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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 j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / o p t c o m
Discussion
Theory and analysis of phase sensitivity-tunable optical sensor based on total internal reflection Jiun-You Lin ⁎ Department of Mechatronics Engineering, National Changhua University of Education, No. 2, Shi-Da Road, Changhua City 50074, Taiwan, ROC
a r t i c l e
i n f o
Article history: Received 8 March 2010 Received in revised form 5 July 2010 Accepted 3 August 2010 Keywords: Sensor Total-internal reflection Sensitivity Refractive index
a b s t r a c t This study presents a theory of a phase sensitivity-tunable optical sensor based on total-internal reflection (TIR). This investigation attempts to design a phase sensitivity-tunable optical sensor consisting of an isosceles right-angle prism, some quarter- and half-wave plates, and a Mach-Zehnder interferometer. When the azimuth angles of the quarter-wave plates are chosen properly, the final phase difference of the two interference signals are associated with the azimuth angle of the fast axis of the half-wave plates, thus creating the controllable phase sensitivity. Numerical analysis demonstrates that the high phase measuring sensitivity and the small measuring range, and the low phase measuring sensitivity and the wide measuring range can be performed by selecting the suitable azimuth angle of the half-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, and precision machinery measurement. © 2010 Elsevier B.V. All rights reserved.
1. Introduction The optical phenomenon of total-internal reflection (TIR) occurs when a ray of light strikes an interface between a denser and a rarer medium at an angle larger than the critical angle with respect to the normal to the surface. This phenomenon leads the reflected wave to suffer the phase shifts, which relates to the angle of incidence and the refractive indices (RI) of the media. Many optical sensing technologies have employed the characteristics in detecting the changes of physical parameters, 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 optical materials [7], angular variation in machine tools [8–11], and displacement in micro-mechanical electronic systems [12]. These methods produced good measurement results. Although some of these methods can measure slight variations of the parameters and achieve high detection sensitivity using a multiple 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 some space-limited area. In addition, to avoid the 2π phase jumping, the systems only provide a narrow measurement range making them unsuitable for tracing large changes in parameters. To detect such
⁎ Tel.: +886 4 7232105x7235; fax: +886 4 7211149. E-mail address: [email protected]. 0030-4018/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2010.08.007
changes, the apparatus must be replaced by one with a few numbers of reflections to expand the measurable range [2–7]. This process causes the setup to be rearranged and recalibrated, and adds the operational complexity. To improve these shortcomings, a method that can be used in various measuring conditions needs to be developed. However, to the author's knowledge, there are no references reported for this purpose. In view of this, we designed a phase sensitivity-tunable optical sensor using total-internal reflection effect and derived some theoretical equations. A linearly polarized beam is guided to propagate through a phase sensitivity-tunable TIR apparatus, consisting of an isosceles rightangle prism, two half-wave plates, and two quarter-wave plates with suitable azimuth angles. The beam exiting from the apparatus enters a Mach-Zehnder interferometer with two acousto-optic modulators separately situated in two arms. Finally, the two linearly orthogonally polarizations of each output beam of the interferometer interfere with each other as they pass through two analyzers, respectively. The final phase difference between the two interference signals is associated with the azimuth angle of fast axis of the half-wave plates, yielding the sensitivity-tunable functionality. The experimental results of phase differences obtained by this technique were well confirmed by the theoretical analysis. Numerical calculations indicate that the refractive index sensitivity of (100–1.2 × 105°/RIU), and the measurement resolution of (9.7 × 10− 5–8.2 × 10−8 RIU) can be achieved. In addition to the tunable phase sensitivity and small size TIR apparatus, this sensor also has the merits of high stability and high resolution due to its common-path configuration and heterodyne phase measurement.
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J.-Y. Lin / Optics Communications 283 (2010) 4899–4906
H1
with
Q1
Q2
H2 Ap =
P i
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ðt cos 2αÞ2 + ðtb sin 2αÞ2 + ta tb sin4α⋅ cosð2Δ + δt Þ ; ð2Þ 2 a
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ðt sin 2αÞ2 + ðtb cos 2αÞ2 −ta tb sin4α⋅ cosð2Δ + δt Þ ; As = 2 a
1
np n
−1
∘ − tan 45 −σ ⋅ tanðΔ + δt = 2Þ ;
ð4Þ
−1
h i ∘ ′ cot 45 −σ ⋅ tanðΔ + δt = 2Þ ;
ð5Þ
ϕp = tan
Fig. 1. Schematic diagram of a phase sensitivity-tunable TIR apparatus. H: half-wave plate, Q: quarter-wave plate, P: isosceles right-angle prism with a refractive index np, n: refractive index of tested sample.
ϕs = tan
2. Principle
δt =2 tan
−1
2.1. Phase difference resulting from a phase sensitivity-tunable TIR apparatus
×
!
cosð2αÞ
sinð2αÞ
sinð2αÞ
−cosð2αÞ
1
0
0
i
!
1 pffiffiffi 2
1
−i
−i 1
0
0
ts ts′ expiðδt = 2Þ
!
cosΔ
iϕp
Ap e As e
iϕs
′
ð8Þ
′
ð9Þ
and 2 cosθi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi tp = 2ffi ; cos θi = np + 1− cos θi = np
ts =
2 cosθi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ffi ; cosθi + n 1− cos θi =np
′
sinΔ
Þ ′
ð1Þ
ts =
A
ð11Þ
ð12Þ
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ffi 2 = np ⋅ 1− cos θi =np rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi;
ð13Þ
1− cos θi = np
2
cosθi + 1 = np ⋅ 1− cos θi = np
H1 Q1 Q2
ð10Þ
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ffi 2 = np ⋅ 1− cos θi =np rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi;
tp = cos θi = np +
1
ð7Þ
tb = ts ts
!
ð
0
9 2 > > sin2 45∘ + sin−1 sinθi = np − n=np = h i h i ; −1 −1 > ∘ ∘ > > sin θi = np ⋅ sin 45 + sin sin θi = np > :tan 45 + sin ;
ta = tp tp
1 cosðΔ + δt = 2Þ⋅ðta cos2α + tb sin2αÞ−isinðΔ + δt = 2Þ⋅ðta cos2α−tb sin2αÞ = pffiffiffi 2 cosðΔ + δt = 2Þ⋅ðta sin2α−tb cos2αÞ−isinðΔ + δt = 2Þ⋅ðta sin2α + tb cos2αÞ =@
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h i ffi
tb t2 ′ tan 2α = b2 tan σ ; ta ta
tan σ =
!
tp tp′ expð−iδt = 2Þ
8 > > <
ð6Þ
Fig. 1 displays the schematic diagram of a phase sensitivity-tunable TIR apparatus. For convenience, the +z-axis is set in the direction of propagation of light and the x-axis is perpendicular to the plane of the paper. A linearly polarized light passes through a half-wave plate H1 (the fast axis at Δ/2 to the x-axis), two quarter-wave plates Q1 and Q 2 (the slow axes at 45° and 0° with respect to the x-axis, respectively), and is then incident at θi on one side of an isosceles right-angle prism P with a refractive index of np. The light beam penetrates into the prism at an angle of incidence θ1 onto the interface between the prism and a medium with a refractive index of n. When θi exceeds θic, which is the angle that makes θ1 equal to the critical angle θ1c, the light is totally reflected at the interface, and then the reflected light beam travels through a half-wave plate H2 (the fast axis at α to the x-axis). The Jones vector of the amplitude Et has the form:
Et =
ð3Þ
2
H2 BS
Laser
Rotational Stage
I
AOM2
n
M1
AOM1
AN1
PBS
D1
M2 AN2 D2 I2
I1
LIA Fig. 2. Schematic diagram of this designed optical sensor. PBS: polarization beamsplitter, M: mirror, AOM: acousto-optic modulator, BS: beamsplitter, AN: analyzer, D: photodetector, LIA: lock-in amplifier.
