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Steady-state performance analysis of fiber-optic ring resonator
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Progress in Quantum Electronics 33 (2009) 1–16 www.elsevier.com/locate/pquantelec
Review
Steady-state performance analysis of fiber-optic ring resonator Faramarz E. Seraji Optical Communication Group, Iran Telecom Research Center, Tehran, Iran
Abstract This paper presents a full steady-state analysis of a fiber-optic ring resonator (FORR). Although in the literature the steady-state response of the FORR has been described, a detailed description of the same is not available. As an understanding of the different steady-state characteristics of the FORR is required to appreciate its characteristic response, in this paper, the expressions for the output and loop intensities, phase angles of the fields, conditions for resonance, output and loop intensities at resonance and off-resonance, finesse, and group delay of the FORR are given for different ideal and practical operating conditions of the resonator. Graphical plots of all the above characteristics are given, by highlighting the important results. The information presented in this paper will be helpful in explaining and understanding the pulse response of the resonator used in different applications of FORR. r 2008 Elsevier Ltd. All rights reserved. Keywords: Fiber-optic ring resonator; Steady state; Group delay; Finesse; Optical filter
Contents 1. 2.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 FORR performance formulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.1. Output and loop intensities of FORR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2. Condition for resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.3. Output and loop intensities at resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.4. Output and loop intensities at off-resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.5. Response of FORR when g0a0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.6. Response of FORR when a ¼ 1 and g0 ¼ 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
E-mail address: [email protected] 0079-6727/$ - see front matter r 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.pquantelec.2008.10.001
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3.
2.7. Phase angles of output and loop fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8. Group delay calculation of FORR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.9. Finesse of FORR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12 12 13 14 15
1. Introduction During the last two decades, optical ring resonators (ORRs) with fiber and waveguide structures [1–6] have been analyzed for different applications such as polarization sensing [7], biosensing [8], optical filters [9–14], fiber dispersion compensation [15,16], optical integration/differentiation and optical triggering [2], optical bistability [17,18], add/drop multiplexer [19], optical switching [20–23], and various other applications [24–30]. In some early works, steady-state [2,31] and dynamic responses of ORR built on fiber were analyzed [32,33] for applications in polarization sensing [7], FM deviation measurement of a laser diode [33], optical triggering, optical integration/differentiation and fiber dispersion compensation [2], and rotation sensing [31]. Recently, dynamic resonance characteristic of a fiber ring resonator has been analyzed for a gyro system [34]. The structure of an ORR basically consists of a 2 2 port directional coupler and a fiber or waveguide loop connecting one of the input ports to one of the output ports, making a ring resonator with a function similar to a Fabry–Perot interferometer. To achieve the resonance effect in an ORR, the loop length could be of the order of few micrometers [10] to tens of meters [32]. The characteristic response of an ORR is determined by parameters such as resonator loop length and delay time, coupling coefficient of the coupler, transmission parameters of the loop fiber, phase angles of the circulating fields, finesse, group delay, and modulation frequency of the circulating field intensity in the resonator [32,33,35]. In this paper, we consider an ORR with a loop made of a single-mode optical fiber.
2. FORR performance formulations A schematic diagram of a fiber-optic ring resonator (FORR) is shown in Fig. 1. Let us assume that the loop is ideal with no polarization coupling or birefringence effects. Also, the coupler is assumed to be polarization insensitive. By referring to Fig. 1, let Ein, Er2, Er1, and E0 be the electric fields of light waves at ports 1, 2, 3, and 4, respectively. Then the complex field amplitudes at ports 3 and 4, under steady-state operation, can be written as [1,35] i pffiffiffiffiffiffiffiffiffiffiffiffiffihpffiffiffiffiffiffiffiffiffiffiffi pffiffiffi E r1 ¼ 1 g0 1 kE in þ kE r2 expðjp=2Þ (1) E0 ¼
pffiffiffiffiffiffiffiffiffiffiffi i pffiffiffiffiffiffiffiffiffiffiffiffiffihpffiffiffi 1 g0 k E in expðjp=2Þ þ 1 kE r2
(2)
where k is the power coupling coefficient of the coupler and g0 is the fractional intensity loss of the coupler.
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Loop delay time = τ
Eo
Er2 (2)
(4)
(1)
(3)
Er1
Ein Fig. 1. Schematic diagram of the fiber-optic ring resonator (FORR) made of a 2 2 directional coupler.
For 100% coupling, k ¼ 1, and for zero coupling, k ¼ 0. We note that the contribution of the input field Ein (or Er2) when cross-coupled to port 4 (or port 3) undergoes a phase shift of p/2 radians, as compared to zero phase shift when straight-coupling to port 3 (or port 4). The assumption of phase shift of p/2 radians [36,37] is required to ensure the law of conservation of energy and will be valid if the loss g0 is very small. From Fig. 1, the electric fields Er2 and Er1 are related by jE r2 j2 ¼ ajE r1 j2 E r2
pffiffiffi ¼ aE r1 expðjoc tÞ
(3) (4)
where t is the loop delay time, oc is the optical angular frequency, and a is the power transmission coefficient of the loop fiber expressed by a ¼ 10ða0 LþzÞ=10
(5)
where a0 is the attenuation coefficient (in dB/m) of the loop fiber, L is the length of the loop (in m), and z is the splice loss (in dB), if any, in the loop fiber. Solving Eqs. (1) and (2) and using Eq. (4) will result in " pffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffi # a 1 g0 1 k0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi E in E r2 ¼ (6) expðjoc tÞ þ j akð1 g0 Þ Substituting the values of Eqs. (1), (4), and (6) in Eq. (2), E0 takes the form "pffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi # pffiffiffiffiffiffiffiffiffiffiffiffiffi a 1 g0 j k expðjoc tÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi E in E 0 ¼ 1 g0 expðjoc tÞ þ j akð1 g0 Þ
(7)
2.1. Output and loop intensities of FORR By some algebraic manipulations of Eq. (7), the real and imaginary parts of the output electric field normalized to the input electric field can, respectively, be expressed as pffiffiffi E0 ð1 kÞð1 g0 Þ a cosðoc tÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (8a) ¼ E 1 þ akð1 g0 Þ þ 2 akð1 g0 Þ sinðoc tÞ in Real
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(pffiffiffi ) pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi E0 k þ að1 g Þ k þ ð1 þ kÞ að1 g Þ sinðo tÞ c 0 0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 1 g0 E 1 þ akð1 g0 Þ þ 2 akð1 g0 Þ sinðoc tÞ in Imag
(8b)
Similarly, using Eq. (6), the real and imaginary parts of the electric field Er2 in the loop, normalized to the input electric field, can be derived as pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi E r2 að1 g0 Þð1 kÞ cosðoc tÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (9a) ¼ E 1 þ akð1 g0 Þ þ 2 akð1 g0 Þ sinðoc tÞ in Real (pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ) pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi E r2 akð1 kÞð1 g0 Þ þ ð1 kÞ sinðoc tÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ að1 g0 Þ E 1 þ akð1 g0 Þ þ 2 akð1 g0 Þ sinðoc tÞ in Imag
(9b)
From Eqs. (8a) and (8b), the output intensity Pout normalized to input intensity can be shown as 2 E0 Pout ¼ (10) E in (
Pout
ð1 kÞ½1 að1 g0 Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ ð1 g0 Þ 1 1 þ akð1 g0 Þ þ 2 akð1 g0 Þ sinðoc tÞ
) (11)
From Eqs. (9a) and (9b), the loop intensity Ploop at port 2 normalized to input intensity can be expressed as 2 E r2 Ploop ¼ (12) E in Ploop ¼
að1 g0 Þð1 kÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ akð1 g0 Þ þ 2 akð1 g0 Þ sinðoc tÞ
(13)
2.2. Condition for resonance The condition for resonance in the FORR is obtained by finding the value of oct at which the loop intensity Ploop builds up to a maximum value and the output intensity Pout reduces to a minimum value. By inspecting Eq. (11), the condition for resonance is found to be oct ¼ p/2, or in general we have oct ¼ 2qpp/2, where q is an integer. Eq. (11) shows that at resonance when oct ¼ 2qpp/2, the value of the output intensity Pout will become a minimum. The additional condition required to make Pout become zero at resonance can be shown as k ¼ að1 g0 Þ
(14)
If the coupler is lossless with g0 ¼ 0, we have k ¼ a. Fig. 2 shows the theoretical output and loop intensities of the FORR as a function of oct for different values of k( ¼ a). It is observed that for all values of k, the output
ARTICLE IN PRESS F.E. Seraji / Progress in Quantum Electronics 33 (2009) 1–16
1.00
20
1.75
15
5
Loop intensity, Ploop
Output intensity, Pout
α=κ
0.50 α=κ κ = 0.95 0.75 0.55 0.35
0.25
κ = 0.95 0.75 0.55 0.35
10
5
(2q-1/2)π
0.0 (2q-2)π
2qπ
0 (2q-2)π
(2q+2)π
ωcτ (rad)
2qπ
(2q+2)π
ωcτ (rad)
Fig. 2. Normalized (a) output intensity and (b) loop intensity versus oct when g0 ¼ 0 and k ¼ a ¼ 0.95, 0.75, 0.55, and 0.35.
