A question about wavelength and refractive index

Text] How to improve the accuracy of optical transmission mode dispersion measurement

Polarization mode dispersion is one of the main factors affecting the performance of the next generation 40Gbps or higher long-distance transmission system. If Improper selection of fiber optic materials or components can lead to high bit error rates even in 10Gbps systems. This article briefly introduces the measurement issues of polarization mode dispersion in optical communication systems and discusses how to improve measurement accuracy.

At the 10Gbps rate, the main cause of polarization mode dispersion (PMD) is the optical fiber (including dispersion compensation fiber) itself; while at the 40Gbps rate, the optical fiber and devices (including erbium-doped fiber amplifiers, optical isolation components such as controllers and connectors) will have an impact on the overall PMD of the system. Therefore, when the transmission rate increases, device design is required to be more stringent to ensure lower PMD. The increase in design requirements also drives test equipment manufacturers to provide more accurate PMD measurement equipment.

Key parameters of PMD

For any given optical device, there is a slowest group speed input main polarization state (PSP-) and a fastest group speed input main polarization state (PSP+), generally there are two input and two (different) output main polarization states (PSP0± and PSP1±), and these main polarization states are usually different from the device intrinsic polarization state. It should be noted that the polarization mode dispersion theory was developed entirely for devices without polarization-dependent loss (PDL). For the case of polarization-related loss PDL>0, the PMD theory is complex and incomplete, so the following part The content does not apply to the case of polarization dependent loss PDL>0.

The main polarization state has characteristics that other polarization states do not have. For devices without polarization-related losses, there is an orthogonal relationship between the main polarization states, and the energy mapped from the input polarization state to the two main polarization states forms two modes (i.e., their primary modes) separated on the link. Harmonics do not exchange energy), so the initial conditions at the input end can be used to describe the signal changes at any point on the device link.

For a given device, the difference in arrival time of fast PSP and slow PSP signals at a specific wavelength λ is called differential group delay DGD(λ). Obviously, this is any two different polarization states Maximum possible delay between signals. Typically DGD on a fiber link is proportional to the square root of the link length, or increases with the number of devices installed. If the link DGD is large, the differential delay will cause a large bit error rate. Therefore, making the DGD much smaller than the bit code length is the key to high-speed long-distance transmission.

Theoretically, the value of DGD is equal to the phase change divided by the frequency increment, that is,

DGD=Δφ/Δω (Δω/ω=-Δλ/λ)

Phase difference refers to the change of the Jones matrix from frequency ω to frequency ω + Δω, so measuring DGD often involves the frequency/wavelength ratio, usually using a tunable laser to achieve wavelength increments. The smaller the DGD, the larger the wavelength increment Δλ must be to ensure that the device operates outside the inherent noise limit. The phase noise determines the lower limit of the DGD resolution of the device. Broadband devices allow larger step sizes, so there are almost no restrictions on measuring small DGD values. Relatively speaking, narrow-band devices are affected by the noise and accuracy distortion of the device itself when the DGD value is small.

A phase change greater than 2π will cause confusion, which also determines the upper limit of the wavelength increment, because if the wavelength increment is too large, Δφ will be greater than 2π and cannot be distinguished from Δφ+2π. This effect limits the maximum measurable DGD for wavelength increment Δλ. Based on experience, we derive a useful rule, that is, the relationship between the maximum measurable delay DGDmax and the wavelength increment Δλmax can be expressed as:

DGDmax·Δλmax＜λ2/2c

At 1,550 nm, use this formula to obtain

DGDmax·Δλmax<4ps·nm Therefore, when measured at 1,550nm and the wavelength increment is 1nm, DGD must be less than 4ps to avoid confusion.

In a sense, correctly selecting the wavelength increment when measuring DGD is a bit like correctly selecting the range of a voltmeter when measuring voltage. If Δλ is too small, it is like trying to use a voltmeter with a range of 3V. Measure the 0.05V voltage instead of using a voltmeter with a range of 0.1V; if it is too large, the corresponding phase change will exceed the upper limit DGDmax. Only by setting Δλ correctly can you effectively take advantage of the accuracy provided by the device.