J.-Y. Lin / Optics Communications 283 (2010) 4899–4906
where the values of (tp, ts) and (t′p, t′s) are the transmission coefficients at the air–prism and the prism–air interface, respectively, and δt are the phase differences between s- and p-polarized total-internal reflections at the interface of the prism-medium [12–14]. In Eqs. (4) and (5), the constant Δ, used to shift the phase level of δt/2, must be set as about − δt,max/4, where δt,max denotes the maximum value of the phase difference δt and is given by [14]
δt;max
2 3 2 np = n −17 −1 6 5: = 2 tan 4 2 np = n
ð14Þ
The level modulation allows the relative curves of θi versus ϕp and ϕs to be more linear than those without modulation. Eqs. (4)–(7) illustrate that the phase shifts ϕp and ϕs clearly depend on the azimuth angle α. If n and θi are specified, ϕp and ϕs can be alerted by selecting α.
4901
interferometer. Two acousto-optic modulators are situated in the two arms of the interferometer, respectively. Light in the interferometer after been split by BS travels in two paths: (a) BS→ AOM1 → M1 → PBS and (b) BS→ AOM2 → M2 → PBS. The s-polarized light of path (a) and the p-polarized light of path (b) at PBS are superimposed to produce the amplitude E1 which is to end at the detector D1: Ap eiðω2 t
1 E1 = pffiffiffi 2
+ ϕp Þ
iðω1 t + ϕs Þ
As e
! ;
ð15Þ
whereas, the p-polarized light of path (a) and the s-polarized light of path (b) are also summed to bear the amplitude E2 which is to end at the detector D2: Ap eiðω1 t
+ ϕp Þ
!
2.2. Principle of phase difference detection
1 E2 = pffiffiffi 2
Fig. 2 shows the schematic diagram of this designed optical sensor. A laser light passes through the phase sensitivity-tunable TIR apparatus and the output beam of the apparatus enters a Mach-Zehnder
where ω1 and ω2 are the angular frequencies of output beams of AOM1 and AOM2, respectively.
a
: Phase difference
50
1
= 23
2
= 25
3 4 5
= 30 = 35 = 45
6 7 8
=0 = 10 = 15
9
= 20
10
= 22
0 -50
8 ic
10
12
io
14
16
18
20
22
4
0.6 0.5 5
0.4
0
4
150 2
= 25
3 4 5
= 30 = 35 = 45
-50
6 7 8
=0 = 10 = 15
-100
9
= 20
10
= 22
0
-150
ic
io
14
16
18
20
14
22
24
i (deg.)
Fig. 3. Phase difference ψ versus incident angle θi for different azimuth angle α. (a) 0° ≤ α ≤ 45° and (b) − 45° ≤ α ≤ 0°.
16
18
20
22
: Amplitude As
1 α = −22°, −23° 2 α = −20°, −25° 3 α = −10°, −15° −30°, −35° 4 α = 0°, −45° 5 α = −10°, −35° 6 α = −15°, −30° 7 α = −20°, −25° 8 α = −22°, −23°
0.6 0.5 5
0.4
6
0.2
7 8
0.1 0
24
θ (deg)
4
0.3
12
12
: Amplitude Ap
0.7
Amplitude
50
10
10
1 2 3
0.8
100
8
8
θio
•
= 23
0.9
6
θic
1 1
4
6
b
: Phase difference
200
7 8
0.1
24
O : Phase difference t
6
0.2
i (deg.)
b
1 α = 22°, 23° 2 α = 20°, 25° 3 α = 10°, 15°, 30°, 35° 4 α = 0°, 45° 5 α = 10°, 35° 6 α = 15°, 30° 7 α = 20°, 25° 8 α = 22°, 23°
0.7
-150
Phase difference (deg.)
: Amplitude As
0.3
6
: Amplitude Ap
1 2 3
0.8
-100
-200
•
0.9
Amplitude
Phase difference (deg.)
100
4
ð16Þ
1
150
-200
:
a
O : Phase difference t
200
iðω2 t + ϕs Þ
As e
4
6
θic
8
θio
10
12
14
16
18
20
22
24
θ (deg)
Fig. 4. Amplitude Ap and As versus incident angle θi for different azimuth angle α. (a) 0° ≤ α ≤ 45° and (b) − 45° ≤ α ≤ 0°.
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J.-Y. Lin / Optics Communications 283 (2010) 4899–4906
After passing through an analyzer AN1 with the transmission axis being at β to the x-axis, E1 becomes E′1 and is detected by D1: 1 ! 0 i ω t + ϕp Þ sin β cos β 1 Ap e ð 2 @ A 2 sin2 β As eiðω1 t + ϕs Þ
2
cos β sin β cos β
1 h i ω t = pffiffiffi cos β⋅Ap e ð 2 2
+ ϕp Þ
iðω1 t + ϕs Þ
+ sin β⋅As e
2
I2 = jE2 j =
i cos β
:
The intensity measured by D1 is therefore 2
I1 = jE1 j =
2 1 2 Ap cos β + ðAs sin βÞ + sin2β⋅Ap As cos ðωt + ϕÞ ; 2
ð18Þ where ϕ = ϕp − ϕs, and ω = ω2 − ω1. On the other hand, E2 passing through an analyzer AN2 (with the transmission axis being at β to the x-axis) becomes E′2 and is detected by D2 with cos2 β
E2 =
sin β cos β
sin β cos β sin2 β
1 h i ω t = pffiffiffi cosβ⋅Ap e ð 1 2
a
+ ϕp Þ
1 ! 0 iðω t + ϕp Þ 1 @ Ap e 1 A 2 As eiðω2 t + ϕs Þ iðω2 t + ϕs Þ
+ sinβ⋅As e
!
ð19Þ
:
sin β
1
1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.
2 3
100
4 5 6
50 0 -50 7
-100
8 9 10 11
-150 -200 6.5
The intensities I1 and I2 are sent to a lock-in amplifier (LIA) for phase analysis [15], enabling the final phase difference ψ = 2ϕ to be accurately determined. Although ψ is independent of β, Eqs. (18) and (20) reveal that the contrast 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 to 22.5° 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 light wavelength 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 the simulated results by plotting the relations of the incident angle θi
a
200 150
ψ (deg.)
i cos β
ð20Þ
ð17Þ
!
sin β
7
7.5
8
θ io
8.5
2 1 2 Ap cos β + ðAs sin βÞ + sin2β⋅Ap As cos ðωt−ϕÞ : 2
22.5°
= = 22.4° = 22.3° = 22.2° = 22.1° = 22.0° = 23.0° = 22.9° = 22.8° = 22.7° = 22.6°
0.07 0.06 0.05 0.04
As
E1 =
The intensity of the beam is
1. 2. 3. 4. 5. 6.
0.03 0.02
1 2 3 4 5 6
0.01 0 6.5
9
7
7.5
θ io
8
i (deg.)
b
200 1
150
3
100
ψ (deg.)
