Organization of this thesis

Chapter 1 Introduction …

1.5 Organization of this thesis

This thesis is organized as following: The principle of diffraction theory is presented in Chapter 2. In Chapter 3, the calculation and the results of the etch-holes array are presented. The processes to fabricate a microreflector which is developed to verify the calculated results are described in Chapter 4. The experimental results and the discussion will be shown in Chapter 5. Finally, the summary of this thesis and future works will be presented in Chapter 6.

Chapter 2

Principle

2.1 Huygens-Fresnel Principle [11]

The Huygens-Fresnel principle describes that every point on the primary wavefront is considered as a source of secondary spherical wavelets, with the same frequency and velocity as the primary wave. The amplitude of the field at any point is the superposition of all these wavelets, taking account of their amplitudes and phases, that is, the amplitudes and phases of secondary wavelets could interfere with each other. The principle is illustrated in Fig. 2.1.

Primary wavefront

Envelope of secondary wavelets

Secondary wavelets

Fig. 2.1 Hygens’ wavefront construction

2.2 Scalar Diffraction Theory

Maxwell’s equation is the beginning for our analysis. Assume that there are no sources present in a dielectric medium which is linear, homogeneous, and isotropic.

The equations are written as

E H where ε is the permittivity and µ is the permeability of the medium. The vector

identity is

(

) ( )

^E ² ^.

∇× ∇×JG = ∇ ∇ ⋅JG − ∇ EJG

(2.5)

Substituting the Maxwell’s equations for E and H, into (2.5), we can obtain two equations, vacuum permittivity. The symbol c mean the velocity of propagation in vacuum.

The components of the electric and magnetic field behave identically and can be described by a single scalar wave equation,

2 2

where u(x, y, z, t) represents any of the scalar field components, and t is a time variable. Assuming a time-harmonic steady-state solution in which the partial time derivative can be replaced by j ω, the Helmholtz equation can be yield as

(

^{∇ +}² ^{k U}²

)

⁼⁰^(2.9)

here U= u(x, y, z, t), and k is the wave number, given by ² n ,

k c

π ω

= λ = where ω is the angular frequency , and λ is the wavelength in the dielectric medium. From Green’s theorem and under some assumptions and derivations, we can obtain the Fresnel-Kirchhoff diffraction formula,

U’(P1) represents the secondary sources with its amplitudes and phases. It can be explain as the spherical wave illuminating from a point source located at P2 to the point P1 on the aperture and weighted its own related obliquity factor, which describes a contribution of angle relation to U(P0). The relationship in space shows in Fig. 2.2, where Σ means the size of the aperture.

Fig. 2.2 Point-source illumination of a plane aperture

p

₂

r G

¹²

r G

n G

p

₀

p

Σ

In general case, r01 and r21 are extremely greater than the size of the aperture, so we assume the obliquity factor [cos( ,n rG G01) cos( ,− n rG G21)] / 2

will be roughly constant,

~1. The Fresnel-Kirchhoff diffraction formula can be rewritten as

( ) ( ) (

)

0 1

1 exp jkr

U P U P ds

jλ _∑ r

∫∫

^(2.12)

where U(P1) means a spherical wave on the aperture diverging from a point source at P2, and is defined as U P

( )

1 = ⎣⎡Aexp

(

jkr21

)

/r²¹⎤⎦. It express the observed field U(P0) as a superposition of diverging spherical wave exp

(

jkr01

)

/r originating from 01

secondary sources located at each and every point P1 within the aperture Σ.

2.3 Fresnel and Fraunhofer Diffraction

2.3.1 Fresnel Diffraction

Before introducing a series of approximations to the scalar diffraction theory, it will be in more explicit form for the case of rectangular coordinates. Fig. 2.3 shows the relation of the object plane and the observed plane on rectangular coordinates in space. According to Eq. (2.12), we can rewrite it as

( ) ( ) (

)

Fig. 2.3 Relation of the scale diffraction theory

By some approximation, the r01 can be replaced by

( ) (

)

The resulting expression for the field at (x,y) therefore becomes

Eq. (2.16) is readily seen to be a convolution, and expressed in the more clear form

where h can be seen as the impulse response in the optical system, and defined as

( )

^, ^e^jkz ^exp ₂^jk

(

² ²

h x y x y

jkz ^⎡ z

)

^⎤^.