Output intensity, Pout
1.00
0.75
0.50 α = 0.85 0.65 0.45
0.25
0.25
0.0 (2q-2)π
2qπ
(2q+2)π
ωcτ (rad) Fig. 3. Normalized output intensity versus oct when g0 ¼ 0, k ¼ 0.85, and a ¼ 0.85, 0.65, 0.45, and 0.25.
response dips to zero, and the loop intensity peaks to a maximum value, indicating the resonance condition. As the value of k( ¼ a) decreases, the sharpness of the resonance response decreases as well. When k ¼ a(1g0), it can be shown that at resonance when oct ¼ 2qpp/2, the maximum developed loop intensityr Ploop,max is Ploop; max ¼
k 1k
(15)
The maximum value of the normalized loop intensity in this case is 19 for k ¼ a ¼ 0.95. A reduction of 0.2 in the value of k( ¼ a) from 0.95 to 0.75 causes the maximum loop intensity to decrease from 19 to 3. We note that in Fig. 2(b) the resonance response is not
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sharp when the value of k( ¼ a) is small. When kaa, the output intensity does not dip to zero, as shown in Fig. 3. 2.3. Output and loop intensities at resonance From Eq. (11), at oct ¼ 2qpp/2 and g0 ¼ 0, the normalized output intensity can be shown by pffiffiffi pffiffiffi2 k a Pout joc t¼p=2 ¼ pffiffiffiffiffiffi2 1 ak
(16)
From Eq. (13), Ploop can be derived as Ploop joc t¼p=2 ¼
að1 kÞ pffiffiffiffiffiffi2 1 ak
(17)
The loop intensity response is shown in Fig. 4 for k ¼ 0.85, and a varying from 0.25 to 0.95. It is seen that for a given value of k the maximum loop intensity increases with the value of a. Eq. (16) gives the minimum value of Pout at oct ¼ 2qpp/2 as a function of k and a. The variation of the minimum value of Pout at oct ¼ 2qpp/2 as a function of a for two fixed values of k ¼ 0.9 and 0.5 is plotted in Fig. 5. The normalized output intensity is equal to k for a ¼ 0, is zero when k ¼ a, and is unity at a ¼ 1. The corresponding loop intensity, given by Eq. (17), is illustrated in Fig. 6 as a function of a. The loop intensity for a given value of k always increases with a, reaching a maximum for a ¼ 1. For a fixed value of a, the response of the FORR varies significantly with changes in the value of k. Fig. 7 shows Pout and Ploop as a function of k at oct ¼ 2qpp/2 for a ¼ 0.9 and g0 ¼ 0. In Fig. 7(a), Pout dips to zero for k ¼ a. As k tends to zero, Pout
Loop intensity, Ploop
6.0
4.5 α = 0.85 0.65 0.45 0.25
3.0
1.5
0.0 (2q-2)π
2qπ
(2q+2)π
ωcτ (rad) Fig. 4. Normalized loop intensity versus oct when g0 ¼ 0, k ¼ 0.85, and a ¼ 0.85, 0.65, 0.45, and 0.25.
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1.00
Output intensity at resonance
κ = 0.50 0.90
0.75
0.50
0.25
0.00 0.00
0.50 Power transmission coefficient, α
1.00
Fig. 5. Normalized output intensity at resonance versus a when g0 ¼ 0, k ¼ 0.5 and 0.9.
40 κ = 0.50
Loop intensity at resonance
0.90
30
20
10
0 0.0
0.50 Power transmission coefficient, α
1.00
Fig. 6. Normalized loop intensity at resonance versus a when g0 ¼ 0, k ¼ 0.5 and 0.9.
increases towards a( ¼ 0.9), and for k40.9, Pout increases towards unity at k ¼ 1. Fig. 7(b) illustrates Ploop as a function of k for a ¼ 0.9 under the resonance condition, where the curve shows a peak at k ¼ a, which is followed by a sharp decline towards zero when k tends to unity. The peak value of Ploop at k ¼ 0.9 is 9. At k ¼ 0 the value of Ploop is equal to a( ¼ 0.9). According to Eq. (17), under the resonance condition, if k ¼ a, the theoretical value of Ploop tends to be large when k tends to unity. But when k ¼ 1, the power coupling in the coupler of the FORR will be 100 percent, which will result in a zero loop intensity. Fig. 8 shows the value of Ploop at the resonance as a function of k, assuming k ¼ a and g0 ¼ 0. For k ¼ a ¼ 0.99, the loop intensity takes a value of 99. A comparison of Figs. 7(b)
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1.00 Loop intensity at resonance
Output intensity at resonance
1.00
0.75
0.50
0.25
0.00
7.5
5.0
2.5
0.0 0.0
0.50 Power coupling coefficient, κ
1.00
0.0
0.5 Power coupling coefficient, κ
1.0
Fig. 7. Normalized (a) output intensity and (b) loop intensity at resonance versus k when g0 ¼ 0 and a ¼ 0.9.
