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1. Analytical approach to polarisation nonreciprocity of Sagnac fibre ring interferometer
Roman Kurbatov
romuald75@mail.ru
Polarisation nonreciprocity (PNR) [1, 2] is the fundamental accuracy limit of Sagnac fibre ring
interferometer (FRI) playing the principal role in high-grade fibre optic gyro (FOG) [3]. Fig. 1 sketches
FRI scheme of commercial small size FOG, which may have highly birefringent (Hi-Bi) optical
components.
Fig. 1. Sagnac FRI scheme of commercial FOG.
At splices, optical axes of components are slightly rotated with respect to each other due to technology
imperfection (imperfect splices, one of PNR sources). Input lightguide and coil fibre may be polarizing
(PZ). Channel waveguides of integrated optic chip (IOC) may be done by proton-exchanged (PE)
technology being PZ, having the intensity polarisation extinction ratio (PER) 𝜺 𝟐
= 𝟏𝟎−𝟓
− 𝟏𝟎−𝟖
, and
possessing the extremely large birefringence order of 0.01 [4]. Polarisation mode coupling (PMC) in coil
fibre, input lightguide, and IOC waveguides also yields PNR, along with mutual interferences of spurious
waves from all these PMC kinds and with those from imperfect splices. Below only coil fibre PMC is taken
into account (i.e., splices input lightguide and IOC waveguides are perfect and, generally, birefringent), as
it is usually done in all known literature.
Fig. 1 explains the concept of minimal FRI, which is equivalent to minimal configuration FRI
from Ref. [1]. Also, minimal FRI means isotropic IOC waveguides. This leads to decoherence absence
within them (decoherence is the x- and y-waves coherence loss). Instead of term “decoherence”, common
for quantum measurement theory [5], term “depolarisation” is used for this phenomenon in the literature,
but it also includes x- and y-waves intensities equalisation, additionally to decoherence.
Quasi-minimal FRI is minimal one plus input lightguide and, generally, IOC waveguides
anisotropy.
Amplitude PNR (APNR) was established in Ref. [2] (APNR ~ 𝜺 𝟏
), along with more expected
smaller intensity PNR (IPNR ~ 𝜺 𝟐
). Both are suppressed when 𝜺 = 𝟎, but this is not the practical case. In
the literature, the following expressions are known for PNR of minimal FRI (Fig. 1):
𝑷𝑵𝑹 𝟏~ 𝒔 𝟏 𝜺 𝟐
𝒉√𝑳 𝒅𝒆𝒄 𝑳 𝑺𝑭⁄ , 𝑷𝑵𝑹 𝟐,𝟑~ 𝒔 𝟐,𝟑 𝜺√𝒉𝑳 𝒅𝒆𝒄 𝑺𝑭⁄ , (1)
where 𝑳 𝒅𝒆𝒄~ 𝝀 𝟎
𝟐 (𝑩𝚫𝝀)⁄ is decoherence length (“depolarisation” length in the literature), 𝝀 𝟎 is the light mean
wavelength, 𝒉 is so called h-parameter of the coil fibre, 𝑺𝑭 ≡ 𝟒𝝅𝑹𝑳 𝝀 𝟎 𝒄⁄ is the scale factor of FRI, 𝒔 𝟎−𝟑
are the light normalised Stokes parameters ( 𝒔 𝟎 = 𝟏), 𝑩 is coil fibre birefringence, 𝑳 and 𝑹 its length and
radius, 𝚫𝝀 is the light spectral bandwidth. For large enough 𝑩 and 𝚫𝝀, one provides 𝑳 𝒅𝒆𝒄 ≪ 𝑳. It is the
purpose of this study to derive consistently Eq. (1), which is still not done in the literature.
2. GENERAL RELATIONSHIPS FOR PNR
Elsewhere [6] it is shown that for practically interesting small PNR, one may yield PNR in the
form of rotation rate error, 𝑷𝑵𝑹 = 𝑷𝑵𝑹 𝟏 + 𝑷𝑵𝑹 𝟐 + 𝑷𝑵𝑹 𝟑, where

2. 𝑷𝑵𝑹 𝟏−𝟑 = 𝒔 𝟏−𝟑
∫ 𝒅𝝀𝑺(𝝀)𝐈𝐦𝑨 𝟎−𝟑(𝝀,𝒕)
𝑺𝑭 ∑ 𝒔 𝒌 ∫ 𝒅𝝀𝑺(𝝀)𝐑𝐞𝑨 𝒌(𝝀,𝒕)𝟑
𝒌=𝟎
. (2)
Here 𝑺(𝝀) is the light spectral intensity, 𝟐𝑨 𝟎 ≡ |𝑴 𝟏𝟏| 𝟐
+ |𝑴 𝟐𝟐| 𝟐
+ 𝟐𝑹𝒆(𝑴 𝟏𝟐 𝑴 𝟐𝟏
∗
), 𝟐𝑨 𝟎 ≡ |𝑴 𝟏𝟏| 𝟐
− |𝑴 𝟐𝟐| 𝟐
+
𝟐𝒊𝑰𝒎(𝑴 𝟏𝟐 𝑴 𝟐𝟏
∗
), 𝟐𝑨 𝟐 ≡ 𝑴 𝟏𝟏
∗
𝑴 𝟏𝟐 + 𝑴 𝟏𝟏 𝑴 𝟐𝟏
∗
+ 𝑴 𝟏𝟐
∗
𝑴 𝟐𝟐 + 𝑴 𝟐𝟏 𝑴 𝟐𝟐
∗
, 2𝑨 𝟑 ≡ 𝒊(𝑴 𝟏𝟏
∗
𝑴 𝟏𝟐 − 𝑴 𝟏𝟏 𝑴 𝟐𝟏
∗
) +
𝒊(𝑴 𝟏𝟐
∗
𝑴 𝟐𝟐 − 𝑴 𝟐𝟏 𝑴 𝟐𝟐
∗
). Here 𝑴 = 𝑴 𝒄𝒘
is Jones matrix for cw-waves (clockwise). For minimal and quasi-
minimal FRI from Fig. 1, one has for counter-clockwise (ccw) waves 𝑴 𝒄𝒄𝒘
= (𝑴 𝒄𝒘) 𝑻
(“T” is transposing
operation). Thus, the problem is to determine the values 𝑨 𝟎−𝟑(𝝎, 𝒕) i.e. 𝑴𝒊,𝒋(𝝎, 𝒕).
