# Passive Optical Network Fiber Length

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## Introduction

Figure 1: Fiber-To-The-Premises Plant Model (Source).

I recently was asked to explain how a Fiber-To-The-Premises (FTTP) system measures the length of fiber between the central office and each residence in the network (Figure 1). This is an interesting question and I thought it would be worthwhile to the describe the measurement process here.

Service providers are interested in the length of their fiber runs because this information is useful in maintaining their fiber optic infrastructure. For example, a common maintenance situation would involve a home where the data service has become unreliable and the service degradation needs to be investigated. My first questions in these cases are (1) how much total signal attenuation is on the fiber, and (2) how long is the fiber run to the residence. I ask these two questions because the most common issue that I uncover is too much attenuation or distance on the fiber, which is usually caused by

• dirty fiber connector or bad splice

These are, by far, the most commonly encountered problems. One common scenario involves a backhoe accidentally cutting a fiber, which means a new fiber must be spliced in. These repairs are often difficult to make cleanly.

• fiber plant design error

These errors usually are accidental.

• fiber length outside of specification
• RF video lasers only meet their quality of service requirements over a given length of fiber

There are dispersion-induced distortions that will occur in analog system.

• extra fiber length adds fiber loss, which may exceed the system budget
• on very long fiber runs, you can see dispersion problems

Low-levels of dispersion will behave like excess fiber loss.

There can be many other sources of trouble, but I begin my troubleshooting by looking for excessive attenuation or fiber length – these are the most common sources of trouble, and they are usually easy to find.

My focus in this post will be on fiber length determination for Gigabit Passive Optical Network (GPON) systems. Other Passive Optical Network (PON) technologies, like EPON, work similarly.

## Background

### Overview

In a PON, data is transferred between a Central Office (CO) and  multiple destinations (called Optical Network Terminals [ONTs]) that are at different distances. In these networks, each ONT will see all the data sent from the CO on the PON, with each ONT assigned a fraction of the CO's transmit time. This means that the ONTs can "pick off" the portion of the data stream that it has been assigned as all the data comes to it.

The situation for transferring data from the ONTs to the CO is more complex. The data must arrive at the CO in a defined order, but each ONT is at a different range. This means that the ONTs must transmit at a time chosen to compensate for the transmission time differences caused by the distance variations. This post is about how this compensation value is chosen.

### Glossary

Central Office (CO)
The CO is the aggregation point for all the data from the residences. The CO will typically have many fiber connections, with each fiber connection serving as many 64 homes. Each home can be different ranges. To be strictly accurate, I should use the term Optical Line Terminal (OLT) instead of CO, however, the acronym CO will be more familiar to the general reader.
Optical Network Terminal (ONT)
Each home will have an ONT, which communicates with the CO and converts the optical signal into Ethernet for distribution within the home.
Downstream (DS)
Downstream describes communication from the CO to the ONT.
Upstream (US)
Upstream describes communication from the ONT to the CO.
Round-Trip Time (τRTT)
The time it takes for data to be sent from the CO down to an ONT. and for the ONT's response to be received by the CO.
Flight Time (τFlight)
The time that an optical signal takes to move from the ONT to the CO or from the CO to the ONT.
Response Time (τResponse)
Each ONT is given a fixed time to process downstream information from the CO before it needs to transmit data back.
Data Frame
To ensure that every ONT gets an opportunity to transmit on a regular basis, data transmission is broken up into frames of 125 μs (i.e. an 8 kHz rate). This means that each ONT gets an opportunity to transmit every 125 μs. To eliminate the possibility of interference, each ONT is assigned a time "slot" within the data frame. The length of the time slot varies with their data needs. I always view the data frame as being like a train with groups of cars assigned to each ONT. It is very important that the ONT data is arranged in the order that the CO expects.
Start Time (NStart)
An ONT's assigned upstream time slot within a data packet. Technically, start "time" is actually a 16-byte increment (NQ) within a data frame. This increment can be converted to a time by using the data rate (RUS) and increment with the formula $\displaystyle {{\tau }_{{Start}}}={{N}_{{Start}}}\cdot {}^{{{{N}_{Q}}}}\!\!\diagup\!\!{}_{{{{R}_{{US}}}}}\;$.
Equalization Delay (τEqD)
Every ONT is assigned an Equalization Delay (τEqD), which is based on on the ONT's range from the CO. The function of the equalization delay can be viewed several ways. My viewpoint is that each ONT must delay its transmission by  τEqD + τStart to ensure that its data arrives back at the CO in the correct position within the data frame. This means that ONTs near the CO have a longer τEqD than ONTs far from the CO. Each ONT must transmit its photons at exactly the correct time to ensure they all arrive at the CO in the correct order. A τEqD is assigned to each ONT during a process called ranging, which I will not be discussing in this post.
Time-Division Multiplexing (TDM)
Since every ONT is sharing access to the fiber, they are each assigned a time "slot" in which to talk to the CO. To ensure that every ONT knows when to send data upstream, the CO periodically sends a "start of frame" signal downstream to all the ONTs. This start of frame signal is used to synchronize the transmissions from the ONTs.