Loop intensity at resonance
100
75
50
25
0 0.00
0.495 Power coupling coefficient, κ
0.990
Fig. 8. Normalized loop intensity at resonance versus k when g0 ¼ 0 and a ¼ k.
and 8 indicates that to get a high loop intensity, the conditions oct ¼ 2qpp/2 and k ¼ a(1g0) should be satisfied simultaneously. 2.4. Output and loop intensities at off-resonance At the off-resonance condition obtained at oct ¼ 2qpp/2, Pout and Ploop can be derived as (pffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 ) k þ að1 g0 Þ Pout joc t¼p=2 ¼ ð1 g0 Þ (18) pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 þ akð1 g0 Þ
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Ploop joc t¼p=2 ¼
að1 g0 Þð1 kÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 þ akð1 g0 Þ
9
(19)
With a special case of k ¼ a and g0 ¼ 0, Eqs. (18) and (19) can, respectively, be simplified as Pout joc t¼p=2 ¼
4k ð1 þ kÞ2
Ploop joc t¼p=2 ¼
(20)
kð1 kÞ ð1 þ kÞ2
(21)
0.1250
1.00 Loop intensity at off-resonance
Output intensity at off-resonance
Eqs. (20) and (21) are plotted as functions of k in Fig. 9 at off-resonance condition. In Fig. 9(a), as k increases to its maximum value of 1, Pout increases to unity. The corresponding Ploop in Fig. 9(b) shows a maximum value of 0.125 at k ¼ a ¼ 0.33 and attains zero value when k ¼ 1 or k ¼ 0. For k and a assumed to be unequal, that is for a fixed value of a ¼ 0.9
0.75
0.50
0.25
0.0
0.0625
0.00 0.0
0.50 Power coupling coefficient, κ
1.0
0.0
0.50 Power coupling coefficient, κ
1.0
Fig. 9. Normalized (a) output intensity and (b) loop intensity at off-resonance versus k when g0 ¼ 0 and a ¼ k.
1.00 Loop intensity at off-resonance
Output intensity at off-resonance
1.00
0.75
0.50
0.25
0.00
0.75
0.50
0.25
0.00 0.0
0.5 Power coupling coefficient, κ
1.0
0.0
0.50 Power coupling coefficient, κ
1.0
Fig. 10. Normalized (a) output intensity and (b) loop intensity at off-resonance versus k for g0 ¼ 0 and a ¼ 0.9.
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with k as a variable, at the off-resonance condition of oct ¼ 2qpp/2, the responses of Pout and Ploop are as shown in Fig. 10. The output intensity Pout varies from a value of Pout ¼ a ¼ 0.9 at k ¼ 0 to a maximum value of unity at k ¼ 1, as shown in Fig. 10(a). Fig. 10(b) shows the off-resonance value of Ploop with a maximum value of 0.9 occurring at k ¼ 0. So far all the results were described for an ideal condition of keeping the fractional intensity loss g0 of the coupler in the FORR equal to zero. In the following we will discuss the response of the FORR when g0 is a non-zero quantity. 2.5. Response of FORR when g0a0 Fig. 11 shows the sketches of output and loop intensities as functions of oct for different values of g0 when k ¼ a ¼ 0.8. As g0 increases, the off-resonance value of Pout reduces and the value of resonance dip decreases by shifting upwards from the zero level, as shown in Fig. 11(a). On the other hand, the loop intensity at the resonance decreases with an increase in g0, as indicated in Fig. 11(b). If we consider the value of a ¼ k/(1g0) and assuming k ¼ 0.8, the plot of Pout will be as shown in Fig. 12 for g0 ¼ 0.03, 0.06, and 0.09, where all the curves dip to zero at resonance. The difference between Fig. 11(a) and 12 is that in the former the condition to get a zero output at resonance is not satisfied while in the latter this condition is always satisfied. 2.6. Response of FORR when a ¼ 1 and g0 ¼ 0
1.00
4
0.75
3
0.50
Loop intensity, Pout
Output intensity, Pout
For an ideal condition of a lossless loop (a ¼ 1) and a lossless coupler (g0 ¼ 0), the output and loop intensities are illustrated in Fig. 13. Irrespective of the value of k, the normalized output intensity Pout is always unity for all values of oct (refer to Eq. (11)). That is, Pout ¼ 1, for all values of oct. This leads to an ‘‘all-pass structure’’ of the FORR, where the output intensity becomes equal to the input intensity for all optical frequencies,
γ0= 0.00 0.03 0.06 0.09
0.25
0.00 (2q-2)π
2qπ ωcτ(rad)
(2q+2)π
γ0= 0.00 0.03 0.06 0.09
2
1
0 (2q-2)π
2qπ
(2q+2)π
ωcτ(rad)
Fig. 11. Normalized (a) output intensity and (b) loop intensity versus oct when k ¼ a ¼ 0.8 and g0 ¼ 0, 0.03, 0.06, and 0.09.
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Output intensity, Pout
1.00
0.75 γ0= 0.00 0.03 0.06 0.09
0.50
0.25
0.00 (2q-2)π
2qπ
(2q+2)π
ωcτ(rad)
1.00
80
0.75
60
Loop intensity, Ploop
Output intensity, Pout
Fig. 12. Normalized output intensity versus oct when k ¼ 0.8 and g0 ¼ 0, 0.03, 0.06, 0.09, and a ¼ k/(1g0).
0.50
0.25
0.00 (2q-2)π
40 20 0
2qπ
(2q+2)π
(2q-2)π
ωcτ(rad)
2qπ
(2q+2)π
ωcτ(rad)
Fig. 13. Normalized (a) output intensity (b) loop intensity versus oct when a ¼ 1.0 with k ¼ 0.95 and g0 ¼ 0.
as shown in Fig. 13(a). The loop intensity, with a ¼ 1 and g0 ¼ 0, is given by Ploop ¼
ð1 kÞ pffiffiffi 1 þ k þ 2 k sinðoc tÞ
(22)
which is not independent of oct. At the resonance when oct ¼ 2qpp/2, the loop intensity builds up to a maximum value expressed as pffiffiffi 1þ k pffiffiffi Ploop joc t¼p=2 ¼ (23) 1 k
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Loop intensity at resonance
400
300
200
100
0 0.0
0.495 Power coupling coefficient, κ
0.990
Fig. 14. Normalized loop intensity at resonance versus k when a ¼ 1.0 and g0 ¼ 0.
For example, the value of Ploop reaches a peak value of 78 at oct ¼ 2qpp/2 for a ¼ 1 and k ¼ 0.95, as depicted in Fig. 13(b). Fig. 14 shows Ploop from Eq. (23) at resonance as a function of k. The loop intensity increases exponentially and reaches a maximum value of 392 when k ¼ 0.99. 2.7. Phase angles of output and loop fields From Eq. (7), the phase angle y0 of the normalized output electric field can be shown as "pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi #
pffiffiffi akð1 g0 Þ þ sinðoc tÞ k cosðoc tÞ y0 ¼ arctan pffiffiffi arctan (24) cosðoc tÞ að1 g0 Þ þ k sinðoc tÞ Similarly, the phase angle yloop of the normalized loop electric field at port 2 can be derived from Eq. (6) and expressed by "pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi # akð1 g0 Þ þ sinðoc tÞ yloop ¼ arctan (25) cosðoc tÞ For an infinite loss in the loop (a ¼ 0), that is when there is no field circulation in the loop, the output phase y0 equals to p/2 radians, confirming the p/2 phase shift introduced by the coupler in a cross-coupled field. 2.8. Group delay calculation of FORR We saw earlier that if a ¼ 1, the FORR works as an all-pass structure (refer to Section 2.6). The delay characteristics of the all-pass structure of the FORR are of interest. The group delay of the FORR is defined as the rate of change of the output phase angle y0
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0.0
–9.5τ
k= 0.30 0.95
Group delay, Γ
Group delay, Γ
0.0
13
α= 1
–19.5τ –28.5τ –38.0τ (2q-2)π
2qπ ωcτ(rad)
(2q+2)π
–2.5τ
k= 0.30 0.95
α= 1
–5.0τ –7.5τ –10.0τ (2q-2)π
2qπ
(2q+2)π
ωcτ(rad)
Fig. 15. Group delay of FORR as a function of oct with g0 ¼ 0 and k ¼ 0.3 and 0.95 (a) a ¼ 1.0 and (b) a ¼ k.
