More accurate fiber-optic Fabry-Pérot sensors modeling · Tests 17 Measured and calculated spectral density of fiber-optic Fabry-Pérot0.0 sensor with empty cavity 0.1 0.2 0.3 0.4

More accurate fiber-optic

Fabry-Pérot sensors modeling

Jerzy Pluciński

Daria Majchrowicz

Katarzyna Karpienko

Department of Metrology and Optoelectronics,

Faculty of Electronics, Telecommunications and Informatics,

Gdańsk University of Technology, Narutowicza 11/12, 80-233 Gdańsk

Presentation outline

2

• Introduction,

• Fabry-Pérot interferometer,

• Fiber-optic Fabry-Pérot sensors,

• Mathematical modeling,

• Measurement system,

• Tests,

• Conclusions.

Introduction

3

Motivations:

1. The Fabry-Pérot cavities used in fiber-optic sensors are very small – width is

equal to the diameter of the fiber, length is from several hundred nanometers to

several hundred micrometers.

2. The fiber-optic Fabry-Pérot sensors can measure many physical, chemical, and

biomedical parameters by their effects on the cavity length or the refractive index

of the substance inside the cavity.

3. These sensors can be highly resistant to disturbances of the optical path (e.g. in

attenuation of fiber-optic path), when information about measurement quantity is

encoded in spectrum.

4. To estimate the cavity length, the refractive index or attenuation coefficient of the

substance inside the cavity with high accuracy, we need an accurate and

effective mathematical model of propagation of the optical radiation inside the

Fabry-Pérot cavity and from the cavity to the core of the optical fiber.

Fabry-Pérot interferometer

4

A typical Fabry-Pérot interferometer uses the cavity inside which

the plane wave propagates inside the cavity and reflects many

times from planar mirrors that forming the cavity.

inin IU

2

outout UI

U0(λ)

U1(λ)

U2(λ)

U3(λ)

Uin(λ)

Uout(λ)

n1(λ) ncav(λ)

n3(λ)

L

r12(λ) r21(λ)

r23(λ)

t12(λ) t21(λ)

…

Iout(λ)

Iin(λ)

µa(λ)

tcav(λ)

,3,2,1,4

exp in

21

2112cavcav2123

iUr

tt

q

LnjtrrU

i

i

in

21

2112

12out 1

Urq

ttqrU

Geometric progression

in

2

21

2112

12out 1

Irq

ttqrI

Fiber-optic Fabry-Pérot sensors

5

Fiber-optic Fabry-Pérot sensors uses the cavity inside which the

other wave than the plane wave propagates along the cavity and

reflects many times from mirrors that forming the cavity.

U0(λ)

U1(λ)

U2(λ)

U3(λ)

Uin(λ)

Uout(λ)

n1(λ)

ncav(λ) n3(λ)

L

…

Iout(λ)

Iin(λ)

n2(λ) cladding

core

cavity

ci(λ)

µa(λ)

Consequences:

• Diffraction of beams

in the cavity;

• Coupling between the

wave in the cavity and

in the optical fiber;

• Extra phase shift (e.g.

as the result of Gouy

effect);

• Curvature of wave

front.

Fiber-optic Fabry-Pérot sensors

6

0

out

i

iUU

in120 UrU

2

outout UI

inin IU

This is not a geometric progression!

This factor depends on the coupling coefficient of a wave

from the cavity to the optical fiber and depends on i!

,3,2,1,

4exp in

21

2112cavcav2123

iUc

r

ttLnjtrrU i

i

i

U0(λ)

U1(λ)

U2(λ)

U3(λ)

Uin(λ)

Uout(λ)

n1(λ)

ncav(λ) n3(λ)

L

…

Iout(λ)

Iin(λ)

n2(λ) cladding

core

cavity

ci(λ)

µa(λ)

Mathematical modeling

7

The key problem of mathematical modeling of

fiber-optic Fabry-Pérot sensors is to find the

coupling coefficient between the wave in the

cavity and in the optical fiber.

What we should to known:

• The electromagnetic field distribution in the optical fiber Ufiber(x, y).

• The electromagnetic field distribution in the cavity Ucavity(x, y).


8

If we know Ufiber(x, y) and Ucavity(x, y) then we can

find the coupling coefficient from:

dxdyyxUyxUdxdyyxUyxU

dxdyyxUyxUc

,,,,

,,

*

fiberfiber

*

cavitycavity

*

fibercavity

coup


9

The electromagnetic field distribution in the optical fiber Ufiber(x, y)

for single-mode optical fiber is given by Bessel functions:

arrkJrU for ~, T0fiber

arrKrU for ~, 0fiber

22

1T

2 nk

22

2

2 n

2

22

1

2 1ln2

a

Vn

NA

aV

2

2

2

2

1 nnNA


10

The electromagnetic field distribution in the cavity Ucavity(x, y) we

can find from Kirchhoff's diffraction formula:

dxdyyxUnr

e

r

e

nyxUyxPU

S

jkrjkr

,,

4

1, fiberfibercavity

Problems:

• Calculation of the coupling coefficient needs huge amount of computer computations.

