COL 11(7), 070502(2013) CHINESE OPTICS LETTERS July 10, 2013
Modal analysis of the 0th order nulled phase masks
Wenting Sun (
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), Peng Lv (
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), Changhe Zhou (
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)
∗
, Jun Wu (
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),
and Shaoqing Wang (
)
Laboratory of Information Optics and Opto-electronic Technology, Shanghai Institute of Optics and Fine Mechanics,
Chinese Academy of Sciences, Shanghai 201800, China
∗
Corresponding author: chazhou@mail.shcnc.ac.cn
Received January 23, 2013; accepted April 12, 2013; posted online July 3, 2013
A simple modal analysis (MA) method t o explain the diffraction process of 0th order nulled phase mask
is presented. In MA, multiple reflections of the grating modes at grating interfaces are considered by
introducing equivalent Fresnel coefficients. Analytical expressions of the diffraction efficiencies and mo dal
guidelines for the 0th order nulled phase grating design are also presented. The phase mask structure,
which comprises a high-index contrast HfO
2
grating and a fused-silica substrate, is optimized using rigorous
coupled-wave analysis around the 800-nm wavelength, after which the modal guideline for cancellation of
the 0th order in a phase mask is verified. The proposed MA method illustrates the inherent physical
mechanism of multiple reflections of the grating modes in the diffraction process, which can help to
analyze and design both low-contrast and high-contrast gratings.
OCIS codes: 050.1950, 050.1960, 050.6624.
doi: 10.3788/COL201311.070502.
One important method for fiber Bragg grating (FBG) in-
scription (writing) is to use 0th order nulled phase masks
to fabricate FBGs. This method
[1]
utilizes the ultraviolet
(UV) beam at normal incidence modulated spatia lly by
the phase ma sk grating made of fused silica, after which
the interference of the diffractive 1st and –1st orders ex-
iting from the phase mas k forms a periodic pattern, with
half the phas e mask gra ting pitch in the photosensitive
fiber immediately behind the phase mask. Then, with
the use of ultrahigh-peak-power femtosecond (fs) laser
radiation for the induction of index change in dielectric
materials to produce waveguide structure
[2]
, FBGs are
written in standard Ge-doped fiber with pulsed 800-nm
laser radiation using a deep-e tch silica phase mask
[3]
.
However, pre vious works
[4,5]
present evidence that stan-
dard fused-silica binary phase masks cannot extinguish
the 0th order, where the period is o nly slightly larger
than the exposure wavelength. In addition, the existence
of the 0th transmitted order has an unfavorable effect
on the formation of a sharp interference pattern with
pulsed 800 -nm las er when a standard fused-silica binary
grating is used. This is because the period of the binary
phase mask is around 1 060 nm, which corresponds to
the Bragg resonance wavelength
[1]
.
The simplified modal method is an imp ortant tool for
analyzing low-contrast gratings
[6−9]
, in which the re flec -
tion at grating interfaces is neglected. Clausnitzer et
al.
[10]
introduced the Fresnel reflection at the interfaces
into the simplified modal method using a symmetrical
encapsulated grating similar to a standard Fabry-Perot
(F-P) cavity. However, only the reflection of one mode
is considered in a previous study
[10]
. For a symmetrical
high contrast grating, previous works
[11,12]
propose a n
analysis and present an explanation based on a coupled
Bloch-mode insight. As far a s we know, modal anal-
ysis (MA) based on the equivalent F-P interference of
dual-mode reflection for an as ymmetrical high-contrast
grating has yet to be conducted in any study.
In this letter, we propose a new MA method based on
the simplified modal method
[6]
and multi-reflection in-
terference effect of propagating grating modes, thus pro-
viding a clear physical image of the diffraction process
in a high-contrast phase grating. Analytical expressions
of the diffractio n efficiencies and modal guidelines for
the 0th order nulled phase grating design are provided.
Rigorous c oupled-wave ana ly sis (RCWA)
[13]
is used to
optimize the phase mas k structure. The optimized high-
contrast phase grating has an extremely low diffraction
efficiency of the 0th order of less than 0.1% under normal
incidence, with an 800-nm fs source. The results using
RCWA coincide with those from the modal guideline for
cancellation of the 0th order in a phase mask. Meanwhile,
the effects of incident angle and incident wavelength on
the phase mask performance are also considered. Some
discussions and conclusions are presented.
Figure 1 shows the schematic of a phase mask structure
illuminated under normal incidence at a wavelength λ of
800 nm; it is composed of a substrate with index n
1
(the
incidence medium) and a binary grating of thickness h,
period Λ, ridge width b, index n
r
, groove width g, and
index n
g
. Duty cycle f is defined as the ratio of ridge
width b to period Λ. Unlike the standard fused-silica bi-
nary phase mask, the phase mask proposed in this letter
comprises a higher index contrast grating on a fused-
silica substra te. The corresponding refractive indices of
the substrate and the groove are 1.45332 and 1.8946 at
800 nm, respectively. The ratio of the grating period
to the wavelength should only be between 1 and 2 to
ensure only three output diffractive orders, namely, 0th,
1st, and –1st. Moreover, only the 0th and 1st diffraction
efficiencies should be considered because the efficiencies
of the 1st and –1st diffr active orders are always the s ame
under no rmal incidence because of their symmetry, i.e.,
θ
1
=θ
−1
.
Although the model method is first developed by Collin
et al.
[14,15]
, it is Botten
[16]
who first applied it to di-
electric gra tings. This model method provides a clea r
1671-7694/2013/070502(5) 070502-1
c
2013 Chinese Optics Letters
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