theory and analysis of phase sensitivity-tunable optical sensor based on total internal reflection

Optics Communications 283 (2010) 4899–4906

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Optics Communications

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Discussion

Theory and analysis of phase sensitivity-tunable optical sensor based on totalinternal reflection

Jiun-You Lin ⁎Department of Mechatronics Engineering, National Changhua University of Education, No. 2, Shi-Da Road, Changhua City 50074, Taiwan, ROC

⁎ Tel.: +886 4 7232105x7235; fax: +886 4 7211149E-mail address: [email protected].

0030-4018/$ – see front matter © 2010 Elsevier B.V. Aldoi:10.1016/j.optcom.2010.08.007

a b s t r a c t
a r t i c l e i n f o
Article history:Received 8 March 2010Received in revised form 5 July 2010Accepted 3 August 2010

Keywords:SensorTotal-internal reflectionSensitivityRefractive index

This study presents a theory of a phase sensitivity-tunable optical sensor based on total-internalreflection (TIR). This investigation attempts to design a phase sensitivity-tunable optical sensorconsisting of an isosceles right-angle prism, some quarter- and half-wave plates, and a Mach-Zehnderinterferometer. When the azimuth angles of the quarter-wave plates are chosen properly, the final phasedifference of the two interference signals are associated with the azimuth angle of the fast axis of thehalf-wave plates, thus creating the controllable phase sensitivity. Numerical analysis demonstrates thatthe high phase measuring sensitivity and the small measuring range, and the low phase measuringsensitivity and the wide measuring range can be performed by selecting the suitable azimuth angle of thehalf-wave plates. The feasibility of the measuring method was demonstrated by the experiment results.The sensor could be applied in various fields, such as chemical, biological, biochemical sensing, andprecision machinery measurement.

.

l rights reserved.

© 2010 Elsevier B.V. All rights reserved.

1. Introduction

The optical phenomenon of total-internal reflection (TIR) occurswhen a ray of light strikes an interface between a denser and a rarermedium at an angle larger than the critical angle with respect to thenormal to the surface. This phenomenon leads the reflected wave tosuffer the phase shifts, which relates to the angle of incidence and therefractive indices (RI) of the media. Many optical sensing technologieshave employed the characteristics in detecting the changes of physicalparameters, such as refractive index change in solutions or gases [1–3],two-dimensional refractive index distribution of optical materials [4],surface and thin-film analysis [5,6], chromatic dispersion in opticalmaterials [7], angular variation in machine tools [8–11], and displace-ment in micro-mechanical electronic systems [12]. These methodsproduced good measurement results. Although some of these methodscan measure slight variations of the parameters and achieve highdetection sensitivity using amultiple total-internal reflection apparatus,the apparatus has the disadvantages of a large volume and weight[1,8–12]. These factors make the TIR apparatus difficult to use in somespace-limited area. In addition, to avoid the 2π phase jumping, thesystems only provide a narrow measurement range making themunsuitable for tracing large changes in parameters. To detect such

changes, the apparatus must be replaced by one with a few numbers ofreflections to expand the measurable range [2–7]. This process causesthe setup to be rearranged and recalibrated, and adds the operationalcomplexity. To improve these shortcomings, a method that can be usedin variousmeasuring conditions needs to be developed. However, to theauthor's knowledge, there areno references reported for this purpose. Inview of this, we designed a phase sensitivity-tunable optical sensorusing total-internal reflection effect and derived some theoreticalequations. A linearly polarized beam is guided to propagate through aphase sensitivity-tunable TIR apparatus, consisting of an isosceles right-angle prism, two half-wave plates, and two quarter-wave plates withsuitable azimuth angles. The beam exiting from the apparatus enters aMach-Zehnder interferometer with two acousto-optic modulatorsseparately situated in two arms. Finally, the two linearly orthogonallypolarizations of each output beam of the interferometer interfere witheach other as they pass through two analyzers, respectively. The finalphase difference between the two interference signals is associatedwiththe azimuth angle of fast axis of the half-wave plates, yielding thesensitivity-tunable functionality. The experimental results of phasedifferences obtained by this technique were well confirmed by thetheoretical analysis. Numerical calculations indicate that the refractiveindex sensitivity of (100–1.2×105°/RIU), and the measurementresolution of (9.7×10−5–8.2×10−8RIU) can be achieved. In additionto the tunable phase sensitivity and small size TIR apparatus, this sensoralso has the merits of high stability and high resolution due to itscommon-path configuration and heterodyne phase measurement.

http://dx.doi.org/10.1016/j.optcom.2010.08.007

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i

H2Q2

Q1H1

P

n

np

1

Fig. 1. Schematic diagram of a phase sensitivity-tunable TIR apparatus. H: half-waveplate, Q: quarter-wave plate, P: isosceles right-angle prismwith a refractive index np, n:refractive index of tested sample.

