OPTICAL REVIEW Vol. 3, No. 4 (1996) 227231 Letter

Nuluerical Method to Fit the RefractiVe IndeX Profile of Planar MicrOlenses Made by lon EXChange Techniques Eva AcosTA,1 Susana Rios,1 Masahiro OIKAWA,2 Akimitsu SAT03 and Kenichi IGA4

*Laboratorio de Q~)tica, Departmento de Fisica Aplicada, Faculted de Fisica Universidad de Santiago de Compostela, E-15706, Santiago de Compostela, Galitia, Spain, 'Nippon Sheet Glass Co.. Ltd.. Sumimoto Fudbsan Shiba Bldg. 11-11, 1-Chome, Shiba, Minato-ku, Tokyo, 105 Japan, 3Nt~pon Sheet Glass Co., Ltd., Fiber C~)tics Division, Tsukuba Research Center, 5-4, Tokodai Tsukuba-city, Ibaraki, 300-26 Japan, ' Tokyo Institute of Technology, 4259, Nagatsuta, Midori-ku,

Yokohama, 227 Japan

(Received August 30, 1995; Accepted April 30, 1996)

We present a fitting method for obtaining a Lunctional form 0L the reLractive index profile of planar microlenses

made by ion exchange techniques from total shearing interferometric measurements. Compared to the usual power series expansion fit, this method allows a reduction in the number of coe~icients needed to characterize a lens.

Key words : niques

planar microlenses, 3-D distributed index profile, shearing interferometry, ion exchange tech-

1. Introduction

Planar microlenses (PMLs) with three-dimensional (3-D) distributed index profile fabricated by the ion-exchange

technique along with photolithographic processes have been widely used in a large variety of devices related to optical communications as well as imaging and informa-tion processing systems. Thus, there is a growing need for this kind of reLractive microoptical element in different devices. There is, however, a lack of methods to determine a simple expression for their refractive index profile. Knowledge on the distributed index shape is required in order to characterize the optical properties such as focal length, numerical aperture, aberrations, and design optical

devices and to optirnize fabrication conditions. Two of the

authors of this work proposed a 3-D representation of the index by fitting to a matrix representation related to the power series expansion with respect to the axial and radial distances.1-3) The fit coefEicients are the matrix elements. It

has also been shown that the 4x4 matrix leads to a good approximation. This representation requires 16 fit coefficients and, therefore, designing the elements and characterizing the fabrication conditions is not an easy task.

In this work we present a method to flt 3-D index profiles of PMLS fabricated by ion exchange to functional

forms obtained from total shearing interferometric measurements.4,5) Although the functional form obtained for a lens may not represent a solution of the diffusion equation, the advantages of this fit lie in the reduction of

the number of fit coef~cients without loss of information on the shape of the equi-index regions, and the change in the index with axial and radial coordinates. The change in

profiles with fabrication conditions and thehrefore the focusing properties of the lens can be studied in terms of

only a few parameters. We describe in Sect. 2 a general method to fit 3-D

distributed index profiles, assumed to be rotationally sym-

metrical about the optical axis, to functional expressions reducing the number of fit coefficients. In Sect. 3 we apply

the method to two different fabricated PML's and discuss the results. Finally we offer some additional remarks on ray tracing calculations.

2. Fit Method

The total shearing interferometric pattern of a thin slide

of a PML, fabricated by ion-exchange technique, repre-sents the isoindicial fringes of the distributed index of the

lens (see Fig. 1).

The value of the index, nh, of the kth fringe is given by:

n* + ~k/d = nh (1) where the fringe number k is counted beginning from the outer one. n* represents the substrate index, ~ the wave-length used in the experiment and d the thickness of the slide. Therefore, the experimentally obtained data is a set

of isoindicial fringes, with index nk, where the shape is given in terms of the radial and axial coordinates, denoted

here by r and z, respectively. In a previous work,6) we described a particular fit of the

refractive index profile of a fabricated microlens to a functional form in terms of only 3 fit coefficients. In this

section we describe a general method to fit this kind of 3-D

distributed index profile to functional expressions further

reducing the number of fit coefficients. The first step of the index representation is to find the

best fit for the shape of each fringe. We propose a 2-D (two-dimensional) polynomial expression of the form:

N* N, ~0.~Iaij(k)ziri~j=0, Vk 1,2, . . . NF (2)

NF being the number of fringes of the interference pat-tern. This equation represents the family of equi-index fringes for the lens under test. Linear algorithms can be used for this fit and the number of coefiicients (NI and N2)

and the significance 0L each coefflcient can be determined by an F test.7)

22 7

228 OPTICAL REVIEW Vol. 3, No. 4 (1996)

The dependence on the index lies, in an implicit way, in the fit coefficients, i.e., for each value of nh we have a set

. . . . . . N2} re-of coefficients {aij(k); i=1,2, Nl; j=1,2, presenting the kth fringe. Some of these coefficients may remain constant for all the fringes (k=1,2, . . . NF) and some of them will vary. We will assume, without any loss 0L generality, that all these coefficients vary with the f ringes.