PMD statistical characteristics

For composite devices composed of multiple components, the total DGD is related to the relative orientation of the PSP of each subcomponent, such as PSPo+(k) of the kth subcomponent The angle αk between PSPi+(k+1). When environmental factors such as pressure or temperature change, the azimuthal stability between PSP(k) will determine the device PMD characteristics. If the orientation changes due to fluctuations in environmental factors, the DGD and the total PSP position of the device will also change with time. Instead, PMD is defined as the time domain average of this DGD value.

If the PSP is stable and does not change with environmental factors, then the PMD will be deterministic, so that even if the environmental factors change or a period of time passes, the DGD and PSP of the device will not change significantly. This is the case for most short-range optical devices.

But if the PSP is to change with environmental factors, the number of subcomponents in the system under test will have a great impact on the PMD. If all initial orientations (αk) and their changes (Δαk) can be determined, the corresponding changes ΔDGD and ΔPSP can theoretically be calculated. But in fact this is only possible when the device consists of only a few sub-components. If the device has thousands of sub-components, it will be impossible to calculate (for example, a length of 1 to 5 meters in a piece of optical fiber must be regarded as independent part). For such subcomponents, the initial orientation cannot be determined, but even if it can be determined accurately, small changes in αk will cause large fluctuations in DGD and total PSP, making actual analysis and prediction impossible.

Because of this, the so-called PMD characteristics of strong mode coupling devices are random and can only be described by statistical methods. Obviously, DGD and PSP change randomly with time (environment), and only predictions from a statistical perspective (such as average DGD or probability distribution) have practical significance. Regardless of the situation, we define the average value of the DGD distribution (a period of time or sample) as PMD, that is, =PMD. Since the terms DGD and PMD are often used interchangeably, it is important to clearly distinguish between the two, remembering that DGD can fluctuate significantly with wavelength and time (environment), while PMD is by definition independent of wavelength and time.

The DGD of broadband devices such as connectors and isolators is deterministic and hardly fluctuates with wavelength and time/environment changes. Therefore, the DGD distribution in a series of measurements is only affected by the accuracy of the measurement process itself, usually A narrow symmetric normal distribution can be obtained, and the width of the distribution is related to the measurement equipment and has nothing to do with the PMD statistical value itself. Since our goal is to design low PMD devices, the general distribution is concentrated in the smaller value range of PMD less than 500fs, and this value is expected to be further reduced in the future.

Due to the internal structure of narrowband devices such as DWDM multiplexers and demultiplexers, the insertion loss and PMD parameters of these devices are significantly different in the pass band and rejection band because the subcomponents are relatively Orientation is generally insensitive to environmental changes, so the PMD characteristics are also certain. The passband of these components is generally narrow, but it is difficult to measure small DGD values ??because large wavelength increments Δλ cannot be used.

For strong mode coupled long fibers, the theoretical distribution of DGD is a Maxwell distribution with only one free parameter γ, which describes the width characteristics of the distribution. Maxwell’s distribution equation can be found in formula (1).

We define polarization mode dispersion (PMD) as the average value of time (see formula (2)).

The above formula shows the concept of defining PMD as the average value of DGD. A larger PMD value indicates a wider distribution, which means that the probability of a larger DGD value is greater, and a larger DGD will seriously affect the chain. The bit error rate of the path. Since the mean of the Maxwell distribution is only a function of the width parameter γ, measuring the PMD (mean) allows us to reconstruct the entire Maxwell distribution and thereby deduce the probability of occurrence of network DGD in a given time.

For homogeneous materials, light wave propagation is theoretically described by the refractive index n, device length L and wavelength λ. Environmental factors mainly affect the refractive index and device length. Since n, L and γ are at the same power index position in the light propagation equation, the wavelength change Δλ has the same effect as the refractive index change Δn or the length change ΔL. Therefore, when DGD samples multiple wavelengths in a time period, the time domain statistics DGD of a device with random characteristics at a certain wavelength will reproduce the same statistical parameters (shape, average, and width). For all PMD instruments, it is a basic assumption that the average DGD values ??sampled by time and wavelength are equal, formula (3).

