Fine structure in Fresnel diffraction patterns and its application in optical measurement

Optics

& Loser

Technology,

Vol. 29, No.

1. pp.

$1’ 1998 Printed

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1997

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0030-3992197 PII:

383-387,

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ELSEVIER

Fine structure in Fresnel diffraction patterns and its application in optical measurement FANG XIAOYONG, CA0 MAOSHENG, BEN YONGZHI, QIN SHIMING

ZHOU YAN,

The function of light intensity distribution of Fresnel diffraction from an opaque fibre is derived, and the fine structure in the diffraction pattern is theoretically described. In this paper, the relationship between fibre diameter and fine structure in the diffraction pattern is given. The fine structure can be used to measure the diameter of fibres, and measuring subscript limits may be greatly squeezed. Several fibres of different diameters are measured by means of the fine interference fringe in the diffraction pattern, and experimental results show the improved diffraction technique is highly reliable with higher precision compared with the original technique. @ 1998 Elsevier Science Ltd. All rights reserved.

KEYWORDS:

Fresnel diffraction, fine structure in the diffraction pattern, optical

measurement

Introduction

fibres between 5 and 500 urn were measured by the two diffraction techniques. Results show that the improved technique is more accurate than the original one, and its error is not greater than 1%.

Some measuring tools, such as vernier calipers, micrometer calipers and micrometer microscopes, are unsuitable for automatic measurement, even though they possess high accuracy. The diffraction technique is widely used in the automatic non-contacting measurement of fine filaments or fibresim3 and it is usually realized according to Babinet’s principle. As Babinet’s principle is only adapted to the far-field state of Fraunhofer diffraction, and the accurate measuring range of the single-slit is from 0.1-200 mm, the measuring subscript limit of diffraction technique is restrained4. Automatic measurement of a fibre which is under a micrometre in diameter, can be realized by a quasi-optical cavity; however, the fibre must be a conductor.

Fine structure in the Fresnel diffraction pattern from an opaque fibre Function

of light

intensity

distribution

An opaque fibre is vertically illuminated by a laser beam of divergent angle 8,, as shown in Fig. 1.

This paper studies the fine structure in the Fresnel diffraction pattern from an opaque fibre, and gives the relationship between the fibre diameter and the fine interference fringe in the diffraction pattern. Using the fine interference fringe, the optical non-contacting measurement of the opaque fibre, where diameter is below 1 urn, may be realized. According to the formula given in this paper, its theoretical subscript limit is predicted to be 50 nm. In our laboratory, several typical

The authors are in the Physics Department, Yanshan University, 066004 Qinhuangdao, China. Received 16 May 1995. Revised 7 May 1997. Accepted 6 June 1997.

Fig. 1 fibre

383

Schematic diagram of Fresnel diffraction

from an opaque

Fine structure

384

in Fresnel

According to the Fresnel-Kirchhoff diffraction formula’, the optical amplitude at P can be written

ss

?,I

lip=+

+cos(0 + I))] dS

diffraction

patterns:

F. Xiaoyong

at P is

So the light intensity as I = ($)2(&)2 (1)

(A)

x

Here, A = 4b(b - a) = 4R*t),(B, radius of the fibre, and b is the the point of the fibre. Under the approximation, cos(8 + $) = 1, written as

et al.

sin*(kbq)

- e,), k = 2rc/L. a is the radius of laser beam at condition of small angle and (1) can now be

+ sin*(kacp) - 2 sin(kbp)sin(kacp)

x cos kg(b2

- a2)]]

As the light phase is not influenced the fraction in (7), the approximate

(8) by the variable value

< of

D(P ’ = 2(R + D) From

Fig. 1, we have can be adopted r = D + RB sin(Q + $) cos 9

From

= D + RB(O + Ic/)

(3)

(I/ - cp)D = RB cos(B + Il/)cos q = RO

(4)

So that the distance

from dS to P is

_

u(p = r, - D = -(r2 - D)

(9)

and from (3), we have

rz - D = ~(-0,

I

RD

Fig. 2, we have

r, - D = a(@, + $,>

’

,-=D+;(R+D)

in (7) and (8).