1. 2. 3.
2 4
5
4. 5. 6. 7. 8. 9. 10. 11.
50 0 -50 6 7 8 9 10 11
-100 -150 -200 6.5
7
7.5
θ io
9
9.5
, 22 , 22.1 , 22.2 , 22.3 , 22.4
10
i (deg.)
8
8.5
= 22.6° = 22.7° = 22.8° = 22.9° = 23.0° = 22.0° = 22.1° = 22.2° = 22.3° = 22.4° = 22.5°
9
i (deg.)
Fig. 5. Phase difference ψ versus incident angle θi for different azimuth angle α around θio. (a) 22° ≤ α ≤ 23° and (b) − 23° ≤ α ≤ − 22°.
0.07 0.06 0.05 0.04
Ap
b
8.5
= 23.0 = 22.9 = 22.8 = 22.7 = 22.6 = 22.5
1.
0.03 0.02
1 2 3 4 5 6
0.01 0 6.5
2. 3.
7
7.5
θ io
4. 5. 6.
8
8.5
9
= -22.0 = -22.1 = -22.2 = -22.3 = -22.4 = -22.5
9.5
, -23.0 , -22.9 , -22.8 , -22.7 , -22.6
10
i (deg.)
Fig. 6. Amplitude Ap and As versus incident angle θi for different azimuth angle α around θio. (a) As with 22° ≤ α ≤ 23° and (b) Ap with − 23° ≤ α ≤ − 22°.
J.-Y. Lin / Optics Communications 283 (2010) 4899–4906
versus the phase difference ψ of the reflected light at different azimuth angles α of the half-wave plate H2. As shown in Fig. 3(a) and (b), operating the angle α between 0° and ±45° can regulate the slope of the curve of θi versus ψ. With α = ±22° or ±23°, ψ notably exhibits sharp variations around the incident angle θio at which the phase
a
shifts (Δ + δt ∕ 2) = 0°. For comparison, the relation of the incident angle θi versus and the phase difference δt between s- and p-polarized light under single total-internal reflection are marked as “o” in Fig. 3. The curve of δt reveals the condition of small, unchangeable phase difference variation. Additionally, Fig. 4 shows the plots of the
b
40 1
30
2 3 4 5
1. θ i = 18.0° 2. θ i = 18.5° 3. θ i = 19.0°
10 0
4. θ i = 19.5° 5. θ i = 20.0°
-10
40
1
30
2
10 0
1. θ i = 15.0 2. θ i = 15.5° 3. θ i = 16.0° 4. θ i = 16.5° 5. θ i = 17.0°
-30 1.34
1.36
1.38
1.4
1.42
1.44
-40 1.32
1.46
1.34
1.36
c
d
80
1.42
1.44
150 1
100
2
1. θ i = 12.0 2. θ i = 12.5° 3. θ i = 13.0°
3
13.5°
5
2
ψ (deg.)
ψ (deg.)
1.4
1
60
0
1.38
n
n
20
4 5
-20
-30
40
3
-10
-20
-40 1.32
50
20
ψ (deg.)
ψ (deg.)
20
4903
4
4. θ i = 5. θ i = 14.0°
50
0
10.0°
1. θ i = 2. θ i = 10.5° 3. θ i = 11.0°
3
4. θ i = 11.5° 5. θ i = 12.0°
5
4
-20 -50 -40 -60 1.33
1.34
1.35
1.36
1.37
1.38
1.39
1.4
-100 1.33 1.335 1.34 1.345 1.35 1.355 1.36 1.365 1.37 1.375
1.41
n
n
e
f
200
200 1 2
150
150
3
50 0 -50
1. θ i = 7.6° 2. θ i = 7.7° 3. θ i = 7.8° 4. θ i = 7.9° 5. θ i = 8.0°
4
50 0 -50
-100
1. θ i = 7.6° 2. θ i = 7.7° 3. θ i =
2
7.8°
4. θ = 7.9° 5. θ i = 8.0°
-100 3
5
-150 -200 1.33
1
100
ψ (deg.)
ψ (deg.)
100
-150 1.3305
1.331
n
1.3315
1.332
-200 1.33
4 5
1.3305
1.331
1.3315
1.332
n
Fig. 7. Phase difference ψ versus refractive index n and incident angle θi (a) α = 0°, (b) α = ± 10°, (c) α = ± 15°, (d) α = ±20°, (e) α = − 22.5° and 22.4°, (f) α = − 22.6° and α = 22.5°, (g) α = ± 25°, (h) α = ±30°, (i) α = ±35°, and (j) α = ± 45°.
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J.-Y. Lin / Optics Communications 283 (2010) 4899–4906
g
h 100
60 40
50
0
-50
4. θ i = 11.5° 5. θ i = 12.0°
5 4
ψ (deg.)
ψ (deg.)
20 1. θ i = 10.0° 2. θ i = 10.5° 3. θ i = 11.0°
1. θ i = 12.0° 2. θ i = 12.5° 3. θ i = 13.0°
0
5
4. θ i = 13.5° 5. θ i = 14.0°
-20
3
4 3
-40
2
2
-100 -60
1
1
-80 1.33
-150 1.33 1.335 1.34 1.345 1.35 1.355 1.36 1.365 1.37 1.375
1.34
1.35
1.36
1.37
n
i
1.38
1.39
1.4
1.41
n
j
40
40 30 30 20
0
20 1. θ i = 15.0 2. θ i = 15.5° 3. θ i = 16.0°
-10 -20
5 4
4. θ i = 16.5° 5. θ i = 17.0°
3
-30
2
-40
1
10
1. θ i = 18.0° 2. θ i = 18.5°
0
3. θ i = 19.0°
-10
4. θ i = 19.5° 5. θ i = 20.0°
ψ (deg.)
ψ (deg.)
10
5 4 3
-20
2
-50 1.32
1.34
1.36
1.38
1.4
1.42
-30
1
-40 1.32
1.44
1.34
n
1.36
1.38
1.4
1.42
1.44
1.46
n Fig. 7 (continued).