4900 J.-Y. Lin / Optics Communications 283 (2010) 4899–4906

2. Principle

2.1. Phase difference resulting from a phase sensitivity-tunable TIRapparatus

Fig. 1 displays the schematic diagram of a phase sensitivity-tunableTIR apparatus. For convenience, the +z-axis is set in the direction ofpropagation of light and the x-axis is perpendicular to the plane of thepaper. A linearlypolarized lightpasses throughahalf-waveplateH1 (thefast axis at Δ/2 to the x-axis), two quarter-wave plates Q1 and Q2 (theslow axes at 45° and 0° with respect to the x-axis, respectively), and isthen incident at θi on one side of an isosceles right-angle prism P with arefractive index of np. The light beam penetrates into the prism at anangle of incidence θ1 onto the interface between the prism and amedium with a refractive index of n. When θi exceeds θic, which is theangle that makes θ1 equal to the critical angle θ1c, the light is totallyreflected at the interface, and then the reflected light beam travelsthrough a half-wave plate H2 (the fast axis at α to the x-axis). The Jonesvector of the amplitude Et has the form:

Et =cos 2αð Þ sin 2αð Þsin 2αð Þ −cos 2αð Þ

!tpt

′pexp −iδt = 2ð Þ 0

0 ts t′s expi δt = 2ð Þ

!

×1 0

0 i

!1ffiffiffi2

p 1 −i

−i 1

!cosΔ

sinΔ

!

=1ffiffiffi2

p ðcos Δ + δt = 2ð Þ⋅ tacos2α + tbsin2αð Þ−isin Δ + δt = 2ð Þ⋅ tacos2α−tbsin2αð Þcos Δ + δt = 2ð Þ⋅ tasin2α−tbcos2αð Þ−isin Δ + δt = 2ð Þ⋅ tasin2α + tbcos2αð ÞÞ

=Ape

iϕp

Aseiϕs

0@

1A ð1Þ

M2

H2

AO

M2

B

I

Q2H1 Q1

Laser

n

Rotational Stage

Fig. 2. Schematic diagram of this designed optical sensor. PBS: polarization beamsplitter, M:mLIA: lock-in amplifier.

with

Ap =

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi12

ta cos 2αð Þ2 + tb sin 2αð Þ2 + tatb sin4α⋅ cos 2Δ + δtð Þ� �r; ð2Þ

As =

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi12

ta sin 2αð Þ2 + tb cos 2αð Þ2−tatb sin4α⋅ cos 2Δ + δtð Þ� �r; ð3Þ

ϕp = tan−1 − tan 45∘−σ� �

⋅ tan Δ + δt = 2ð Þ� �; ð4Þ

ϕs = tan−1 cot 45∘−σ ′� �

⋅ tan Δ + δt = 2ð Þh i

; ð5Þ

δt=2tan−1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffisin2 45∘ + sin−1 sinθi = np

� �h �i− n=np

� �2r

tan 45∘ + sin−1 sin θi = np

� �h i⋅ sin 45∘ + sin−1 sin θi = np

� �h i8>><>>:

9>>=>>;;

ð6Þ

tanσ =tbtatan 2α =

t2bt2a

tanσ ′; ð7Þ

ta = tpt′p ð8Þ

tb = tst′s ð9Þ

and

tp =2cosθi

cos θi = np

� �+

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1− cos θi =np

� �2r ; ð10Þ

ts =2cosθi

cosθi + n


� �2r ; ð11Þ

t′p =2= np

� �⋅ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1− cos θi =np

� �2r

cos θi = np

� �+


� �2r ; ð12Þ

t′s =2 = np


� �2r

cosθi + 1= np


� �2r ; ð13Þ

M1

D1AN1

I2 I1

AOM1

D2

AN2

S

PBS

LIA

irror, AOM: acousto-optic modulator, BS: beamsplitter, AN: analyzer, D: photodetector,

