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A full set of double-sided power constraints are derived from the single-sided constraints at only two points of each section of a point to point optical link. It is shown that this can be obtained if the input power of the link remains within a feasible range. The two single-sided constraints provide the upper bounds for the gain values of the amplifiers at each module. Therefore, double-sided constraints at all of the points along the link can be obtained by satisfying the gain limits of the amplifiers. Furthermore, unlike the conventional way of considering a unified set of constraints for the entire link, different minimum and maximum power constraints are considered for each section of the link.
© 2012 Optical Society of America
1. Introduction
The power level of an optical signal needs to be managed across any optical networking architectures including point to point optical links. The lower limit of the optical power is dictated by the design requirements including the sensitivities of the optical components [1]. On the other hand, excessive level of power creates nonlinearity [2] and saturates the optical amplifiers [3]. Therefore, the power of the optical signal should be maintained within the maximum and the minimum limits. Different power constraints at each part of the link are imposed by the different manufacturing specifications of fiber spans and optical components. Unlike the conventional method of considering a unified set of Pmin and Pmax for the entire link, in this paper different power constraints, Pmin,i and Pmax,i, are considered for each section i of the link. This provides a better utilization of the signal power which is otherwise wasted by considering a unified worst case values. Furthermore, to capture both the attenuation and the dispersion impairments, the point to point link is modeled as a chain of the modules that includes dispersion compensating fibers (DCFs) and amplifiers.
It is shown in this paper that if the gain of the modules follow an optimal map [4], and the input power remains within a feasible range, a full set of double-sided power constraints can be obtained from the single-sided constraints only at two points of each section of the link. Furthermore, the upper limits for the gain of the two amplifiers at each module are obtained from the two single-sided constraints at the input and at the output of DCF. This is used to propose a simplified design process for satisfying the power constraints: If the gain of amplifiers satisfy the proposed upper limits and the input power remains within the proposed feasible range, then the double-sided power constraints will be satisfied everywhere along the link.
2. The link model
In order to address both the attenuation and the dispersion impairments, a combination of amplifiers and DCF is often utilized. For the point to point links, this module is repeatedly located at the end of fiber spans. Figures 1(a) and 1(b) illustrate a link model with modules including amplifiers and DCFs where the first amplifier in the module amplifies the signal attenuated along the fiber span, the DCF compensates the signal dispersion after traveling along the fiber and the second amplifier regains the signal power attenuated over the DCF. Pi and Pi,mid are the signal levels respectively at the input and at the middle of the fiber span i. Pi,mod is the signal level at the input of the module i and Pi,G and Pi,DCF are the signal levels at the output of the first amplifier and the DCF of module i. An optimal gain map is obtained if the average power and the accumulated amplifier spontaneous emission (ASE) noise of a chain of only amplifiers are optimized [4]. Such a result can be extended to the point to point cascaded connection of any module with an active overall transfer function of GM,i = GiΓiG′i, if GM,i follows the same optimal map. A chain of such modules is modeled as Fig. 1(b) where the optimal gain map of module i is:
Γi = exp(−αiLi) is the loss and is the effective length of the span i. Also, αi is the loss coefficient and Li is the length of span i. Assuming αi = α for all spans gives: Λi = (1 − exp(−αLi))/α. By using this result, Eq. 1 simplifies to: If the input power of the link is PIn and the signal power at the middle of the first span is Pmid,1 then: . By using 2 and for the second span we have: . This can be consequently repeated for all sections and therefore for i ∈ {1, 2, ..., N}. This result is used in the following subsections where the power constraints Pi ≤ Pmax and Pi,mod ≥ Pmin are derived by having Pi ≤ Pmax,i and Pi,mod ≥ Pmin,i and assuming that the overall gain of the modules follows (1).3. Obtaining the constraints
If the signal power at the input and output of the amplifiers, DCFs and the fiber spans are constrained, the entire link is constrained. Therefore, we should show that the four points of Pi, Pi,Mod, Pi,G and Pi,DCF that are illustrated in Fig. 1(a) are constrained by double-sided limits. We assume that Pi ≤ Pmax,i and Pi,mod ≥ Pmin,i and start to find the the condition required to satisfy the constraints which are not obtainable using link budget formulations.