reflection coefficients Ap and As of the reflected beam. Clearly, these results indicate that the values of As with α = 22° or 23° and Ap with α = −22° or −23° is small. Figs. 5 and 6 further illustrate the sharp variation 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 dip around θio, and correspondingly, the phase difference ψ changes dramatically when 22° ≤ α ≤ 23° or −23° ≤ α ≤ −22°. Since Eqs. (4)–(9) exhibit that the phase difference ψ depends on the refractive index n of the contact medium, this condition suggests that the sensing system can be designed for measuring phase variations with respect to changes in the refractive index of the sample. Fig. 7 shows the relations of n versus ψ at various α and θi. The simulated results have the tendency for sharp phase difference variations in the region of 22° ≤ α ≤ 23° or − 23° ≤ α ≤ − 22°, especially in Fig. 7(e) and (f) indicating that phase difference ψ with α = − 22.6°, 22.4°, or ± 22.5° at θi = 7.8° has more noticeable phase difference changes and better linearity than those at other θi. The condition can lead the sensor to exhibit very high measurement sensitivity for determining the slight variation of n but limit the measurable range. Fig. 7(a)–(d) and (g)–(j) indicates that the measurement range of n is extended and the phase difference variation of ψ is decreased, as α approaches 0° or ± 45°. The
operations for expanding measurement range are suitable for estimating the large variation in the refractive index of n. To estimate the RI sensitivity of this sensor, we differentiate the equation ψ = 2ϕ = 2(ϕp − ϕs), and, thus, the sensitivity S can be related as follows: S=
j j
∂ψ : ∂n
ð21Þ
According to the results of Fig. 7(a), (e), (f), and (j), the relations of the sensitivity S versus n are plotted in Fig. 8. Fig. 8(a) and (b) shows that the 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 lowest sensitivity is about 100°/RIU with α = 0° or ±45° at θi = 18°. Besides, the measurement resolution of this system can be defined as Δnerr =
j j
∂n Δψ; ∂ψ
ð22Þ
where Δnerr and Δψ are the errors in n and ψ, respectively. When the second harmonic error and the polarization-mixing error [16–19] are considered, 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
J.-Y. Lin / Optics Communications 283 (2010) 4899–4906
a 12
a
x 104
3.5
x 10-6 1
1
2.5
6
Δnerr
1. θ i = 7.9° 2. θ i = 7.8° 3. θ i = 8.0°
8
1. θ i = 7.6° 2. θ i = 7.7° 3. θ i = 8.0°
3
10
S (deg./RIU)
4905
7.7°
4. θ i = 5. θ i = 7.6°
4
4. θ i = 7.8° 5. θ i = 7.9°
2 1.5
2
1 2
2
4 5
0 1.33
1.3305
1.331
1.3315
3
0.5
3
4 5
0 1.33
1.332
1.3305
1.331
n
b
b
3.5
4
14
x 10
1. θ i = 7.6° 2. θ i = 7.7° 3. θ i = 8.0°
1
8
Δnerr
S (deg./RIU)
1. θ i = 7.9° 2. θ i = 7.8° 3. θ i = 8.0° 4. θ i = 7.7° 5. θ i = 7.6°
6
1
4. θ i = 7.8° 5. θ i = 7.9°
2.5 10
2 1.5
2
1
4
3
0.5
2
2
4
3
1.3305
1.331
1.3315
5
0 1.33
4 5
0 1.33
1.3305
1.331
c
550
1.3315
1.332
n
1.332
n
c
1.332
x 10-6
3 12
1.3315
n
1
x 10-5
10
500 8
400
2
1. θ i = 18.0° 2. θ i = 18.5° 3. θ i = 19.0°
350
3
19.5°
4. θ i = 5. θ i =i 20.0°
300 250
Δnerr
S (deg./RIU)
450
6
1. θ i = 18.0° 2. θ i = 18.5°
4
3. θ i = 19.0°
4
5
200
4. θ i = 19.5° 5. θ i = 20.0°
5 4 3 2 1
2
150 100 1.32
1.32 1.34
1.36
1.38
1.4
1.42
1.44
1.34
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°.
1.38
1.4
1.42
1.44
1.46
n
1.46
n
1.36
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°.
3.2. Experimental results Eq. (22), the relations of the resolutions Δnerr versus n depicted in Fig. 9. Fig. 9(a) and (b) indicates that the best resolution Δnerr ≅ 8.2 × 10− 7 can be achieved when α = −22.6°, 22.4°, or ±22.5° at θi = 7.8°, and the lowest resolution is about 9.7× 10− 5 as α = 0° or ±45° at θi = 18°. The results of Figs. 8 and 9 are also summarized in Table 1.
To validate the approach, a SF11 isosceles right-angle prism with np = 1.7786 was used as a TIR apparatus, and pure water was injected into a cell on the base of the prism. The apparatus and the tested sample were mounted tighter on high-precision rotational stage (Model M-URM100PP, New focus) with an angular resolution 0.001°.
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J.-Y. Lin / Optics Communications 283 (2010) 4899–4906
Table 1 The RI sensitivity, the measurement resolution, and the measurable range for α = (−22.6°, 22.4°, ±22.5°), and α = (0°, ±45°). S (°/RIU)
Δnerr (×10− 6)
Measurable range
7.8°
25,000–120,000
0.082–0.356
1.33–1.3317
20°
100–246
41–97
x : α = 0° ∗ : α = −20° o : α = −20°
150 100
ψ (deg.)
α = − 22.6°, 22.4°, ±22.5° α = 0°, ± 45°
θi
200
1.33–1.44
50 0 -50
The 632.8 nm line from a He–Ne laser served as the light source. In the Mach-Zehnder interferometer, the two linearly orthogonally polarizations modulated by two acousto-optic modulators (Model AOM40, IntraAction) of each output beam have the frequency difference 60 kHz. A lock-in amplifier (LIA) (Model SR830, Stanford) with an angular 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 θic occurred at the abrupt change of the phase difference ψ. The angle was measured with θic = 48.529°, and n = 1.3327 can be obtained based on Snell's law. Using Eq. (14), the constant Δ ≅ − δt,max / 4 = − 8.2° can also be estimated. Next, the azimuth angle of H1 was set to meet the constant Δ ≅ − 8.2°, and the phase difference ψ versus incident angle θi was measured with α = 0°, −15°, and −22°. In order to increase the contrast, β = 45°, 20°, and 1° were also chosen, respectively. Fig. 10 plots the measurements and theoretical results, respectively, where “○”, “×”, and “*” represent the measured data, and the solid lines represent the theoretical calculations. The experimental results confirm the theoretically predicted curve that the sensitivity is tunable by controlling the orientation of fast axis of H2. 4. Conclusion This study derived the equations of the totally reflected light phase differences of p- and s-polarizations in a phase sensitivity-tunable optical sensor. The sensor is composed of an isosceles right-angle prism, some wave plates, and a Mach-Zehnder interferometer. Based on the phase difference equations, this study examined the relation of incident angle and refractive index of medium versus the phase difference of the two output beams at various azimuth angles of fast axis of the half-wave plate in this system. The sensitivity of refractive index was found to be altered by the azimuth angle of the half-wave plate. The experimental results matched well with the theoretical analysis. The extent of the RI sensitivity and the measurement resolution can reach 100–1.2 ×105°/RIU, and 9.7 ×10− 5–8.2 ×10−8 RIU, respectively.
-100 -150 -200
5
10
15
20
θ i (deg.) Fig. 10. Measurement results and theoretical curves of ψ versus θi.
Acknowledgment The author would like to thank the National Science Council of the Republic of China, Taiwan, for financially supporting this research under Contract No. 97-2221-E-018-003.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19]
M.H. Chiu, J.Y. Lee, D.C. Su, K.H. Lee, Precis. Eng. 23 (1999) 260. S. Patskovsky, M. Meunier, A.V. Kabashin, Opt. Express 15 (2007) 12523. S. Sainov, V. Sainov, G. Stoilov, Appl. Opt. 34 (1995) 2848. Z.C. Jian, P.J. Hsieh, H.C. Hsieh, H.W. Chen, D.C. Su, Opt. Commun. 268 (2006) 23. M. Poksinski, H. Arwin, Sens. Actuator B 94 (2003) 247. H. Arwin, M. Poksinski, K. Johansen, Appl. Opt. 43 (2004) 3028. M.H. Chiu, C.W. Lai, S.F. Wang, D.C. Su, S. Chang, Appl. Opt. 45 (2006) 6781. W. Zhou, L. Cai, Appl. Opt. 37 (1998) 5957. W. Zhou, L. Cai, Appl. Opt. 38 (1999) 1179. A. Zhang, P.S. Huang, Appl. Opt. 40 (2001) 1617. M.H. Chiu, S.F. Wang, R.S. Chang, Appl. Opt. 43 (2004) 5438. S.F. Wang, M.H. Chiu, W.W. Chen, F.H. Kao, R.S. Chang, Appl. Opt. 48 (2009) 2566. E. Hecht, Optics, fourth ed, Addison Wesley, San Francisco, 2002, p. 113. S.S. Huard, Polarization of Light, John Wiley & Sons, New York, 1997, p. 212. D.C. Su, M.H. Chiu, C.D. Chen, J. Opt. 27 (1996) 19. N.M. Oldham, J.A. Kramar, P.S. Hetrick, E.C. Teague, Precis. Eng. 15 (1993) 173. J.M. De Freitas, M.A. Player, Meas. Sci. Technol. 4 (1993) 1173. W. Hou, G. Wilkening, Precis. Eng. 14 (1992) 91. A.E. Rosenbluth, N. Bobroff, Precis. Eng. 12 (1990) 7.
