Non-contact optical profilometer with linearized response and high sensitivity

Non-contact optical profilometer with linearized response and high sensitivity J. Shamir, A. Brunfeld and G. Toker An optical profilometer is described based on the direct determination of a surface distance from a focal plane. An advanced mathematical analysis, taking into account the effects of finite lens apertures, has led to the design of a system of extremely high resolution and wide, linear dynamic range. Demonstrated sensitivity exceeds 3 nm, with linear response through a dynamic range of 50/~m.

Keywords: non-contact optical profilometer, linearized response, high sensitivity

Progress in high technology places an increasing demand on the accuracy of geometrical shapes and surface quality. Extremely strict tolerance requirements exist in a variety of fields such as the production of mechanical parts, optical components, optical and magnetic storage media, very large scale integration (VLSI) technology and integrated optics. The initial purpose of our work was to develop a simple and easily applicable method for determining the flatness Of polished substrates used in the manufacture of integrated optical devices. A non-contact optical system for the absolute measurement of surface profiles is described. The system has an accuracy comparable to interferometric methods but does not suffer from the ambiguity that arises in interferometric systems due to their periodic nature. The measuring system is based on a detection method that can determine the distance of a scanned surface from the focal plane of an optical microscope with extremely high sensitivity. Being a non-contact measuring procedure, the system is non-invasive, fast and reliable, and is suitable for use on a production line. An optical profilometer has previously been described with high sensitivity and large dynamic range m. This could be used for measurements on machined mechanical parts with an accuracy of 1 #m. The method was based on a differential optical configuration, for detecting the focal plane of a lens, and a feedback system to linearize the measurement. The nominal accuracy was not limited by the optical sensor but by the feedback system employing mechanical translators.

optical configuration. Preliminary investigation i2~attained a sensitivity of 20 nm and an approximately linear dynamic range of 50 #m. More exact calculations, taking into account the finite extent of the apertures in the system, are shown to lead to an improved design with a measuring sensitivity exceeding 3 nm. Some recent publicationst3 7] have also treated profile measurements by focus sensing, but no one was capable of combining such high sensitivities with a wide dynamic range. The measuring method is reviewed, and Fourier optics is used in the notation of the operator algebra ]8-1°j (see summary in the appendix) to complement the earlier [~ geometrical analysis of an ideal system. New mathematical expressions are used to calculate the system response, including the effects of finite apertures. The theoretical results are then shown and compared with experiment.

System configuration The optical system is essentially a laser scanning microscope, as shown in Figure 1. The laser beam, filtered by D1

Da -

1

L

' •

The objective of the present work is to exploit the capabilities of the focus-sensing system and to replace the feedback arrangement by response linearization. Bearing in mind the proposed applications, we also limited the analysis to polished samples that could be approximated as specular reflectors. The measuring procedure was also simplified by using a more compact

430

~

D2

BS/

L1 :

~/~

~

L2

\

S ,'HI>u J / I / i / / ' Fig. 1 The experimental system: L is the laser beam; BS are beam splitters; L~ and L2 are lenses; D represents detectors with masks; I is the image of spatial filter, SF; S is the sample surface; M is the optional mirror for interferometric measurements

0308-9126/88/060430-05 O 1988 Butterworth & Co (Publishers) Ltd NDT International Volume 21 Number 6 December 1988

a spatial filter, SF, which plays the role of an illuminating point source, is focused on the sample by the lens, L t. The light reflected from the sample is collected by the same lens, forming an image of the spatial filter at some point, I, within the detection system, which consists of a beam-splitter, BS, and two detectors, D 1 and D2, each provided with a specially designed spatial filter (mask)J21, as discussed below. In the experimental system the lens, L 2, and the mirror, M, provided a reference path for a calibration interferometer that could also be used to complement the profilometer during measurement. Axial movement of the sample surface induces a displacement of I along the detector's optical axis. Since I represents the waist of a Gaussian beam, this translation changes the spot size on the masks and, consequently, the power transmitted to the detectors. Various approaches have been discussed for the filter design tl'2J. In Reference (1), a small circular aperture was used, while in Reference (2) it was shown that a ring aperture performs better. Adopting the ring aperture for the present analysis, one has to integrate the power over the area of the ring. Assuming a Gaussian beam of width, w, symmetrically incident on a ring of radius, R, and width, AR, the detected optical power will be given as a function of w by R+AR

