grating film; 5, receiving window (filter); 6, photodetector.

rectangular receiving window is fixed onto a photodetector, which is connected to a digital power meter. The photodetector has a detection resolution of 0.1 µW. The experimental result is found to be in general agreement with the theoretical one, although there is a difference in absolute values between theory and experiment. The difference is mainly because the influences of longitudinal parameters of sensor structure, such as the gap between gratings, grating depths, distance between light source and gratings etc are not considered in the analysis. However the longitudinal parameters are beyond the scope of this paper. Figure 9 shows experimental and simulated results using a low-density grating. The F ∼ RX curve in figure 9(a) is a result for a single grating with pitch of 1 mm and open ratio R = 0.5, and with the receiving window size 5 × 0.7 mm2 . The two curves from the simulated and experimental results show a similar tendency. This similarity demonstrates that the analysis on a single-grating sensor in this paper is approaching the real case. Figure 9(b) shows the results of a grating-pair test. The gratings and the receiving window are identical to 314

(i) Both sensors can attain their best properties: constant sensitivity, linear response and maximum dynamic response range. (ii) The response of optical flux to grating displacement is always linear for a single-grating sensor, but not for a grating-pair sensor. (iii) For a single-grating sensor, the dynamic response range will dramatically decrease as the relative receiving window width RW increases, but it is less sensitive to RW1 and RW2 for a grating-pair sensor. (iv) Generally, with a grating-pair sensor, we can obtain a larger dynamic response range, or higher modulation depth. (v) The system design and manufacturing of a grating-pair sensor is generally more complicated than that of a singlegrating one. (vi) For a single-grating sensor, near-field receiving is a necessary condition, if the grating density is higher than 25 lines mm−1 . In this paper we have presented an analysis that describes the properties of a linear and angular displacement transducer based on a moving grating and photodetector. The optical transmitting function technique with the Fourier series method was used to characterize some basic sensor devices. It should be emphasized that this technique could be employed for any

Designing and modelling of a grating-based displacement micro-transducer

(a) Experimental data

F / I0S

1

Simulated data

0.8 0.6 0.4

R X = ∆ X/P

0.2 0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Here, f = 1/β is the spatial angle frequency of the grating, An are the coefficients of the Fourier series; they can be expressed as  1 β An = T (θ) exp(−i2π nf θ) dθ β 0  1 kβ+α/2 = exp(−i2π nf θ ) dθ β kβ−α/2

1 = sin πnf α πn = R sinc(nR ) (A.3) where R = α/β = αf , and A0 = R . Upon substituting equation (A.3) into (A.2), we obtain T (θ ) = R + 2R

(b)

n=∞ 

sinc(nR ) cos(2π nf θ ).

(A.4)

n=1

Experimental data

F / I0S

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

Simulated data

R X = ∆ X/P 0

0.2

0. 4

0.6

0.8

reduced type of grating-based displacement transducer. The proposed technique provides a powerful tool for optimizing the design of displacement transducers. The analysis indicates that we can expect a high-sensitivity, wide-dynamic-range and linear-response transducer.

Appendix. Transmission function of radial grating Considering the binary radial grating shown in figure 7, the optical transmitting function can be expressed as  1 kβ − 0.5α
θ
kβ + 0.5α T (θ ) = (A.1) 0 other region where k = 0, ±1, ±2, . . . , β is the angle period of the grating; α is the angle of the open slot. T (θ ) can be expanded as the following Fourier series: n=∞  n=−∞

An exp(i2πnf θ ).

T (θ) = R +2R

n=∞ 

  sinc(nR ) cos 2π nf (θ − θ ) . (A.6)

n=1

1

Figure 9. Comparison of simulated and experimental results: (a) single grating testing, grating pitch 1 mm, open ratio 0.5, receiving window size 5 × 0.7 mm; (b) grating pair testing; grating parameters and receiving window are the same as (a).

T (θ ) =

If the grating is rotated through an angle of θ (see figure 7(b)), the transmitting function of the grating becomes  1 kβ − 0.5α + θ
θ
kβ + 0.5α + θ T (θ ) = 0 other region. (A.5) Its corresponding Fourier series is

References [1] Norton and Harry N 1989 Handbook of Transducers (Englewood Cliffs, NJ: Prentice-Hall) [2] Westbrook M H and Turner J D 1994 Automotive Sensors (Bristol: Institute of Physics Publishing) [3] Abbaspour-Sani E, Huang R S and Kwok C Y 1995 A wide-range linear optical accelerometer Sensors Actuators A 49 149–54 [4] Bochobza-Degani O, Seter D J and Socher E et al 1999 Comparative study of novel micromachined accelerometers employing MIDOS Proc. 12th IEEE Int. Conf. on Micro Electro Mechanical Systems, MEMS’ 99 (Orlando, FL, 1999) pp 66–71 [5] Neviere M, Popov E and Bojhkov B et al 1999 High-accuracy translation–rotation encoder with two gratings in a Littrow mount Appl. Opt. 38 67–76 [6] Sene D E, Bright V M, Comtois J H and Grantham J W 1996 Polysilicon micromechanical gratings for optical modulation Sensors Actuators A 57 145–51 [7] Wronkowski L 1988 Fresnel images of a binary diffraction grating with open ratio 0.5 J. Mod. Opt. 35 1417–22 [8] Goodman J W 1996 Introduction to Fourier Optics (New York: McGraw Hill) [9] Creath K and Wyant J C 1992 Moir´e and fringe projection techniques Optical Shop Testing 2nd edn, ed D Malacara (New York: Wiley) ch 16 [10] Yokozeki S 1982 Moir´e fringes Opt. Lasers Eng. 3 15–27

(A.2)

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