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Optics Communications j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / o p t c o m
Discussion
Theory and analysis of phase sensitivity-tunable optical sensor based on total internal reflection Jiun-You Lin ⁎ Department of Mechatronics Engineering, National Changhua University of Education, No. 2, Shi-Da Road, Changhua City 50074, Taiwan, ROC
a r t i c l e
i n f o
Article history: Received 8 March 2010 Received in revised form 5 July 2010 Accepted 3 August 2010 Keywords: Sensor Total-internal reflection Sensitivity Refractive index
a b s t r a c t This study presents a theory of a phase sensitivity-tunable optical sensor based on total-internal reflection (TIR). This investigation attempts to design a phase sensitivity-tunable optical sensor consisting of an isosceles right-angle prism, some quarter- and half-wave plates, and a Mach-Zehnder interferometer. When the azimuth angles of the quarter-wave plates are chosen properly, the final phase difference of the two interference signals are associated with the azimuth angle of the fast axis of the half-wave plates, thus creating the controllable phase sensitivity. Numerical analysis demonstrates that the high phase measuring sensitivity and the small measuring range, and the low phase measuring sensitivity and the wide measuring range can be performed by selecting the suitable azimuth angle of the half-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, and precision machinery measurement. © 2010 Elsevier B.V. All rights reserved.
1. Introduction The optical phenomenon of total-internal reflection (TIR) occurs when a ray of light strikes an interface between a denser and a rarer medium at an angle larger than the critical angle with respect to the normal to the surface. This phenomenon leads the reflected wave to suffer the phase shifts, which relates to the angle of incidence and the refractive indices (RI) of the media. Many optical sensing technologies have employed the characteristics in detecting the changes of physical parameters, 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 optical materials [7], angular variation in machine tools [8–11], and displacement in micro-mechanical electronic systems [12]. These methods produced good measurement results. Although some of these methods can measure slight variations of the parameters and achieve high detection sensitivity using a multiple 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 some space-limited area. In addition, to avoid the 2π phase jumping, the systems only provide a narrow measurement range making them unsuitable for tracing large changes in parameters. To detect such
⁎ Tel.: +886 4 7232105x7235; fax: +886 4 7211149. E-mail address: [email protected]. 0030-4018/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2010.08.007
changes, the apparatus must be replaced by one with a few numbers of reflections to expand the measurable range [2–7]. This process causes the setup to be rearranged and recalibrated, and adds the operational complexity. To improve these shortcomings, a method that can be used in various measuring conditions needs to be developed. However, to the author's knowledge, there are no references reported for this purpose. In view of this, we designed a phase sensitivity-tunable optical sensor using total-internal reflection effect and derived some theoretical equations. A linearly polarized beam is guided to propagate through a phase sensitivity-tunable TIR apparatus, consisting of an isosceles rightangle prism, two half-wave plates, and two quarter-wave plates with suitable azimuth angles. The beam exiting from the apparatus enters a Mach-Zehnder interferometer with two acousto-optic modulators separately situated in two arms. Finally, the two linearly orthogonally polarizations of each output beam of the interferometer interfere with each other as they pass through two analyzers, respectively. The final phase difference between the two interference signals is associated with the azimuth angle of fast axis of the half-wave plates, yielding the sensitivity-tunable functionality. The experimental results of phase differences obtained by this technique were well confirmed by the theoretical analysis. Numerical calculations indicate that the refractive index sensitivity of (100–1.2 × 105°/RIU), and the measurement resolution of (9.7 × 10− 5–8.2 × 10−8 RIU) can be achieved. In addition to the tunable phase sensitivity and small size TIR apparatus, this sensor also has the merits of high stability and high resolution due to its common-path configuration and heterodyne phase measurement.
4900
J.-Y. Lin / Optics Communications 283 (2010) 4899–4906
H1
with
Q1
Q2
H2 Ap =
P i
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ðt cos 2αÞ2 + ðtb sin 2αÞ2 + ta tb sin4α⋅ cosð2Δ + δt Þ ; ð2Þ 2 a
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ðt sin 2αÞ2 + ðtb cos 2αÞ2 −ta tb sin4α⋅ cosð2Δ + δt Þ ; As = 2 a
1
np n
−1
∘ − tan 45 −σ ⋅ tanðΔ + δt = 2Þ ;
ð4Þ
−1
h i ∘ ′ cot 45 −σ ⋅ tanðΔ + δt = 2Þ ;
ð5Þ
ϕp = tan
Fig. 1. Schematic diagram of a phase sensitivity-tunable TIR apparatus. H: half-wave plate, Q: quarter-wave plate, P: isosceles right-angle prism with a refractive index np, n: refractive index of tested sample.
ϕs = tan
2. Principle
δt =2 tan
−1
2.1. Phase difference resulting from a phase sensitivity-tunable TIR apparatus
×
!
cosð2αÞ
sinð2αÞ
sinð2αÞ
−cosð2αÞ
1
0
0
i
!
1 pffiffiffi 2
1
−i
−i 1
0
0
ts ts′ expiðδt = 2Þ
!
cosΔ
iϕp
Ap e As e
iϕs
′
ð8Þ
′
ð9Þ
and 2 cosθi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi tp = 2ffi ; cos θi = np + 1− cos θi = np
ts =
2 cosθi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ffi ; cosθi + n 1− cos θi =np
′
sinΔ
Þ ′
ð1Þ
ts =
A
ð11Þ
ð12Þ
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ffi 2 = np ⋅ 1− cos θi =np rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi;
ð13Þ
1− cos θi = np
2
cosθi + 1 = np ⋅ 1− cos θi = np
H1 Q1 Q2
ð10Þ
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ffi 2 = np ⋅ 1− cos θi =np rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi;
tp = cos θi = np +
1
ð7Þ
tb = ts ts
!