S(w)=A f exp(-2r212rcrdr w']

(1)

R

where A is a constant factor depending on the total power and the optical system. Denoting the respective power incident on the two detectors by St (i = 1, 2), normalization on the system response may be performed by measuring the normalized quantity

I_$1 -$2 Si + S2

(2)

The value of ! uniquely determines the position of I relative to the two detectors and, consequently, the distance of the observed surface, S, from the focal plane of the optical system. Our aim is to linearize this response function with respect to object surface position over as large a range as possible. To achieve this linearization one has to take into account the variation in magnification during measurement that originates from the wide dynamic range (of the order of 50/am in our case). This large dynamic range causes the spot size on the object surface to vary from one end to the other. For optimal adjustment the object surface should be in the focal plane for the central region of the measuring range. Using this approach, the minimum spot size in the system was 1/am, increasing to 12/am at the edge of the measuring range. These values are adequate for most applications since, usually, no such wide dynamic range is necessary where high spatial resolution is required.

G

r~

v

so

z = 2s1-2x

s

Fig. 2 Unfolded representation of the relevant part of Fig. 1 with definition of system parameters

sample may be situated at the image plane or displaced by a distance x from this plane. Our final objective is to determine this displacement from the measurement of the power detected by the two detectors. In Figure 2 a Gaussian beam emerging from the pinhole is incident on the first lens, L 1, of aperture, P1, and is propagated a distance, z, to the next lens, L2, of aperture, P2, to be transmitted towards the detector at a distance, s. Generally, two different lenses are considered, although in this system these are two views of the same lens. Using the convenient operator notation [s-t°], we may derive the system operator T starting from the entrance pupil of the first lens and ending at the detector surface. Referring to the summary of the operator notation in the appendix, the lens operators are given by

L,=PiQ[~j

( i = 1,2)

(3)

where f~ is the focal length of the lens and Q denotes the quadratic phase factor (see Equation (A-1 )). Propagation through the two sections of free space is described by the Fresnel-Kirchhoffintegral with its operator representation given by the free-space propagation operator (FPO) shown in Equations (A-7) and (A-8). Thus the complete system operator can be written as

[,,,

,

,4,

To investigate the first-order effects of the apertures, complementary apertures are defined by the relation

(5)

e'=l-e

Substituting into Equation (3) and neglecting secondorder diffraction effects involving the product P~ x P~, we obtain

-1

,

-1

General system analysis The optical path between the light source (SF in Figure 1) and each detector can be viewed as a separate system, the essential part of which may be described by an unfolded equivalent optical configuration, as illustrated in Figure 2. The sample, S, is assumed to act as a plane mirror, thus a two-lens system is being considered. The

NDT International December 1988

The first term is the ideal infinite aperture case, while the rest constitute the correction terms. The operator of Equation (6) operates on the input Gaussian beam that may be represented within the operator notation by a quadratic phase factor of complex label tS]

431

[I]A

u,. = Q ~

(7)

where A is the amplitude (a constant) and qi is the Gaussian beam parameter. Thus the complex amplitude distribution at the detector plane is given by Uout = T u i n = Uo -

ul -- U2

(8)

where

uo=R[s]Q[--~]R[z]Q[1]A; L Y J

(9)

Lq~J

u.

k J J Lq~J In the above equations we used the fact that the two lenses are the same, ie f~ = ./2 = f ;

P', = P~ = e'

(12)

and we defined 1

1

. . . . .

1

(13)

ql qi fl Using some simple operator algebra rules 16 s] (see appendix, in particular Equation (A-21)) and ignoring constant amplitude and phase factors, the zero-order term reduces to a new Gaussian beam that has been considered in an earlier publication t21

Uo

q-S-+S

(14)

' 1

where 1 -- = q2

ql

1

1

+z

f

(15)

The two correction terms may be written in the form

u, =R[s]P'Q[~]