with respect to optical frequency oc. That is, from Eq. (24) we can write 9 8 pffiffiffi pffiffiffi pffiffiffi pffiffiffi k= a sinðoc tÞ 1 þ k= a sinðoc tÞ þ ðk=aÞcos2 ðoc tÞ > > > > > > pffiffiffi pffiffiffi 2 > > > > 2 ðo tÞ = < sinðo 1 þ k = a tÞ þ ðk=aÞcos c c dy0 p ffiffi ffi p ffiffiffi ¼t G¼ 2 > > ð1=aÞcos ðoc tÞ þ sinðoc tÞ k= a þ ð1=aÞ sinðoc tÞ doc > > > > pffiffiffi 2 > > pffiffiffi > > 2 ; : k þ ð1= aÞ sinðoc tÞ þ ð1=aÞcos ðoc tÞ
(26)
Eq. (26) gives the group delay under steady-state operation of the FORR. Under ideal conditions of a ¼ 1 and g0 ¼ 0, the group delay in Eq. (26) at resonance reduces to
pffiffiffi 1þ k pffiffiffi G ¼ t (27) 1 k which is t times the corresponding loop intensity Ploop at the resonance as per Eq. (23). The group delay as a function of oct is shown graphically in Fig. 15(a) for a case when a ¼ 1, g0 ¼ 0, and k ¼ 0.3 and 0.95. The plot shows that, when the value of k is high, around resonance, the FORR introduces a large group delay. The group delay changes with the optical frequency oc. This property of the FORR can be utilized for optical dispersion equalization [15]. Fig. 15(b) illustrates the group delay G as a function of oct when a ¼ k. For k ¼ 0.95 and 0.3, it is observed that the maximum group delay obtained is smaller than that of Fig. 15(a). Also, when a ¼ 1, the output intensity does not change with oct, whereas when a ¼ k, the output will dip to zero at the resonance. 2.9. Finesse of FORR The sharpness of the resonance peak in the loop intensity and the resonance dip of the output intensity expressed in terms of finesse of the FORR are defined as [1,35] F¼
FSR Df
(28)
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8
300
6
225 Finesse, F
Finesse, F
14
4
2
150
75
0 0.20
0.35 Power transmission coefficient, α
0.50
0 0.200
0.595 Power transmission coefficient, α
0.990
Fig. 16. Finesse of FORR as a function of a when k ¼ a and g0 ¼ 0, (a) low values and (b) high values.
where FSR is the free spectral range of the FORR expressed as c 1 ¼ (29) nL t where c is the velocity of light in free space, n is the refractive index of the loop fiber, and L is the loop length. Df is the full-width at half-maximum of the loop intensity and can be found by equating Eq. (13) to half of Ploop,max. That is, Df can finally be obtained as ( "rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
#) 1 1 1 að1 g0 Þ 2 1 1 (30) kþ Df ¼ sin 4þ t 2 p 4k k að1 g0 Þ FSR ¼
Therefore, from Eqs. (28) and (29), the finesse can be expressed as F¼
1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ðað1 g0 ÞÞ=4k ðk þ 2=kÞ 4 þ ð1=að1 g0 ÞÞ ð1=tÞ ð1=2Þ ð1=pÞsin
(31) For a special case when k ¼ a and g0 ¼ 0, Eq. (31) can be simplified as 1 F ¼ ð1=2Þ ð1=pÞsin1 1 ð1 aÞ2 2a
(32)
Eq. (32) indicates that the finesse of the FORR has a dependency on the value of a. Fig. 16 shows the finesse as a function of a, for k ¼ a and g0 ¼ 0. Up to a ¼ 0.5, the finesse increases (but not significantly) with a ¼ k ¼ 0.5, as indicated in Fig. 16(a). For a ¼ k ¼ 0.5, the finesse is only about 4.3. When a increases beyond 0.5, the finesse increases steeply, as shown in Fig. 16(b). For a ¼ 0.99, the calculated finesse is about 294. 3. Conclusion This paper dealt with the basic properties of an FORR under steady-state condition. A theoretical analysis on the steady-state response of the FORR was presented by considering the output and loop intensities for various conditions of a and k. The
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expressions for the output and the loop intensities were derived at resonance and offresonance conditions and the results were illustrated graphically for ideal and practical conditions. The expressions for phase angles of the output and the loop fields were formulated. The expression for the finesse, which describes the quality of resonance in the FORR, was derived and the result was sketched graphically. The analyses have shown that all the FORR characteristic parameters, such as resonance condition, output and loop intensities, group delay, and finesse, as functions of loop phase can mainly be controlled and adjusted by the power coupling coefficient, fractional intensity loss of the coupler, and transmission coefficient of the loop. The analytical results given in this article will be useful to understand the dynamic and pulse responses of the FORR, used in different applications. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
[11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33]
L.F. Stokes, M. Chodorow, H.J. Shaw, Opt. Lett. 7 (1982) 288–290. G. Soundra Pandian, Faramarz E. Seraji, IEE Proc. J. 138 (3) (1991) 235–239. M. Sumetsky, Y. Dulashko, A. Hale, Opt. Express 12 (2004) 3521–3531. M. Sumetsky, Opt. Express 12 (2004) 2303–2316. Huilian Ma, Shijin Wang, Zhonghe Jin, Opt. Commun. 281 (2008) 2509–2512. N. Talebi, M. Shahabadi, Prog. Electromagn. Res., PIER 66 (2006) 287–299. G. Soundra Pandian, Faramarz E. Seraji, J. Mod. Optics 39 (5) (1992) 991–1001. R.W. Boyd, J.H. Heebner, Appl. Optics 40 (2001) 5742–5747. Sai T. Chu, B.E. Little, Wugen Pan, Taro Kaneko, Shinay Sato, Yasuo Kokubun, IEEE Photonic Technol. Lett. 11 (6) (1999) 691–693. D.G. Rabus, M. Hamacher, H. Heidrich, Active and passive microring resonator filter applications on GaInAsP/InP, in: Proceedings of the 13th International Conference on Indium Phosphide and Related Materials, IPRM 14–18 May 2001, Nara, Japan, Pap. ThA1-3, 2001, pp. 477–480. Fufei Pang, Feng Liu, Xiuyou Han, Haiwen Cai, Ronghui Qu, Zujie Fang, Opt. Commun. 260 (2006) 567–570. Sanjoy Mandal, Kamal Dasgupta, T.K. Basak, S.K. Ghosh, Opt. Commun. 264 (2006) 97–104. P.P. Yupapin, P. Saeung, C. Li, Opt. Commun. 272 (2007) 81–86. P. Saeung, P.P. Yupapin, Optik 119 (10) (2008) 465–472. S. Dilwali, G.S. Pandian, IEEE Photonic Technol. Lett. 4 (8) (1992) 942–944. Hao Shen, Jian-Ping Chen, Xin-Wan Li, Yi-Ping Wang, Opt. Commun. 262 (2006) 200–205. Y. Dumeige, C. Arnaud, P. Fe´ron, Opt. Commun. 250 (2005) 376–383 (Erratum: Opt. Commun. 272 (2007) 279). P.P. Yupapin, Optik 119 (10) (2008) 492–494. Otto Schwelb, Opt. Commun. 265 (2006) 175–179. Junqing Li, Li Li, Jiaqun Zhao, Chunfei Li, Opt. Commun. 256 (2005) 319–325. A. Rostami, Microelectr. J. 37 (2006) 976–981. P.P. Yupapin, P. Chunpang, Opt. Laser Technol. 40 (2008) 273–277. J.L.S. Lima, K.D.A. Sabo´ia, J.C. Sales, J.W.M. Menezes, W.B. de Fraga, G.F. Guimara˜es, A.S.B. Sombra, Opt. Fiber Technol. 14 (2008) 79–83. S.L. McCall, A.F.J. Levi, R.E. Slusher, S.J. Pearton, R.A. Logan, Appl. Phys. Lett. 60 (1992) 289–291. D. Garus, R. Hereth, F. Schliep, Design considerations for Brillouin–Ringlaser gyroscopes, in: Conference Proceedings Symposium on Gyro Technology (DGON), Stuttgart, 1994, p. 2.0. K. Kalli, D.A. Jackson, Opt. Lett. 17 (15) (1992) 1090–1092. B.E. Little, S.T. Chu, J.V. Hryniewicz, P.P. Absil, Opt. Lett. 25 (5) (2000) 344–346. T. Kominato, Y. Ohmori, N. Takato, H. Okazaki, M. Yasu, J. Lightwave Technol. 12 (1992) 1781–1788. R. Adar, Y. Shani, C.H. Henry, R.C. Kistler, G.E. Blonder, N.A. Olsson, Appl. Phys. Lett. 58 (5) (1991) 444–445. K. Oda, S. Suzuki, H. Takahashi, H. Toba, IEEE Photonics Technology. Letters 6 (8) (1994) 1031–1034. Faramarz E. Seraji, Japanese Journal of Applied Physics, Part 1 32 (4) (1993) 1661–1667. Faramarz E. Seraji, G.S. Pandian, J. Mod. Optics 38 (1991) 671–676. G.S. Pandian, Faramarz E. Seraji, Jpn. J. Appl. Phys. 29 (10) (1990) 1967–1973.
