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Supporting Information
Integrated optical gyroscope using active Long-range surface
plasmon-polariton waveguide resonator
*Tong Zhang1, 2, Guang Qian1, 2, Yang-Yang Wang1, 2, Xiao-Jun Xue1, Feng Shan1, 2,
Ruo-Zhou Li1, 2, Jing-Yuan Wu1, 2 & Xiao-Yang Zhang1, 2
1School
of Electronic Science and Engineering, Southeast University, and Key Laboratory of
Micro-Inertial Instrument and Advanced Navigation Technology, Ministry of Education, Nanjing,
210096, People’s Republic of China
2
Suzhou Key Laboratory of Metal Nano-Optoelectronic Technology, Suzhou Research Institute of
Southeast University, Suzhou, 215123, People’s Republic of China
Theoretical modeling of resonant optical gyroscope’s sensitivity
Generally speaking, in a gyroscope with gain medium, the spontaneous emission’s
impact on the sensitivity is greater than that of shot noise, becoming a major limit to
gyroscope’s sensitivity1. But for the situation with deep loss compensation, the shot
noise also exposes non-neglectful effects on the gyroscope’s limited sensitivity.
The sensitivity of resonator gyro limited by SNR is expressed as2
 
a L
2 f FWHM
4A
SNR
;
(1)
where a is input light wavelength, L is the perimeter of the resonator, A is the area
enclosed by the ring, f FWHM is the full width of half maximum (FWHM) of the
resonant peak and SNR is signal to noise ratio.
When the limiting noise of the system is shot noise, the SNR of shot noise is given
by2
SNR 
 D D Pin
hv

Tmax - Tmin
(2)
Tmax
where  D is quantum efficiency of photo detector,  D is integration time of photo
detector , Pin is input optical power, h is the Planck's constant, v is the frequency of
the signal, T represents the transmittance and Tmax is maximum and Tmin is the
minimum value of T. Combining Eq. (1) and Eq. (2) yields the shot noise limited
sensitivity (SNSL), i.e.,
  (
a L
4A
)
2f FWHM
 D D Pin

Tmax  Tmin
hv
(3)
Tmax
When the gain medium is added to the gyroscope, the main limit of the
gyroscope’s sensitivity is the spontaneous emission1. For simplicity, following
assumptions are applied.
Firstly, with 982 nm pump light, the atomic system of Er3+ is a three level system,
i.e. 4 I11/ 2 , 4 I13/ 2 and 4 I15 / 2 3. The photon’s lifetime at 4 I11/ 2 is about 1 ns, which is far
smaller than that 10 ms of 4 I13/ 2 . Thus, the atomic system is approximate to a two-level
system. The population of atomics at 4 I13/ 2 and 4 I15 / 2 is N2 and zero respectively
according to Ref. [4].
Secondly, the total gain is approximately equal to the total loss (including the
propagation loss of the waveguide and the insert loss of the coupler) by carefully
controlling the pumping power. According to Ref. [4]
E2  e

 net
2
L
e
 j L
E3
(4)
where E2 is the amplitude of the output electric field of the straight waveguide, E3 is
the amplitude of the electric field coupled from the ring resonator into the straight
waveguide,  is the propagation constant and  net is the coefficient of net loss
which represents the propagation loss of the LRSPP waveguide with gain. The x in
Ref.4 should be rewritten as follow:
xe

 net
2
L
(5)
Finally, the atomic transition assumed to be purely homogeneously broadened
with a center frequency of fa. Thus, the transition line shape is the Lorentzian line
shape with a FWHM of Δωa and the stimulated emission cross section is donated
by  e , the photon lifetime of spontaneous emission is  se .
Note that the loss of the LRSPP ring resonator without gain
 intr
is not assumed to
be very low though the bending loss can be neglected when the ring’s radius is big
enough5,6, which is different from Ref. [4].
Based on the assumption above, the expression of the total change of phase of the
optical signal’s field after M spontaneous emissions in a duration time of  D is
   ( D )  
2
M ( D )
(6)
2n
where  n  is the average number of photons within the ring. Next, we obtained 4
M ( D ) 
 D N2
 sp s
 n 
3
，s 
N eff Pc L
2hf a c
2 N V
eff
a
2
， Pc 
c
1

a
FPin
(7)
(8)
where V is the volume of the ring resonator, N eff is the effective refractive index of
the ring, Pc is the power of light in the ring, F is the finesse of the gyroscope. As
mentioned above, for LRSPP ring resonator, the loss of the waveguide without gain,
intrinsic loss, is not neglected and gain is needed to compensate the loss. Under
vertical pumping 7,8. The expression of net loss is given by
G net  exp(  net L)  exp(  intr   e N 2 / V ) L)  1
(9)
We obtained
N2 
V
e
 intr
(10)
with 4
e 
1 1
a
2
(11)
2  sp N eff a
Combining Eq. (6), (7), (8), (10), and (11), we can obtain
   ( D )  
2
 D c hf a intr
2
(12)
2
N eff LPin F
Finally, the spontaneous emission limited sensitivity (SESL) is given by
  intr Lhf a 
se 


4 2 AN eff  FPin D 
a c
1.
1/ 2
(13)
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