4901J.-Y. Lin / Optics Communications 283 (2010) 4899–4906

where the values of (tp, ts) and (t′p, t′s) are the transmission coefficientsat the air–prism and the prism–air interface, respectively, and δt arethe phase differences between s- and p-polarized total-internalreflections at the interface of the prism-medium [12–14]. In Eqs. (4)and (5), the constantΔ, used to shift the phase level of δt/2,must be setas about −δt,max/4, where δt,max denotes the maximum value of thephase difference δt and is given by [14]

δt;max = 2 tan−1np =n� �2−1

2 np = n� �

264

375: ð14Þ

The level modulation allows the relative curves of θi versus ϕp andϕs to be more linear than those without modulation. Eqs. (4)–(7)illustrate that the phase shiftsϕp andϕs clearly depend on the azimuthangleα. If n and θi are specified,ϕp andϕs can be alerted by selectingα.

2.2. Principle of phase difference detection

Fig. 2 shows the schematic diagramof this designed optical sensor. Alaser light passes through the phase sensitivity-tunable TIR apparatusand the output beam of the apparatus enters a Mach-Zehnder

4 6 8 10 12 14 16 18 20 22 24-200

-150

-100

-50

0

50

100

150

200

Phas

e di

ffer

ence

(de

g.)

Phas

e di

ffer

ence

(de

g.)

i (deg.)io ic

1

2

345

678

9

= 23

= 25

= 30= 35= 45

10

= 0= 10= 15

= 20

= 22

O : Phase difference t

: Phase difference

4 6 8 10 12 14 16 18 20 22 24-200

-150

-100

-50

0

50

100

150

200

i (deg.)io ic

1

2

345

678

9

= 23

= 25

= 30= 35= 45

10

= 0= =

=

=

1015

20

22

O : Phase difference t

: Phase difference

a

b

Fig. 3. Phase difference ψ versus incident angle θi for different azimuth angle α.(a) 0°≤α≤45° and (b) −45°≤α≤0°.

interferometer. Two acousto-optic modulators are situated in the twoarms of the interferometer, respectively. Light in the interferometerafter been split by BS travels in two paths: (a) BS→AOM1→M1→PBSand (b) BS→AOM2→M2→PBS. The s-polarized light of path (a) andthe p-polarized light of path (b) at PBS are superimposed to produce theamplitude E1 which is to end at the detector D1:

E1 =1ffiffiffi2

p Apei ω2t + ϕpð Þ

Asei ω1t + ϕsð Þ

!; ð15Þ

whereas, the p-polarized light of path (a) and the s-polarized light ofpath (b) are also summed to bear the amplitude E2 which is to end atthe detector D2:

E2 =1ffiffiffi2

p Apei ω1t + ϕpð Þ


!: ð16Þ

where ω1 and ω2 are the angular frequencies of output beams ofAOM1 and AOM2, respectively.

4 6 8 10 12 14 16 18 20 22 240

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Am

plitu

de

θ (deg)

• : Amplitude Ap

: Amplitude As

123

4

5

6

78

123

4

α = 22°, 23°α = 20°, 25°α = 10°, 15°,

30°, 35°

8 α = 22°, 23°

567 α = 20°, 25°

α = 15°, 30°α = 10°, 35°α = 0°, 45°

θioθic

4 6 8 10 12 14 16 18 20 22 240

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1• : Amplitude As

: Amplitude Ap

123

45678 α = −22°, −23°

α = −22°, −23°

α = −20°, −25°

α = −20°, −25°

α = −15°, −30°

α = −10°, −15°−30°, −35°

α = −10°, −35°α = 0°, −45°

Am

plitu

de

θ (deg)

123

4

5

6

78

θioθic

a

b

Fig. 4. Amplitude Ap and As versus incident angle θi for different azimuth angle α. (a)0°≤α≤45° and (b) −45°≤α≤0°.


