Файл: Ersoy O.K. Diffraction, Fourier optics, and imaging (Wiley, 2006)(ISBN 0471238163)(427s) PEo .pdf

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THE FILTERED-BACKPROJECTION ALGORITHM

 

 

 

337

The discretized impulse response at t ¼ n t is given by

 

 

 

 

 

 

8

0

 

n even

 

 

 

 

 

 

>

F02

 

n ¼ 0

 

 

 

 

 

 

 

4F

0

 

 

 

 

ð

 

Þ ¼

<

 

 

 

ð

 

Þ

h

n t

 

>

 

 

 

 

 

18:8-12

 

 

 

 

>

 

 

 

 

 

 

 

 

 

 

>

 

 

 

 

 

 

 

 

 

 

:

 

 

 

 

 

 

 

 

 

 

>

n2p2

n odd

 

 

 

Equation (18.4-5) as a convolution in the time-domain is given by

 

 

 

 

 

 

1

 

 

 

 

 

 

 

^pðuÞ ¼

ð

pðtÞhðu tÞdt

ð18:8-13Þ

 

 

1

 

 

 

 

 

 

 

Its discrete version is given by

X

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

^pðn Þ ’

1

pð mÞhððn mÞ Þ

ð18:8-14Þ

 

 

 

m¼ 1

 

 

 

 

 

Since pð tmÞ is zero for jmj > N2,

 

 

 

 

 

 

 

 

 

¼X

 

 

 

 

 

 

 

 

 

N=2

pð tmÞhððn mÞ tÞ

ð18:8-15Þ

^pðn tÞ ’ t

 

 

 

m

N=2

 

 

 

 

 

This linear convolution can now be computed with DFTs of size 2N after zeropadding both pð Þ and hð Þ. Further improvement is possible by properly windowing the frequency domain results. The whole procedure can be written as follows:

1. Zero-pad pð Þ and hð Þ in the form

 

 

8

pðnÞ

0 n

 

N

 

 

 

 

 

2

 

 

 

 

 

 

>

 

 

 

 

 

 

 

 

 

 

 

 

 

 

>

 

 

 

 

 

 

 

 

 

 

 

 

 

 

>

 

 

N

 

 

3N

 

 

 

 

 

<

 

 

 

 

 

 

 

ð

Þ ¼

>

 

 

 

 

ð

 

Þ

>

 

 

2

 

 

 

2

 

 

p n

 

>

0

 

 

< n

<

 

 

 

 

18:8-16

 

 

 

>

 

 

 

n > 3N

 

 

 

 

 

>

 

 

 

 

 

 

 

 

>

 

 

 

 

 

 

 

 

>

 

2N

 

 

 

 

 

 

> p n

 

 

 

 

 

 

:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

> ð

 

Þ

 

 

2

 

 

 

 

 

 

and similarly for hð Þ.

2.Compute the size 2N DFTs of pð Þ and hð Þ.

3.Do the transform domain operations.

4.Window the results of step 3 by a proper window.

5.Compute the size 2N inverse DFT.

19

Dense Wavelength Division Multiplexing

19.1INTRODUCTION

In recent years, optical Fourier techniques have found major applications in optical communications and networking. One such application area is arrayed waveguide grating (AWG) technology used in dense wavelength division multiplexing

(DWDM) systems. DWDM provides a new direction for solving capacity and flexibility problems in optical communications and networking. It offers a very large transmission capacity and novel network architectures [Brackett, 90], [Brackett, 93]. Major components in DWDM systems are the wavelength multiplexers and demultiplexers. Commercially available components are based on fiber-optic or microoptic techniques [Pennings, 1995], [Pennings, 1996].

Research on integrated-optic (de)multiplexers has increasingly been focused on grating-based and phased-array-based (PHASAR) devices (also called arrayed waveguide gratings) [Laude, 1993], [Smit, 1988]. Both are imaging devices, that is, they image the field of an input waveguide onto an array of output waveguides in a dispersive way. In grating-based devices, a vertically etched reflection grating provides the focusing and dispersive properties required for demultiplexing. In phased-array-based devices, these properties are provided by an array of waveguides, whose lengths are chosen to satisfy the required imaging and dispersive properties.

As phased-array-based devices are realized in conventional waveguide technology and do not require the vertical etching step needed in grating-based devices, they appear to be more robust and fabrication tolerant. The first devices operating at short wavelengths were reported by Vellekoop and Smit [1989] [Verbeek and Smit, 1995]. Takahashi et al. [1990] reported the first devices operating in the long wavelength window around 1.6 micron. Dragone [1991] extended the phased-array concept from 1 N to N N devices.

This chapter consists of six sections. The principle of arrayed waveguide grating is discussed in Section 19.2. A major issue is that the number of available channels is limited due to the fact that each focused beam at a particular wavelength repeats at periodic locations. Section 19.3 discusses a method called method of irregularly sampled zero-crossings (MISZC) developed to significantly reduce this problem.

Diffraction, Fourier Optics and Imaging, by Okan K. Ersoy

Copyright # 2007 John Wiley & Sons, Inc.

338

ARRAY WAVEGUIDE GRATING

339

Section 4 provides detailed computer simulations. Section 19.4 provides an analysis of the properties of the MISZC. Computer simulations with the method in 2-D and 3-D are described in Section 19.5. Implementational issues in 2-D and 3-D are covered in Section 19.6.