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[34] Diqing Ying, Huilian Ma, Zhonghe Jin, Dynamic resonance characteristic analysis of fiber ring resonator, Optical Fiber Technology, doi:10.1016/j.yofte.2008.03.005, in press. [35] Faramarz E. Seraji, Analysis of fibre-optic ring resonator and sensing systems, Ph.D. Thesis, EE Department, Indian Institute of Technology, Delhi, India, 1991. [36] R.C. Youngquist, L.F. Stokes, H.J. Shaw, IEEE J. Quantum Elect. QE-19 (1983) 1888–1896. [37] T. Mizumoto, Y. Naito, IEEE J. Lightwave Technol. 8 (10) (1990) 1571–1576.
Progress in Quantum Electronics 33 (2009) 1–16 www.elsevier.com/locate/pquantelec
Review
Steady-state performance analysis of fiber-optic ring resonator Faramarz E. Seraji Optical Communication Group, Iran Telecom Research Center, Tehran, Iran
Abstract This paper presents a full steady-state analysis of a fiber-optic ring resonator (FORR). Although in the literature the steady-state response of the FORR has been described, a detailed description of the same is not available. As an understanding of the different steady-state characteristics of the FORR is required to appreciate its characteristic response, in this paper, the expressions for the output and loop intensities, phase angles of the fields, conditions for resonance, output and loop intensities at resonance and off-resonance, finesse, and group delay of the FORR are given for different ideal and practical operating conditions of the resonator. Graphical plots of all the above characteristics are given, by highlighting the important results. The information presented in this paper will be helpful in explaining and understanding the pulse response of the resonator used in different applications of FORR. r 2008 Elsevier Ltd. All rights reserved. Keywords: Fiber-optic ring resonator; Steady state; Group delay; Finesse; Optical filter
Contents 1. 2.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 FORR performance formulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.1. Output and loop intensities of FORR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2. Condition for resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.3. Output and loop intensities at resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.4. Output and loop intensities at off-resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.5. Response of FORR when g0a0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.6. Response of FORR when a ¼ 1 and g0 ¼ 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
E-mail address: [email protected] 0079-6727/$ - see front matter r 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.pquantelec.2008.10.001
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3.
2.7. Phase angles of output and loop fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8. Group delay calculation of FORR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.9. Finesse of FORR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12 12 13 14 15
1. Introduction During the last two decades, optical ring resonators (ORRs) with fiber and waveguide structures [1–6] have been analyzed for different applications such as polarization sensing [7], biosensing [8], optical filters [9–14], fiber dispersion compensation [15,16], optical integration/differentiation and optical triggering [2], optical bistability [17,18], add/drop multiplexer [19], optical switching [20–23], and various other applications [24–30]. In some early works, steady-state [2,31] and dynamic responses of ORR built on fiber were analyzed [32,33] for applications in polarization sensing [7], FM deviation measurement of a laser diode [33], optical triggering, optical integration/differentiation and fiber dispersion compensation [2], and rotation sensing [31]. Recently, dynamic resonance characteristic of a fiber ring resonator has been analyzed for a gyro system [34]. The structure of an ORR basically consists of a 2 2 port directional coupler and a fiber or waveguide loop connecting one of the input ports to one of the output ports, making a ring resonator with a function similar to a Fabry–Perot interferometer. To achieve the resonance effect in an ORR, the loop length could be of the order of few micrometers [10] to tens of meters [32]. The characteristic response of an ORR is determined by parameters such as resonator loop length and delay time, coupling coefficient of the coupler, transmission parameters of the loop fiber, phase angles of the circulating fields, finesse, group delay, and modulation frequency of the circulating field intensity in the resonator [32,33,35]. In this paper, we consider an ORR with a loop made of a single-mode optical fiber.
2. FORR performance formulations A schematic diagram of a fiber-optic ring resonator (FORR) is shown in Fig. 1. Let us assume that the loop is ideal with no polarization coupling or birefringence effects. Also, the coupler is assumed to be polarization insensitive. By referring to Fig. 1, let Ein, Er2, Er1, and E0 be the electric fields of light waves at ports 1, 2, 3, and 4, respectively. Then the complex field amplitudes at ports 3 and 4, under steady-state operation, can be written as [1,35] i pffiffiffiffiffiffiffiffiffiffiffiffiffihpffiffiffiffiffiffiffiffiffiffiffi pffiffiffi E r1 ¼ 1 g0 1 kE in þ kE r2 expðjp=2Þ (1) E0 ¼
pffiffiffiffiffiffiffiffiffiffiffi i pffiffiffiffiffiffiffiffiffiffiffiffiffihpffiffiffi 1 g0 k E in expðjp=2Þ þ 1 kE r2
(2)
where k is the power coupling coefficient of the coupler and g0 is the fractional intensity loss of the coupler.
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3
Loop delay time = τ
Eo
Er2 (2)
(4)
(1)
(3)
Er1
Ein Fig. 1. Schematic diagram of the fiber-optic ring resonator (FORR) made of a 2 2 directional coupler.