After passing through an analyzer AN1 with the transmission axisbeing at β to the x-axis, E1 becomes E′1 and is detected by D1:

E1 =cos2 β sin β cos β

sin β cos β sin2 β

!12

Apei ω2t + ϕpð Þ


0@

1A

=1ffiffiffi2

p cos β⋅Apei ω2t + ϕpð Þ + sin β⋅Ase

i ω1t + ϕsð Þh i cos β

sin β

!:

ð17Þ

The intensity measured by D1 is therefore

I1 = jE1j2 =12

Ap cosβ� �2

+ As sinβð Þ2 + sin2β⋅ApAs cos ωt + ϕð Þ

;

ð18Þ

where ϕ=ϕp−ϕs, and ω=ω2−ω1. On the other hand, E2 passingthrough an analyzer AN2 (with the transmission axis being at β to thex-axis) becomes E′2 and is detected by D2 with

E2 =cos2 β sin β cos β

sin β cos β sin2 β

!12

Apei ω1t + ϕpð Þ


0@

1A

=1ffiffiffi2

p cosβ⋅Apei ω1t + ϕpð Þ + sinβ⋅Ase

i ω2t + ϕsð Þh i cos β

sin β

!:

ð19Þ

6.5 7 7.5 8 8.5 9-200

-150

-100

-50

0

50

100

150

200

1

23

46

5

78

910

11

1. = 22.5°

2. = 22.4°

3. = 22.3°

4. = 22.2°

5. = 22.1°

6. = 22.0°

7. = 23.0°

8. = 22.9°

9. = 22.8°

10. = 22.7°

11. = 22.6°

io

6.5 7 7.5 8 8.5 9-200

-150

-100

-50

0

50

100

150

200

1

23

4

6

5

78

910

11

1.

2.

3.

5.

6.

7.

8.

9.

10.

11.

i (deg.)

= 22.6°

= 22.7°

= 22.8°

= 22.9°

= 23.0°

= 22.0°

= 22.1°

= 22.2°

= 22.3°

= 22.4°

= 22.5°

4.

i (deg.)

ψ(d

eg.)

ψ(d

eg.)

a

b

θ

io θ

Fig. 5. Phase difference ψ versus incident angle θi for different azimuth angle α aroundθio. (a) 22°≤α≤23° and (b) −23°≤α≤−22°.

The intensity of the beam is

I2 = jE2j2 =12

Ap cosβ� �2

+ As sinβð Þ2 + sin2β⋅ApAs cos ωt−ϕð Þ

:

ð20Þ

The intensities I1 and I2 are sent to a lock-in amplifier (LIA) forphase analysis [15], enabling the final phase difference ψ=2ϕ to beaccurately determined.

Although ψ is independent of β, Eqs. (18) and (20) reveal that thecontrast of I1 and I2 depends on β. To increase the contrast of I1 and I2,either of the two following conditions should apply: (i) α is closed to22.5° and β closed to 90°; (ii) α is closed to−22.5° and β closed to 0°.

3. Results and discussions

3.1. Results of numerical analysis

This section describes numerical analysis. The incident lightwavelength was 633 nm, and np=1.7786 and n=1.33 were chosen,respectively. Using Eq. (14), δt,max=32.8° can be derived and Δ=−8.2° was hence selected in the simulation. Fig. 3 depicts thesimulated results by plotting the relations of the incident angle θi

6.5 7 7.5 8 8.5 9 9.5 100

0.01

0.02

0.03

0.04

0.05

0.06

0.07

A s

123456

1. = 23.0

2. = 22.9 , 22

3. = 22.8 , 22.1

4. = 22.7 , 22.2

5. = 22.6 , 22.3

6. = 22.5 , 22.4

6.5 7 7.5 8 8.5 9 9.5 100

0.01

0.02

0.03

0.04

0.05

0.06

0.07

123456

1.

2.

3.

4.

5.

6.

A p

i (deg.)

i (deg.)

= -22.0

= -22.1 , -23.0

= -22.2 , -22.9

= -22.3 , -22.8

= -22.4 , -22.7

= -22.5 , -22.6

a

b

io θ

io θ

Fig. 6. Amplitude Ap and As versus incident angle θi for different azimuth angle α aroundθio. (a) As with 22°≤α≤23° and (b) Ap with −23°≤α≤−22°.


versus the phase difference ψ of the reflected light at differentazimuth angles α of the half-wave plate H2. As shown in Fig. 3(a) and(b), operating the angleα between 0° and±45° can regulate the slopeof the curve of θi versus ψ. With α=±22° or ±23°, ψ notably exhibitssharp variations around the incident angle θio at which the phase