Below, simplified analytical models are treated for perfect splices (Fig. 1) and for input lightguide
without polarisation mode coupling (PMC), which is perfect input lightguide (treating the model with
imperfect splices is the future step). Thus, only PMC in fibre coil is assumed as the source of PNR here.
More complex problems could be treated only numerically, while presented below analytical approach,
besides the illustrative purposes, provides the basis for numerical models checking.
3. Polarisation mode coupling in PZ-fibre and its Jones matrix
Fibre Jones matrix could be derived from polarisation mode coupling (PMC) equations in the form
of [7], generalised for the case of PZ-fibre:
[𝒅 𝒅𝒛⁄ + 𝝃 𝒙 𝟐⁄ + 𝒊𝜷 𝒙(𝝀)]𝒂 𝒙(𝝀, 𝒛) = 𝒊𝒄 𝒙,𝒚(𝒛)𝒂 𝒚(𝝀, 𝒛), [ 𝒅 𝒅𝒛⁄ + 𝝃 𝒚
𝟐⁄ + 𝒊𝜷 𝒚
( 𝝀)] 𝒂 𝒚( 𝝀, 𝒛) = 𝒊𝒄 𝒚,𝒙( 𝒛) 𝒂 𝒙( 𝝀, 𝒛).
(3)
In the case of 𝝃 𝒙,𝒚 = 𝟎, the total optical power |𝒂 𝒙(𝝀, 𝒛)| 𝟐
+ |𝒂 𝒚(𝝀, 𝒛)|
𝟐
= 𝒄𝒐𝒏𝒔𝒕, so one must set 𝒄 𝒙,𝒚(𝒛) =
𝒄 𝒚,𝒙
∗ (𝒛). Obviously, dichroism do not break this condition, because it does not influence the PMC sources
(i.e., the functions 𝒄 𝒙,𝒚(𝒛) and 𝒄 𝒚,𝒙(𝒛)). Also, for weak-guiding fibres, which are used in FOG, their two
polarisation fundamental modes have almost the same electrical (and magnetic) fields with quasi-Gaussian
distributions, 𝒆 𝒙(𝒙, 𝒚) ≈ 𝒆 𝒚(𝒙, 𝒚) ≈ 𝒆(𝒙, 𝒚), which is the real function, so
𝒄 𝒙,𝒚(𝒛) = 𝒄 𝒚,𝒙
∗ (𝒛) ≈ 𝒄(𝒛) ≡ ∫ 𝒅𝒙𝒅𝒚𝜹𝒏 𝟐(𝒙, 𝒚, 𝒛)𝒆 𝟐(𝒙, 𝒚) ∫ 𝒅𝒙𝒅𝒚𝒆 𝟐(𝒙, 𝒚)⁄ ,
where 𝜹𝒏 𝟐(𝒙, 𝒚, 𝒛) is dielectric tensor perturbation due to external or internal sources [7], varying with 𝒛
randomly, and leading to randomly distributed PMC. Thus, the latter could be described by real random
function 𝒄(𝒛) for both equations (3). For their solution, one may represent the amplitudes 𝒂 𝒙,𝒚(𝒛) as
𝒂 𝒙,𝒚(𝝀, 𝒛) = 𝑨 𝒙,𝒚(𝝀, 𝒛)𝒆𝒙𝒑{−[𝝃 𝒙,𝒚 𝟐⁄ + 𝒊𝜷 𝒙,𝒚(𝝀)]𝒛},
so equations (3) will be rewritten as 𝒅𝑨 𝒙(𝝀, 𝒛) 𝒅𝒛⁄ = 𝒊𝒄(𝒛)𝑨 𝒚(𝝀, 𝒛)𝒆𝒙𝒑{[− 𝝃 𝟐⁄ + 𝒊∆𝜷(𝝀)]𝒛}, 𝒅𝑨 𝒚(𝝀, 𝒛) 𝒅𝒛⁄ =
𝒊𝒄(𝒛)𝑨 𝒙(𝝀, 𝒛)𝒆𝒙𝒑{[𝝃 𝟐⁄ − 𝒊∆𝜷(𝝀)]𝒛}, and solved as
𝑨 𝒙(𝝀, 𝒛) = 𝑨 𝒙(𝝀, 𝟎) + 𝒊 ∫ 𝒅𝒔𝒄(𝒔)𝑨 𝒚(𝝀, 𝒔)𝒆𝒙𝒑{[− 𝝃 𝟐⁄ + 𝒊∆𝜷(𝝀)]𝒔}
𝒛
𝟎
,
𝑨 𝒚(𝝀, 𝒛) = 𝑨 𝒚(𝝀, 𝟎) + 𝒊 ∫ 𝒅𝒔𝒄(𝒔)𝑨 𝒙(𝝀, 𝒔)𝒆𝒙𝒑{[𝝃 𝟐⁄ − 𝒊∆𝜷(𝝀)]𝒔}
𝒛
𝟎
,
(4)
where 𝝃 ≡ 𝝃 𝒚 − 𝝃 𝒙 is the fibre dichroism ( 𝝃 𝒙,𝒚 are x- and y-modes losses), ∆𝜷(𝝀) ≡ 𝜷 𝒙(𝝀) − 𝜷 𝒚(𝝀) is the fibre