For 100% coupling, k ¼ 1, and for zero coupling, k ¼ 0. We note that the contribution of the input field Ein (or Er2) when cross-coupled to port 4 (or port 3) undergoes a phase shift of p/2 radians, as compared to zero phase shift when straight-coupling to port 3 (or port 4). The assumption of phase shift of p/2 radians [36,37] is required to ensure the law of conservation of energy and will be valid if the loss g0 is very small. From Fig. 1, the electric fields Er2 and Er1 are related by jE r2 j2 ¼ ajE r1 j2 E r2
pffiffiffi ¼ aE r1 expðjoc tÞ
(3) (4)
where t is the loop delay time, oc is the optical angular frequency, and a is the power transmission coefficient of the loop fiber expressed by a ¼ 10ða0 LþzÞ=10
(5)
where a0 is the attenuation coefficient (in dB/m) of the loop fiber, L is the length of the loop (in m), and z is the splice loss (in dB), if any, in the loop fiber. Solving Eqs. (1) and (2) and using Eq. (4) will result in " pffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffi # a 1 g0 1 k0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi E in E r2 ¼ (6) expðjoc tÞ þ j akð1 g0 Þ Substituting the values of Eqs. (1), (4), and (6) in Eq. (2), E0 takes the form "pffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi # pffiffiffiffiffiffiffiffiffiffiffiffiffi a 1 g0 j k expðjoc tÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi E in E 0 ¼ 1 g0 expðjoc tÞ þ j akð1 g0 Þ
(7)
2.1. Output and loop intensities of FORR By some algebraic manipulations of Eq. (7), the real and imaginary parts of the output electric field normalized to the input electric field can, respectively, be expressed as pffiffiffi E0 ð1 kÞð1 g0 Þ a cosðoc tÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (8a) ¼ E 1 þ akð1 g0 Þ þ 2 akð1 g0 Þ sinðoc tÞ in Real
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(pffiffiffi ) pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi E0 k þ að1 g Þ k þ ð1 þ kÞ að1 g Þ sinðo tÞ c 0 0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 1 g0 E 1 þ akð1 g0 Þ þ 2 akð1 g0 Þ sinðoc tÞ in Imag
(8b)
Similarly, using Eq. (6), the real and imaginary parts of the electric field Er2 in the loop, normalized to the input electric field, can be derived as pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi E r2 að1 g0 Þð1 kÞ cosðoc tÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (9a) ¼ E 1 þ akð1 g0 Þ þ 2 akð1 g0 Þ sinðoc tÞ in Real (pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ) pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi E r2 akð1 kÞð1 g0 Þ þ ð1 kÞ sinðoc tÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ að1 g0 Þ E 1 þ akð1 g0 Þ þ 2 akð1 g0 Þ sinðoc tÞ in Imag
(9b)
From Eqs. (8a) and (8b), the output intensity Pout normalized to input intensity can be shown as 2 E0 Pout ¼ (10) E in (
Pout
ð1 kÞ½1 að1 g0 Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ ð1 g0 Þ 1 1 þ akð1 g0 Þ þ 2 akð1 g0 Þ sinðoc tÞ
) (11)
From Eqs. (9a) and (9b), the loop intensity Ploop at port 2 normalized to input intensity can be expressed as 2 E r2 Ploop ¼ (12) E in Ploop ¼
að1 g0 Þð1 kÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ akð1 g0 Þ þ 2 akð1 g0 Þ sinðoc tÞ
(13)
2.2. Condition for resonance The condition for resonance in the FORR is obtained by finding the value of oct at which the loop intensity Ploop builds up to a maximum value and the output intensity Pout reduces to a minimum value. By inspecting Eq. (11), the condition for resonance is found to be oct ¼ p/2, or in general we have oct ¼ 2qpp/2, where q is an integer. Eq. (11) shows that at resonance when oct ¼ 2qpp/2, the value of the output intensity Pout will become a minimum. The additional condition required to make Pout become zero at resonance can be shown as k ¼ að1 g0 Þ
(14)
If the coupler is lossless with g0 ¼ 0, we have k ¼ a. Fig. 2 shows the theoretical output and loop intensities of the FORR as a function of oct for different values of k( ¼ a). It is observed that for all values of k, the output
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1.00
20
1.75
15
5
Loop intensity, Ploop
Output intensity, Pout
α=κ
0.50 α=κ κ = 0.95 0.75 0.55 0.35
0.25
κ = 0.95 0.75 0.55 0.35
10
5
(2q-1/2)π
0.0 (2q-2)π
2qπ
0 (2q-2)π
(2q+2)π
ωcτ (rad)
2qπ
(2q+2)π
ωcτ (rad)
Fig. 2. Normalized (a) output intensity and (b) loop intensity versus oct when g0 ¼ 0 and k ¼ a ¼ 0.95, 0.75, 0.55, and 0.35.
Output intensity, Pout
1.00
0.75
0.50 α = 0.85 0.65 0.45
0.25
0.25
0.0 (2q-2)π
2qπ
(2q+2)π
ωcτ (rad) Fig. 3. Normalized output intensity versus oct when g0 ¼ 0, k ¼ 0.85, and a ¼ 0.85, 0.65, 0.45, and 0.25.
response dips to zero, and the loop intensity peaks to a maximum value, indicating the resonance condition. As the value of k( ¼ a) decreases, the sharpness of the resonance response decreases as well. When k ¼ a(1g0), it can be shown that at resonance when oct ¼ 2qpp/2, the maximum developed loop intensityr Ploop,max is Ploop; max ¼
k 1k
(15)
The maximum value of the normalized loop intensity in this case is 19 for k ¼ a ¼ 0.95. A reduction of 0.2 in the value of k( ¼ a) from 0.95 to 0.75 causes the maximum loop intensity to decrease from 19 to 3. We note that in Fig. 2(b) the resonance response is not
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sharp when the value of k( ¼ a) is small. When kaa, the output intensity does not dip to zero, as shown in Fig. 3. 2.3. Output and loop intensities at resonance From Eq. (11), at oct ¼ 2qpp/2 and g0 ¼ 0, the normalized output intensity can be shown by pffiffiffi pffiffiffi2 k a Pout joc t¼p=2 ¼ pffiffiffiffiffiffi2 1 ak
(16)
From Eq. (13), Ploop can be derived as Ploop joc t¼p=2 ¼
að1 kÞ pffiffiffiffiffiffi2 1 ak
(17)
The loop intensity response is shown in Fig. 4 for k ¼ 0.85, and a varying from 0.25 to 0.95. It is seen that for a given value of k the maximum loop intensity increases with the value of a. Eq. (16) gives the minimum value of Pout at oct ¼ 2qpp/2 as a function of k and a. The variation of the minimum value of Pout at oct ¼ 2qpp/2 as a function of a for two fixed values of k ¼ 0.9 and 0.5 is plotted in Fig. 5. The normalized output intensity is equal to k for a ¼ 0, is zero when k ¼ a, and is unity at a ¼ 1. The corresponding loop intensity, given by Eq. (17), is illustrated in Fig. 6 as a function of a. The loop intensity for a given value of k always increases with a, reaching a maximum for a ¼ 1. For a fixed value of a, the response of the FORR varies significantly with changes in the value of k. Fig. 7 shows Pout and Ploop as a function of k at oct ¼ 2qpp/2 for a ¼ 0.9 and g0 ¼ 0. In Fig. 7(a), Pout dips to zero for k ¼ a. As k tends to zero, Pout
Loop intensity, Ploop
6.0
4.5 α = 0.85 0.65 0.45 0.25
3.0
1.5
0.0 (2q-2)π
2qπ
(2q+2)π
ωcτ (rad) Fig. 4. Normalized loop intensity versus oct when g0 ¼ 0, k ¼ 0.85, and a ¼ 0.85, 0.65, 0.45, and 0.25.