1.32 1.34 1.36 1.38 1.4 1.42 1.44 1.46-40

-30

-20

-10

0

10

20

30

40a

1. = 18.0°

2. = 18.5°

3. = 19.0°

4. = 19.5°

5. = 20.0°

1

2

345

n

1.33 1.34 1.35 1.36 1.37 1.38 1.39 1.4 1.41-60

-40

-20

0

20

40

60

80

n

1

2

3

4

5

1.33 1.3305 1.331 1.3315 1.332-200

-150

-100

-50

0

50

100

150

200

n

12

3

4

5

θi

θi

θi

θi

θi

1. = 12.0

2. = 12.5°

3. = 13.0°

4. = 13.5°

5. = 14.0°

θi

θi

θi

θi

θi

1. = 7.6°

2. = 7.7°

3. = 7.8°

4. = 7.9°

5. = 8.0°

θi

θi

θi

iθθi

ψ(d

eg.)

ψ(d

eg.)

ψ(d

eg.)

c

e

Fig. 7. Phase difference ψ versus refractive index n and incident angle θi (a) α=0°, (b) αα=22.5°, (g) α=±25°, (h) α=±30°, (i) α=±35°, and (j) α=±45°.

shifts (Δ+δt∕2)=0°. For comparison, the relation of the incidentangle θi versus and the phase difference δt between s- and p-polarizedlight under single total-internal reflection are marked as “o” in Fig. 3.The curve of δt reveals the condition of small, unchangeable phasedifference variation. Additionally, Fig. 4 shows the plots of the

1.32 1.34 1.36 1.38 1.4 1.42 1.44-40

-30

-20

-10

0

10

20

30

40

50

1

2

3

4

5

n

1.33 1.335 1.34 1.345 1.35 1.355 1.36 1.365 1.37 1.375-100

-50

0

50

100

150

1

2

3

4

5

n

1.33 1.3305 1.331 1.3315 1.332-200

-150

-100

-50

0

50

100

150

200

1

2

3

45

n

1. = 15.0

2. = 15.5°

3. = 16.0°

4. = 16.5°

5. = 17.0°

θi

θi

θi

θi

θi

1. = 10.0°

2. = 10.5°

3. = 11.0°

4. = 11.5°

5. = 12.0°

θi

θi

θi

θi

θi

1. = 7.6°

2. = 7.7°

3. = 7.8°

4. = 7.9°

5. = 8.0°

θi

θi

θi

θθi

ψ(d

eg.)

ψ(d

eg.)

ψ(d

eg.)

b

d

f

=±10°, (c) α=±15°, (d) α=±20°, (e) α=−22.5° and 22.4°, (f) α=−22.6° and

1.33 1.335 1.34 1.345 1.35 1.355 1.36 1.365 1.37 1.375-150

-100

-50

0

50

100

5

4

3

2

1

n1.33 1.34 1.35 1.36 1.37 1.38 1.39 1.4 1.41

-80

-60

-40

-20

0

20

40

60

5

4

3

2

1

n

1.32 1.34 1.36 1.38 1.4 1.42 1.44-50

-40

-30

-20

-10

0

10

20

30

40

n

5

4

3

2

1

1.32 1.34 1.36 1.38 1.4 1.42 1.44 1.46-40

-30

-20

-10

0

10

20

30

40

n

54

3

2

1

1. = 10.0°

2. = 10.5°

3. = 11.0°

4. = 11.5°

5. = 12.0°

θi

θi

θi

θi

θi

1. = 12.0°

2. = 12.5°

3. = 13.0°

4. = 13.5°

5. = 14.0°

θi

θi

θi

θi

θi

1. = 15.0

2. = 15.5°

3. = 16.0°

4. = 16.5°

5. = 17.0°

θi

θi

θi

θi

θi

1. = 18.0°

2. = 18.5°

3. = 19.0°

4. = 19.5°

5. = 20.0°

θi

θi

θi

θi

θi

ψ(d

eg.)

ψ(d

eg.)

ψ(d

eg.)