By writing Eq. (2) in the form:

N* N, ast(k)zSrst+ ~ ~ aij(k)ziri~j=0 Vk 1,2, N (3)

i=0 j=0 i~sj~t

and fitting each coefiicient aij, i~s, j~t, versus ast we can

transform Eq. (3) into the form:

N* N, ast(k)zSrS tl~0.~ ,JLast(k)]ziri~j~O Vk 1,2, N (4)

i~sj~t

where the f.j's represent fit functions for the set of values {aij(k); k=1,2, . . . NF} versus {ast(k); k=1,2, . . . NF}. A

simple expression as a function of only one coefficient has

been found for the family of isoindicial fringes given in Eq. (2), and thus the shape of each fringe is represented by

Eq. (4) for the different values of the coefficient ast'

If the fit functions, f.j, can be chosen in such a way that

Eq. (4) can be rewritten in an explicit expression of ast Of

the form:

ast(k)=F(r,z) (5) where F represents a function containing the radial and axial coordinates as well as the fit coefficients of the f.j

functions, a functional form of the refractive index profile

can be obtained by choosing a suitable fit trial function relating ast and nk: n/'=T(ast(k)). ThereL0re, a functional

form for the refractive index profile is given by:

n(r,z)= T LF(r,z)] (6) In this final expression F contains information about

the shape of the equi-index fringes and T provides the rate

of change of the index from one fringe to another. In the case of PML's fabricated by ion-exchange tech-

niques, a linear dependence between the refractive index profile and the ion concentration distributions can be assumed from the Lorentz-Lorentz formula. Therefore, this method can lead to a representation of the concentra-tion distribution in terms of a small number of coefiicients

(see, for instance, Ref. 8), where 16 fit coefEcients were

214 um

Fig. I . Interferometric pattern of the Microlens I obtained by total

shearing interferometry.

E. ACOSTA et al.

needed) .

On the other hand, from the refraction index, the lens properties such as focal length, N.A., aberrations. . . . can be predicted, and taking into account the relationship of the fit coefEcients and the fabrication conditions through the concentration distribution, we believe this method will

be useful for optimizing the fabrication process. Several functional forms for the refractive index proflle

can usually be found by this method. To try to discrimi-nate which is the best fit for the lens we compare the final

fits in terms of the following parameters: the standard error of the fit coe~icients and the coeflicient of determina-

tion, rd2, given by:

N i~1[yi- yp]2

rd =1- N i~_1[ yi- y~]2

where N is the number of fitted data, {yi; i=1, . . . N} the

set of experimental data, {yp; i=1, . . . N} the predicted

data and y~ the mean value of the fitted data. A coefficient of determination close to unity is needed to

ensure that the measured data are close to the fltting function. Furthermore, the fit coefficients must have small standard errors in order to have high significance in the fit.

To illustrate the method we will apply it to the recovery

of the index profile of two PML's differently fabricated.

3. Experimental Results

Two fabricated PML's were characterized with this method. The first one was made by diffusing T1+ ions in a BZS-4 substrate from TlN03 salts. The interference pattern obtained by the total shearing interferometric technique is shown in Fig. 1. The second lens was also made by diffusion of T1+ ions, assisted by an electric field,

in the ZS-2 substrate. The interference pattern for this microlens is shown in Fig. 2. 3. 1. Microlens I

The interference pattern consists of 10 isoindicial fringes. The substrate index was n~=1.54, the wavelength

used in the interferometric measurements was I =0.633 ,clm and thickness of the PML was d=40 /Im.

When the isoindicial fringes were fitted to polynomial expressions as in Eq. (2), we found that two different fits

could represent the family of fringes accurately. Each one

400 um

Fig. 2. Interferometric pattern of the Microlens 11 obtained by total

shearing interferometry.

OPTICAL REVIEW Vol. 3, No. 4 (1996)

a2 2

o. 6

o. 5

0.4

03

0.2

(a ;)' = ( b+c(a: y)/(1+d(a: )')

.l

<)

, ~'

)1.