= ⎢⎣ + ⎥⎦ (2.17)

In fact, h(x,y) provide a parabolic phase weight. We can explain Eq. (2.16) that the diffraction field U(x,y) is the field U(s,t) taken convolution with a parabolic pahse exponential, the impulse response of a free space system. Moreover, the other form of Eq. (2.16) is found:

which we recognize to be the Fourier transform of the product of the complex field just to the right of the aperture and a parabolic phase exponential.

2.3.2 Fraunhofer Diffraction

Referring to Eq. (2.16) and (2.19) as the Fresnel diffraction formula, which is valid in the near field of the object plane. For the far field diffraction, the condition will be met if the distance z satisfies

(

² ²

)

_max

2 . k s t

z +

(2.19)

Eq. (2.19) can be further simplified as

( )

^, ^e^jkz ^j²^k^z⁽^x² ^y²⁾

( )

^{, e} ^j²^z⁽^{xs yt}⁾

which is well known as the Fraunhofer diffraction formula. It relates the diffraction field to the object field by Fourier transform. We note that Fourier transform remains valid for Fresnel and Fraunhofer diffraction. It means that we can calculate the diffracted fields in the fraunhofer region with accuracy of the Fresnel approximation.

Fig. 2.4 shows the relation of the two approximations along the optical axis with the different distances from an aperture.

Coherent illumination

Fresnel region

Fraunhofer region

Fig. 2.4 The relation of the two approximations along the optical axis

2.4 Analysis of Etch-holes Array with Babinet’s Principle [12]

The Babinet’s principle can be used to analyzing the diffraction of the etch-holes array on MEMS devices. We can assume the field outputted from a mirror as a 2-dimension grating with periodic. This reflected field is the same with a transmitted field. It is more convenience for us to create the model for different kinds of devices, either reflection type or transmission type.

From the Fresnel-Kirchhoff diffraction formula, the Babinet’s principle can be developed. Two masks like those of Fig. 2.5, in which clear and opaque regions are simply reversed, are called complementary apertures. If one of the mask, say A, and then the other, B, are put into place and the amplitude at some point of the screen is determined for each, the sum of these amplitudes must equal the unobstructed amplitude there. The diffracted fields of A and B can be expressed as

A B u,

E +E =E (2.20)

with A and B representing any two complementary apertures. An interesting special case of Babinet’s principle is a point where Eu=0. Then, by Eq. (2.22),

A B

E = −E (2.20)

and

A B

I = (2.21) I

at the point.

＋＝

A B 1

Fig. 2.5 Two complementary aperture

In practice, Fresnel diffraction dose not produce amplitudes Eu=0 without an aperture. Fraunhofer diffraction does in the case of the pattern formed by a point source and a lens. For the region outside the small Airy disc, Eu is zero, essentially.

Thus positive and negative transparencies of the same pattern produce the same diffraction pattern in Fraunhofer diffraction.

Chapter 3

Simulation

3.1 Introduction

As we described in Section 1.2, the etch holes are inevitable for free space features under surface micromachining technologies. Chapter 3 describes the analysis of diffraction from etch-holes array. The array with regular spacing, the standard case in MEMS design rule, will be discussed first. Moreover, the random distribution and circular distribution are then proposed for different purposes.

In the typical design, the sizes of each elements (d) and the distances between the holes (s) should be kept in a range of d= 2~5μm and s= 20~30μm, respectively Smaller size of etch holes will cause the etchant can’t immediately permeate the sacrificial layer under the microstructures. Longer distances between the holes will cause incomplete release, whereas closer distances will cause the film to crack easily.

We can arrange different layouts from the traditional case in these design rules.