ψ(d

eg.)

g h

i j

Fig. 7 (continued).


reflection coefficients Ap and As of the reflected beam. Clearly, theseresults indicate that the values of As with α=22° or 23° and Ap withα=−22° or −23° is small. Figs. 5 and 6 further illustrate the sharpvariation ranges of phase difference ψ and reflection coefficients (Ap,As) in the vicinity of θio, respectively. As indicated in the two figures,the reflection coefficients Ap, and As evidently have a strong diparound θio, and correspondingly, the phase difference ψ changesdramatically when 22°≤α≤23° or −23°≤α≤−22°.

Since Eqs. (4)–(9) exhibit that the phase difference ψ depends onthe refractive index n of the contact medium, this condition suggeststhat the sensing system can be designed for measuring phasevariations with respect to changes in the refractive index of thesample. Fig. 7 shows the relations of n versus ψ at various α and θi.The simulated results have the tendency for sharp phase differencevariations in the region of 22°≤α≤23° or −23°≤α≤−22°,especially in Fig. 7(e) and (f) indicating that phase difference ψwith α=−22.6°, 22.4°, or ±22.5° at θi=7.8° has more noticeablephase difference changes and better linearity than those at other θi.The condition can lead the sensor to exhibit very high measurementsensitivity for determining the slight variation of n but limit themeasurable range. Fig. 7(a)–(d) and (g)–(j) indicates that themeasurement range of n is extended and the phase differencevariation of ψ is decreased, as α approaches 0° or ±45°. The

operations for expanding measurement range are suitable forestimating the large variation in the refractive index of n.

To estimate the RI sensitivity of this sensor, we differentiate theequation ψ=2ϕ=2(ϕp−ϕs), and, thus, the sensitivity S can berelated as follows:

S = j ∂ψ∂n j: ð21Þ

According to the results of Fig. 7(a), (e), (f), and (j), the relations ofthe sensitivity S versusn are plotted in Fig. 8. Fig. 8(a) and (b) shows thatthe highest RI sensitivity can reach about 1.2×105°/RIU as α=−22.6°,22.4°, or ±22.5° at θi=7.8°. Fig. 8(c) also reveals that the lowestsensitivity is about100°/RIUwithα=0°or±45° at θi=18°. Besides, themeasurement resolution of this system can be defined as

Δnerr = j ∂n∂ψ jΔψ; ð22Þ

where Δnerr and Δψ are the errors in n and ψ, respectively. When thesecond harmonic error and the polarization-mixing error [16–19] areconsidered, the net phase difference error Δψ declines to about 0.01°.Substituting Δψ=0.01° and the results of Fig. 7(a), (e), (f), and (j) into

1.33 1.3305 1.331 1.3315 1.3320

2

4

6

8

10

a12

x 104

1

2

3

45

n

1.33 1.3305 1.331 1.3315 1.3320

2

4

6

8

10

12

14

1

2

3

45

n

1.32 1.34 1.36 1.38 1.4 1.42 1.44 1.46100

150

200

250

300

350

400

450

500

550 1

2

3

4

5

n

S (d

eg./R

IU)

S (d

eg./R

IU)

S (d

eg./R

IU)

x 104

1. = 7.9°

2. = 7.8°

3. = 8.0°

4. = 7.7°

5. = 7.6°

θi

θi

θi

θi

θi

1. = 7.9°

2. = 7.8°

3. = 8.0°

4. = 7.7°

5. = 7.6°

θi

θi

θi

θi

θi

1. = 18.0°

2. = 18.5°

3. = 19.0°

4. = 19.5°

5. = 20.0°

θi

θi

θi

θi

iθ i

b

c

Fig. 8. Sensitivity S versus refractive index n and incident angle θi (a) α=±22.5°, (b)α=−22.6° and 22.4°, (c) α=0° and ±45°.

1.33 1.3305 1.331 1.3315 1.3320

0.5

1

1.5

2

2.5

3

3.51

2

3

4

5

n

1.33 1.3305 1.331 1.3315 1.3320

0.5

1

1.5

2

2.5

3

3.5

1

2

3

4

5

n

1.32 1.34 1.36 1.38 1.4 1.42 1.44 1.46

2

4

6

8

10

1

2345

n

x 10-6

x 10-6

x 10-5

Δner

rΔn

err

Δner

r

1. = 7.6°

2. = 7.7°

3. = 8.0°

4. = 7.8°

5. = 7.9°

θi

θi

θi

θi

θi

1. = 7.6°

2. = 7.7°

3. = 8.0°

4. = 7.8°

5. = 7.9°

θi

θi

θi

θi

θi

1. = 18.0°

2. = 18.5°

3. = 19.0°

4. = 19.5°

5. = 20.0°

θi

θi

θi

θi

iθ

a

b

c

Fig. 9. Resolution Δnerrversus refractive index n and incident angle θi (a) α=±22.5°,(b) α=−22.6° and 22.4°, (c) α=0° and ±45°.


