本論文針對主題「微光學元件之製作與測量」作深入的探討。
在微直角錐反射鏡方面，我們對反射鏡所製做的路標其對比度會比 傳統的塗漆式路標其對比度要好上 1.64﹪~4.12﹪，因此可以證明 出微光學反射直角稜鏡的確優於傳統式的路標。在微透鏡方面，我 們完成微米（μm）級的微小光學元件的設計與製作，而微透鏡陣列 光點收斂的聚焦點直徑也略大於繞射極限。在其它微光學元件方面，
我們也製作出微光柵繞射元件以及各種微柱狀鏡，並且利用 Matlab 分析其各元件之粗糙度，包括以小波函數轉換，目的只為了加強我 們微光學元件的實用性。未來將朝非球面鏡製作，並研究如何設計 更準確地控制加工深度，及如何研磨微光學元件，使本系統朝向更 高解析度之製程邁進，以滿足將來更高品質之光學微系統需求。
本文中介紹的新穎 LIGA-like 微加工技術，目前已整合在 雷射光刻電鑄及模造技術中，近年來準分子雷射加工技術，由於其 相較於一般蝕刻技術擁有較精確和較高深寬比（aspect ratio）的 特性，且相較於電子束和 X-ray 光刻（lithography）技術在成本 效益的考量上便宜許多，已廣泛地應用在許多微小結構加工成型與 微細裝置光刻上。以準分子雷射作為微光學陣列元件之製程技術，
既快速又準確，足以取代現行之標準半導體製程模式，而且更短波
長（193nm）之準分子雷射加工系統，目前也純熟地不斷被開發出來，
未來更具彈性之微光學設計製作環境逐漸地實現。
此外，未來在 DVD 讀寫頭（picked up head）、雷射二極體陣列
（diode laser arrays）、偵測器陣列（detector arrays）、光計算
（optical computing）、光連接（optical interconnection）與光 通訊（optical communication）等等應用或結合平面光學
（planar-optics）與固態雷射在微光電系統上將有更彈性之設計空 間與應用領域等待開發，尤其在消費性光電子產品中的 CGH（computer generated holograms）或 DOE（diffractive optical elements）
更迫切地需要以塑膠射出、壓克力壓膜、樹脂翻製等大量製造微光 學元件與系統組件時，準份子雷射微加工製程方式提供了迅速、彈 性與廉價等優勢。
展望未來，世界即將進入所謂「寬頻網路革命」的時代，就是 整合網際網路、電信以及有線電視，透過光纖網路高速傳輸大量影 音資訊。在此同時，當前主流的矽半導體積體化已經瀕臨極限，光 積體電路可望取而代之，未來光積體電路應用在光纖通訊以及光學 信號傳輸之後，速度、效率與傳輸量將遠超過目前的晶片技術，其 中微光學技術的重要性不言可喻，因此下一個世紀將是光電產業大 放異彩的新世紀。
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1 2 ,..., 2 , 1 ,
0 −
= p
k
附錄
附錄（一）：小波函數之理論
1. THE WAVELET AND DISCRETE WAVELET TRANSFORM
To define the Daubechies wavelets dbp(p >0), consider the two functions, )
φ(x and ψ(x)appearing as solution to the equations
( ) (2 )
1 2
0
k x c x
p
k
k −
=
∑
−=
φ
φ ,
(2)
∑
−=
−
=2 1
0
) 2 ( )
(
p
k
k x k
d
x ψ
ψ ,
(3) with
∫
φ(x)dx =1,∫
ψ( dxx) =0.Let
, (x) 22 (2jx k),
j k
j = φ −
φ , (x) 22 (2jx k),
j k
j = ψ −
ψ
where j,k∈Zdenote the dilation and translation of the scaling function (φj,k(x)) and the wavelet (ψj,k(x)) respectively.The set, {c }(scaling coefficient) andk
{d }(wavelet coefficient), are related ask dk =(−1)kc2p−1−kfor ， )
φ(x and ψ(x) form an orthonomal basis on L2(R)and
<φj,k(x),φj,m(x)>=δk,m, (4)
<ψj,k(x),ψj,m(x)>=δk,m, (5)
whereδ is the Kronecker delta function , k ,m ψ(x)=ψ0,0(x),satisfies
∀m∈[0,1,2,...,p−1],
∫
ψ(x)xmdx=0. (6)It is usual to let the spaces spanned by φ(x)and ψ(x) over the parameter k , with j fixed, be denoted by V (coarse space) and j W (detail space), i.e.,j
Vj =span{φj,k(x)}k∈Z, Wj =span{ψj,k(x)}k∈Z.