(16)

and

u2=Q

R[oe]P Q

(17) 1

f~

procedure, the effects of the finite apertures may be determined by calculating the diffraction integrals for a Gaussian beam propagated through the complementary apertures, ie the RP'Q operation. The errors obtained are then used to modify the system parameters and improve performance. Numerical

analysis

The introduction of complementary apertures simplifies the procedure for calculating the diffraction integrals. A general, detailed analysis will be given in a subsequent publication, while the results relevant for the present system are described here. The two correction terms (Equations (16) and (17)) are introduced by the two respective apertures. In most optical systems one of the apertures is the limiting aperture, and therefore one of the correcting terms will usually dominate. This assumption was made in Reference 1 l, where only the limiting aperture was taken into account. However, the second aperture may have an appreciable effect; thus both apertures were included in the present optimization process. Figure 3 shows the power received by each detector as a function of displacement (x in Figure 2). Figure 4 is the complete system response with the combination of the two detectors according to Equation (2) for two values of the lens aperture. The figure shows that a system optimized for a given aperture size is no longer optimal even for a small deviation from this size. Our experimental system was optimized for a 7 mm aperture. The experimental values noted on the figure indicate good agreement with theory (only one point deviates from the calculated value). These measurements were made using a piezoelectric transducer calibrated with the help of the interferometer incorporated in the measuring configuration (see Figure 1). Figure 5 shows the results of exciting the transducer by a saw-tooth signal and by a sinusoidal signal, demonstrating linearity and sensitivity better than 3 nm. Measurement results are shown in Figures 6-8, where the sample was translated using an extremely smooth translation stage made of gauge blocks. One advantage of this measuring system is shown in Figure 6 for the measurement of the height difference between two Johanson blocks, where the gap does not introduce any ambiguity as it would for an interferometric or a stylus

where

a=s+z---SZ

f The zero-order term (Equation (14)) is used for first-order calculations and basic filter design, while the influence of the apertures is determined by the correction terms (Equations (16) and (I 7)). Returning to Equation ( 1), we have to use the Gaussian beam parameter given by Equation (14) to calculate the values of the powers S 1 and $2 to be substituted in Equation (2). Parameters to be considered for the optimization process are the dimensions of the two ring apertures, which are different for each detector, and the distance, s, of Equation (14) for each detector. The first approximation for the system parameters is obtained by a computer-aided optimization, considering the required sensitivity and dynamic range. For a given set of parameters evaluated by the above

432

1.00

(18)

0.90 r-

0.80

0.70 o.

0.60

"o

0,50

•~

0.40

c:

0.30

"~

0.20 0.10

0

~"

0 0.15

0.16

0.17

0.18

0.19

0.20

0.21

0.22

0.23

Object s u r f a c e position (mrn) Fig. 3 Power (on arbitrary scale) transmitted to each detector as a function of object surface position

N DT International December 1988

Aperture ~ 6 mm 1.00 t.t. oo

0.80 060

~'~

0.q0

Aperture

N 7 mrr

Nit"

020

~

o.oo

-o ~ -0.20 ~ -0.401-

lib

Experimental

~'~

~

Calculated

|

-0.601-

E u-I

3, 080 I

01 --

0.15

Fig. 4 points

I

I

I

I

I

I

I

0.16

0.17 0.18 0.19 0.20 0.21 0.22 0.23 Object surface position (mm) Calculated normalized system response with measured

Fig. 6 Scan caross the border between t w o Johanson blocks with a nominal height difference of 5 #m

•

a

•

+

1- 2 seconds-I

Fig. 7

Surface scan of a Johanson block

.,~--2seconds

b Fig. 5

Sensitivity test (see text for details)

J

measurement that may even damage the system. The surface of one such block is scanned in Figure 7 on a more sensitive scale, revealing surface roughness and a scratch. Figure 8 illustrates the surface of an InPh substrate prepared for integrated optical devices, showing a regular oscillatory profile. Fig. 8

TE

~-lO0.m~l

Surface scan of an InPh substrate

Conclusions The performance of a focus-sensing profilometer has been optimized using design parameters derived by an advanced mathematical procedure and taking into account the effects of finite apertures in the optical system. The demonstrated sensitivity of 3 nm was limited by mechanical and electronic noise. Repeatability measurements were

NDT International

December

1988

in the same range, indicating that with a more stable construction and proper calibration an accuracy of the same order can be expected. This system proved to be accurate, reliable, fast and inexpensive, and is suitable for a wide range of applications in the laboratory and industry•

433

Appendix

Q[a]Q[b] = Q[a + b]

(A-13)

The definitions and relevant relations of operator algebra lz-4] are summarized. All constant factors that cancel in the normalization process of Equation (1) and which are thus irrelevant for the measurement have been ignored. For all operations a general complex function, f ( x , y), is assumed and the basic operators are defined as follows.

