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The Optimum Even-phase Four-beam Multiplier
R. Borghi,§† F. Frezza,¶ L. Pajewski,¶ M. Santarsiero,† and G. Schettini§
July 11, 2002
¶
Dip. di Ingegneria Elettronica, Università La Sapienza
Via Eudossiana 18, I-00184 Rome, Italy.
§
Dip. di Ingegneria Elettronica, Università Roma Tre
Via della Vasca Navale 84, I-00146 Rome, Italy.
Dip. di Fisica, Università Roma Tre
Via della Vasca Navale 84, I-00146 Rome, Italy.
†
Istituto Nazionale di Fisica della Materia
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Abstract
The even-phase diffractive optical element that produces four equi-intense
diffraction orders with the maximum efficiency is derived. A full-wave electromagnetic analysis is performed to study the behavior of the device in the
resonance domain. Moreover, such full-wave analysis agrees with the scalar
prediction in the limit of very large values of the grating period. Design curves
for the transmission efficiency as a function of the grating period and of the
incidence angle are presented. The effects of a quantization of the phase profile to a discrete number of phase levels are considered. The optimization of a
simple structure with only four phase levels is reported, showing good results
in terms of diffraction efficiency and angular bandwidth.
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Introduction
Beam multipliers are very popular diffractive optical elements (DOEs for short), which
divide an input beam into a certain number of output beams with equal power. They
are used in many applications, for example in optical signal processing, laser manufacturing optical interconnections (one-to-many array coupling, beam path switching,
etc.), and more [1, 2, 3, 4].
Maximization of the diffraction efficiency (fraction of the incident beam power
that is converted into the power of the desired output beams) is a fundamental target
in designing a beam multiplier, and optimal methods for the design and analysis of
diffracting structures have been developed [4, 5, 6].
In this paper we prove that an optimum-efficiency even-phase four-beam multiplier, or quadruplicator, exists, and we give its phase transmittance in an analytic
form (Sect. 2).
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As well known, in the paraxial domain the vectorial nature of light can be neglected
and the influence of the diffractive element on the illuminating wave front can be
described by its transmission function [7]. In the resonance domain, instead, when the
grating-profile features have transverse dimensions comparable with the wavelength
of the impinging radiation, the scalar diffraction theory fails. To study the properties
of the element correctly, a rigorous application of the electromagnetic theory becomes
necessary. The rigorous diffraction theory has been already applied to the analysis
of beam multipliers: see, for example, Ref. [8] and references therein. A detailed
analysis of the even-phase optimum quadruplicator, by using the electromagnetic
theory of gratings, is presented in Sect. 3. Such a treatment of the problem allows
us, for example, to understand the operational limits of the quadruplicator, to study
its angular response, or to find solutions that are not predicted by the scalar theory.
A grating’s surface profile can either be multilevel or continuous. In the former
case the structure typically consists of M = 2N discrete depth values, where N
is the number of fabrication steps, while in the latter the shape of the profile is
continuous. One can approximate the continuous profile of a grating by dividing it
into a large number of thin planar binary gratings: if each layer is thin enough, the
original structure can be analyzed to a high degree of accuracy by using mathematical
techniques developed for multilevel gratings. The quantization of the continuous
phase profile in a discrete number of steps affects the uniformity and the efficiency of
the output beams of the quadruplicator, as discussed in Sect. 3. We also investigate on
how many steps are necessary to approximate the continuous profile of the optimum
quadruplicator in a satisfactory way. This is interesting from a practical point of
view, too: depending on the required uniformity of the output beams, it is possible
to choose the minimum number of fabrication steps. Finally, in Sect. 4, we design of
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an even-phase four-level quadruplicator, which is simple to manufacture and shows
very good characteristics.
2
Even-phase optimum quadruplicator
In this section we will obtain the transmission function of an even-phase DOE producing four, equally intense, replicas of an incident field, with the maximum efficiency
(see the inset in Fig. 1: note that the incidence is normal).
Let us consider a grating having the transmission function τ (x) = eiΦ(x) , where
Φ(x) denotes the phase profile. Without loss of generality, in the formulas of this
Section we assume a unitary value (in suitable units) of the period d. Due to its
periodicity, the following Fourier series expansion holds:
∞
τ (x) =
τn ei2πnx ,
(1)
n=−∞
where
τn =
1/2
−1/2
τ (x)e−i2πnx dx.
(2)
On assuming an even phase profile [9], i.e., Φ(x) = Φ(−x), we have τ−m = τm . In
particolar, the expression of τ±m turns out to be
1/2
eiΦ(x) cos (2πmx) dx,
τ±m = 2
(3)
0
where m = 1, 3. It is to be noted that the choice of the ±1 and ±3 orders does
not represent a restriction: indeed, any different choice of contiguous orders can be
reconducted to this one, by adding to Φ(x) a suitable linear function of x. It is
also to be noted that an odd-phase function corresponds to hermitian transmission
functions: this entails that the Fourier coefficients are real, so that the degree of
4
We applied the full-wave theory for multilevel gratings also to the analysis of such
device. Figure 10 shows the transmission efficiencies η±1 (solid line), η±3 (dashed line)
and η (dotted line), vs. d/λ, for TE polarization. In Fig. 11, the same as in Fig. 10 is
reported, for TM polarization. As the angular bandwidth of the device is concerned,
Fig. 12 shows the transmission efficiencies of the various orders, as a function of the
incidence angle ϑ, when d/λ = 20.
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Conclusions
In this work, we presented the synthesis and a full-wave analysis of the optimum
even-phase four-beam multiplier. We studied the behavior of the device and its
performances in terms of diffraction efficiency as well as angular responses, in the
resonance domain and in the scalar limit. Furthermore, the effects of the quantization
of the continuous phase profile of the optimum quadruplicator in a discrete number
of steps have been discussed. The optimization of a simple to manufacture structure
with only four phase levels is reported, showing good results, much better than the
ones obtainable with Dammann gratings. The proposed devices may be used in many
applications such as optical interconnections and signal processing. The approach
employed may be generalized to the case of N-beam multipliers.
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