(5)

According

+ $*)

(10) I

to (9) and (lo), we have

4(R + D)”

On the outside of the geometrical source, we have”

(h’ - 0’) = (u’ - I)(; + ;)a

g

image of the light

+(6)

=

1

1)

@2

_

u

1)

40,

=(u2 -

Suppose

+ $1)

2

d-0,

-

+ $2)

2

I

l)aq

q=k;(R+D),

<=Hf

Dq 2(R + D) ’

and according to (2), (5) and (6), the optical at P can now be written as 2iAb u, = - Texp(ik(R

X

+ D))

R+D exp(ik - RD

amplitude

w s 2iq< - 1

So the function of light intensity distribution of the Fresnel diffraction from an opaque fibre can now be written as sin* ux

I = 1, --+-_2!!y~

[

sin* x

X2

I (11)

cos(u2 - 1)x

2 b + ikbql

- exp(ik- R+D RD

2 a + ikacp)

1 (7)

= gexp{ik(R

+ D))

2iqt2 - 1

sin(kbp)expiikib2\

-sin(katp)exp(

ik%a’}} Fig. 2

Schematic

diagram

of derivation

of interf-+n-=

far*nr

Fine structure in Fresnel diffraction patterns: F. Xiaoyong et al.

Here Z,,= (4Aab/;lRD)2; x = 27ca(p/1,cp is the diffraction angle at P; and u = b/a.

385

Here, d is the diameter of the opaque fibre. For the Gaussian beam waist of half-width wO,

Theoretical analysis of the fine structure in a Fresnel diffraction pattern

2

(2b)= = (2~4~ +

From (1 l), we have n7c ___ 2(u - 1) 2(u + 1) n=+2m ( 29n77 - 1>2

sin2 5~0s 41”

sin2 5~0s 410 2(u - 1)

2

nn 22(u + 1)

(17)

Substitute (17) into (16), and the formula of the diameter can now be written as

d = ((2wo)’ + (~~+(~)2)“2-f$

2

(18)

n= +_(2m+ 1) Here S stands for the width of the fine interference fringe at P.

( LIZ n7c - 1> m = 0,1,2...

(12)

Equation (12) shows that a series of bright belts will appear in each diffraction grade, and the belts are regarded as the first-order interference fringes of the Fresnel diffraction from opaque fibres, its interference minimum can be defined by

As the major error came from the measurement of the fine fringe width, known by (1 S), the error is D/id Od = S((2bS)2 + (DL)=)“=66

(19)

Here cr stands for standard error of measurement. n= n=

k/(24+1) +(21--l)(u-1)

u = 2m u=2m+l

1=1,2,...

Measuring

(13) Furthermore, considering the influence of the secondorder non-linear term in (5) the fine structure factor cos(u2 - 1)x exists in (1 l), so fine interference fringes will appear in each bright belt and its angle width is Aq =

/? a(29 - 1)

(14)

Equation (14) shows the relationship between the radius of the fibre and the fine structure in the diffraction pattern. In addition, the fine interference fringe number within a k-grade diffraction zone can be written as’ u = 2m

Several fibres with diameters between 500 and 5 urn are measured by the fine structure, and results are shown in Table 1. Here, the light source adopted is a HeNe laser of mode TEM,,, the power is 2.2 mW, o. is 200 pm. Fine interference fringes are determined by a JCW-1 micrometer eyepiece, and its minimum graduation is 10 urn. When a collimated HeNe laser is substituted for the light source in Fig. 3, the diameter of the same sample can be measured by (20) Here, 6 stands for the width of the O-grade diffraction zone, and the measuring results are shown in Table 2.

N= u = 2m + 1

results

(15)

k = 1,2, 3..

In Tables 1 and 2, the standard diameter of the sample is denoted by do, and is defined by an OBO-1 optical micrometer. The minimum graduation of the optical micrometer is 1 urn, and the counter error of display values is not greater than 0.1 urn.

Equation (1.5) can successfully explain the fine structure which is observed in the Fresnel diffraction pattern’. Application measuring Measuring

of the fine structure fibre diameter principle

The fine structure in the Fresnel diffraction pattern can be used to measure the diameter of fibres. Its light circuit is shown in Fig. 3. From (14), we have d =

((2,,:+

(+-)=)‘I’-6

(16) Fig. 3

Measuring light circuit

Fine structure in Fresnel diffraction

386 Measuring

Table 1.

results

6 (v-n)

gd (v-n)

d (v-n)

1 2 3 4 5

5884 2408 640 211 51

1 1 1 1 1

440.19 251 .18 75.06 24.98 6.05

Measuring

results

by diffraction

et al.