3.2 Diffraction from Etch-holes Array with Regular Spacing

The diffraction caused by a two-dimensional (2D) etch-holes array is analogous to the case in which a uniform plane wave is diffracted by a crossed grating. In this case, the features on the perforated film are treated as scalar amplitude objects. This means that the incident beam is simply modulated by a complex transmission (or reflection) coefficient introduced by the properties of microstructure with etch holes array. Here we ignore the physics of interaction between the etch hole and the

incident beam in the conformity with the relief pattern and assume aperture part merely imports to the incident beam an amplitude modulation. Despite the fact that solving the Maxwell equation is most exact way for diffraction analysis, approximating etch holes as amplitude objects holds well for our case where typical aperture sizes of MEMS structure are much larger than the working wavelength. In addition, we can avoid entailed massive computations and get a fast calculation. The wavelength λ for the simulation is 633 nm. However, as long as two conditions are met (1) the diffracting etch hole must large compared with wavelength, and (2) the diffracting fields must not be observed too close to the aperture [11], the calculated values can be applied to different wavelengths by scaling all the geometry parameters involved proportionally. Fig. 3.1 illustrates the transmission function B(x,y) of a perforated die and geometric parameters for diffraction calculation. If the overall size of the die is limited to c ×n, etch holes have a size of a ×l and are spaced b and m apart in the x and y directions, respectively, then the transmission function of the etch-holes array can be expressed as

( , ) ( )x ( )x ( )x ( )y (y) ( ),

B x y rect comb rect rect comb rect

a b c l m

⎡ ⎤ ⎡ ⎤

=⎢⎣ ⊗ ⎥⎦ ×⎢⎣ ⊗ ⎥⎦

n (3.1)

where rect(x/a) is a rectangular function of width a. The function for such an aperture is defined as

and the comb function is defined as

( ) ( )

Note that etch-holes array with regular spacing can be generated by convolving the rectangular function rect(x) with comb(x).

x y

Fig. 3.1 The transmission function B(x,y)

For a typical MEMS device where propagation distance z is much larger compared with x’ and y’ (the axes on image plane), the diffraction by 2D etch holes array can be estimated using Fraunhofer diffraction formula, by which the diffracted pattern is obtained as the Fourier transform of the transmission function, that is F{B(x)}= E(fx, fy), where the argument fx = sin θx/ λ ≈ x’/ λz and fy = sin θy/ λ ≈ y’/ λz, as shown in Fig. 3.2. The Fourier transform of the comb function is

{

^{( , )}

}

⁽ x^{, ).}¹

F comb x b b comb f b

∞

=−∞

∑

^(3.4)

Fig. 3.2 Geometric parameters of diffraction calculation

Consequently, we can obtain the amplitude distribution of the wave diffracted through the perforated film

{ }

F rect comb rect rect comb rect

a b c l m n

asinc af csinc cf comb f lsinc lf nsinc nf comb f

b m

where Nx and Ny are the number of etch holes in the x and y directions, respectively.

Fig. 3.3 shows the irradiance distribution of a square etch-holes array with a = l = 5µm, b = m = 20 µm and c = n = 200 µm. Because the order of the dimension in the transmission pattern is a < b < c, the order of the diffracted pattern which is the Fourier transform of the etch holes array has an inverse relationship; namely,

1 1 1

c< <b a (3.6)

where 1/c is the size of an individual spot, 1/b is the spacing between each spikes, and 1/a is the overall size of the diffraction pattern [13]. When the ratio of the aperture width a and the period b is an integer, the minimum of the envelope will exactly coincide with some of the spikes of the array pattern. In this case, b/a = 4, the fourth spikes overlaps with the first null of side lobe and the intensity of the fourth spike thus disappears. Since the intensity distribution is obtained as

( _x, _y) *( _x, _y) ( _x, _y)

I f f =E f f ⋅E f f (3.7)

accordingly, we can calculate the intensity

which can be verified through the serial expansion

1 (

Fig. 3.3 (a) The membrane perforated by a square etch-holes array, (b) irradiance cross-section, where the order of the pattern has an inverse relationship of the etch-holes layout.