modal birefringence at optical frequency 𝝀. Here an optical frequency 𝝀-dependence is introduced
explicitly in order to stress the polychromatic nature of light and the waveguide dispersion of optical
components. However, the dichroism 𝝃 is left independent on 𝝀, because practical PZ-fibre is the
component with dichroism only within the finite spectral range (window), being often in complicated
dependence on 𝝀, so some minimal 𝝃-value within this window is assumed. In the first order of PMC, (4)
could be rewritten as
𝑨 𝒙(𝝀, 𝒛) = 𝑨 𝒙(𝝀, 𝟎) + 𝒊𝑨 𝒚(𝝀, 𝟎) ∫ 𝒅𝒔𝒄(𝒔)𝒆𝒙𝒑{[− 𝝃 𝟐⁄ + 𝒊∆𝜷(𝝀)]𝒔}
𝒛
𝟎
,
𝑨 𝒚(𝝀, 𝒛) = 𝑨 𝒚(𝝀, 𝟎) + 𝒊𝑨 𝒙(𝝀, 𝟎) ∫ 𝒅𝒔𝒄(𝒔)𝒆𝒙𝒑{[𝝃 𝟐⁄ − 𝒊∆𝜷(𝝀)]𝒔}
𝒛
𝟎
,
(5)

3. Here, under integrals in (4), 𝑨 𝒚(𝝀, 𝒔) ≈ 𝑨 𝒚(𝝀, 𝟎) is set. For PZ-fibre, one, thus, yields the following Jones
matrix:
𝑭(𝝀) = (
𝒂(𝝀, 𝑳) 𝒊𝒂(𝝀, 𝑳)𝒌 𝟏(𝝀)
𝒊𝑮(𝑳)𝒂∗(𝝀, 𝑳)𝒌 𝟐(𝝀) 𝑮(𝑳)𝒂∗(𝝀, 𝑳)
), (6)
where 𝒌 𝟏(𝝀) = ∫ 𝒅𝒛𝒄(𝒛)𝑮(𝒛)𝒂(𝝀, −𝟐𝒛)
𝑳
𝟎
, 𝒌 𝟐(𝝀) = ∫ 𝒅𝒛𝒄(𝒛)𝑮−𝟏(𝒛)𝒂(𝝀, 𝟐𝒛)
𝑳
𝟎
, 𝑮(𝒙) = 𝒆𝒙𝒑(− 𝝃𝒙 𝟐⁄ ), 𝒂(𝝀, 𝒙) =
𝒆𝒙𝒑[−𝒊 𝜟𝜷(𝝀)𝒙 𝟐⁄ ]. A multiplier 𝒆𝒙𝒑{−𝒊[𝜷 𝒙(𝝀) + 𝜷 𝒚(𝝀)] 𝑳 𝟐⁄ − 𝝃 𝒙 𝑳 𝟐⁄ } is omitted, the same for all 𝑭𝒊,𝒋(𝝀), so
it does not lead to PNR. For 𝝃 𝒙,𝒚 = 𝟎 (usual case of lossless polarisation maintaining fibre) one has the
matrix elements from [8].
4. Hi-Bi fibre characterising by h-parameter
Conventional characteristic of Hi-Bi fibre is h-parameter [9], which characterises the fibre when
only one input field component is excited, say, 𝑬 𝒙(𝝀). This parameter could be measured as the ratio of
output intensities 〈𝑷 𝒙,𝒚〉 = 〈|𝑬 𝒙,𝒚(𝝀)|
𝟐̅̅̅̅̅̅̅̅̅̅̅̅̅
〉, where the angle brackets are the averaging over ensemble of large
number of fibres and
(… )̅̅̅̅̅ ≡ ∫ 𝒅𝝀(… )𝑺(𝝀)
∞
𝟎
is the averaging over spectrum. For the case 〈𝑷 𝒚〉 ≪ 〈𝑷 𝒙〉, which should correspond to small PNR, h-
parameter is defined in the following approximate form [9]:
𝒉 ≈ 𝑳−𝟏 〈𝑷 𝒚〉 〈𝑷 𝒙〉⁄ . (7)
Using (6), one may yield 〈𝑷 𝒙〉 ≈ 〈|𝒂(𝝀, 𝑳)| 𝟐̅̅̅̅̅̅̅̅̅̅̅̅〉 = ∫ 𝒅𝝀𝑺(𝝀)
∞
𝟎
and the following:
〈𝑷 𝒚〉 ≈ 〈|𝒌 𝟏(𝝀)| 𝟐̅̅̅̅̅̅̅̅̅̅̅〉 = ∫ 𝒅𝒛𝒅𝒙〈𝒄(𝒛)𝒄(𝒙)〉Г(𝒛 + 𝒙)𝒆𝒙𝒑[𝒊𝜟𝜷(𝝀)(𝒛 − 𝒙)]̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅
𝑳
𝟎
=
= ∫ 𝒅𝒛𝒅𝒙𝑪(𝒛 − 𝒙)Г(𝒛 + 𝒙)𝜪(𝒛 − 𝒙)
𝑳
𝟎
, (8)