,

coeffilcients Std. error r~=0 991

b= 22 O O l c=1 6e-4 2e-05

d=-2 . I e-4 4e-5

1~rl~TI ~~ll~1~~ rlrrl o 1000 2000 3000 4000 5000 6000 a2

Fig. 3. Dependence of th~ parameters of the isoindicial elliptical fringes for PML I. Dots are the experimental data. a2 (dimensionless)

represents the ellipticity of the fringes and al ('clm) the minor axis size. The coefficient of determination of the fit is rd2=0.991 and the

fit coefiicients: b=0.22~0.01, c=1.6e-4~2e-5, d=-2.1e-4~4e-5.

led to a different functional form for the index in terms of

the r and z coordinates (developed in Sect. 3.1.1 and 3.1.2).

The final fits are compared in Sect. 3.1.3.

3. 1.1 First trial fit for the family of isoindicial fringes

From the interference pattern data we found that the shape of each isoindicial fringe can be fitted to centered

ellipses given by:

z2+(a2(k))2r2=(al(k))2 Vk=1,2 . . . NF (7)

For the fits of all the Lringes the mean coe{ficient of determination was rd2=0.995 and the standard error of the ai coefficients did not exceed 3% of their values.

In the next stop we fitted al versus a2 for each value of k, which allowed us to experess Eq. (7) in terms of al alone

(a2-f2(al))' As the best fit was found for a relationship

between al and a2 of the form

(a2(k))2=(b+c(al(k))2)/(1+d(al(k))2) (8) the farnily of ellipses in Eq. (7) could be represented in

terms of al' alone by:

(al(k))2=(dz2+cr2-1)+ (dz2+cr2-1)+4d(z2+br2) /2d (9)

(The fit represented by Eq. (8) and the statistics of the fit

coefiicients b, c and d are shown in Fig. 3). To find a functional form for the reLractive index of the

PML we used some trial functions to flt al(k) versus nk. The best fit was found for the following relationship:

In(n02~n2)=1nq+pln(al2) (10) where no' q and p are the fit coefHcients. (no represents the

maximum index obtained after diffusion). Finally, the expression for the refractive index as a

function of the r and z coordinates, obtained by inserting Eq. (9) in Eq. (10), is given by:

E. ACOSTA et al. 229

2 - l)+ r---n2 = - qtkdz2 +cr 12d f for n ~ ns n2 +cr I +4d br +z

rd2=0.98, n02=2.869:k0.005, q= 8652:~2225, p = I .32~0~03,

b=0.22i0.01, c= 1.6 10-4i2 10-5, d=-2.1 10-4~4 10-5

/

/ / /

/

/

75 50 25

/

/

/ /

/ /

c'()

L- c) c_ ~ ' )~~i~~

~

,

1~

=~L) f~ -

\ I~ ')( /i{

r'lj ' (, n ~

( ~ o 1 c

c e;= l l ' /f;J_ ', t: l]L ~ [

-/-__ O _25 50 _75 _100

(

)

1)

T)

T)

')

/ 2.9

28

27

-26

25

, 24 -/lO / ~7 20

/ 30 l' 7~-, 40 50

60

c~~

Fig. 4. Refractive index profile representation of PML I (first trial flt). Dots are the experimental data. r (,u mX1.5) represents the radial

coordinate and z (,clmxl.5) the axial distance.

n2= n02~q{ (dz2+cr2- 1)

+ (dz2+cr2-1)2Hr4d(br2+z2)/2d}p for n~ns (11)

Figure 4 shows the fit of Eq. (11) along with the values of the coefficients. Only 7 fit coefficients were needed for

the best final fit. Non-1inear Marquard9) algorithms were

employed. 3.1.2 Second trial fitfor thefamily of isoinditialfringes

We found that the fringes can also be represented by a family of non-centered ellipses. The accuracy of this fit is

comparable to the previous one but an additional coeflicient is needed. The mean value for the coefficient of

determination of all the ellipses was rd2=0.998 and the standard error of the ai coefflcients did not exceed 20/0 of

their values.