These space, V and j W are related asj, … … … … . ⊂V−1 ⊂V0 ⊂V1 ⊂..., and
Vj =Vj−1⊕Wj−1, <φj−1,k(x),ψj−1,m(x)>=δk,m , (7)
i.e., the Wj−1 is the orthogonal complement of Vj−1 in V .Utilizing orthonormalityj
of the wavelets ,ψj,k(x),we obtain the important statement13,14
L2(R)=⊕Wj,j∈Z,
i.e., the wavelet basis is complete . Hence, any f(x) ∈L2(R) can be written as
f(x) v , j,k(x),
Z
j k Z
k
j ψ
∑∑
∈ ∈= (8)
with the set of expansion coefficients , {vj,k}, appearing as a result of orthogonality
>
=<
=
∫
( ) , ( ) ( ), , ( ), f x x dx f x x
vjk ψjk ψjk .
(9)
Naturally, infinite sums and integrals are meaningless when one begins to implement a wavelet expansion on a computer and we must limit the range of the scale parameter
j and the location k .
In practice, in the DWT, there is a limit to how small the smallest structure can be depending on, e.g., the numerical grid resolution or the sampling frequency in a signal processing. Hence, an approximation would be constructed in a finite space such as
0 0 2
1 W ... W V W
VJ = J− ⊕ J− ⊕ ⊕ ⊕ ,
(10)
Eq.(9) can be written as
∑ ∑ ∑∑
−=
>
<
+
>
<
=
>
<
=
k k
J
j k
k j k
j k
k k
J k
J
J f x f x x x f x x x f x x x
P
1
0
, ,
, 0 ,
0 ,
, ( ) ( ) ( ), ( ) ( ) ( ), ( ) ( )
), ( )
( φ φ φ φ ψ ψ
(11)
2. WAVE-FRONT REPRESENTATION WITH DISCRETE WAVELET TRANSFORM
In this paper, the one-dimension DWT were developed as a convenient set for representing wave-front aberrations over a circular pupil. Further they are easily related to the wave-front aberrations.
We will show how the orthogonality of the one-dimension scaling function and wavelet that simplifies fitting the aberration polynomial to measured data.
The scaling function and wavelet are complete, this means that any function (aberration wave-front) in x direction with the approximation on a computer can be expressed as a linear combination of as follows:
=
∑
=∑
< >k
k j k
j k
j k
k
jW x u x W x x x
P ( ) φ, ( ) ( ),φ, ( ) φ, ( ). (12)
By taking one step decomposition13:
Vj(x)=Vj−1(x)⊕Wj−1(x), i.e.,
PW(x) u , (x) v 1, (x) w j 1,k(x)
k k
k k
j k k
j k
k
j =
∑
φ =∑
φ− +∑
ψ − .(13)
Consider aberration function expanded in the basis φj,k(x) (at scale j ), the
sample points of function are given as a vector:
u =(u0,u1,...,uN−1), N =2j.
From Eq. (14), we obtain:
( ) ( ) ( )
1 2
0
, 1 ,
1 1
0
1 2
0
, x v x w x
u
N
k
k j k k
j N
k
N
k k k
j
k
∑
∑ ∑
−
= −
−
−
=
−
=
+
= φ ψ
φ ,
(14)
from Eqs. (3)、 (4)、 (5) and (6), we obtain:
=
∑
< − >=∑
+k
n k k n
j k j k
k
n u x x cu
v , 1, 2
2 ) 1 ( ), ( φ
φ ,
(15)
=
∑
< − >=∑
+k
n k k n
j k j k
k
n u x x d u
w , 1, 2
2 ) 1 ( ), ( ψ
φ .
(16)
i.e., the aberration signal, W(x), passes through two complementary filters (lowpass and highpass) and emerges as two signal15,16, Fig 1 shows one-stage filtering. We consider the defocus aberration 、 astigmatism and 3rd spherical aberration functionW(x,y) = A(x2 +y2)2+C(x2+3y2)+D(x2+ y2)only in x or y direction.