where 𝑪(𝒛 − 𝒙) = 〈𝒄(𝒛)𝒄(𝒙)〉 is PMC autocorrelation function ( 𝒄(𝒛) is assumed stationary). Also, 𝜪(𝒚) ≡
𝒆𝒙𝒑[𝒊𝜟𝜷(𝝀)𝒚]̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅. For its calculating, one may implement the decomposition 𝜟𝜷(𝝀) ≈ 𝜟𝜷 𝟎 + 𝑫(𝝀 − 𝝀 𝟎) [9],
where 𝜟𝜷 𝟎 = 𝜟𝜷(𝝀 𝟎), 𝑫 = (𝒅𝜟𝜷 𝒅𝝀⁄ )] 𝝀 𝟎
. For Panda-type fibres, one has 𝜟𝜷(𝝀) ≈ 𝟐𝝅𝑩 𝝀⁄ ( 𝑩 ≡ 𝒏 𝒙 − 𝒏 𝒚 is
fibre material birefringence), so 𝑫 ≈ − 𝟐𝝅𝑩 𝝀⁄ 𝟎
𝟐
. As a result,
𝜪(𝒚) ≈ 𝒆𝒙𝒑(𝒊𝜟𝜷 𝟎 𝒚)𝑮(𝒚), (9)
where 𝑮(𝒚) = 𝒆𝒙𝒑[𝒊𝑫(𝝀 − 𝝀 𝟎)𝒚]̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅ is complex coherence function. Thus, one may rewrite (7) in the form
𝒉 ≈
𝟏
𝑳
∫ 𝒅𝒖𝑪(𝒖)𝜪(𝒖) ∫ 𝒅𝒗Г(𝒗)
𝑳
𝟎
𝑳
−𝑳
≈ [
𝟏−𝑮 𝟐(𝑳)
𝝃𝑳
] ∫ 𝒅𝒖𝑪(𝒖)𝒐(𝒖)
+∞
−∞
, (10)
where in (8), variables were changed as 𝒖 = 𝒛 − 𝒙 and 𝟐𝒗 = 𝒛 + 𝒙. Usually, their integration limits are
determined according to some a little bit complex procedure (see [10]), but here it is set that 𝒖 is range
from – 𝑳 (minimal possible value of 𝒛 − 𝒙) to +𝑳 (maximal possible value of 𝒛 − 𝒙), while 𝒗 ranges from 𝟎
to +𝑳. Also, 𝒐(𝒚) ≡ 𝑶(𝒚) ∫ 𝒅𝝀𝑺(𝝀)
∞
𝟎
⁄ , so a normalised coherence function could be introduced:
𝒈(𝒚) ≡ 𝑮(𝒚) ∫ 𝒅𝝀𝑺(𝝀)
∞
𝟎
⁄ = ∫ 𝒅𝝀𝒆𝒙𝒑[𝒊𝑫(𝝀 − 𝝀 𝟎)𝒚]𝑺(𝝀)
∞
𝟎
∫ 𝒅𝝀𝑺(𝝀)
∞
𝟎
⁄ . (11)
so 𝒐(𝒖) = 𝒆𝒙𝒑(𝒊𝜟𝜷 𝟎 𝒖)𝒈(𝒖). For PM-fibre ( 𝝃 = 𝟎) and for practical PZ-fibre ( 𝑮 𝟐(𝑳) ≈ 𝟎) one has the
following:

4. 𝒉 𝑷𝑴 ≈ ∫ 𝒅𝒖𝑪(𝒖)𝒐(𝒖)
+∞
−∞
, 𝒉 𝑷𝒁 ≈ 𝒉 𝑷𝑴 𝝃𝑳⁄ ,
which means that 𝒉 𝑷𝑴 𝒉 𝑷𝒁 ≫ 𝟏⁄ . This was observed experimentally in the form of extremely small h-
parameter which could belong only to PZ-fibre [11]. Also, this means that 〈𝑷 𝒚〉-value of PM-fibre is
proportional to 𝑳, while for PZ-fibre it is constant, because the feeding of 〈𝑷 𝒚〉 by 〈𝑷 𝒙〉 through PMC is
counter-balanced by attenuation of 〈𝑷 𝒚〉.
Expression (10) contains the normalised coherence function 𝒈(𝒖). For symmetrical spectra relative
to the central frequency 𝝎 𝟎, from (11), one has
𝐈𝐦𝒈(𝒚) =
∫ 𝒅𝝀𝒔𝒊𝒏[𝑫(𝝀 − 𝝀 𝟎)𝒚]𝑺(𝝀)
∞
𝟎
∫ 𝒅𝝀𝑺(𝝀)
∞
𝟎
= [𝑺(𝝀) → 𝑺(𝝀 − 𝝀 𝟎)] =
∫ 𝒅𝝀𝒔𝒊𝒏(𝑫𝝀𝒚)𝑺(𝝀)
∞
−𝝀 𝟎
∫ 𝒅𝝀𝑺(𝝀)
∞
𝟎
≈
≈ (𝝀 𝟎 ≫ 𝜹𝝀) ≈ 𝜹𝝀
∫ 𝒅𝒖𝐬𝐢𝐧(𝑫𝒖𝜹𝝀𝒚)𝑺(𝒖)
∞
−∞
∫ 𝒅𝝀𝑺(𝝀)
∞
𝟎
≈ 𝟎,
because sinus is the odd function, while symmetrical spectrum 𝑺(𝝀) is the even function. In this case,
𝒈(𝒚) =
∫ 𝒅𝝀𝐜𝐨𝐬[𝑫(𝝀−𝝀 𝟎)𝒚]𝑺(𝝀)
∞
𝟎
∫ 𝒅𝝀𝑺(𝝀)
∞
𝟎
≈
∫ 𝒅𝒖𝐜𝐨𝐬(𝑫𝒖𝜹𝝀𝒚)𝑺(𝒖)
∞
−∞
∫ 𝒅𝒖𝑺(𝒖)
∞
−∞
.