Normally, system designers are only interested in the change of DGD over time in a certain channel at a specific wavelength. All PMD instruments using wavelength sampling technology can obtain measurement results immediately. The equality of the above equation is assumed. System operators are assured of accurate results. This equation has been tested on application transmission lines and has shown to be correct, which is a good thing since it is very difficult to generate all possible statistical states (various environmental conditions) in such experiments.

Obviously, the measurement accuracy of DGD and PMD is different, and the characteristics of statistical PMD must be considered. The uncertainty of PMD measurement of random devices (such as optical fibers) is higher than that of DGD measurement accuracy of deterministic devices (such as isolators). There are more questions.

Analysis of factors affecting accuracy

DGD accuracy

DGD uncertainty can be calculated by formula (4):

If there is no wavelength error ( That is, δ(Δλ)=0), then the DGD error is caused by the equipment’s inability to resolve the small phase change Δφ. Any device has some internal phase noise, which affects the accuracy of the device.

For example, when measuring a short patch of single-mode fiber with almost no DGD, most commercial Jones Matrix Eigen Analysis (JME) equipment uses a wavelength increment of Δλ = 10 nm, and the measured noise is 3 to 5 fs. For such a large step size, the relative uncertainty δ(Δλ)/Δλ is actually negligible, so a DGD of 3 to 5 fs actually corresponds to 2°Δφ [calculated as follows: Δω(10nm)=7,854×109

1/sec; Δφ=DGD×Δω=5fs×7,854×109

1/sec=4×10-2 rad=2°]. It can be seen that in such cases, relatively accurate results can be obtained only when the phase shift changes around 5° to 10°.

All DGD measurement techniques such as Jones matrix eigenanalysis use tunable lasers. The best tunable lasers currently have a δ (Δλ) of ±10pm, so the relative wavelength is uncertain at a step size of 100pm. The characteristic is 20%. As long as the phase shift is much larger than 20°, its effect can be ignored relative to δ(Δλ)/Δλ. If an external wavelength meter with an uncertainty of only δ (Δλ) = 1 to 3 pm is used to measure the wavelength, the accuracy of DGD can be greatly improved.

Since the wavelength step can be increased without restriction, phase is not the primary limiting factor for broadband deterministic devices even when measuring small DGD values. However, for narrowband devices, the wavelength step Δλ is limited by the passband structure. The passband of a multiplexer with a channel spacing of 100GHz is 50 to 60GHz. Assuming that the PMD phase shift is 10° (5 times larger than the internal phase of the instrument), Then the minimum DGD value that can be accurately measured is formula (5).

Or almost 0.5ps. For low PMD components used in 40Gb/s systems, this value appears to be too large. When the wavelength increment is greater than Δλmax, the maximum measurable DGD is determined by the uncertainty of the phase measurement.

Tunable laser sources often have the same error when generating the same wavelength increment, that is, the wavelength errors are generally repetitive. The wavelength error is usually shifted in one direction and is usually not symmetrically dispersed around the specified wavelength increment, which causes the DGD or PMD value to deviate from the average value. For this reason, we strongly recommend using an external wavelength meter to measure the wavelength step when using small wavelength steps.

PMD accuracy

We know that for broadband and narrowband deterministic devices, DGD is almost independent of wavelength, so we can get many DGD samples by scanning a specific wavelength range , and then calculate the average value, which is the PMD value. At this time, the DGD distribution can be assumed to conform to the Gaussian distribution, and the PMD measurement uncertainty is 1/√n times the usual standard deviation σDGD, where n represents the number of DGD samples.