ð
0
9 2 > > sin2 45∘ + sin−1 sinθi = np − n=np = h i h i ; −1 −1 > ∘ ∘ > > sin θi = np ⋅ sin 45 + sin sin θi = np > :tan 45 + sin ;
ta = tp tp
1 cosðΔ + δt = 2Þ⋅ðta cos2α + tb sin2αÞ−isinðΔ + δt = 2Þ⋅ðta cos2α−tb sin2αÞ = pffiffiffi 2 cosðΔ + δt = 2Þ⋅ðta sin2α−tb cos2αÞ−isinðΔ + δt = 2Þ⋅ðta sin2α + tb cos2αÞ =@
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h i ffi
tb t2 ′ tan 2α = b2 tan σ ; ta ta
tan σ =
!
tp tp′ expð−iδt = 2Þ
8 > > <
ð6Þ
Fig. 1 displays the schematic diagram of a phase sensitivity-tunable TIR apparatus. For convenience, the +z-axis is set in the direction of propagation of light and the x-axis is perpendicular to the plane of the paper. A linearly polarized light passes through a half-wave plate H1 (the fast axis at Δ/2 to the x-axis), two quarter-wave plates Q1 and Q 2 (the slow axes at 45° and 0° with respect to the x-axis, respectively), and is then incident at θi on one side of an isosceles right-angle prism P with a refractive index of np. The light beam penetrates into the prism at an angle of incidence θ1 onto the interface between the prism and a medium with a refractive index of n. When θi exceeds θic, which is the angle that makes θ1 equal to the critical angle θ1c, the light is totally reflected at the interface, and then the reflected light beam travels through a half-wave plate H2 (the fast axis at α to the x-axis). The Jones vector of the amplitude Et has the form:
Et =
ð3Þ
2
H2 BS
Laser
Rotational Stage
I
AOM2
n
M1
AOM1
AN1
PBS
D1
M2 AN2 D2 I2
I1
LIA Fig. 2. Schematic diagram of this designed optical sensor. PBS: polarization beamsplitter, M: mirror, AOM: acousto-optic modulator, BS: beamsplitter, AN: analyzer, D: photodetector, LIA: lock-in amplifier.
J.-Y. Lin / Optics Communications 283 (2010) 4899–4906
where the values of (tp, ts) and (t′p, t′s) are the transmission coefficients at the air–prism and the prism–air interface, respectively, and δt are the phase differences between s- and p-polarized total-internal reflections at the interface of the prism-medium [12–14]. In Eqs. (4) and (5), the constant Δ, used to shift the phase level of δt/2, must be set as about − δt,max/4, where δt,max denotes the maximum value of the phase difference δt and is given by [14]
δt;max
2 3 2 np = n −17 −1 6 5: = 2 tan 4 2 np = n
ð14Þ
The level modulation allows the relative curves of θi versus ϕp and ϕs to be more linear than those without modulation. Eqs. (4)–(7) illustrate that the phase shifts ϕp and ϕs clearly depend on the azimuth angle α. If n and θi are specified, ϕp and ϕs can be alerted by selecting α.
4901
interferometer. Two acousto-optic modulators are situated in the two arms of the interferometer, respectively. Light in the interferometer after been split by BS travels in two paths: (a) BS→ AOM1 → M1 → PBS and (b) BS→ AOM2 → M2 → PBS. The s-polarized light of path (a) and the p-polarized light of path (b) at PBS are superimposed to produce the amplitude E1 which is to end at the detector D1: Ap eiðω2 t
1 E1 = pffiffiffi 2
+ ϕp Þ
iðω1 t + ϕs Þ
As e
! ;
ð15Þ
whereas, the p-polarized light of path (a) and the s-polarized light of path (b) are also summed to bear the amplitude E2 which is to end at the detector D2: Ap eiðω1 t
+ ϕp Þ
!
2.2. Principle of phase difference detection
1 E2 = pffiffiffi 2
Fig. 2 shows the schematic diagram of this designed optical sensor. A laser light passes through the phase sensitivity-tunable TIR apparatus and the output beam of the apparatus enters a Mach-Zehnder
where ω1 and ω2 are the angular frequencies of output beams of AOM1 and AOM2, respectively.
a
: Phase difference
50
1
= 23
2
= 25
3 4 5
= 30 = 35 = 45
6 7 8
=0 = 10 = 15
9
= 20
10
= 22
0 -50
8 ic
10
12
io
14
16
18
20
22
4
0.6 0.5 5
0.4
0
4
150 2
= 25
3 4 5
= 30 = 35 = 45
-50
6 7 8
=0 = 10 = 15
-100
9
= 20
10
= 22
0
-150
ic
io
14
16
18
20
14
22
24
i (deg.)
Fig. 3. Phase difference ψ versus incident angle θi for different azimuth angle α. (a) 0° ≤ α ≤ 45° and (b) − 45° ≤ α ≤ 0°.
16
18
20
22
: Amplitude As
1 α = −22°, −23° 2 α = −20°, −25° 3 α = −10°, −15° −30°, −35° 4 α = 0°, −45° 5 α = −10°, −35° 6 α = −15°, −30° 7 α = −20°, −25° 8 α = −22°, −23°
0.6 0.5 5
0.4
6
0.2
7 8
0.1 0
24
θ (deg)
4
0.3
12
12
: Amplitude Ap
0.7
Amplitude
50
10
10
1 2 3
0.8
100
8
8
θio
•
= 23
0.9
6
θic
1 1
4
6
b
: Phase difference
200
7 8
0.1
24
O : Phase difference t
6
0.2
i (deg.)
b
1 α = 22°, 23° 2 α = 20°, 25° 3 α = 10°, 15°, 30°, 35° 4 α = 0°, 45° 5 α = 10°, 35° 6 α = 15°, 30° 7 α = 20°, 25° 8 α = 22°, 23°
0.7
-150
Phase difference (deg.)
: Amplitude As
0.3
6
: Amplitude Ap
1 2 3
0.8
-100
-200
•
0.9
Amplitude
Phase difference (deg.)
100
4
ð16Þ
1
150
-200
:
a
O : Phase difference t
200
iðω2 t + ϕs Þ
As e
4
6
θic
8
θio
10
12
14
16
18
20
22
24
θ (deg)
Fig. 4. Amplitude Ap and As versus incident angle θi for different azimuth angle α. (a) 0° ≤ α ≤ 45° and (b) − 45° ≤ α ≤ 0°.
4902
J.-Y. Lin / Optics Communications 283 (2010) 4899–4906
After passing through an analyzer AN1 with the transmission axis being at β to the x-axis, E1 becomes E′1 and is detected by D1: 1 ! 0 i ω t + ϕp Þ sin β cos β 1 Ap e ð 2 @ A 2 sin2 β As eiðω1 t + ϕs Þ
2
cos β sin β cos β
1 h i ω t = pffiffiffi cos β⋅Ap e ð 2 2
+ ϕp Þ
iðω1 t + ϕs Þ
+ sin β⋅As e
2
I2 = jE2 j =
i cos β
:
The intensity measured by D1 is therefore 2
I1 = jE1 j =
2 1 2 Ap cos β + ðAs sin βÞ + sin2β⋅Ap As cos ðωt + ϕÞ ; 2
ð18Þ where ϕ = ϕp − ϕs, and ω = ω2 − ω1. On the other hand, E2 passing through an analyzer AN2 (with the transmission axis being at β to the x-axis) becomes E′2 and is detected by D2 with cos2 β
E2 =
sin β cos β
sin β cos β sin2 β
1 h i ω t = pffiffiffi cosβ⋅Ap e ð 1 2
a
+ ϕp Þ
1 ! 0 iðω t + ϕp Þ 1 @ Ap e 1 A 2 As eiðω2 t + ϕs Þ iðω2 t + ϕs Þ
+ sinβ⋅As e
!
ð19Þ
:
sin β
1
1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.