For example, Gaussian spectrum yields
𝒈(𝒚) ≈
∫ 𝒅𝒖𝒄𝒐𝒔(𝑫𝒖𝚫𝝀𝒚)𝒆𝒙𝒑(−𝒖 𝟐)
∞
−∞
∫ 𝒅𝒖𝒆𝒙𝒑(−𝒖 𝟐)
∞
−∞
= (
𝟐
𝑫𝚫𝝀
≡ 𝑳 𝒅𝒆𝒄) = 𝒆𝒙𝒑[−(𝒚 𝑳 𝒅𝒆𝒄⁄ ) 𝟐].
For Panda-type fibres, one has 𝑫 ≈ − 𝟐𝝅𝑩 𝝀⁄ 𝟎
𝟐
, so 𝑳 𝒅𝒆𝒄 ≈ 𝝀 𝟎
𝟐 (𝝅𝜟𝝀𝑩)⁄ . Thus, 𝒈(𝒚) could be set 𝒈(𝒚) ≈ 𝟎 for
|𝒚| > 𝟑𝑳 𝜸. For modern typical values 𝜟𝝀 = 𝟑𝟎 nm and 𝑩 = 𝟒×𝟏𝟎−𝟒
, one has only 𝑳 𝜸 ≈ 𝟔𝟒 mm. This is the
reason why PNR was managed to be reduced dramatically, according to Eq. (1).
Similar to this, one may yield the function 𝒈(𝒚) for some other spectra, also yielding non-zero
values only in the small region near 𝒚 = 𝟎.
As for PMC correlation function 𝑪(𝒖), the simplest case is delta-correlated 𝑪(𝒖) = 𝑪 𝟎 𝜹(𝒖), so
𝒉 𝑷𝑴 ≈ 𝑪 𝟎 ≈ 𝝃𝑳𝒉 𝑷𝒁. There is no dependence on birefringence and light spectrum parameters. This is the
case of anisotropic Rayleigh scattering, which is the fundamental limit of fibre polarisation maintaining
(PM) ability [12]. However, experimental results reveal considerably lower PM ability, depending on
birefringence as 𝒉 𝑷𝑴 ≈ 𝟐𝑪 𝟎 (𝟏 + 𝜟𝜷 𝟎
𝟐
𝒍 𝒄
𝟐
)⁄ [13], and for Hi-Bi fibres with 𝜟𝜷 𝟎 𝒍 𝒄 ≫ 𝟏, so 𝒉 𝑷𝑴 ≈ 𝟐𝑪 𝟎 𝜟𝜷 𝟎
−𝟐
𝒍 𝒄
−𝟐
.
Thus, for 𝑪(𝒖) one should use another model with correlation length 𝒍 𝒄 larger than the beat length. Assume
that 𝑪(𝒖) is independent on 𝝃, so the case 𝝃 = 𝟎 is enough (PM-fibre). Consider (10) for two extreme cases
of 𝒍 𝒄 ≪ 𝑳 𝒅𝒆𝒄 and 𝒍 𝒄 ≫ 𝑳 𝒅𝒆𝒄:
𝒉(𝒍 𝒄 ≪ 𝑳 𝜸) ≈ 𝟐 ∫ 𝒅𝒖𝑪(𝒖)𝒄𝒐𝒔(𝜟𝜷 𝟎 𝒖)
+∞
𝟎
, 𝒉(𝒍 𝒄 ≫ 𝑳 𝜸) ≈ 𝑪 𝟎 𝒍 𝒄
−𝟏
∫ 𝒅𝒖𝒈(𝒖)𝒆𝒙𝒑(𝒊𝜟𝜷 𝟎 𝒖)
+∞
−∞
.
First expression is presented in [13, 14, 15], being conventional until in [12] a coherence function 𝒈(𝒖) was
taken into account. For exponential correlator [12]
𝑪(𝒖) = 𝒍 𝒄
−𝟏
𝒆𝒙𝒑(− |𝒖| 𝒍 𝒄⁄ ) (12)
with correlation length 𝒍 𝒄 one yields
𝒉(𝒍 𝒄 ≪ 𝑳 𝜸) ≈ 𝟐𝑪 𝟎 (𝟏 + 𝜟𝜷 𝟎
𝟐
𝒍 𝒄
𝟐
)⁄ , 𝒉(𝒍 𝒄 ≫ 𝑳 𝜸) ≈ (𝑪 𝟎 𝑳 𝜸 𝒍 𝒄⁄ )[𝑺(𝟎)𝝅𝚫𝝀 ∫ 𝒅𝝀𝑺(𝝀)
∞
𝟎
⁄ ]. (13)
In particular, for Gaussian spectrum, 𝒉(𝒍 𝒄 ≫ 𝑳 𝜸) ≈ (𝑪 𝟎 𝑳 𝜸 𝒍 𝒄⁄ )𝒆𝒙𝒑[−(𝝀 𝟎 𝚫𝝀⁄ ) 𝟐], which is too unrealistic,
unlike the expression for 𝒉(𝒍 𝒄 ≪ 𝑳 𝜸), already appeared above as the experimental result. Moreover, another
possible functions 𝑪(𝒖) (Gaussian, Lorenzian, rectangular etc.) also will not lead to reasonable results
neither for 𝒍 𝒄 ≪ 𝑳 𝜸, nor for 𝒍 𝒄 ≫ 𝑳 𝜸.