Eq. (22), the relations of the resolutionsΔnerr versus n depicted in Fig. 9.Fig. 9(a) and (b) indicates that the best resolutionΔnerr≅8.2×10−7 canbe achieved when α=−22.6°, 22.4°, or ±22.5° at θi=7.8°, and thelowest resolution is about 9.7×10−5 as α=0° or ±45° at θi=18°. Theresults of Figs. 8 and 9 are also summarized in Table 1.

3.2. Experimental results

To validate the approach, a SF11 isosceles right-angle prism withnp=1.7786 was used as a TIR apparatus, and pure water was injectedinto a cell on the base of the prism. The apparatus and the testedsample were mounted tighter on high-precision rotational stage(Model M-URM100PP, New focus) with an angular resolution 0.001°.

Table 1TheRI sensitivity, themeasurement resolution, and themeasurable range forα=(−22.6°,22.4°, ±22.5°), and α=(0°, ±45°).

θi S (°/RIU) Δnerr (×10−6) Measurable range

α=−22.6°, 22.4°,±22.5°

7.8° 25,000–120,000 0.082–0.356 1.33–1.3317

α=0°, ±45° 20° 100–246 41–97 1.33–1.44

5 10 15 20-200

-150

-100

-50

0

50

100

150

200

θi (deg.)

ψ(d

eg.)

x : α = 0°

∗ : α = −20°

o : α = −20°

Fig. 10. Measurement results and theoretical curves of ψ versus θi.


The 632.8 nm line from a He–Ne laser served as the light source. In theMach-Zehnder interferometer, the two linearly orthogonally polar-izations modulated by two acousto-optic modulators (Model AOM-40, IntraAction) of each output beam have the frequency difference60 kHz. A lock-in amplifier (LIA) (Model SR830, Stanford) with anangular resolution 0.01° was applied to measure the phase difference.In order to estimate the value of Δ used to shift the phase level of δt/2,the conditions of Δ=0 and α=0° were first chosen (from Eqs. (4),(5), and ψ=2ϕ, the phase difference ψ can be simplified as ψ=−2δt).Then, the sample was slowly rotated to identify the angle of θicoccurred at the abrupt change of the phase difference ψ. The angle wasmeasuredwith θic=48.529°, and n=1.3327 can be obtained based onSnell's law. Using Eq. (14), the constant Δ≅−δt,max /4=−8.2° canalso be estimated. Next, the azimuth angle of H1 was set to meet theconstantΔ≅−8.2°, and the phase differenceψ versus incident angle θiwas measured with α=0°, −15°, and −22°. In order to increase thecontrast, β=45°, 20°, and 1° were also chosen, respectively. Fig. 10plots the measurements and theoretical results, respectively, where“○”, “×”, and “*” represent the measured data, and the solid linesrepresent the theoretical calculations. The experimental resultsconfirm the theoretically predicted curve that the sensitivity istunable by controlling the orientation of fast axis of H2.

4. Conclusion

This study derived the equations of the totally reflected light phasedifferences of p- and s-polarizations in a phase sensitivity-tunable opticalsensor. The sensor is composed of an isosceles right-angle prism, somewave plates, and a Mach-Zehnder interferometer. Based on the phasedifference equations, this study examined the relation of incident angleand refractive index of medium versus the phase difference of the twooutput beams at various azimuth angles of fast axis of thehalf-wave platein this system. The sensitivity of refractive index was found to be alteredby the azimuth angle of the half-wave plate. The experimental resultsmatchedwellwith the theoretical analysis. Theextent of theRI sensitivityand the measurement resolution can reach 100–1.2×105°/RIU, and9.7×10−5–8.2×10−8RIU, respectively.

Acknowledgment

The author would like to thank the National Science Council of theRepublic of China, Taiwan, for financially supporting this researchunder Contract No. 97-2221-E-018-003.

References

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theory and analysis of phase sensitivity-tunable optical sensor based on total internal reflection

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