(A = 632.8 nm, R = 300.0 mm, D = 3500.0

by fine structure

N

Table 2.

patterns: F. Xiaoyong

technique

$

(pm) 0.04 0.08 0.1 1 0.12 0.12

mm)

do (pm)

e (%)

440.0 251 .O 75.0 25.0 6.0

0.04 0.07 0.08 0.08 0.83

(A = 632.8 nm, D = 3500.0

mm)

N

6 (mm)

d (pm)

do (pm)

e (%)

1 2 3 4 5

10.130 17.876 60.870 197.126 889.218

437.28 247.80 72.81 22.47 4.98

440.0 251 .O 75.0 25.0 6.0

0.62 1.3 2.9 10 17

In Tables 1 and 2, r is used to compare the accuracies the two diffraction techniques, and is defined by

of

(21)

Comparison diffraction Measuring

between techniques

the

As seen from Fig. 4, if the OBO-1 optical micrometer is used to measure the width of the fine interference fringes, under the condition D = 500 cm, the measuring subscript limit from the theoretical estimation is 50 nm, and its theoretical error is not greater than 10%.

two

accuracy

Tables 1 and 2 show, for samples under 0.5 mm in diameter, that the accuracies of measurements using the fine structure are higher than the diffraction technique. Some major causes include the following.

(1) With the decrease of the fibre diameter,

(2)

the width of the O-grade diffraction zone is increased, and the error produced from computing formula (20) will become greater and greater. Known by (1 l), the function of the light intensity distribution of the Fraunhofer diffraction from an opaque fibre can be written as7 I = I,

sin’ ux _+-.Y [

sin’ x .Y\-z

sin ux sin x 2 -x X I

subscript

According to (19) the measuring accuracy of the fine structure technique may be improved by the increased values of D, for example, the error of the sample which is about 0.3 urn in diameter, when D = 1000 mm, is about 10%; when D = 5000 mm, it is about 2%.

(22)

Equation (22) shows, under the condition of Fraunhofer diffraction, that a series of interference fringes will also appear in the diffraction pattern. The edges of the O-grade diffraction zone cannot be decided under the influence of interference fringes, and so produce a great error. (3) The improved diffraction technique only depends on the width of the fine interference fringe, consequently the condition of the small angle approximation is satisfied to a great extent, and there is no theoretical error produced from computing formula (18). Measuring

For the fine structure technique, its measuring subscript limit d,,, is mainly dependent on R, D and the minimum graduation of the micrometer 6,. When an opaque fibre is laid close to the head of a HeNe laser, D and 6, influence dm, as shown in Fig. 4.

3

2

I

limit

The analysis as stated above shows that the measuring subscript limit of the original diffraction technique is restrained4, and this is proved by the experimental data in Table 2.

0

I

I

I

I

2

3

I

1 4

1

I 5

D(m)

Fig. 4

Influence

of D and 6, on measuring the subscript limit

387

Fine structure in Fresnel diffraction patterns: F. Xiaoyong et al.

Conclusions

2

In summary, as the second-order non-linear term exists in the laser beam, the fine structure appears in the Fresnel diffraction pattern from an opaque fibre. In particular, the fine structure can be used to measure the diameter of opaque fibres. Also, this technique has the advantages of high accuracy and reliability, and its experimental facilities are simple.

3

References 1

Guoguan,

Publishing

Y. Modern Optical Technology. Machine

House,

Beijing (1986) 230-257

Industry

4

5

Carson, R.W. Laser measures diameter Product Engineering, 39 (1968) 50-52 West, P. Automatic non-contacting filaments, Research, 4 (1971) 6-8

of moving

measurement

strand

of wire,

of fine

Guilin, S. Measurement correctness by Fraunhofer diffraction, Optical Technique, (2) (1995) 36-38 Hache, E., Zajac, A. Optics, Addison-Wesley, New York (1974) 365-392

6

Xiaoyong, F., Maosheng, C., Keqin, R. Deduction for function of light-intensity distribution of filament Fresnel diffraction, CoNege Physics, 14 (1995) 5-l

7

Xiaoyong, F., Maosheng, C., Keqin, R. Study of fine structure of interference fringe on diffraction pattern by filament. Applied Laser. 16 ( 1996) 30-32

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