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1.00
Output intensity at resonance
κ = 0.50 0.90
0.75
0.50
0.25
0.00 0.00
0.50 Power transmission coefficient, α
1.00
Fig. 5. Normalized output intensity at resonance versus a when g0 ¼ 0, k ¼ 0.5 and 0.9.
40 κ = 0.50
Loop intensity at resonance
0.90
30
20
10
0 0.0
0.50 Power transmission coefficient, α
1.00
Fig. 6. Normalized loop intensity at resonance versus a when g0 ¼ 0, k ¼ 0.5 and 0.9.
increases towards a( ¼ 0.9), and for k40.9, Pout increases towards unity at k ¼ 1. Fig. 7(b) illustrates Ploop as a function of k for a ¼ 0.9 under the resonance condition, where the curve shows a peak at k ¼ a, which is followed by a sharp decline towards zero when k tends to unity. The peak value of Ploop at k ¼ 0.9 is 9. At k ¼ 0 the value of Ploop is equal to a( ¼ 0.9). According to Eq. (17), under the resonance condition, if k ¼ a, the theoretical value of Ploop tends to be large when k tends to unity. But when k ¼ 1, the power coupling in the coupler of the FORR will be 100 percent, which will result in a zero loop intensity. Fig. 8 shows the value of Ploop at the resonance as a function of k, assuming k ¼ a and g0 ¼ 0. For k ¼ a ¼ 0.99, the loop intensity takes a value of 99. A comparison of Figs. 7(b)
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1.00 Loop intensity at resonance
Output intensity at resonance
1.00
0.75
0.50
0.25
0.00
7.5
5.0
2.5
0.0 0.0
0.50 Power coupling coefficient, κ
1.00
0.0
0.5 Power coupling coefficient, κ
1.0
Fig. 7. Normalized (a) output intensity and (b) loop intensity at resonance versus k when g0 ¼ 0 and a ¼ 0.9.
Loop intensity at resonance
100
75
50
25
0 0.00
0.495 Power coupling coefficient, κ
0.990
Fig. 8. Normalized loop intensity at resonance versus k when g0 ¼ 0 and a ¼ k.
and 8 indicates that to get a high loop intensity, the conditions oct ¼ 2qpp/2 and k ¼ a(1g0) should be satisfied simultaneously. 2.4. Output and loop intensities at off-resonance At the off-resonance condition obtained at oct ¼ 2qpp/2, Pout and Ploop can be derived as (pffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 ) k þ að1 g0 Þ Pout joc t¼p=2 ¼ ð1 g0 Þ (18) pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 þ akð1 g0 Þ
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Ploop joc t¼p=2 ¼
að1 g0 Þð1 kÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 þ akð1 g0 Þ
9
(19)
With a special case of k ¼ a and g0 ¼ 0, Eqs. (18) and (19) can, respectively, be simplified as Pout joc t¼p=2 ¼
4k ð1 þ kÞ2
Ploop joc t¼p=2 ¼
(20)
kð1 kÞ ð1 þ kÞ2
(21)
0.1250
1.00 Loop intensity at off-resonance
Output intensity at off-resonance
Eqs. (20) and (21) are plotted as functions of k in Fig. 9 at off-resonance condition. In Fig. 9(a), as k increases to its maximum value of 1, Pout increases to unity. The corresponding Ploop in Fig. 9(b) shows a maximum value of 0.125 at k ¼ a ¼ 0.33 and attains zero value when k ¼ 1 or k ¼ 0. For k and a assumed to be unequal, that is for a fixed value of a ¼ 0.9
0.75
0.50
0.25
0.0
0.0625
0.00 0.0
0.50 Power coupling coefficient, κ
1.0
0.0
0.50 Power coupling coefficient, κ
1.0
Fig. 9. Normalized (a) output intensity and (b) loop intensity at off-resonance versus k when g0 ¼ 0 and a ¼ k.
1.00 Loop intensity at off-resonance
Output intensity at off-resonance
1.00
0.75
0.50
0.25
0.00
0.75
0.50
0.25
0.00 0.0
0.5 Power coupling coefficient, κ
1.0
0.0
0.50 Power coupling coefficient, κ
1.0
Fig. 10. Normalized (a) output intensity and (b) loop intensity at off-resonance versus k for g0 ¼ 0 and a ¼ 0.9.
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with k as a variable, at the off-resonance condition of oct ¼ 2qpp/2, the responses of Pout and Ploop are as shown in Fig. 10. The output intensity Pout varies from a value of Pout ¼ a ¼ 0.9 at k ¼ 0 to a maximum value of unity at k ¼ 1, as shown in Fig. 10(a). Fig. 10(b) shows the off-resonance value of Ploop with a maximum value of 0.9 occurring at k ¼ 0. So far all the results were described for an ideal condition of keeping the fractional intensity loss g0 of the coupler in the FORR equal to zero. In the following we will discuss the response of the FORR when g0 is a non-zero quantity. 2.5. Response of FORR when g0a0 Fig. 11 shows the sketches of output and loop intensities as functions of oct for different values of g0 when k ¼ a ¼ 0.8. As g0 increases, the off-resonance value of Pout reduces and the value of resonance dip decreases by shifting upwards from the zero level, as shown in Fig. 11(a). On the other hand, the loop intensity at the resonance decreases with an increase in g0, as indicated in Fig. 11(b). If we consider the value of a ¼ k/(1g0) and assuming k ¼ 0.8, the plot of Pout will be as shown in Fig. 12 for g0 ¼ 0.03, 0.06, and 0.09, where all the curves dip to zero at resonance. The difference between Fig. 11(a) and 12 is that in the former the condition to get a zero output at resonance is not satisfied while in the latter this condition is always satisfied. 2.6. Response of FORR when a ¼ 1 and g0 ¼ 0
1.00
4
0.75
3
0.50
Loop intensity, Pout
Output intensity, Pout
For an ideal condition of a lossless loop (a ¼ 1) and a lossless coupler (g0 ¼ 0), the output and loop intensities are illustrated in Fig. 13. Irrespective of the value of k, the normalized output intensity Pout is always unity for all values of oct (refer to Eq. (11)). That is, Pout ¼ 1, for all values of oct. This leads to an ‘‘all-pass structure’’ of the FORR, where the output intensity becomes equal to the input intensity for all optical frequencies,
γ0= 0.00 0.03 0.06 0.09
0.25
0.00 (2q-2)π
2qπ ωcτ(rad)
(2q+2)π
γ0= 0.00 0.03 0.06 0.09
2
1
0 (2q-2)π
2qπ
(2q+2)π
ωcτ(rad)
Fig. 11. Normalized (a) output intensity and (b) loop intensity versus oct when k ¼ a ¼ 0.8 and g0 ¼ 0, 0.03, 0.06, and 0.09.
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Output intensity, Pout
1.00
0.75 γ0= 0.00 0.03 0.06 0.09
0.50
0.25
0.00 (2q-2)π
2qπ
(2q+2)π
ωcτ(rad)
1.00
80
0.75
60
Loop intensity, Ploop
Output intensity, Pout
Fig. 12. Normalized output intensity versus oct when k ¼ 0.8 and g0 ¼ 0, 0.03, 0.06, 0.09, and a ¼ k/(1g0).