3.1. Maximum limit for Pi
For the maximum limit of Pi, it is assumed that PIn is less than the lowest possible and by following a worst case scenario, it is shown that Pi is limited by Pmax,i. As a worst case scenario, the lowest possible value of occurs when its numerator takes the smallest Γi and its denominator takes the largest Γ1. These two boundary values are noted as Γi,LB and Γ1,UB where indices LB and UB represents the lower bound and the upper bound limits. Therefore as the worst case situation we have:
Since (3) satisfies the worst case boundary values, it satisfies any other values of Γi and Γ1 where Γi ≥ Γi,LB and Γ1 ≤ Γ1,UB for i ∈ {1, 2, ..., N}. Therefore (3) gives: Multiplying both sides of (4) by and replacing the equivalent term for , gives , which is equivalent to Pi ≤ Pmax,i.3.2. Minimum limit for Pi,Mod
For the minimum limit of Pi,Mod, is taken as the lower limit of PIn. As the worst case scenario, PIn is greater than the largest possible lower bound which is obtained by taking the smallest possible Γi and Γ1, Γi,LB and Γ1,LB. This is formulated as:
From (5) and for any other Γi and Γ1 where i ∈ {1, ..., N}, we have: By multiplying both sides of (6) by and replacing which is known from (2), we have: . The minimum limit for Pi,mod is then found by substituting its equivalent, , which gives: Pi,mod ≥ Pmin,i.Inequalities 3 and 5 indicate the minimum and the maximum limits for the input power of the link. In order to have a feasible PIn, the lower limit should be less than or equal to the upper limit, or:
which simplifies to: The right hand side of (8) only depends on the loss coefficient and the maximum and minimum power limits and determines the feasible span lengths which guarantee the existence of a PIn in order to satisfy the sufficiency of Pi,G ≤ Pmax,i and Pi,DCM ≥ Pmin,i for obtaining Pi ≤ Pmax,i and Pi,mod ≥ Pmin,i. A three dimensional illustration of the minimum and maximum feasible values for PIn is plotted in Fig. 2. Axis X is the length of fiber span i in kilometers and axis Y is length of the first fiber span in kilometers. The vertical axis Z represents the power value in milliWatts (mW). In Fig. 2, Pmin,i and Pmax,i are respectively 0.01 mW and 10 mW for all sections. A two dimensional cross section of Fig. 2 is illustrated in Fig. 3(a) where Pmin,i and Pmax,i are same as above. When Li changes over the range of 55 km to 110 km, for both L1 = 55 km and L1 = 110 km, there is a feasible PIn value. However, the feasible range is smaller for L1 = 55 km compared to L1 = 110 km that provides a wider range for PIn. It should be noted here that designing the length of L1 is very convenient from a practical point of view. Since L1 is the length of the first span, it is easily accessible at the transmitter side and therefor its length can be easily modified. In Fig. 3(b), L1 = 80 km and Pmin,i and Pmax,i take different values. When Pmin,i = 0.05 mW and Pmax,i = 2 mW for all sections, the feasible region that is illustrated using black vertical shade lines ends at L = 63.5 km. For a wider band of power constraints, when Pmin,i = 0.01 mW and Pmax,i = 10 mW, there is always a feasible PIn value for the range of Li from 55 km to 110 km. This area is shown using gray vertical shaded lines.