It can be seen from Eq. (3.8) that the diffracted pattern is contributed by two factors of different natures. The factors outside the square brackets are determined solely by the shape of aperture element and are called the shape factor, which appears as the envelope of the diffraction pattern. On the other hand, the factors inside the square brackets are determined solely by the period and die size of etch holes array, are called the array factor. The diffracted pattern is the product of these two factors.

Sometimes the actual shape of etch holes after plasma etching tends to be circular rather than square due to imperfect photolithography and etching process. In this case, sinc function caused by the rectangular element in Eq. (3.5) can be substituted by the Bessel function, which is the Fourier transform of the circular aperture. Similarly, when the surface area of a microstructure is relatively larger than the illuminated spot size, the overall size is determined by the spot size rather than the external border of the device. Therefore, sinc function caused by the rectangular die should be replaced by the Bessel function as well.

3.3 Diffraction from Etch-holes Array with Random Distribution

According to the preceding study, we note that periodicity of the etch holes array leads to considerable amount of high order beams. In the above case, the normalized intensity of the lowest few orders are 1 [0th order], 0.8113[(1,0), (0,1) order], 0.6582 [(1,1) order], 0.4072 [(2,0), (0,2) order]. Such high order diffraction beams are often undesirable to the optical performance by generating noise and erroneous crosstalk signals in most optical systems. Let us examine an etch holes layout, where etch holes are randomly arranged but without any rotation of the individual elements, as shown in Fig. 3 4 (a), where the dash lines represent the original position of each etch hole and solid lines represent the random offset from the

original one. For simplicity, we ignore the external border and only consider one dimensional case. If each square Aperture is translated in the x direction by a random distance bk from its original position, then the transmission function can be represented by

where the translated distance bk are generated by the computer according to discrete uniform distribution [14]. Using the shift property of the Fourier transform,

{

⁽ k⁾

}

^j² ^{f b}^{x k} ⁽

F h x b± =e^∓ ^π H f_x).

(3.11)

The amplitude distribution in far field can be expressed by

(3.12)

since bj and bk are random in this case, the translation corresponds to a superposition of cosine functions with a random period, which leads the interference term in Eq. (16) to zero. Such analysis can be extended to 2D case and the intensity of the diffracted wave thus becomes NxNysinc²(afx)sinc²(lfx), as shown in Fig. 3.4 (b), where the grid pattern due to the array factor is average out and the diffraction pattern is solely determined by the form factor of each element but with N times the intensity.

x y

Fig. 3.4 (a) Geometric parameters of the translated etch-holes layout, dash lines represent the original position and solid lines represent the random offset by discrete uniform distribution, and its (b) calculated irradiance.

(a) (b)

Although random spacing is an effective approach to eliminate the high order diffracted beams and thus alleviate the crosstalk effect, one more requirement should be noted when failure mechanisms are taken into consideration [15].Because typical surface micromachining uses polysilicon thin film as the structure layer and the silicon oxide thin film as the sacrificial layer. If the spacing of etch holes on the polysilicon layer is too far, the etchant is hard to access the center of the large area micromechanical structure, thereby failing to release the component. On the contrary, while the spacing is too close, the stress concentration is accumulated around the etch holes and the polysilicon film is easily cracked during the fabrication or assembly process. Therefore, one more condition is needed that the spacing between each etch hole bk should be limited in the range of 20±5 µm in this case. We can further randomize each aperture element with respect to rotation angle θ as well as translation, as shown in Fig. 3.5 (a). Since the form factor of etch holes has been randomized with center symmetry, the calculated irradiance in Fig. 3.5 (b) looks like one obtained by rotating the diffracted field of a single square aperture about it’s center.

x y

Fig. 3.5 (a) Geometric parameters of the translated etch-holes layout, dash lines represent the original position and solid lines represent the position after random translation and rotation, and its (b) calculated irradiance.