19.2ARRAY WAVEGUIDE GRATING

The AWG-based multiplexers and demultiplexers are essentially the same. Depending on the direction of light wave propagation, the device can be used as either a multiplexer or a demultiplexer due to the reciprocity principle. For the sake of simplicity, the demultiplexer operation is discussed here.

The AWG consists of two arrays of input/output waveguides, two focusing slab regions, and the array grating waveguides. This is illustrated in Figure 19.1. A single fiber containing the multiwavelength input is connected to the array of input waveguides. The input multiwavelength signal is evenly split among the input waveguides, and the signal propagates through the input waveguides to reach the input focusing slab region. The light wave travels through the focusing slab and is coupled into the array grating waveguides. A linear phase shift occurs in the light wave traveling through each array grating waveguide due to the path length differences between the array grating waveguides.

The light wave is subsequently coupled into the output focusing slab, in which the multiwavelength input signal is split into different beams according to their wavelengths due to diffraction and wavelength dispersion. The length of the array waveguides and the path length difference L between two adjacent waveguides are chosen in such a way that the phase retardation for the light wave of the center wavelength

Figure 19.1. Schematic for the arrayed waveguide grating device [Courtesy of Okamoto].

340

DENSE WAVELENGTH DIVISION MULTIPLEXING

passing through every array waveguide is 2pm, m being the diffraction order equal to an integer. The phase retardations of the light waves of wavelengths other than the center wavelength are different from the phase retardation of the center wavelength. As a result, a unique phase front is created for each wavelength component which is then focused to a different position on the output side of the output focusing slab region. Each wavelength component is then fed into an output waveguide.

An approximate analysis of the demultiplexing operation is discussed below. The definitions in the input and output focusing slab regions are as follows:

Input focusing slab region:

D1: the spacing between the ends of adjacent waveguides,

d1: the spacing between the ends of adjacent waveguides on the output side, x1: distance measured from the center of the input side,

f1: the radius of the output curvature.

Output focusing slab region:

d: the spacing between the ends of adjacent waveguides connected to the array waveguides,

D: the spacing between the ends of adjacent waveguides connected to the output waveguides,

f: the radius of the output curvature.

We reiterate that the path difference between two adjacent waveguides is L, and the corresponding phase retardation is 2pm with respect to the center wavelength.

Consider the light beams passing through the ith and (i-1)th array waveguides. In order for the two light beams to interfere constructively, their phase difference should be a multiple of 2p as they reach the output side of the focusing slab region.

The condition for constructive interference is then given by

 

 

 

 

 

d1x1

 

dx

 

bsðl0Þ f1

 

þ bcðl0Þ½Lc þ ði 1Þ L& þ bsðl0Þ f þ

 

 

 

 

 

2f1

 

2f

 

ð19:2-1Þ

 

 

d1x1

dx

 

 

¼ bsðl0Þ f1 þ

 

þ bcðl0Þ½Lc þ i L& þ bsðl0Þ f

 

 

2pm

2f1

2f

where bs and bc denote the propagation constants (wave numbers) in the slab region and the array waveguide, respectively, m is the diffraction order, l0 is the center wavelength of the multiple wavelength input, and Lc is the minimum array waveguide length. Subtracting common terms from Eq. (19.2-1), we obtain

bsðl0Þ

d1x1

bsðl0Þ

dx

þ bcðl0

Þ L ¼ 2pm

ð19:2-2Þ

f1

f

 

When the condition

 

 

 

 

 

 

 

 

 

bcðl0Þ L ¼ 2pm

 

ð19:2-3Þ

METHOD OF IRREGULARLY SAMPLED ZERO-CROSSINGS (MISZC)

341

is satisfied for l0, the light input position x1 and the output position x satisfy the condition

d1x1

¼

dx

ð19:2-4Þ

f1

f

The above equation means that when light is coupled into the input position x1, the output position x is determined by Eq. (19.2-4).

The path length difference L can be shown to be

L ¼

nsdDl0

 

ð19:2-5Þ

ncf l

The spatial separation (free spectral range) of

the mth and (m þ l)th

focused beams for the same wavelength can be derived from Eq. (19.2-2) as [Okamoto]

l0f

ð19:2-6Þ

XFSR ¼ xm xmþ1 ¼ nsd

The number of available wavelength channels Nch is obtained by dividing XFSR by the output waveguide separation D as

Nch ¼

XFSR

¼

l0f

ð19:2-7Þ

D

nsdD

 

In practice, achieving the layout of the waveguides in a planar geometry such that the length difference between two waveguides is ml is not a trivial task. Professional computer-aided design programs are usually used for this purpose. An example is shown in Figure 19.2 in which the BeamPROP software package by Rsoft Inc. was used to carry out the design [Lu, Ersoy].

Two examples of the results obtained with PHASAR simulation at a center wavelength of 1.55 m and channel spacing of 0.8 m are shown in Figures 19.3 and 19.4 to illustrate how the number of channels are limited [Lu and Ersoy et al., 2003]. In Figure 19.3, there are 16 channels, and the second order channels on either side of the central channels do not overlap with the central channels. On the other hand, in Figure 19.4, there are 64 channels, and the second order channels on either side of the central channels start overlapping with the central channels. In this particular case, the number of channels could not be increased any further. Currently, PHASAR devices being marketed have of the order of 40 channels.

19.3 METHOD OF IRREGULARLY SAMPLED ZERO-CROSSINGS (MISZC)

In this method, the design of a DWDM device is undertaken such that there is only one image per wavelength so that the number of channels is not restricted due to the distance between successive orders as discussed with Eq. (19.2-6) above

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