## Abstract

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.

© 2007 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 *P*_{min} and *P*_{max} for the entire link, in this paper different power constraints, *P*_{min,}* _{i}* and

*P*

_{max,}

*, are considered for each section*

_{i}*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. *P _{i}* and

*P*

_{i}_{,mid}are the signal levels respectively at the input and at the middle of the fiber span

*i. P*

_{i}_{,mod}is the signal level at the input of the module

*i*and

*P*

_{i}_{,G}and

*P*

_{i}_{,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

*G*

_{M,}

*=*

_{i}*G*Γ

_{i}*′*

_{i}G*, if*

_{i}*G*

_{M,}

*follows the same optimal map. A chain of such modules is modeled as Fig. 1(b) where the optimal gain map of module*

_{i}*i*is:

*= exp(−*

_{i}*α*) is the loss and ${\mathrm{\Lambda}}_{i}={\int}_{0}^{{L}_{i}}\text{exp}(-{\int}_{0}^{z}{\alpha}_{i}({z}^{\prime})d{z}^{\prime})dz$ is the effective length of the span

_{i}L_{i}*i*. Also,

*α*is the loss coefficient and

_{i}*L*is the length of span

_{i}*i*. Assuming

*α*=

_{i}*α*for all spans gives: Λ

*= (1 − exp(−*

_{i}*αL*))/

_{i}*α*. By using this result, Eq. 1 simplifies to: If the input power of the link is

*P*

_{In}and the signal power at the middle of the first span is

*P*

_{mid,1}then: ${P}_{1,\text{mid}}={P}_{\text{In}}/\sqrt{{\mathrm{\Gamma}}_{1}}$. By using 2 and for the second span we have: ${P}_{2,\text{mid}}={P}_{1,\text{mid}}\times \sqrt{{\mathrm{\Gamma}}_{1}}\times {G}_{1,\text{M}}\times \sqrt{{\mathrm{\Gamma}}_{2}}={P}_{1,\text{mid}}$. This can be consequently repeated for all sections and therefore ${P}_{i,\text{mid}}={P}_{\text{In}}/\sqrt{{\mathrm{\Gamma}}_{1}}$for

*i*∈ {1, 2, ...,

*N*}. This result is used in the following subsections where the power constraints

*P*≤

_{i}*P*

_{max}and

*P*

_{i}_{,mod}≥

*P*

_{min}are derived by having

*P*≤

_{i}*P*

_{max,}

*and*

_{i}*P*

_{i}_{,mod}≥

*P*

_{min,}

*and assuming that the overall gain of the modules follows (1).*

_{i}## 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 *P _{i}*,

*P*

_{i}_{,Mod},

*P*

_{i}_{,G}and

*P*

_{i}_{,DCF}that are illustrated in Fig. 1(a) are constrained by double-sided limits. We assume that

*P*≤

_{i}*P*

_{max,}

*and*

_{i}*P*

_{i}_{,mod}≥

*P*

_{min,}

*and start to find the the condition required to satisfy the constraints which are not obtainable using link budget formulations.*

_{i}#### 3.1. Maximum limit for P_{i}

For the maximum limit of *P _{i}*, it is assumed that

*P*

_{In}is less than the lowest possible ${P}_{\text{max},i}\sqrt{{\mathrm{\Gamma}}_{i}}/\sqrt{{\mathrm{\Gamma}}_{1}}$ and by following a worst case scenario, it is shown that

*P*is limited by

_{i}*P*

_{max,}

*. As a worst case scenario, the lowest possible value of ${P}_{\text{max},i}\sqrt{{\mathrm{\Gamma}}_{i}}/\sqrt{{\mathrm{\Gamma}}_{1}}$ occurs when its numerator takes the smallest Γ*

_{i}*and its denominator takes the largest Γ*

_{i}_{1}. These two boundary values are noted as Γ

_{i}_{,LB}and Γ

_{1,UB}where indices LB and UB represents the lower bound and the upper bound limits. Therefore as the worst case situation we have:

*and Γ*

_{i}_{1}where Γ

*≥ Γ*

_{i}

_{i}_{,LB}and Γ

_{1}≤ Γ

_{1,UB}for

*i*∈ {1, 2, ...,

*N*}. Therefore (3) gives:

*P*≤

_{i}*P*

_{max,}

*.*

_{i}*3.2. Minimum limit for P*_{i,Mod}

For the minimum limit of *P _{i}*

_{,Mod}, ${P}_{\text{min},i}/(\sqrt{{\mathrm{\Gamma}}_{i}}\times \sqrt{{\mathrm{\Gamma}}_{1}})$ is taken as the lower limit of

*P*

_{In}. As the worst case scenario,

*P*

_{In}is greater than the largest possible lower bound which is obtained by taking the smallest possible Γ

*and Γ*

_{i}_{1}, Γ

_{i}_{,LB}and Γ

_{1,LB}. This is formulated as:

*and Γ*

_{i}_{1}where

*i*∈ {1, ...,

*N*}, we have:

*P*

_{i}_{,mod}is then found by substituting its equivalent, ${P}_{i,\text{mid}}\times \sqrt{{\mathrm{\Gamma}}_{i}}$, which gives:

*P*

_{i}_{,}

*≥*

_{mod}*P*

_{min,}

*.*

_{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 *P*_{In}, the lower limit should be less than or equal to the upper limit, or:

*P*

_{In}in order to satisfy the sufficiency of

*P*

_{i}_{,G}≤

*P*

_{max,}

*and*

_{i}*P*

_{i}_{,DCM}≥

*P*

_{min,}

*for obtaining*

_{i}*P*≤

_{i}*P*

_{max,}

*and*

_{i}*P*

_{i}_{,mod}≥

*P*

_{min,}

*. A three dimensional illustration of the minimum and maximum feasible values for*

_{i}*P*

_{In}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,

*P*

_{min,}

*and*

_{i}*P*

_{max,}

*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*

_{i}*P*

_{min,}

*and*

_{i}*P*

_{max,}

*are same as above. When*

_{i}*L*changes over the range of 55 km to 110 km, for both

_{i}*L*

_{1}= 55 km and

*L*

_{1}= 110 km, there is a feasible

*P*

_{In}value. However, the feasible range is smaller for

*L*

_{1}= 55 km compared to

*L*

_{1}= 110 km that provides a wider range for

*P*

_{In}. It should be noted here that designing the length of

*L*

_{1}is very convenient from a practical point of view. Since

*L*

_{1}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),

*L*

_{1}= 80 km and

*P*

_{min,}

*and*

_{i}*P*

_{max,}

*take different values. When*

_{i}*P*

_{min,}

*= 0.05 mW and*

_{i}*P*

_{max,}

*= 2 mW for all sections, the feasible region that is illustrated using black vertical shade lines ends at*

_{i}*L*= 63.5 km. For a wider band of power constraints, when

*P*

_{min,}

*= 0.01 mW and*

_{i}*P*

_{max,}

*= 10 mW, there is always a feasible*

_{i}*P*

_{In}value for the range of

*L*from 55 km to 110 km. This area is shown using gray vertical shaded lines.

_{i}#### 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, *P _{i}*

_{,G}≤

*P*

_{max,}

*and*

_{i}*P*

_{i}_{,DCM}≥

*P*

_{min,}

*, by applying link budget calculations:*

_{i}**Maximum Limit for** *P _{i}*

_{,DCM}: The maximum limit for

*P*

_{i}_{,DCM}can be obtained from

*P*

_{i}_{,G}≤

*P*

_{max,}

*and knowing that DCF gain is less than unity. Therefore,*

_{i}*P*

_{i}_{,G}Γ′

*≤*

_{i}*P*

_{max,}

*or equivalently*

_{i}*P*

_{i}_{,DCM}Γ′

*≤*

_{i}*P*

_{max,}

*.*

_{i}**Minimum Limit for** *P _{i}*

_{,G}: The minimum limit for

*P*

_{i}_{,G}is obtained from

*P*

_{i}_{,DCM}≥

*P*

_{min,}

*and knowing that Γ′*

_{i}

_{i}_{−1}> 1. Therefore

*P*

_{i}_{−1,DCM}/Γ′

_{i}_{−1}≥

*P*

_{min,}

*. Replacing the equivalent term of*

_{i}*P*

_{i}_{−1,DCM}/Γ′

_{i}_{−1}, gives

*P*

_{i}_{,G}≥

*P*

_{min,}

_{i}**Minimum Limit for** *P _{i}*: The minimum power constraint for the input power value of span

*i*,

*P*can be obtained from lower bound constraint of the DCM output power in section (

_{i}*i*−1). From

*P*

_{i}_{−1,DCM}≥

*P*

_{min,}

*and*

_{i}*G*′

_{i}_{−1}> 1 we have

*P*

_{i}_{−1,DCM}

*G*′

_{i}_{−1}≥

*P*

_{min,}

*which is equal to*

_{i}*P*≥

_{i}*P*

_{min,}

*.*

_{i}**Maximum Limit for** *P _{i}*

_{,Mod}: In order to obtain

*P*

_{i}_{,Mod}≤

*P*

_{max,}

*, we have:*

_{i}*P*

_{i}_{,G}

*≤ P*

_{max,}

*and*

_{i}*G*≥ 1. Therefore,

_{i}*P*

_{i}_{,G}/

*G*≤

_{i}*P*

_{max,}

*and replacing the the equivalent value for*

_{i}*P*

_{i}_{,G}/

*G*gives:

_{i}*P*

_{i}_{,Mod}≤

*P*

_{max,}

*.*

_{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 *P _{i}*

_{,G}≤

*P*

_{max,}

*and*

_{i}*P*

_{i}_{,G}=

*P*

_{In}Γ

*we have:*

_{i}G_{i}*P*

_{i}_{,DCF}≥

*P*

_{min,}

*and*

_{i}*P*

_{i}_{,DCF}=

*P*

_{In}Γ

*Γ′*

_{i}G_{i}*and the overall gain value of each module*

_{i}*G*

_{M,}

*=*

_{i}*G*Γ′

_{i}*′*

_{i}G*, we have:*

_{i}*G*

_{M,}

*from (2) gives:*

_{i}## 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 *P*_{in}, 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, *L*_{UB,}* _{i}*. The upper and the lower limits for

*P*

_{in}due to constraints at each section of the link, are found using (4) and (6) and shown in Table 1. The minimum of the

*P*

_{In,UB}over ten spans is 7.6 mW and is taken as the maximum feasible input power. Likewise, the maximum of the

*P*

_{In,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).

## 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

**1. **S. D. Personick, “Fundumental Limits in Optical Communication” Proceedings of the IEEE **69**, 2, 262–266 (1981). [CrossRef]

**2. **R-J. Essiambre and P. J. Winzer “Fibre nonlinearities in electronically pre-distorted transmission” ECOC Proceedings **2**, 191–192 (2005).

**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]