2 3
100
4 5 6
50 0 -50 7
-100
8 9 10 11
-150 -200 6.5
The intensities I1 and I2 are sent to a lock-in amplifier (LIA) for phase analysis [15], enabling the final phase difference ψ = 2ϕ to be accurately determined. Although ψ is independent of β, Eqs. (18) and (20) reveal that the contrast 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 to 22.5° 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 light wavelength 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 the simulated results by plotting the relations of the incident angle θi
a
200 150
ψ (deg.)
i cos β
ð20Þ
ð17Þ
!
sin β
7
7.5
8
θ io
8.5
2 1 2 Ap cos β + ðAs sin βÞ + sin2β⋅Ap As cos ðωt−ϕÞ : 2
22.5°
= = 22.4° = 22.3° = 22.2° = 22.1° = 22.0° = 23.0° = 22.9° = 22.8° = 22.7° = 22.6°
0.07 0.06 0.05 0.04
As
E1 =
The intensity of the beam is
1. 2. 3. 4. 5. 6.
0.03 0.02
1 2 3 4 5 6
0.01 0 6.5
9
7
7.5
θ io
8
i (deg.)
b
200 1
150
3
100
ψ (deg.)
1. 2. 3.
2 4
5
4. 5. 6. 7. 8. 9. 10. 11.
50 0 -50 6 7 8 9 10 11
-100 -150 -200 6.5
7
7.5
θ io
9
9.5
, 22 , 22.1 , 22.2 , 22.3 , 22.4
10
i (deg.)
8
8.5
= 22.6° = 22.7° = 22.8° = 22.9° = 23.0° = 22.0° = 22.1° = 22.2° = 22.3° = 22.4° = 22.5°
9
i (deg.)
Fig. 5. Phase difference ψ versus incident angle θi for different azimuth angle α around θio. (a) 22° ≤ α ≤ 23° and (b) − 23° ≤ α ≤ − 22°.
0.07 0.06 0.05 0.04
Ap
b
8.5
= 23.0 = 22.9 = 22.8 = 22.7 = 22.6 = 22.5
1.
0.03 0.02
1 2 3 4 5 6
0.01 0 6.5
2. 3.
7
7.5
θ io
4. 5. 6.
8
8.5
9
= -22.0 = -22.1 = -22.2 = -22.3 = -22.4 = -22.5
9.5
, -23.0 , -22.9 , -22.8 , -22.7 , -22.6
10
i (deg.)
Fig. 6. Amplitude Ap and As versus incident angle θi for different azimuth angle α around θio. (a) As with 22° ≤ α ≤ 23° and (b) Ap with − 23° ≤ α ≤ − 22°.
J.-Y. Lin / Optics Communications 283 (2010) 4899–4906
versus the phase difference ψ of the reflected light at different azimuth angles α of the half-wave plate H2. As shown in Fig. 3(a) and (b), operating the angle α between 0° and ±45° can regulate the slope of the curve of θi versus ψ. With α = ±22° or ±23°, ψ notably exhibits sharp variations around the incident angle θio at which the phase
a
shifts (Δ + δt ∕ 2) = 0°. For comparison, the relation of the incident angle θi versus and the phase difference δt between s- and p-polarized light under single total-internal reflection are marked as “o” in Fig. 3. The curve of δt reveals the condition of small, unchangeable phase difference variation. Additionally, Fig. 4 shows the plots of the
b
40 1
30
2 3 4 5
1. θ i = 18.0° 2. θ i = 18.5° 3. θ i = 19.0°
10 0
4. θ i = 19.5° 5. θ i = 20.0°
-10
40
1
30
2
10 0
1. θ i = 15.0 2. θ i = 15.5° 3. θ i = 16.0° 4. θ i = 16.5° 5. θ i = 17.0°
-30 1.34
1.36
1.38
1.4
1.42
1.44
-40 1.32
1.46
1.34
1.36
c
d
80
1.42
1.44
150 1
100
2
1. θ i = 12.0 2. θ i = 12.5° 3. θ i = 13.0°
3
13.5°
5
2
ψ (deg.)
ψ (deg.)
1.4
1
60
0
1.38
n
n
20
4 5
-20
-30
40
3
-10
-20
-40 1.32
50
20
ψ (deg.)
ψ (deg.)
20
4903
4
4. θ i = 5. θ i = 14.0°
50
0
10.0°
1. θ i = 2. θ i = 10.5° 3. θ i = 11.0°
3
4. θ i = 11.5° 5. θ i = 12.0°
5
4
-20 -50 -40 -60 1.33
1.34
1.35
1.36
1.37
1.38
1.39
1.4
-100 1.33 1.335 1.34 1.345 1.35 1.355 1.36 1.365 1.37 1.375
1.41
n
n
e
f
200
200 1 2
150
150
3
50 0 -50
1. θ i = 7.6° 2. θ i = 7.7° 3. θ i = 7.8° 4. θ i = 7.9° 5. θ i = 8.0°
4
50 0 -50
-100
1. θ i = 7.6° 2. θ i = 7.7° 3. θ i =
2
7.8°
4. θ = 7.9° 5. θ i = 8.0°
-100 3
5
-150 -200 1.33
1
100
ψ (deg.)
ψ (deg.)
100
-150 1.3305
1.331
n
1.3315
1.332
-200 1.33
4 5
1.3305
1.331
1.3315
1.332
n
Fig. 7. Phase difference ψ versus refractive index n and incident angle θi (a) α = 0°, (b) α = ± 10°, (c) α = ± 15°, (d) α = ±20°, (e) α = − 22.5° and 22.4°, (f) α = − 22.6° and α = 22.5°, (g) α = ± 25°, (h) α = ±30°, (i) α = ±35°, and (j) α = ± 45°.
4904
J.-Y. Lin / Optics Communications 283 (2010) 4899–4906
g
h 100
60 40
50
0
-50
4. θ i = 11.5° 5. θ i = 12.0°
5 4
ψ (deg.)
ψ (deg.)
20 1. θ i = 10.0° 2. θ i = 10.5° 3. θ i = 11.0°
1. θ i = 12.0° 2. θ i = 12.5° 3. θ i = 13.0°
0
5
4. θ i = 13.5° 5. θ i = 14.0°
-20
3
4 3
-40
2
2
-100 -60
1
1
-80 1.33
-150 1.33 1.335 1.34 1.345 1.35 1.355 1.36 1.365 1.37 1.375
1.34
1.35
1.36
1.37
n
i
1.38
1.39
1.4
1.41
n
j
40
40 30 30 20
0
20 1. θ i = 15.0 2. θ i = 15.5° 3. θ i = 16.0°
-10 -20
5 4
4. θ i = 16.5° 5. θ i = 17.0°
3
-30
2
-40
1
10
1. θ i = 18.0° 2. θ i = 18.5°
0
3. θ i = 19.0°
-10
4. θ i = 19.5° 5. θ i = 20.0°
ψ (deg.)
ψ (deg.)