0.50
0.25
0.00 (2q-2)π
40 20 0
2qπ
(2q+2)π
(2q-2)π
ωcτ(rad)
2qπ
(2q+2)π
ωcτ(rad)
Fig. 13. Normalized (a) output intensity (b) loop intensity versus oct when a ¼ 1.0 with k ¼ 0.95 and g0 ¼ 0.
as shown in Fig. 13(a). The loop intensity, with a ¼ 1 and g0 ¼ 0, is given by Ploop ¼
ð1 kÞ pffiffiffi 1 þ k þ 2 k sinðoc tÞ
(22)
which is not independent of oct. At the resonance when oct ¼ 2qpp/2, the loop intensity builds up to a maximum value expressed as pffiffiffi 1þ k pffiffiffi Ploop joc t¼p=2 ¼ (23) 1 k
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Loop intensity at resonance
400
300
200
100
0 0.0
0.495 Power coupling coefficient, κ
0.990
Fig. 14. Normalized loop intensity at resonance versus k when a ¼ 1.0 and g0 ¼ 0.
For example, the value of Ploop reaches a peak value of 78 at oct ¼ 2qpp/2 for a ¼ 1 and k ¼ 0.95, as depicted in Fig. 13(b). Fig. 14 shows Ploop from Eq. (23) at resonance as a function of k. The loop intensity increases exponentially and reaches a maximum value of 392 when k ¼ 0.99. 2.7. Phase angles of output and loop fields From Eq. (7), the phase angle y0 of the normalized output electric field can be shown as "pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi #
pffiffiffi akð1 g0 Þ þ sinðoc tÞ k cosðoc tÞ y0 ¼ arctan pffiffiffi arctan (24) cosðoc tÞ að1 g0 Þ þ k sinðoc tÞ Similarly, the phase angle yloop of the normalized loop electric field at port 2 can be derived from Eq. (6) and expressed by "pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi # akð1 g0 Þ þ sinðoc tÞ yloop ¼ arctan (25) cosðoc tÞ For an infinite loss in the loop (a ¼ 0), that is when there is no field circulation in the loop, the output phase y0 equals to p/2 radians, confirming the p/2 phase shift introduced by the coupler in a cross-coupled field. 2.8. Group delay calculation of FORR We saw earlier that if a ¼ 1, the FORR works as an all-pass structure (refer to Section 2.6). The delay characteristics of the all-pass structure of the FORR are of interest. The group delay of the FORR is defined as the rate of change of the output phase angle y0
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0.0
–9.5τ
k= 0.30 0.95
Group delay, Γ
Group delay, Γ
0.0
13
α= 1
–19.5τ –28.5τ –38.0τ (2q-2)π
2qπ ωcτ(rad)
(2q+2)π
–2.5τ
k= 0.30 0.95
α= 1
–5.0τ –7.5τ –10.0τ (2q-2)π
2qπ
(2q+2)π
ωcτ(rad)
Fig. 15. Group delay of FORR as a function of oct with g0 ¼ 0 and k ¼ 0.3 and 0.95 (a) a ¼ 1.0 and (b) a ¼ k.
with respect to optical frequency oc. That is, from Eq. (24) we can write 9 8 pffiffiffi pffiffiffi pffiffiffi pffiffiffi k= a sinðoc tÞ 1 þ k= a sinðoc tÞ þ ðk=aÞcos2 ðoc tÞ > > > > > > pffiffiffi pffiffiffi 2 > > > > 2 ðo tÞ = < sinðo 1 þ k = a tÞ þ ðk=aÞcos c c dy0 p ffiffi ffi p ffiffiffi ¼t G¼ 2 > > ð1=aÞcos ðoc tÞ þ sinðoc tÞ k= a þ ð1=aÞ sinðoc tÞ doc > > > > pffiffiffi 2 > > pffiffiffi > > 2 ; : k þ ð1= aÞ sinðoc tÞ þ ð1=aÞcos ðoc tÞ
(26)
Eq. (26) gives the group delay under steady-state operation of the FORR. Under ideal conditions of a ¼ 1 and g0 ¼ 0, the group delay in Eq. (26) at resonance reduces to
pffiffiffi 1þ k pffiffiffi G ¼ t (27) 1 k which is t times the corresponding loop intensity Ploop at the resonance as per Eq. (23). The group delay as a function of oct is shown graphically in Fig. 15(a) for a case when a ¼ 1, g0 ¼ 0, and k ¼ 0.3 and 0.95. The plot shows that, when the value of k is high, around resonance, the FORR introduces a large group delay. The group delay changes with the optical frequency oc. This property of the FORR can be utilized for optical dispersion equalization [15]. Fig. 15(b) illustrates the group delay G as a function of oct when a ¼ k. For k ¼ 0.95 and 0.3, it is observed that the maximum group delay obtained is smaller than that of Fig. 15(a). Also, when a ¼ 1, the output intensity does not change with oct, whereas when a ¼ k, the output will dip to zero at the resonance. 2.9. Finesse of FORR The sharpness of the resonance peak in the loop intensity and the resonance dip of the output intensity expressed in terms of finesse of the FORR are defined as [1,35] F¼
FSR Df
(28)
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8
300
6
225 Finesse, F
Finesse, F
14
4
2
150
75
0 0.20
0.35 Power transmission coefficient, α
0.50
0 0.200
0.595 Power transmission coefficient, α
0.990
Fig. 16. Finesse of FORR as a function of a when k ¼ a and g0 ¼ 0, (a) low values and (b) high values.
where FSR is the free spectral range of the FORR expressed as c 1 ¼ (29) nL t where c is the velocity of light in free space, n is the refractive index of the loop fiber, and L is the loop length. Df is the full-width at half-maximum of the loop intensity and can be found by equating Eq. (13) to half of Ploop,max. That is, Df can finally be obtained as ( "rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
#) 1 1 1 að1 g0 Þ 2 1 1 (30) kþ Df ¼ sin 4þ t 2 p 4k k að1 g0 Þ FSR ¼
Therefore, from Eqs. (28) and (29), the finesse can be expressed as F¼
1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ðað1 g0 ÞÞ=4k ðk þ 2=kÞ 4 þ ð1=að1 g0 ÞÞ ð1=tÞ ð1=2Þ ð1=pÞsin
(31) For a special case when k ¼ a and g0 ¼ 0, Eq. (31) can be simplified as 1 F ¼ ð1=2Þ ð1=pÞsin1 1 ð1 aÞ2 2a
(32)
Eq. (32) indicates that the finesse of the FORR has a dependency on the value of a. Fig. 16 shows the finesse as a function of a, for k ¼ a and g0 ¼ 0. Up to a ¼ 0.5, the finesse increases (but not significantly) with a ¼ k ¼ 0.5, as indicated in Fig. 16(a). For a ¼ k ¼ 0.5, the finesse is only about 4.3. When a increases beyond 0.5, the finesse increases steeply, as shown in Fig. 16(b). For a ¼ 0.99, the calculated finesse is about 294. 3. Conclusion This paper dealt with the basic properties of an FORR under steady-state condition. A theoretical analysis on the steady-state response of the FORR was presented by considering the output and loop intensities for various conditions of a and k. The
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expressions for the output and the loop intensities were derived at resonance and offresonance conditions and the results were illustrated graphically for ideal and practical conditions. The expressions for phase angles of the output and the loop fields were formulated. The expression for the finesse, which describes the quality of resonance in the FORR, was derived and the result was sketched graphically. The analyses have shown that all the FORR characteristic parameters, such as resonance condition, output and loop intensities, group delay, and finesse, as functions of loop phase can mainly be controlled and adjusted by the power coupling coefficient, fractional intensity loss of the coupler, and transmission coefficient of the loop. The analytical results given in this article will be useful to understand the dynamic and pulse responses of the FORR, used in different applications. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
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