The fringes, fitted to non-centered ellipses, can be re-

presented by:

(z+a3(k))2~(a2(k))2r2=(al(k))2 (12) The next step was to find the relationship arnong the ai

coef~cients. In these flts, we observed that coefiicient a3(k)

remained practically constant for all the fringes. The mean

value of a3' designated henceforth by a, was 10.425~0.015

/1 m. We also noted that a2(k) remained practically constant

with k with a mean value of a2=0.7701!~0.005. The last step for obtaining the final fit was to find a functional form to

fit al(k) versus nl<. For this purpose we chose the following

functional expression:

l -2b2ln(n2/ n02) (13) a 2=

By inserting Eq. (13) in Eq. (12) we obtained the final functional form for the refractive index profile:

n2=n02exp{ -(1/b2) [(z+a)2+a22r2] } for n ~ ns (14)

where we can see that only four fit coeflicients define the lens profile. Figure 5 represents the fit and the statistics of

the coefiicients.

230 OPTICAL REVIEW Vol. 3, No. 4 (1996)

J ( 2~ n = no cxpl-kl/b )[[(z+a)2 +a2 r21JJ forn~n,

r,i2=0.95, n 2=2 9g4iO.005, b=198, 141!:1 .28,

a=10.425i0.015, a2=0.770iO 005

/' ~-- ___ / ___ _~~}/~ ._ '~ 3 / '~~. / / // ~\_ ___ *\ / ,~ _.,. '[ t ' /

\ (:]*

i ~l t' L

/ / '\?? ~_ / 27 ~ (r'c ' , t~)")

/ // c~ ' 2 6 ~ I) l

l)1 '

/ r ( '} (l ~ ..~ -_ 2 4 " *1 ~ i.' --- In _- 7' Iv _~h , (~~~:] ~12() 30 75 50 25 o _25 _ 40 z \

50 50 75 _100 60

J"ig. 5. Refractive index profile representation 0L PML I (second trial fit). Dots are the experimental data. r (,umX1.5) represents the

radial coordinate and z (p mxl.5) the axial distance.

3.1.3 Analysis of both final fits for PML I Now we attempt to discriminate which is the best fit for

the PML. The first fit requires 7 coefflcients and the coefficients of determination of the final fits are better than

in the second one, which requires only 4 fit coefficients.

However, in order to choose the best profile we will analyze the physical meaning of the fit coefficients, and then their statistics in the final fit.

a) For the first:

no represents the maximum index value obtained with the diffusion process (relative standard error (RSE) of 0.17%).

b represernts the ellipticity of the fringes on the surface

(RSE of 4.5%). c/d represent the upper limit for the ellipticity of the

fringes (RSE of 12.5% for c and 19% for d). q represents the decreasing rate for the index with r and

z (RSE of 25%). b) For the second fit:

no represents the maximum index value obtained in the diffusion process (RSE of 0.17%).

a represents the mean center of the ellipses in the z axis

(RSE of 0.01%). a2 represents the mean ellipticity for the isoindicial

fringes (RSE of 0.005%). b represents the decreasing rate 0L the index with r and

z (RSE of 0.65%). Let us analyze the statistics of the fits. The coeflicient of

determination for the first final fit was rd2=0.98 and for the

second one rd2=0.95. The value of the residuals expressed in % did not exceed 2% for the first and 2.5% for the second.

Although both fits represented the experimental data accurately, the high value of the standard error of the coefficients in the first fit (up to 25% in some coefficients)

means that they have low significance. Thus, in this case we chose the second fit since the coefficients obtained were

determined with much bett~r accuracy. Therefore, when

E. ACOSTA et al.

using this method to characterize profiles or to relate the obtained fit pararneters to the fabrication conditions, it is

preferable to choose the fit with the smallest uncertainty in

its coeLicients.

Taking Eq. (14) into account it can be easily shown that the depth of the planar microlens obtained after diffusion, zmax' is given in terms of the fit coefficients by:

( 02 ) s l/2 n zma* = b~ log 2 - a ( 15) n

and the aperture radius by:

_ ~~lpg(~/Z~~~~~ 1/2 ( a where ns represents the refractive index of the substrate.

Therefore, the fit coeflicients are closely related to the

lens shape.

3.2 Microlens II The interference pattern consists of 14 equi-index

fringes. The substrate index was ns=1.53, the wavelength used in the interferometric measurements was ~ =0.633 ,clm, and the thickness of the PML was d=40 ,um.

The isoindicial fringes were fitted to a polynomial expression as in Eq. (2) and it was found that they could be represented with good accuracy by a farnily of centered ellipses.

