Non-contact profile measurement of a reflecting diaphragm surface

Non-contact profile measurement of a reflecting diaphragm surface T. Kumezawa, T. Sakamoto end S. Shida*

An interference pattern produced by a simple optical set-up irradiating a curved surface with coherent light is presented. The pattern is formed where reflected light beams intersect, even far from the curved surface. This reflected light pattern can be used for measuring curved diaphragm profiles which are expressed by mathematical functions with points of inflection.

Keywords: optical measurement, diaphragm surface, interference pattern

Small thin plates and foils made of silicon, glass and metal are playing an important role in precision machines, semiconductor devices and office automation equipment. These small parts are not always flat; they are sometimes deformed stati- cally and dynamically by external or piezo-electric forces. Curved surfaces deformed by such forces generally determine the characteristics and reliabil- ity of the machines. Thus, a method for measuring these diaphragms easily, quickly and precisely, without contact, is desirable.

The most simple optical method of measuring small curved surfaces is to use a Fizeau interferometer to form Newton rings. However, it is not easy to set an optical flat on the object to form the high contrast fringes required. Medium size objects of diameter about 30 mm are suitable for use with the Fizeau interferometer; for objects of diameter 1 or 2 ram, a magnification system is required.

In this paper, a method of reflected light interferometry is described, which is simpler than the Fizeau interferometry measurement method.

Profile of curved surface

A curved surface generally has numerous curva- tures. Some surfaces have been expressed by equations, although many have not. Table 1 shows some typical curved surfaces which can be expressed by equations, but which are not composed of combinations of flat surfaces. The surface of Table 1 (a) is obtained when a thin circular plate is fixed all around and deformed by lateral external forces such as compressed gas or oil pressure. This equation can represent a point of inflection. The gradient of the curvature is zero at the centre, and at the fixed points. The surface in Table 1 (b) is formed when the thin plate, bonded to a rigid body, is bent and then bonded to another rigid body. Such surfaces are also formed when one side of the plate has moved laterally after both sides of the plate have been bonded to two bodies. In these cases, the surfaces are represented by the equation, in

* Hitachi Ltd., Mechanical Engineering Research Laboratory, 502, Kandatsu-machi, Tsuchiura-shi, Ibaraki 300, Japan

which x is the distance parameter. The equation representing this type of surface also describes a point of inflection with the gradient of the curve zero at bonded points. The corrugated surface shown in Table 1 (c) is expressed by a sinusoidal function. This surface also has a point of inflection and zero gradient at bonded points.

Reflected light from a curved surface

The curved surface discussed here is the deflected surface of a thin circular plate with clamped edges deformed by external pressure, as shown in Table 1 (a). The deflection W(x) of the plate in the z-x coordinate system can be expressed as 1:

P W(x) = [a 4 - (a 2 --x2) 2] 64D

Table 1. Typical curved surfaces expressed by an equation

(1)

a b c

<3= <3=

Pressurizing Bonding M o v i n g Forming

(a) W = A(a 2 - x2) z

(b) W= Ax3 + Bx2 + Cx + D

(c) W = A s i n x + B

where W = deflection, x = distance from a side or centre, a = radius, A,B,C,D = constants

OCTOBER 1985 VOL 7 NO 4 0141~359/85/040201~5 $03.00 (~) 1985 Butterworth & Co (Publishers) Ltd 201

Kumazawa, Sakamoto and Shida - - measurement of diaphragm surface

where:

W(x) Deflection at radius x p Pressure a Radius D Plate rigidity = (Eh3)/12(1 - v 2) h Plate thickness v Poisson's ratio

When a surface is i l luminated with a beam parallel to the z axis, light is reflected according to the curvature. The surface is assumed to have a mirror finish. The point at which incident light intersects the surface is (Zo, xo). The angle formed by incident light with the normal is 0. If 0 is small enough, the gradient of reflected light wil l be approximately 20:

~ 8.4

Sample Y H e - Ne laser beam

Screen Fig 2 Sample and testing method

aW(xo) 2 - - - 2 t a n O ~ 2 8 (2)

Using Eq (2), the locus of reflected light from point (Zo, Xo)is expressed by:

X --X 0 - - ( z - Zo)

~W(xo) 2 - -

ax

(3)

A screen is placed at distance L from the curved surface perpendicular to the z-axis. Substituting L for z in Eq (3), point xL on the screen is obtained by:

~)W(xo) X L = --2 (L --Zo) + Xo (4)