5. Exponential correlator means the following: if singular perturbation of fibre occurs at some point
𝒛 𝟎 ≫ 𝒍 𝒄 within the fibre, corresponding fibre distortion 𝒄 𝟎 decays as 𝒄 𝟎 𝒆𝒙𝒑[− (𝒛 − 𝒛 𝟎) 𝒍 𝒄⁄ ]. Such distortion
could be yielded from ODE with singular imperfection source at point 𝒛 𝟎 in the right-hand side
(𝒅 𝒅𝒛⁄ + 𝒍 𝒄
−𝟏)𝒄(𝒛) = 𝒄 𝟎 𝒍 𝒄
−𝟏
𝜹(𝒛 − 𝒛 𝟎).
In practice, one has continuously distributed PMC random source along the whole fibre 𝜻(𝒛). Similar ODE
with such noise source in right-hand side has the form
(𝒅 𝒅𝒛⁄ + 𝒍 𝒄
−𝟏)𝒄(𝒛) = 𝒄 𝟎 𝒍 𝒄
−𝟏
𝜻(𝒛).
If noise source is delta-correlated, i.e. 〈𝜻(𝒛)𝜻(𝒙)〉 = 〈𝜻(𝒛 − 𝒙)𝜻(𝟎)〉 = 𝜻 𝟎
𝟐
𝜹(𝒛 − 𝒙), one yields required auto-
correlator (12) [16]. All of this will be used below for PNR estimations.
Note also that the condition 𝒍 𝒄 ≪ 𝑳 𝜸 should be correct for those of Hi-Bi solid-core microstructured
fibres that are already implemented in FOG, because they don’t demonstrate really exotic features that
more sophisticated fibre microstructures do for some other applications.
PNR calculating
Consider FRI with perfect input lightguide and IOC (without PMC), along with perfect splices.
Coil PM-fibre, input lightguide and IOC Jones matrices have the form [12], according to (6)
𝑭(𝝀) = [
𝒂(𝑳, 𝝀) 𝒊𝒌(𝝀)
𝒊𝒌∗(𝝀) 𝒂(−𝑳, 𝝀)
], 𝑭𝒊𝒏(𝝀) = [
𝒂𝒊𝒏(𝑳𝒊𝒏, 𝝀) 𝟎
𝟎 𝑮𝒊𝒏(𝑳𝒊𝒏)𝒂𝒊𝒏(−𝑳𝒊𝒏, 𝝀)
],
𝑰𝑶𝑪(𝝀) =
𝟏
√𝟐
[
𝒂 𝑰𝑶𝑪(𝑳 𝑰𝑶𝑪, 𝝀) 𝟎
𝟎 𝜺𝒂 𝑰𝑶𝑪(−𝑳 𝑰𝑶𝑪, 𝝀)
].
Here, according to the above, it is assumed that only coil fibre possesses PMC. FRI Jones matrix for cw-
wave, thus, could be written as
𝑴 = 𝑭𝒊𝒏
𝑻
×𝑰𝑶𝑪 𝑻
×𝑭×𝑰𝑶𝑪×𝑭𝒊𝒏
with elements 𝑴 𝟏𝟏(𝝀) = 𝑨(𝝀)𝒂(𝑳, 𝝀), 𝑴 𝟏𝟐(𝝀) = 𝒊𝜺𝑮𝒊𝒏(𝑳𝒊𝒏)𝒌(𝝀), 𝑴 𝟐𝟏(𝝀) = −𝒊×𝑴 𝟏𝟐
∗ (𝝀), 𝑴 𝟐𝟐(𝝀) =
𝜺 𝟐
𝑨(𝝀)𝑮𝒊𝒏
𝟐
(𝑳𝒊𝒏)𝑴 𝟏𝟏
∗
(𝝀), 𝑨(𝝀) ≡ 𝒂𝒊𝒏
𝟐
(𝑳𝒊𝒏, 𝝀)𝒂 𝑰𝑶𝑪
𝟐
(𝑳 𝑰𝑶𝑪, 𝝀). Value of 𝑷𝑵𝑹 𝟏 is determined from Eq. (2), and for
its mean value one has
𝑷𝑵𝑹 𝟏 = 𝟐𝜺 𝟐
𝑮𝒊𝒏
𝟐
(𝑳𝒊𝒏) ∫ 𝒅𝝀𝑺(𝝀)𝑰𝒎𝒌(𝝀)𝑹𝒆𝒌(𝝀) =
= −𝜺 𝟐
𝑮𝒊𝒏
𝟐
(𝑳𝒊𝒏)𝑰𝒎 ∫ 𝒅𝒛𝒅𝒙𝒄(𝒛)𝒄(𝒙)𝐞𝐱𝐩[𝟐𝒊𝚫𝜷 𝟎(𝑳 − 𝒛 − 𝒙)]Г(𝑳 − 𝒛 − 𝒙)
𝑳
𝟎
. (14)
For RMS error, the following could be written:
𝝈(𝑷𝑵𝑹 𝟏) = 𝟐𝜺 𝟐
𝑮𝒊𝒏
𝟐
(𝑳𝒊𝒏)√∫ 𝒅𝝀𝒅𝝀/ 𝑺(𝝀)𝑺(𝝀/)〈𝑰𝒎𝑲(𝝀)𝑹𝒆𝑲(𝝀)𝑰𝒎𝑲(𝝀/)𝑹𝒆𝑲(𝝀/)〉, (15)
where 𝑲(𝝀) ≡ 𝒂(𝑳, 𝝀)𝒌(𝝀). In Ref. [8], it is mentioned that 𝑰𝒎𝑲(𝝀) and 𝑹𝒆𝑲(𝝀) are uncorrelated. This
could be proved by above methods implemented for h-parameter, yielding 〈𝑰𝒎𝑲(𝝀)𝑹𝒆𝑲(𝝀)〉 ≈ 𝟎. Thus,
〈𝑰𝒎𝑲(𝝀)𝑹𝒆𝑲(𝝀)𝑰𝒎𝑲(𝝀/
)𝑹𝒆𝑲(𝝀/
)〉 ≈ 〈𝑰𝒎𝑲(𝝀)𝑰𝒎𝑲(𝝀/
)〉〈𝑹𝒆𝑲(𝝀)𝑹𝒆𝑲(𝝀/
)〉.