In this case the expression of these ellipses is given by:

z2+(m2(k))2r2=(ml(k))2 V k=1,2 . . . (17)

We found that the ellipticity of the fringes, m2(k) remains practically constant with k and thus have used the mean value (m2)2=0.624~0.003. Finally, we fit coeffcients ml(k) as a function of nh in the form:

n2=n02exp(bml2) (18) and then, inserting Eq. (17) in Eq. (18), the following expression for the index profile was obtained:

n2=n02exp[b(z2+m22r2)] for n~ ns (19) where ns represents the substrate index, n02 the maximum index after diffusion, b is the gradient parameter in the axial direction, m2 the mean ellipticity of the equi-index

fringes and bm2 represents the gradient parameter in the radial direction.

Figure 6 represents the fit of Eq. (19) and the statistics of

the coefficients. We can see that for this microlens the index profile can be represented in terms of only three accurately determined coefficients.

Here, the depth of the lens is given by:

( :: )1/2 1 n zma*= ~ Iog n (20) and the aperture radius:

( :: 22 ) 1 n rmax= bm log n (21) The Gaussian profile is therefore a good approximation

for the refractive index distribution of PML's fabricated by

ion exchange technique. Furthermore, the fit coefficients can provide not only the maximum index after diffusion but also the microlens diffusion depth and aperture radius.

OPTICAL REVIEW Vol. 3, No. 4 (1996)

n exp[Lb(z2 + m2r2)] n2 = for n~ns r~=0.98, n02=2.754:!:0.001, b=-5. 12 10~5~3. I l0-8 m 2=0.624:1:O.003

/

*~

o

) )

,

,

l '

2 85

2.75

2.65

2.55

2.45

2.35

(~F{

15105

50 25 35302520 o _25 _50 _75 4540

Fig. 6. Refractive index profile representation of PML II. Dots are the experimental data. r (pmX4) represents the radial coordinate and z (/ImX4) the axial distance.

4. Additional Remarks

To analyze the possibility of using this representation for the index profile to evaluate some optical properties of

the microlenses, we performed ray tracing for this lens with a Gaussian distribution of the index as shown in Eq. (19) and compared it to the experimental data. The focal length obtained from ray tracing, measured from the point of maximum light intensity,8) was 2100 pm. With a 0.633 nm He-Ne laser we measured the focal length of the lens and obtained a value of 1970 /e m. The diarneter of the focal

spot size from ray tracing was about 20 p m and the experi-

mental one 15 pm. (Diffraction of light was not taken into account in the calculation).

On the other hand, we found that in computer calcula-tions for ray tracing about OO% of CPU time is saved when using Gaussian functions to represent the index profile 0L the PML instead of the usual matrix fit.

Since we L0und good agreement between ray tracing results and experimental datra, we think the method can

also

E. ACOSTA et al.

be useful to optimize the fabrication conditions.

231

5. Conclusions

We have proposed a method to fit the refractive index profile of planar microlenses fabricated by ion exchange techniques to analytical expressions resulting in a reduc-tion of the number of fit coefficients. In the experimental

results for two fabricated microlenses we found that 3 or 4 coefiicients are sufficient to fit the index shape with the

same accuracy as the 16 used in the matrix fit. We also discovered that the Gaussian profile (Green's function of the diffusion equation) is suitable to express the diffusion

profile of such microlenses. Although this profile is not a solution of the diffusion equation, it represents the experi-

mental data of the index with good accuracy. Optical properties calculated by ray tracing, on the other hand,

showed good agreement with the experimental data. Furthermore, the computing time for ray tracing purposes can also be significantly reduced. Since the PML can be represented by fewer pararneters, the change of the fit coefiicients under fabrication conditions such as mask radius, diffusion time, temperature and salt composition can also be determined, Ieading to an easier characteriza-tion of the fabrication process as well as a faster design of

optical components.

Ref erences

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3)

4)

5)

6)

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D.T. Moore: J. Opt. Soc. Am. 65 (1975) 451. K. Iga, Y. Kokubun and M. Oikawa: Fundamentals of Mi-crooptics, (Academic Press Inc., Tokyo, 1984) Chap. 2, Y. Kokubun and K. Iga: Appl. Opt. 21 (1982) 1030. Y. Kokubun and K. Iga: Appl. Opt. 19 (1980) 846. E. Acosta, M. Oikawa and K. Iga: Proc. of the 4th Microoptics

Conference and 1lth Topical Meeting on Gredient Index Q~,tical Systems, B2 (Jpn. Soc. Appl. Phys, 1993) p. 18.

7) P. Bevington: Data Reduetion and Error Analysis for the Physical Stiences (MCGraw Hill, 1969). X. Zhu and K. Iga: Appl. Opt. 27 (1988) 121. D.W. Marquard: J. Soc. Ind. Appl. Math. 2 (1963) 431.

8)

9)