;)x

Reflected light behaviour on the screen was investigated by calculating xL. The curved surface is assumed to be in the first quadrant and to be illumi- nated by a beam parallel to the z-axis. When the point of incident light is moved sequentially from the centre of the deflected surface towards the edge as shown in Fig 1, reflected light on the screen moves from the centre (position ( ~ ) towards the negative direction in accordance with Eq (4). Reflected light returns to ( ~ after it reaches a minimum value (position ( ~ ) . Reflected light values are accurately represented by a single z = L line, but the line ( ~ - ~ ) - (~l is shown in Fig 1 to aid understanding of reflected light movement on

idean~ Z= L ® Curved su rface

Screen Fig 1 Curved surface and reflected light locus

---~Z

L=4m

10mm L = 8m Fig 3 Fringe patterns observed 4 m and 8 m from curved surface

the screen. Point ~ is the light on the screen which is reflected from an inflection point (x = a/~/3). This shows that two rays reflected from different points on the deformed surface take the same x value, resulting in overlapping of rays and pro- ducing interference fringes. Since light is continuously projected along the (~ - ® line, reflected light is continuously produced, causing sequential interference.

O b s e r v a t i o n o f f r i n g e s

A simple optical system for fringe observation is shown in Fig 2. A curved diaphragm was illumi- nated from a slightly raised position by a colli- mated parallel laser beam - - t h i s is simpler than the normal i l lumination method using a half mirror. A 1 mW He-Ne laser was used. The fringe pattern was observed at any point where the reflected light beams were propagated and intersected. The curved surface sample was made of silicon and was polished to a mirror finish. Diameter and thickness of the sample were 8.4 mm and 0.185 mm, respectively. Young's modulus and Poisson's ratio were 1 x 10 s MPa and 0.2, respectively. A concave surface was formed by reducing the inner pressure using a vacuum pump. The difference between the inner and outer pressure was approximately 0.098 MPa.

The fringe pattern observed on screens 4 m and 8 m from the plate is shown in Fig 3. The circular interference pattern is clear, and the visibility of fringes is high even in high fringe number regions.

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Fringe diameter increases with distance from the surface and the spacing between fringes decreases as the number of fringes increases, from the out- side fringe to the inside fringes. The maximum fringe number obtained on the screen 4 m from the plate was 14.5.

Optical path difference and intensity An optical path difference exists between two rays meeting on the screen, due to the two rays being reflected from different points on the surface. The two rays which meet are in fact, those reflected from either side, not the same side, of an inflection point. Let 5 be the optical path difference between light waves which meet on the observation screen. The interference intensity I is expressed by3:

[2,8 ] = COS ~--~-- ] where ;k is wavelength (5)

The optical path difference & in Eq (5) can now be calculated for a thin circular plate of the same dimensions as above. The external pressure de- forming the plate is 0.098 MPa; the plate is illumi- nated normally with a parallel laser beam and the fringe pattern is observed on a screen 4 m from the plate. A computer was used to calculate the & value. The resultant optical path difference gradually increases toward the fringe pattern centre (Fig 4) and the space between fringes is reduced towards the centre of the fringe pattern. The maximum fringe number is 14.9. This calculated value is in fair agreement with the experimental result of 14.5.

Characteristics of fringes Deflection derived from fringes

The relationship between deflection and maximum fringe number is discussed here. The centre fringe yields the maximum fringe number and is obtained

== tc :5 5

~- 0 5 10 15 o

a Distance from centre, mm

t -

._>

ec 0 0 5 10

I

15

b Distance from centre, mm Fig 4 (a) Optical path difference and (b) light intensity

x

a

x 0 I Q

G m 4

I II P (L,0) Z

Screen Fig 5 Coordinate system for maximum optical path difference

from rays I and II which meet at the centre of the screen, as shown in Fig 5. Let point (L, O) be the intersection point of rays I and II on the screen and point Q(zo, Xo) be the illumination point of ray II on the surface. The maximum deflection of the deformed diaphragm is 8r,. The values for 8r, andz were derived from Eq (1):

Pa 4 P 8m - Zo = $ m -- - - ( a2 -- x2) 2 (6)

64D 64D

The optical path was determined by calculating from a base line z =&m to the screen in Fig 5. The optical path difference between the loci of rays I and II was obtained as follows:

8=((Sm+L)--((Sm--Zo+%/(L--zo)2+x 2) (7)

where zo is in micrometres, L is in metres, and (/. - Zo) is nearly equal to L. From Eqs (6) and (7), optical path difference can be derived, as follows:

The maximum optical path difference in Eq (8) was calculated under the same conditions as above.

The maximum optical path difference was also derived experimentally as the product of wavelength and maximum fringe number. These results are shown in Fig 6. The experimental results agree well with calculated results. The optical path difference divided by twice the maximum deflection (8/28rn) equals 80.6% for a distance L = 4 m. This ratio increases to 90.1% when L= 8 m. By using this 8/28m value and the maximum fringe number, the deflection value can be derived. In this case, the value of deflection will be more precise when the maximum fringe number is measured at a place where the value 8/28r, is near 100%.