Consequently, the integral from Eq. (15) may be written as
∫ 𝒅𝒛𝒅𝒙𝒅𝒒𝒅𝒑𝑪(𝒛 − 𝒙)𝑪(𝒒 − 𝒑)𝑰(𝒛 − 𝒒)𝑰(𝒙 − 𝒑)
𝑳
𝟎
,

6. where 𝑰(𝒔) ≡ 𝐈𝐦[𝒆𝒙𝒑(𝒊𝜟𝜷 𝟎 𝒔)Г(𝒔)] (values with 𝒛 + 𝒒 and 𝒙 + 𝒑 yield zeros). Because of 𝒍 𝒄 ≪ 𝑳 𝜸 (see
above), correlators 𝑪(𝒛 − 𝒙) and 𝑪(𝒒 − 𝒑) are similar to Dirac delta-functions for Г(𝒛 − 𝒒) and Г(𝒙 − 𝒑), so
it is possible to write Г(𝒙 − 𝒑) ≈ Г(𝒛 − 𝒒). Unfortunately, this is not the case for values like 𝒂(𝒛 − 𝒒, 𝝀) and
𝒂(𝒙 − 𝒑, 𝝀), because 𝒍 𝒄 𝚫𝜷 𝟎 ≫ 𝟏. For symmetric spectrum, one, thus, has the following:
∫ 𝒅𝝀𝒅𝝀/
𝑺(𝝀)𝑺(𝝀/
)〈𝑰𝒎𝑲(𝝀)𝑰𝒎𝑲(𝝀/
)〉〈𝑹𝒆𝑲(𝝀)𝑹𝒆𝑲(𝝀/
)〉 ≈
≈
𝟏
𝟐
𝑹𝒆 ∫ 𝒅𝒛𝒅𝒒
𝑳
𝟎
Г 𝟐(𝒛 − 𝒒)𝑯(𝒛)𝑯∗(𝒒) ≈
𝟏
𝟐
𝑳 ∫ 𝒅𝒖
∞
𝟎
Г 𝟐(𝒖). (16)
Here the following was used:
𝑯(𝒛) ≡ ∫ 𝒅𝒙𝑪(𝒛 − 𝒙)𝒆𝒙𝒑[𝒊𝜟𝜷 𝟎(𝒛 − 𝒙)]
𝑳
𝟎
≈ 𝒉 − 𝒉(𝟏 + 𝒊𝜟𝜷 𝟎 𝒍 𝒄)𝒆𝒙𝒑[−(𝒍 𝒄
−𝟏
− 𝒊𝜟𝜷 𝟎)𝒛] ≈ 𝒉. (17)
This integral could be taken directly, i.e. substituting Eq. (12) for 𝑪(𝒛 − 𝒙) and dividing the integration as
∫ 𝒅𝒙(… )
𝒛
𝟎
+ ∫ 𝒅𝒙(… )
𝑳
𝒛
, so for the first of them 𝑪(𝒛 − 𝒙) = 𝒍 𝒄
−𝟏
𝒆𝒙𝒑[− (𝒛 − 𝒙) 𝒍 𝒄⁄ ], while for the second one
𝑪(𝒛 − 𝒙) = 𝒍 𝒄
−𝟏
𝒆𝒙𝒑[− (𝒙 − 𝒛) 𝒍 𝒄⁄ ]. The last approximation in (17) is made in the sense of integrating in Eq.