Fig. 2 Three dimensional demonstration of the minimum and the maximum limits for the input power. Pmin,i = 0.01 mW and Pmax,i = 10 mW for all i from 1 to N
3.3. Constraints obtained using link budget
As presented in the following subsections, the remaining constraints for the rest of the network do not require any additional conditions and can be obtained from the two initial assumed constraints, Pi,G ≤ Pmax,i and Pi,DCM ≥ Pmin,i, by applying link budget calculations:
Maximum Limit for Pi,DCM: The maximum limit for Pi,DCM can be obtained from Pi,G ≤ Pmax,i and knowing that DCF gain is less than unity. Therefore, Pi,GΓ′i ≤ Pmax,i or equivalently Pi,DCMΓ′i ≤ Pmax,i.
Minimum Limit for Pi,G: The minimum limit for Pi,G is obtained from Pi,DCM ≥ Pmin,i and knowing that Γ′i−1 > 1. Therefore Pi−1,DCM/Γ′i−1 ≥ Pmin,i. Replacing the equivalent term of Pi−1,DCM/Γ′i−1, gives Pi,G ≥ Pmin,i
Minimum Limit for Pi: The minimum power constraint for the input power value of span i, Pi can be obtained from lower bound constraint of the DCM output power in section (i−1). From Pi−1,DCM ≥ Pmin,i and G′i−1 > 1 we have Pi−1,DCMG′i−1 ≥ Pmin,i which is equal to Pi ≥ Pmin,i.
Maximum Limit for Pi,Mod: In order to obtain Pi,Mod ≤ Pmax,i, we have: Pi,G ≤ Pmax,i and Gi ≥ 1. Therefore, Pi,G/Gi ≤ Pmax,i and replacing the the equivalent value for Pi,G/Gi gives: Pi,Mod ≤ Pmax,i.
4. Feasible gain values
The upper limits for the gain values of the two amplifiers located at each module along the link can be obtained using the single sided power constraints at the input and at the output of the DCF. From Pi,G ≤ Pmax,i and Pi,G = PInΓiGi we have:
Similarly, from Pi,DCF ≥ Pmin,i and Pi,DCF = PInΓiGiΓ′i and the overall gain value of each module GM,i = GiΓ′iG′i, we have: Replacing, GM,i from (2) gives: Inequalities 9 and 11 are the upper bound of the feasible values for the gain of the two amplifiers at each repeating module. Since these two equations are interchangeably equivalent to the two single-sided constraints at the input and at the output of the DCF, satisfying (9) and (11) are sufficient to obtain the full set of double-sided constraints for the entire link.5. A design application
The application of the analytical derivations presented so far is shown using a test example. A link with ten spans is considered for simulation purposes. The length of each fiber span as well as the minimum and the maximum power limits for each section of the link are given. These values are respectively shown in the first, second and the third rows of Table 1. The first step for finding the feasible range of Pin, is to make sure that the power limits and the span lengths satisfy (8). As seen, all of the span lengths satisfy their maximum constraints, LUB,i. The upper and the lower limits for Pin due to constraints at each section of the link, are found using (4) and (6) and shown in Table 1. The minimum of the PIn,UB over ten spans is 7.6 mW and is taken as the maximum feasible input power. Likewise, the maximum of the PIn,LB is, 2.4 mW and is selected as the minimum feasible input power. Therefore, input power swings over a feasible range of 5.2 mW. Finally, the last two rows of the table show the upper limits for the gain of the amplifiers at each span obtained from (9) and (11).
Table 1. Finding PIn,UB, PIn,LB, GLB,i and GUB,i from given span lengths
6. Conclusion
We showed that if the gain values of the amplifiers satisfy the proposed upper limits and the input power remains within the suggested feasible range, the entire point to point optical link is constrained by the minimum and maximum power limits.
Acknowledgments
Author would like to thank Dr. K. Hinton and Prof. A. Mecozzi for their useful comments.
References and links
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3. S. S. Wagner “Optical Amplifier Applications in Fiber Optic Local Networks” IEEE Trans. on Commun. 35, 4419–426 (1987). [CrossRef]
4. A. Mecozzi “ On Optimization of the Gain Distribution of Transmission Lines with Unequal Amplifier Spacing” Photon. Technol. Lett. 10, 7, 1033–1035 (1998). [CrossRef]