• We need to calculate of the coupling coefficient for many wavelength.


11

Question:

Is it an easier method to estimate Ufiber(x, y) and

Ucavity(x, y) needed to find the coupling coefficient?

dxdyyxUyxUdxdyyxUyxU

dxdyyxUyxUc

,,,,

,,

*

fiberfiber

*

cavitycavity

*

fibercavity

coup

Answer:

Yes, if we approximate the field distribution at the

end of fiber Ufiber(x, y) by Gaussian distribution

and the field distribution in the cavity Ucavity(x, y)

by Gaussian beam.


12

Approximation of the field distribution at the end

of the fiber Ufiber(x, y) by the Gaussian distribution:

The fundamental-mode size width w0 of a step-index waveguide (or optical fiber) of normalized frequency V (0.8 < V < 2.5) is given by:

D. Marcuse, "Loss analysis of single-mode fiber splices," Bell. Syst.

Tech. J., vol. 56, pp. 703-718, May-June 1977.

2

0

2

0fiber exp,w

BU r

62/30

879.2619.165.0

VVaw

2

2

2

1 nnNA

NA

aV

2

222 yx

2

0

in0

2

w

IB


13

The field distribution in the cavity (Gaussian beam):

B. E. A. Saleh, M. C. Teich: Fundamentals of Photonics, 2nd Ed.,

John Wiley & Sons, NY, 2007.

,

,2exp

,exp

,,

2

2

2

00cavity zj

zRjkzjk

zwzw

wAU i

iii

ii

2

0

0

2 1,

z

zwzwi

0

arctan,z

zzi

2

01,z

zzzRi

cav

2

00

nwz

cav2 n

k

iLz 2

Phase shift that

is responsible

for Goue effect


14

The coupling coefficient between the wave in the

cavity and in the optical fiber

2

0

coup/1

exp

wkjz

zjkc

i

,3,2,1,

4exp in

21

2112cavcav2123

iUc

r

ttLnjtrrU i

i

i

iLz 2

where:

So:

cav2 n

k

2

0cav2/21

1

wniLjci

J. Pluciński, K. Karpienko: Fiber-optic Fabry-Pérot sensors –modeling versus measurements results. SPIE Proc., 2016, in print.

Measurement system

15

Tests

16

Measured spectral density of source 1290 nm

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1200 1220 1240 1260 1280 1300 1320 1340 1360 1380

Norm

aliz

ed s

pectr

al density

Wavelength [nm]

Tests

17

Measured and calculated spectral density of fiber-optic

Fabry-Pérot sensor with empty cavity

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1200 1220 1240 1260 1280 1300 1320 1340 1360 1380

Norm

aliz

ed s

pectr

al d

ensity

Wavelength [nm]

Calculated

Measured

Calculated for L=144.820 μm

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.10

1230 1232 1234 1236 1238 1240 1242

Norm

aliz

ed s

pectr

al density

Wavelength [nm]

Calculated

Measured


0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.10

1328 1330 1332 1334 1336 1338 1340

Norm

aliz

ed s

pectr

al density

Wavelength [nm]

Calculated

Measured


The calculated and measured spectra are well

matched for L=144.820 μm.

This value was calculated by fitting the

calculated and measured spectra by the least

squares method (global minimum).

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.10

1328 1330 1332 1334 1336 1338 1340

Norm

aliz

ed s

pectr

al density

Wavelength [nm]

Calculated

Measured


0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.10

1230 1232 1234 1236 1238 1240 1242

Norm

aliz

ed s

pectr

al density

Wavelength [nm]

Calculated

Measured


0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1200 1220 1240 1260 1280 1300 1320 1340 1360 1380

Norm

aliz

ed s

pectr

al density

Wavelength [nm]

Calculated

Measured


Tests



The distance between the fringes of the calculated

spectrum is larger than the distance between the

fringes of the measured spectrum, so obtained

L=144.175 μm is too small (local minimum).