Comparison between fringes obtained with Fizeau and reflected light methods It is a simple procedure to use the Fizeau interferometer to measure the profile of a curved diaphragm, though an optical parallel is needed as the testing instrument. Fizeau fringes are shown in Fig 7 to compare with the fringe pattern obtained by the proposed method. A pressurized diaphragm

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was used as in the previous example. The spacing 40 - between fringes is wide at the centre and also in the outer part of the circular patterns. This is because the gradient of deflection at the centre 30 - and outer sides of the patterns is smaller than near the inflection point. In this case, the outer fringe is formed by the light reflected from the outer fixed 20 - end, so the information obtained from this fringe concerns the fixed end of the curved plate.

Spacing between fringes in the proposed 10 - reflected light method decreases as the fringe E order increases. Furthermore, the outer fringe is E formed by the interference of light reflected from "-" 0 I near the inflection point. Therefore, the shape and E -1 width information related to the area near the ._~ inflection point is obtained from the outer fringe, a Although the pattern in Fig 7 was measured care- 10 - fully, noise fringes appeared. It was found that to

~ Theoretical

E 0 Sample 1 / Experimental • Sample 2

¢<3

= 100 ¢3

:6 50

0 0 0 1 2 3 4 5

Distance, m

J t ' t

w

I I I 6 7 8

Fig 6 Maximum optical path difference according to the distance of the screen from curved surface

Theoretical

• "0 . . . . 0" " Experimental

. 1 J 0 0 0.5 1

i

I Pressure, atm

20 / i ' " ~ i'

3 0 ~ I ~ C o n c a v e C o n v e x surface surface

Fig 8 Pressure dependency of the diameter of outer fringe

create the Fizeau fringe without noise is not as easy as with the proposed reflected light method.

Change of fr inge pattern by pressure

The deflection of the plate is changed by pressure and the fringe pattern changes in accordance with deflection of the plate. The relationship between pressure and fringes observed at 4 m from a curved surface was investigated using the sample shown in Fig 2. The relation between pressure and the diameter of the outer fringe of the interference fringe pattern is shown in Fig 8. The diameters were measured where the fringes were large, omit- ting cases where pressure was approximately zero and reflected light was bright. There were some small differences between the experimental and calculated results, thought to be due to differences in original parameter values.

I I 5ram

Fig 7 Fizeau fringe pattern for pressurized curved surface

Discussion and conclusion

Reflected l ight interference fringes were formed when a parallel i l luminating l ightwas projected onto the surface of a sample. The light source used in this method was a coherent light such as a laser. It was shown that the diameter of the fringe changed according to the face profile deformed by pressure: the diameter became larger with increase in pressure. Dynamic changes in fringe pattern with deviation in pressure were clearly visi- ble with the human eye, which meant one could determine the degree of pressure by sight. The fringe pattern also fol lowed Young's Modulus, with plate thickness determining the surface profile, when the pressure was constant. Therefore

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Kumazawa, Sakamoto and Shida - - measurement o f d iaphragm surface

observation of the fringe pattern leads to a simple measurement method of such parameters.

Let the concave surface be at the centre of the x - z coordinate system. If the face is il luminated in the first quadrant, the reflected light meets and forms fringes in the fourth quadrant. If the face is convex instead of concave, the same treatment is applicable, but the reflected light meets and makes fringes in the first quadrant 2. The fol lowing conclusions may be drawn: • It has been verified that an interference pattern

is produced by a simple optical set-up irradiating a curved surface by coherent light, in which the curved surface has a mirror finish and a point of inflection.

• An interference pattern is produced wherever reflected light beams intersect and this is clear even with a large number of fringes. The fringe

number increases towards the centre of fringe patterns. The spacing between fringes gradually decreases with higher fringe numbers.

The parameters affecting face deflection can be estimated from the fringe pattern. Information about the surface near the point of inflection was obtained from the outer fringe. The precise value of deflection was obtained from the fringes reflecting over a greater distance.

References 1 Timoshenko S. and Woinouwsky-Krieger S, Theory of Plates

and Shells. McGraw-Hill, New York, 1959 2 Kumazawa T. et el. Interference of l ight reflected from a con-

vex face. Applied Optics, 1984 23, (14)/(15), 2426 3 Kumezewa T. et el. Profile measurement of a concave surface

using the interference fringe of reflected light. J. Japan. Soc. Precis. Eng. 1984, 50, (2), 406

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Non-contact profile measurement of a reflecting diaphragm surface