10
5 4 3
-20
2
-50 1.32
1.34
1.36
1.38
1.4
1.42
-30
1
-40 1.32
1.44
1.34
n
1.36
1.38
1.4
1.42
1.44
1.46
n Fig. 7 (continued).
reflection coefficients Ap and As of the reflected beam. Clearly, these results indicate that the values of As with α = 22° or 23° and Ap with α = −22° or −23° is small. Figs. 5 and 6 further illustrate the sharp variation 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 dip around θio, and correspondingly, the phase difference ψ changes dramatically when 22° ≤ α ≤ 23° or −23° ≤ α ≤ −22°. Since Eqs. (4)–(9) exhibit that the phase difference ψ depends on the refractive index n of the contact medium, this condition suggests that the sensing system can be designed for measuring phase variations with respect to changes in the refractive index of the sample. Fig. 7 shows the relations of n versus ψ at various α and θi. The simulated results have the tendency for sharp phase difference variations in the region of 22° ≤ α ≤ 23° or − 23° ≤ α ≤ − 22°, especially in Fig. 7(e) and (f) indicating that phase difference ψ with α = − 22.6°, 22.4°, or ± 22.5° at θi = 7.8° has more noticeable phase difference changes and better linearity than those at other θi. The condition can lead the sensor to exhibit very high measurement sensitivity for determining the slight variation of n but limit the measurable range. Fig. 7(a)–(d) and (g)–(j) indicates that the measurement range of n is extended and the phase difference variation of ψ is decreased, as α approaches 0° or ± 45°. The
operations for expanding measurement range are suitable for estimating the large variation in the refractive index of n. To estimate the RI sensitivity of this sensor, we differentiate the equation ψ = 2ϕ = 2(ϕp − ϕs), and, thus, the sensitivity S can be related as follows: S=
j j
∂ψ : ∂n
ð21Þ
According to the results of Fig. 7(a), (e), (f), and (j), the relations of the sensitivity S versus n are plotted in Fig. 8. Fig. 8(a) and (b) shows that the 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 lowest sensitivity is about 100°/RIU with α = 0° or ±45° at θi = 18°. Besides, the measurement resolution of this system can be defined as Δnerr =
j j
∂n Δψ; ∂ψ
ð22Þ
where Δnerr and Δψ are the errors in n and ψ, respectively. When the second harmonic error and the polarization-mixing error [16–19] are considered, 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
J.-Y. Lin / Optics Communications 283 (2010) 4899–4906
a 12
a
x 104
3.5
x 10-6 1
1
2.5
6
Δnerr
1. θ i = 7.9° 2. θ i = 7.8° 3. θ i = 8.0°
8
1. θ i = 7.6° 2. θ i = 7.7° 3. θ i = 8.0°
3
10
S (deg./RIU)
4905
7.7°
4. θ i = 5. θ i = 7.6°
4
4. θ i = 7.8° 5. θ i = 7.9°
2 1.5
2
1 2
2
4 5
0 1.33
1.3305
1.331
1.3315
3
0.5
3
4 5
0 1.33
1.332
1.3305
1.331
n
b
b
3.5
4
14
x 10
1. θ i = 7.6° 2. θ i = 7.7° 3. θ i = 8.0°
1
8
Δnerr
S (deg./RIU)
1. θ i = 7.9° 2. θ i = 7.8° 3. θ i = 8.0° 4. θ i = 7.7° 5. θ i = 7.6°
6
1
4. θ i = 7.8° 5. θ i = 7.9°
2.5 10
2 1.5
2
1
4
3
0.5
2
2
4
3
1.3305
1.331
1.3315
5
0 1.33
4 5
0 1.33
1.3305
1.331
c
550
1.3315
1.332
n
1.332
n
c
1.332
x 10-6
3 12
1.3315
n
1
x 10-5
10
500 8
400
2
1. θ i = 18.0° 2. θ i = 18.5° 3. θ i = 19.0°
350
3
19.5°
4. θ i = 5. θ i =i 20.0°
300 250
Δnerr
S (deg./RIU)
450
6
1. θ i = 18.0° 2. θ i = 18.5°
4
3. θ i = 19.0°
4
5
200
4. θ i = 19.5° 5. θ i = 20.0°
5 4 3 2 1
2
150 100 1.32
1.32 1.34
1.36
1.38
1.4
1.42
1.44
1.34
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°.
1.38
1.4
1.42
1.44
1.46
n
1.46
n
1.36
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°.
3.2. Experimental results Eq. (22), the relations of the resolutions Δnerr versus n depicted in Fig. 9. Fig. 9(a) and (b) indicates that the best resolution Δnerr ≅ 8.2 × 10− 7 can be achieved when α = −22.6°, 22.4°, or ±22.5° at θi = 7.8°, and the lowest resolution is about 9.7× 10− 5 as α = 0° or ±45° at θi = 18°. The results of Figs. 8 and 9 are also summarized in Table 1.
To validate the approach, a SF11 isosceles right-angle prism with np = 1.7786 was used as a TIR apparatus, and pure water was injected into a cell on the base of the prism. The apparatus and the tested sample were mounted tighter on high-precision rotational stage (Model M-URM100PP, New focus) with an angular resolution 0.001°.
4906
J.-Y. Lin / Optics Communications 283 (2010) 4899–4906
Table 1 The RI sensitivity, the measurement resolution, and the measurable range for α = (−22.6°, 22.4°, ±22.5°), and α = (0°, ±45°). S (°/RIU)
Δnerr (×10− 6)
Measurable range
7.8°
25,000–120,000
0.082–0.356
1.33–1.3317
20°
100–246
41–97
x : α = 0° ∗ : α = −20° o : α = −20°
150 100
ψ (deg.)
α = − 22.6°, 22.4°, ±22.5° α = 0°, ± 45°
θi
200
1.33–1.44
50 0 -50
The 632.8 nm line from a He–Ne laser served as the light source. In the Mach-Zehnder interferometer, the two linearly orthogonally polarizations modulated by two acousto-optic modulators (Model AOM40, IntraAction) of each output beam have the frequency difference 60 kHz. A lock-in amplifier (LIA) (Model SR830, Stanford) with an angular 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 θic occurred at the abrupt change of the phase difference ψ. The angle was measured with θic = 48.529°, and n = 1.3327 can be obtained based on Snell's law. Using Eq. (14), the constant Δ ≅ − δt,max / 4 = − 8.2° can also be estimated. Next, the azimuth angle of H1 was set to meet the constant Δ ≅ − 8.2°, and the phase difference ψ versus incident angle θi was measured with α = 0°, −15°, and −22°. In order to increase the contrast, β = 45°, 20°, and 1° were also chosen, respectively. Fig. 10 plots the measurements and theoretical results, respectively, where “○”, “×”, and “*” represent the measured data, and the solid lines represent the theoretical calculations. The experimental results confirm the theoretically predicted curve that the sensitivity is tunable by controlling the orientation of fast axis of H2. 4. Conclusion This study derived the equations of the totally reflected light phase differences of p- and s-polarizations in a phase sensitivity-tunable optical sensor. The sensor is composed of an isosceles right-angle prism, some wave plates, and a Mach-Zehnder interferometer. Based on the phase difference equations, this study examined the relation of incident angle and refractive index of medium versus the phase difference of the two output beams at various azimuth angles of fast axis of the half-wave plate in this system. The sensitivity of refractive index was found to be altered by the azimuth angle of the half-wave plate. The experimental results matched well with the theoretical analysis. The extent of the RI sensitivity and the measurement resolution can reach 100–1.2 ×105°/RIU, and 9.7 ×10− 5–8.2 ×10−8 RIU, respectively.
-100 -150 -200
5
10
15
20
θ i (deg.) Fig. 10. Measurement results and theoretical curves of ψ versus θi.
Acknowledgment The author would like to thank the National Science Council of the Republic of China, Taiwan, for financially supporting this research under Contract No. 97-2221-E-018-003.
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