(a) (b)

3.4 Diffraction from etch holes array with concentric layout

Since most optical elements have a circular aperture, compare with rectangular grid pattern, it is a more natural arrangement to discretize a circular boundary by a grid of concentric circles shown in Fig. 3.6. To fit the geometric symmetry, here we design circular holes, and let the rectangular grid be replaced by a family of concentric circles with radius rm given by

rm =md m=0,1, 2,…, (3.14)

with d the spacing. If we design the same spacing d between adjacent elements along any circle, then

2πr_m=2m dπ =N_md (3.15)

In which Nm is the number of elements on the m-th ring. It is clear that the values of Nm that satisfy Eq. (3.15) are not integers, but one can round the results to the nearest integer. Taking the typical case, R = 333μm and d = 40μm, for example, there are 4 concentric circles, and the numbers of etching holes per ring are given in Table-3.1, under the restriction that Nm be divided by 6, so that there exists a quadratic symmetry. Here the factor θm means the angle of the sector which separate one element from the other one corresponding to the origin on m-th ring, and it is clear that θm is different magnitude for different m. This gives 61 elements in the array, exactly the same density as were used in the rectangular grid array. Therefore, the circular grid pattern would not cause any difficulty in the consideration of wet etching process.

Table 3.1 the geometric parameters of each ring on the concentric array

Fig. 3.6 (a) The concentric array, (b) 1/6 circle of the layout with its geometric parameters

(a) (b)

To describe the diffraction patterns of the circular multiaperture systems, a geometric analysis was performed for the concentric rings of identical circular etching holes. The transmittance t(r) can be written as the convolution (⊗ ) of the transmittance of the unit circular element circ(r/a) with an array of delta functions δ(r), which describe the position of each etching holes.

( ) (

where a is the radius of each subapertures, and M is the maximum ring numbers within a given pupil size. The diffraction pattern can be analyzed from the transmittance function illuminated by a normally incident laser beam. Based on Fraunhofer diffraction theory, the analysis is consisted of examining the far-field diffraction patterns produced by imaging a point source through multiaperture system.

The following expression is used to calculate the diffraction patterns propagating a distance z

This is the impulse response, where represents the scaled coordinates in the image plane, and (f

) , (s t

s, ft) are their spatial frequency. The symbol ρ represent the radial spatial frequency and can be replaced by . Excluding nonessential proportionality factors, it is proportional to the Fourier transform of the transmittance of the exit pupil, which is formed by the combination of each exponential component

with a 1st Bessel function as the envelope. We can also get its intensity distribution from I=h*h. Fig. 3.7 illustrates the relation of the symbols between the object plane and the image plane in free space. Fig. 3.7 The relationship between the object plane and the

image plane in free space in the polar coordinates

The purpose of this study was to design a circular etching holes layout with transmission (or reflectance) that exhibit the central beam widths and side picks irradiances less than that of the conventional rectangular arrangement. The amplitude function can be expanded as the superposition of the diffraction patterns caused by each concentric circle grids. It is noted that the complex field amplitudes generated by each ring is determined by the argument of each exponential.

The normalized irradiance distribution of this symmetric circular array we calculated is shown in Fig. 3.8. It is seen to consist of a main beam additional to a family of side picks. In order to quantify the results, we define the diffraction efficiency asE =I_max / I₀, where I0 and Imax are the intensity of the central main beam and the high order beam with the maximum magnitude, respectively. From the results,

在文檔中自由空間微光機電元件蝕刻孔陣列之光學繞射分析 (頁 15-0)

Chapter 1 Introduction …

1.5 Organization of this thesis

Chapter 2

Principle

(

) ( )

(

)

p

r G

r G

n G

p

p

Σ

( ) ( ) (

)

∫∫

( )

(

)

(

)

( ) ( ) (

)

( ) (

)

( )

(

)

(

)

( )

( )

＋ ＝

Chapter 3

Simulation

{

}

∑

{ }

{

}

( ) (

＋＝