From Eq. (15), we obtain:
( ) ( ) ( ) ( )
1 2
0
, 1 ,
1 1
0
1 2
0
, x v x w x
u x
W P
N
k
k j x k k
j N
k
N
k x k k
j x k
j
∑ ∑ ∑
−
= −
−
−
=
−
=
+
=
= φ φ ψ
(17)
( ) ( ) ( ) ( )
1 2
0
, 1 ,
1 1
0
1 2
0
, y v y w y
u y
W P
N
k
k j y k k
j N
k
N
k y k k
j y k
j
∑ ∑ ∑
−
= −
−
−
=
−
=
+
=
= ϕ φ ψ
(18) with
wkx =<W(x),ψj−1,k(x)>= A< x4,ψj−1,k(x)> +(C+D)< x2,ψj−1,k(x) >, (19)
>
<
+ +
>
<
>=
=<W(y), −1, (y) A y4, −1, (y) (3C D) y2, −1, (y)
wky ψj k ψj k ψj k .
(20)
According to Daubechies waveletsdb3、db and Eq. (7), the coefficient A2 、C and D can be calculated, which are given as:
= < >
− ( ) , 1,
4 x
x A w
k j x
k ψ , = < >
− ( ) , 1,
4 y
y A w
k j y
k ψ , calculated by
db wavelet,3
+ = − < >< >
−
−
) ( , )
( ,
, 1 2 ,
1 4
x x
x x
A D w
C
k j k
j x
k ψ ψ
, calculated bydb wavelet,2
+ = − < >< >
−
−
) ( , )
( 3 ,
, 1 2 ,
1 4
y y
y y
A D w
C
k j k
j y
k ψ ψ
, calculated bydb wavelet.2
3. NUMERICAL SIMULATION WITH SPECKLE NOISE
To test the performance of the DWT method and to compare it with conventional method, we conducted a numerical simulation with speckle noise. In this simulation, an interferogram is given with the following primary aberration terms:
(1). spherical aberration contributes 3λ.
(2) astigmatism contributes 4λ.
(3) defocusing contributes 5λ.
For the case of aberration function this expression was given by:
W(x,y)=−3(x2 +y2)2 +4(x2+3y2)−5(x2+ y2)+noise, (21)
the interferogram fringe patterns G(x,y)are of the form:
G(x,y) =1+cos{2π[−3(x2+ y2)2 +4(x2+3y2)−5(x2+ y2)+noise]}. (22)
This interferogram is digitized over a 128*128 point grid of the circular aperture, as shown in Fig2. We consider the aberration function only in x or y direction. By limiting to one direction, the phase variation in the test section can be written as:
cos−1[G(x,0)−1)]=2π(−3x4 −x2)+noisex (23)
cos−1[G(0,y)−1)]=2π(−3y4+7y2)+noisey (24)
The simulation results of the DWT method and the least-squares matrix inversion method in x and y direction are shown in Table 1, The computer simulation results of the aberration coefficient are shown in Table 2.
The polynomial fit of interferogram in x and y direction are shown in figures 3 and 4 by using the least-squares matrix inversion method and DWT method respectively.
Compared with the conventional method (least-squares), the DWT method finds the solution quickly and accurately.
4. CONCLUSION
There are some sources of errors in the polynomial fit, such as fit error, digitization error, round off error, and finite sampling error. The round off error becomes negligible if orthonormal polynomials are used, the coefficients of the orthonormal polynomials have less error than those of the nonorthonormal polynomials.
Zernike polynomials are not orthogonal over the digitized points for the undersampled or nonuniformly digitized circular aperture interferograms and are not orthogonal over a non- circular aperture.
There are several numerical techniques to solve for the value of Zernike coefficients, the least-squares matrix inversion method and the Gram-Schmidt method would become ill-conditioned17 due to an improper data sampling. The coefficients determined through the least-squares fit depend on the order of the polynomial expansion.
In this paper, we present the one-dimension (DWT) technique to find primary aberration coefficient. The method offers great improvement in the accuracy and calculating speed of the fitting aberration coefficients better than the least-squares matrix inversion method and the Gram-Schmidt orthogonalization method.
Furthermore, the result of solving aberration coefficients through the one-dimension (DWT) is independent of the order of the polynomial expansion. So we can find a true value from the datum of fitting.
5. ACKNOWLEDGEMENT
This work has been founded by National Science Council, Taiwan, R.O.C. under project grants NSC87-2215-E-008-016、project NSC87-2314-B-008-004-M01.