(16), i.e. 𝒉 will contribute into integral much larger than 𝒉(𝟏 + 𝒊𝜟𝜷 𝟎 𝒍 𝒄)𝒆𝒙𝒑[−(𝒍 𝒄
−𝟏
− 𝒊𝜟𝜷 𝟎)𝒛], because the
latter is zero for 𝒛 > 𝒍 𝒄. Thus, for Gaussian spectrum, Eq. (16) yields
𝝈(𝑷𝑵𝑹 𝟏) = 𝜺 𝟐
𝑮𝒊𝒏
𝟐
(𝑳𝒊𝒏)√𝟐𝑳 ∫ 𝒅𝒖
∞
𝟎
Г 𝟐(𝒖) = 𝜺 𝟐
𝑮𝒊𝒏
𝟐
(𝑳𝒊𝒏)√𝟐𝑳 ∫ 𝒅𝒖
∞
𝟎
𝒆𝒙𝒑[−𝟐(𝒖 𝑳 𝒅𝒆𝒄⁄ ) 𝟐] =
= 𝟎. 𝟓𝜺 𝟐
𝑮𝒊𝒏
𝟐
(𝑳𝒊𝒏)√√ 𝝅𝑳𝑳 𝒅𝒆𝒄 ≈ 𝟎. 𝟔𝟕𝜺 𝟐
𝑮𝒊𝒏
𝟐
(𝑳𝒊𝒏)√𝑳𝑳 𝒅𝒆𝒄
This agrees well with result from (1) in the absence of input lightguide ( 𝑮𝒊𝒏
𝟐
(𝑳𝒊𝒏) = 𝟏). For other spectra,
coefficient 0.67 will be replaced by some other ones, order of unity.
For 𝑷𝑵𝑹 𝟐 value from Eq. (2), using the above described method, one has
𝑷𝑵𝑹 𝟐 = 𝑷𝑵𝑹 𝟐
(𝒃𝒆𝒈)
+ 𝑷𝑵𝑹 𝟐
(𝒆𝒏𝒅)
,
𝑷𝑵𝑹 𝟐
(𝒃𝒆𝒈)
≡ 𝜺𝑮𝒊𝒏(𝑳𝒊𝒏)𝑹𝒆 ∫ 𝒅𝒛𝒄(𝒛)𝒆𝒙𝒑(−𝒊𝜟𝜷 𝟎 𝒛)Г(𝒛 + 𝒛 𝟎)
𝑳
𝟎
,
𝑷𝑵𝑹 𝟐
(𝒆𝒏𝒅)
= −𝜺𝑮𝒊𝒏(𝑳𝒊𝒏)𝑹𝒆 ∫ 𝒅𝒛𝒄(𝒛)𝒆𝒙𝒑(−𝒊𝜟𝜷 𝟎 𝒛)Г(𝑳 − 𝒛 − 𝒛 𝟎)
𝑳
𝟎
,
where 𝒛 𝟎 ≡ (𝑩𝒊𝒏 𝑳𝒊𝒏 + 𝑩 𝑰𝑶𝑪 𝑳 𝑰𝑶𝑪) 𝑩⁄ . These are two independent contributions to 𝑷𝑵𝑹 𝟐 from coil fibre initial
and final sections of the length order of 𝑳 𝒅𝒆𝒄 ≪ 𝑳 (basic coherence zones) where Г(𝒛 + 𝒛 𝟎) and Г(𝑳 − 𝒛 − 𝒛 𝟎)
may be non-zero. For 𝒛 𝟎 ≫ 𝑳 𝒅𝒆𝒄, one may write Г(𝒛 + 𝒛 𝟎) ≈ Г(𝑳 − 𝒛 − 𝒛 𝟎) ≈ 𝟎 (basic coherence zones
switching off [17], when x- and y-waves enter the coil being completely incoherent), and, thus, 𝑷𝑵𝑹 𝟐 ≈ 𝟎.
For 𝒛 𝟎 = 𝟎, using the above described procedures, one yields
𝑷𝑵𝑹 𝟐,𝟑~ 𝒔 𝟐,𝟑 𝜺𝑮𝒊𝒏(𝑳𝒊𝒏)√𝒉𝑳 𝒅𝒆𝒄 𝑺𝑭⁄
which agrees with that from Eq. (1) for minimal FRI ( 𝑮𝒊𝒏(𝑳𝒊𝒏) = 𝟏).
Similar to this, one may treat minimal FRI with PM-coil and 𝜽 𝟏,𝟐 = 𝟒𝟓 𝟎
at output splices (Fig. 1),
yielding the following:
𝝈 𝒎𝒊𝒏,𝟒𝟓(𝑷𝑵𝑹 𝟏) ~ 𝟒𝒔 𝟏 𝜺 𝟐
√𝒉𝑳 𝒅𝒆𝒄 𝑺𝑭⁄ , 𝝈 𝒎𝒊𝒏,𝟒𝟓(𝑷𝑵𝑹 𝟐) ~ 𝟒𝒔 𝟐 𝜺𝒉√𝑳 𝒅𝒆𝒄 𝑳 𝑺𝑭⁄ , 𝝈 𝒎𝒊𝒏,𝟒𝟓(𝑷𝑵𝑹 𝟑) ~ 𝟐𝒔 𝟑 𝜺√𝒉𝑳 𝒅𝒆𝒄 𝑺𝑭⁄ .
For 𝝈 𝒎𝒊𝒏,𝟒𝟓(𝑷𝑵𝑹 𝟏,𝟐), one yields smaller values than in Eq. (1), but 𝝈 𝒎𝒊𝒏,𝟒𝟓(𝑷𝑵𝑹 𝟑) = 2𝝈 𝒎𝒊𝒏(𝑷𝑵𝑹 𝟑), where
𝝈 𝒎𝒊𝒏(𝑷𝑵𝑹 𝟑) is from Eq. (1). This may be treated as PNR reduction for 𝑷𝑵𝑹 𝟏,𝟐, but not for 𝑷𝑵𝑹 𝟑.
References
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