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.10

1328 1330 1332 1334 1336 1338 1340

Norm

aliz

ed s

pectr

al density

Wavelength [nm]

Calculated

Measured


0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.10

1230 1232 1234 1236 1238 1240 1242

Norm

aliz

ed s

pectr

al density

Wavelength [nm]

Calculated

Measured


0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1200 1220 1240 1260 1280 1300 1320 1340 1360 1380

Norm

aliz

ed s

pectr

al density

Wavelength [nm]

Calculated

Measured


Tests



The distance between the fringes of the calculated

spectrum is smaller than the distance between the

fringes of the measured spectrum, so obtained

L=145.464 μm is too big (local minimum).

Tests


Fabry-Pérot sensor with cavity filled by distillated water

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1200 1220 1240 1260 1280 1300 1320 1340 1360 1380

No

rma

lize

d s

pe

ctr

al d

en

sity

Wavelength [nm]

Calculated

Measured

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1284 1286 1288 1290 1292 1294

Norm

aliz

ed s

pectr

al density

Wavelength [nm]

Calculated

Measured

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.10

1230 1232 1234 1236 1238 1240 1242

Norm

aliz

ed s

pectr

al density

Wavelength [nm]

Calculated

Measured

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.10

1328 1330 1332 1334 1336 1338 1340

Norm

aliz

ed s

pectr

al density

Wavelength [nm]

Calculated

Measured

a) b)

c)

d)

Calculated and measured spectra of water assuming that the refractive index of

measured water is 0.0221% smaller than given in literature, where L = 144.820 µm.

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1200 1220 1240 1260 1280 1300 1320 1340 1360 1380

Norm

aliz

ed s

pectr

al density

Wavelength [nm]

Calculated

Measured

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1284 1286 1288 1290 1292 1294

Norm

aliz

ed s

pectr

al density

Wavelength [nm]

Calculated

Measured

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.10

1230 1232 1234 1236 1238 1240 1242

No

rma

lize

d s

pe

ctr

al d

en

sity

Wavelength [nm]

Calculated

Measured

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.10

1328 1330 1332 1334 1336 1338 1340

No

rma

lize

d s

pe

ctr

al d

en

sity

Wavelength [nm]

Calculated

Measured

a) b)

c)

d)

Tests

Measured and calculated spectral density of fiber-optic Fabry-

Pérot sensor with for 10% aqueous solution of ethylene glycol

The measured refractive index of 10% aqueous solution of ethylene glycol in the studied

wavelength range was about 0.74% higher than refractive index of distilled water.

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1200 1220 1240 1260 1280 1300 1320 1340 1360 1380

No

rma

lize

d s

pe

ctr

al d

en

sity

Wavelength [nm]

Calculated

Measured

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1284 1286 1288 1290 1292 1294

Norm

aliz

ed

sp

ectr

al d

en

sity

Wavelength [nm]

Calculated

Measured

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.10

1230 1232 1234 1236 1238 1240 1242

No

rma

lize

d s

pe

ctr

al d

en

sity

Wavelength [nm]

Calculated

Measured

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.10

1328 1330 1332 1334 1336 1338 1340

No

rma

lize

d s

pe

ctr

al d

en

sity

Wavelength [nm]

Calculated

Measured

a) b)

c)

d)

Tests


Fabry-Pérot sensor with for 20% aqueous ink solution

The measured refractive index of 20% aqueous solution of ink in the studied wavelength

range was about 0.3852% higher than refractive index of distilled water, μa = 6.25 mm–1.

Measurement accuracy

and resolution

23

• The optical spectrum analyzer can measure the spectrum with wavelength accuracy Δλ = ±0.02 nm and with 0.001 nm resolution.

• For that accuracy, the measurement accuracy of the length of the Fabry-Pérot cavity and the refractive index of the tested substance are ΔL = ±12 nm and 10–5 RIU (refractive index unit), respectively.

• It is possible to measure the length of the Fabry-Pérot cavity and the refractive index of tested substances with the resolution of 1 nm and 10–6 RIU, respectively

• The measurement accuracy of the reflectance metallic surface of the cavity that is about 1%. For this condition the measurement accuracy of the absorption coefficient is ±0.015 mm–1.

Conclusions

24

• Modeling of the operation of the sensing interferometer, including the mode field diameter of a single mode fiber, dependence of the diameter of the laser beam, phase shift from the Gouy effect, the curvature of the wave front, refractive index and absorption of the medium inside the cavity of the interferometer, was conducted.

• Performed measurements of the length of the Fabry-Pérot cavity and the refractive index of liquids are characterized by a remarkable accuracy.

• Despite the fact that developed interferometer is optimized for measurements of refractive index, it can be also used for measurements of the absorption coefficient.

25

Thank you for yourkind attention !!!

Documents

More accurate fiber-optic Fabry-Pérot sensors modeling · Tests 17 Measured and calculated spectral density of fiber-optic Fabry-Pérot0.0 sensor with empty cavity 0.1 0.2 0.3 0.4