If it is assumed that the DGD of the narrowband device does not fluctuate significantly with the wavelength, then a series of DGD measurements can be performed at the center wavelength position within the passband. Unlike insertion loss, since the transmission channel does not operate in the suppression band, DGD is only meaningful for the pass band, and the suppression band is only used to suppress signal crosstalk between adjacent channels. The wavelength increment Δλ should be as large as possible, so that the maximum phase shift can be achieved for the specified passband DGD, so the wavelength increment is only slightly smaller than the passband bandwidth. In addition, since the results come from standard measurement procedures, and the uncertainty of PMD measurement is determined by σDGD, the DGD distribution can be considered to conform to the normal distribution. It should be noted that any large incremental shift in system wavelength will appear as a systematic error δ (Δλ), and will immediately cause the entire DGD distribution function to shift, and the PMD value will also shift, so this type of device is relatively The measurement of small PMD values ??must have high wavelength measurement accuracy.

For devices with many polarized mode couplings, such as optical fibers, DGD appears to vary randomly at different times (environments) and wavelengths, but even so, measured at two wavelengths that are very close together The DGD value is still relevant to some extent. This correlation means that if the DGD at λ1 is known, the probability of the DGD value at λ2 can be appropriately predicted, provided that λ2-λ1 is smaller than the typical wavelength separation. This correlation is somewhat similar to near-term and medium- to long-term weather forecasts, where weather forecasts are generally more reliable for the next day, but somewhat vague for the following week. The wavelength (frequency) interval where there is correlation is called the PMD bandwidth ΔBλ. For a device that conforms to the Maxwell distribution, the PMD bandwidth is obtained by ΔBλ=0.64/PMD, which is inversely proportional to the PMD value. At a wavelength of 1,550nm, this equation simplifies to ΔBλ = (5.1/PMD), where PMD is expressed in ns.

The larger the PMD, the smaller the PMD bandwidth, and the faster the changes in DGD, PSP and polarization state will be in a given wavelength range. Since the PMD bandwidth represents the wavelength range where DGD changes significantly, the wavelength increment Δλ used for a single DGD measurement should be much smaller than the PMD bandwidth ΔBλ, otherwise a single DGD measurement is just a smoothing of the DGD.

Obviously, to accurately reproduce the Maxwell distribution, multiple DGD values ??under different environmental conditions must be sampled, otherwise the estimation of the PMD value will be inaccurate.

In terms of correlation, the two values ??(DGD(λ1) and DGD(λ2)) are statistically independent only if the wavelength interval (λ2-λ1) is large enough, so for random mode coupling devices, The wavelength separation between adjacent DGD values ??should be slightly larger than ΔBλ.

However, there is a problem because the PMD bandwidth limits the upper limit of the number of statistically independent samples that can be taken in the measurement of the specified scan range. Since the actual scanning range is limited by λstart and λstop, the number of independent samples is approximately between [ωstop-ωstart]/ΔBλ~ωstop-ωstart]×PMD, so the reduction of the scanning range and PMD bandwidth will affect the PMD accuracy, which can Theoretically use formula (6) to verify:

Even if the independent DGD measurement accuracy is very high, it cannot exceed this limitation, because this is the first principle, and it is only assumed that DGD conforms to the Maxwell distribution, so it is Any PMD measurement technology is suitable.

For a device with a PMD of 10ps, the ΔPMD uncertainty obtained with an adjustable range of 10nm is relatively good, ±10% or 1ps; but for a device with a PMD of 1ps, the uncertainty rate is using a 10nm range. is ±30%, which is relatively large (in percentage terms), so it is necessary to scan the 100nm wavelength to reduce the expected error to about 10%. Compared with these relatively large internal uncertainties, equipment errors due to wavelength or phase errors are negligible in most cases. Conclusion of this article: DGD uncertainty is related to many factors, including wavelength errors caused by wavelength increment changes and internal phase noise of the equipment. Wavelength uncertainty can be significantly improved by using an external wavelength meter rather than relying on the internal step accuracy of the tunable laser. The internal phase noise of the device PMD will affect the lower limit of the minimum DGD value. The passband bandwidth of the narrow-band device limits the increment of wavelength change, which has now become a huge obstacle for such models to obtain lower PMD values. For broadband optical fiber devices with random characteristics, PMD accuracy is mainly determined by the reduced adjustable range and PMD bandwidth, and only in few cases can an uncertainty rate better than ±10% be achieved.