Via] V[b] = V[ab]

(A-14)

V[a]Q[b] = Q[a2b] Via]

(A-15)

V[b]S[m] = S [ b ] V [ b

(A-16)

The quadratic phase factor:

Q[a] = e x p ~ a ( x 2 + y2)

]

Fourier analysis shows that

V[b]F=FV[~]

(A-17)

FG[s] = s [ S-2] F

(A-18)

FS[m] = G[ - 2 m ] F

(A-19)

(A-I)

The linear phase factor: G [ s ] = expjk(sxx

+

syy)

(A-2)

With these and the basic relation it can also be shown that

The scaling operator is defined by

V[a]f(x, y) = f(ax, ay)Vial

(A-3)

The Fourier transform operator:

Ff(x)

+

f f ( x , y) exp 2nj(xx' + yy') dx' dy'

(A-4)

V[b]R[d] = R[d/b ~] V[b]

The shift operator:

S [ m ] f ( x ) = f ( x - m x, y - my)Sire]

(A-5)

The transfer operator of an ideal thin lens of focal length f is the quadratic phase:

L[f]=Q[-Z~-]

(A-6>

Free-space propagation, ie the Fresnel-Kirchhoffintegral, is described by the free-space propagation operator ( F P O ) which can be expressed in various ways by the basic operators:

Rid] = F - X Q [ - g 2 d ] F = F Q [ - 2 2 d ] F

-1

(A-7)

where d is the propagation distance. Another useful expression is

The F P O satisfies the cascading property:

R [ a ] R [ b ] = R[a + b]

(A-9)

A simple optical system containing a single lens may satisfy the Fourier transforming condition

(A-lO) or the imaging condition 1

1 t-

a

b

1 -

(A-20)

(A-II)

f

which yields

tA-21]

Acknowledgements We are pleased to thank I. Levavi and G. Zettzer for helping with the experimental work. This work was partially supported by a grant from the National Council for Research and Development, Israel, and the Heinrich Hertz Institute, Berlin, FRG.

References 1 Fainman,Y., Leaz, E. and Shamir, J. 'Optical profilometer- a new method for high sensitivity and wide dynamic range' Appl Opt 21 (1982) pp 3200-3208 2 Bnmfeld,A., Toker, G. and Shamir, J. 'High resolution optical profilometer' Proc SPIE 680 (t986) pp 118-123 3 Cork, T.R. et al 'Depth response of confocal optical microscope" Opt Letters 11 (1986) pp 770-772 4 Wilson,T. aad Carliai, A.R. 'The image of thick step objects in the confocal scanning optical microscope' Optik 72 (1986) pp 109-114 5 Code,T.R. et al 'Distance measurement by differentialconfocal optical ranging' Appl Opt 26 (1987) pp 2416-2420 6 Shep0ard, C.J.R. and Matthews, H.J. 'The extended-focus,autofocus and surface-profilingtechniques of confocal microscopy' J Mod Opt 35 (1988) pp 145-154 7 Quereiolly,F. et ai 'Optical surface profile transducer' Opt Eno 27 (1988) pp 135-142 8 Nazarathy, M. and Skamir, J. 'Fourier optics described by operator algebra' J Opt Soc Am 70 (1980) pp 150-158 9 Nazarathy, M. and Shamir, J. 'Holography describedby operator algebra' J Opt $oc Am 71 (1981) pp 529-541 10 Nazarathy,M. and Shamir, J. 'First-order optics - a canonical operator representation:tossless systems'J Opt Soc Am 72 (1982) pp 356-364 11 Toker,G., Brufdd,/i.D. and Shamir, J. °Deterioratingeffectsof beam truncation on focus sensing systems' Proc SPIE 897 l0 (1988)

Authors

R[a]Q[-~-]R[b]=Q[~(I+;)]V[-;] (A-12) Some basic relations are evident from the definitions of the basic operators:

J. Shamir and A. Brunfeld are in the Department of Electrical Engineering, Technion - Israel Institute of Technology, Haifa 32000, Israel, G: Toker is at EI-Op Electro-Optic Industries, Kiriat Weizmann, P O Box 1 165 Rehovot, 76110, Israel.

Paper received 1 J u l y 1988

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NDT International December 1988