### System Architecture

For the purposes of this post, here is what you need to know about a TDM FTTP system:

• The ONTs are all assigned equalization delays that will make them respond with the same timing as an ONT at the maximum fiber range.
• A "start of frame" message is sent down regularly to synchronize their clocks.
• The CO can measure the round-trip time for sending down a packet and receiving a response.
• The ONTs have a specific amount of time that they are allowed to process a data request.
• The CO knows the round-trip time, the ONT processing time, and the equalization delay.

### Distance Calculation

Most engineering calculations have a "bookkeeping" aspect to them and determining the length of a fiber optic cable is no exception. We will determine the fiber distance by measuring the time required to send data from the CO to the ONT and for the ONT to respond back. Equation 1 is the key relationship, which basically says that distance equals rate (speed of light) multiplied by time.

 Eq. 1 ${{d}_{{Fiber}}}={{c}_{{Fiber}}}\cdot {{\tau }_{{Flight}}}$

where

• cFiber is the speed of light on the fiber. I discuss the nuances of this calculation in Appendix A.
• τ Flight is time required for the signal to travel from CO to the ONT, or from the ONT to the CO. Technically, the two times are different because the speed of light varies with wavelength on a fiber, but the speeds are so close that I assume them to be equal.
• dFiber is the fiber distance.

## Analysis

### Bookkeeping

Figure 2 shows the delays that make up the round-trip time, which is what the CO can measure. The CO knows every number shown in Figure 2 but the flight time – and we have two flight times, CO→ONT and ONT→CO.

With a little algebra, we can compute the flight time.

Figure 2: Delays that must be accounted for.

### Distance Formula

The distance formula is derived in Figure 3.

Figure 3: Derivation of Distance Equation.

The parenthetical term in the highlighted equation is the flight time. We can compute the constant factor (cFiber/2) as shown in Figure 4.

Figure 4: Calculate the Speed of Light on the Fiber.

### Example Calculation

Figure 5 shows an example calculation.

Figure 5: Fiber Length Calculation Example.

## Conclusion

This question has come up before and it is now time to write down the answer in detail. It provides a nice illustration of the myriad bookkeeping details associated with what is a very simple concept. Remember – all this to estimate a distance using the grade­-school formula distance equal rate multiplied by time.

The discussion above uses my notation. Here is an excerpt from the GPON specification that gives the official formula. It is the same formula, just notationally different.

Figure 6: Fiber Distance Formula from the NGPON2 Specification.

where

• FDi is the fiber distance for the ith ONT.
• RTTi is the round-trip time measured when communicating with ith ONT.
• EqDi is the equalization time assigned to the ith ONT.
• RspTimei is the response time of the ith ONT.
• StartTime is the number of bits within the data frame at which the ith ONT begins to respond. This variable also should have an i subscript, but they forgot it.
• Rnom is the bit rate of the upstream transmission.

## Appendix A: Speed of Light on a Fiber

Equation 2 is used to model the speed of light on the fiber.

 Eq. 2 ${{c}_{{Fiber}}}=\frac{c}{n_{Fiber}\left(\lambda\right)}$

where

• nFiber(λ) is the fiber's index of refraction, which is a function of wavelength.
• cFiber is the speed of light on the fiber.
• c is the speed of light in a vacuum.

The fiber's index of refraction as a function of wavelength is given by Figure 7.

Figure 7: Effective Fiber Index of Refraction.

Figure 7 is from this Corning document.

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