CN105182470A

CN105182470A - Polarization dependency of multi-mode optical fiber main mode and derivation method therefor

Info

Publication number: CN105182470A
Application number: CN201510112568.7A
Authority: CN
Inventors: 白昀; 毛蔚
Original assignee: Individual
Current assignee: Bai Yun
Priority date: 2015-03-16
Filing date: 2015-03-16
Publication date: 2015-12-23

Abstract

The invention relates to a polarization dependency of multi-mode optical fiber main mode. When a curvature variance and an optical fiber length are small, namely, a low coupling mechanism is generated, group delay is very close to a non-coupling value and is in linear proportion to the optical fiber length, the main mode still generates high polarization, polarization dependence of pulse response reappears in this mechanism, and the polarization dependence can be observed in a multi-mode optical fiber made of a silicon material. When the curvature variance and an optical fiber length are large enough, namely a high coupling mechanism is generated, then group delay of propagation can be reduced and is in proportion to the square root of the optical fiber length, the main mode is depolarized, and in this model, the group delay of the model is consistent with reduction of propagation group delay observed in an multi-mode optical fiber MMF made of a plastic material. A parameterized physical method is provided for a gradual change refractive index MMF, calculation of space and polarization mode coupling coefficients can be carried out through the method, and PMs and time delay can be calculated.

Description

A kind of polarization independent relation of multimode optical fiber master mode and derivation method thereof

Technical field

The present invention relates to a kind of gradually changeable index multimode fiber, particularly a kind of polarization independent relation based on graded index multimode fiber master mode under space and polarization mode coupling condition and derivation method thereof.

Background technology

In multimode optical fiber (MMF), different mode is generally propagated with different group delays (GDs), and this phenomenon is called as modal dispersion.The defect of optical fiber, as refractive index inhomogeneity, fibre core ellipticity and eccentricity, bends, can produce the coupling between mode, namely be called as the effect of modal coupling.Due to modal coupling, even if light pulse emission is in a single mode, it also tends to other mode that are coupled, and causes the Multiple pulse superposition that MMF exports.

Traditionally, the modal dispersion in MMF and coupling are described by use coupling power model.This model implicitly supposes that ideal state and its GDs are not revised by modal coupling.Coupling only causes the redistribution of the power between mode, and describes by coupling coefficient, and this coupling coefficient is that real number, non-negative and phase place are independent.These models are very effective in the distribution of description mode power, and the distribution of mode power is the function of time and fiber lengths, and these models also contribute to understanding distorted signals, as the pulse strenching of the function of fiber lengths and fibre loss.

The impact of phase place do not considered by this model, but they are only suitable to incoherent source, as light emitting diode.Found by contrast, in single-mode fiber (SMF), describe polarization modal coupling and polarization modal dispersion (PMD) by the field coupling model of major state model.In this model, the coupling between polarization modal field amplitude is that the recombination coefficient passing through relied on phase place describes.This coupling have modified ideal state and its GDs, makes they and frequency dependence.There is pair of orthogonal polarization state, be called the major state of polarization, it is the characteristic modes of GD operational symbol, and it has field amplitude independent of fundamental frequency and GDs.

In recent years, many experiments that adopted the method for coherent source and High Speed Modulation to carry out in MMF, these experiments show some effect can not explained with power coupling model, and such as impulse response is to the dependence of launching polarization.Fan and Kahn introduces a kind of field coupling model, and this model is the simplified summary to single-mode fiber (SMF) the major state model for PMD.Especially, their model prediction one group is called as the orthogonal modes of master mode (PMs), and it is the characteristic modes of GD operational symbol, and is the amplitude independent of fundamental frequency and GDs.In other words, PMs is the constraint not by modal dispersion and fundamental frequency.The light signal of low order PMs launched by the using adaptability optical instruments such as Shen, and to reduce modal dispersion, make to transmit in two-forty-distance product, even if in the optical fiber having modal coupling, result can not adopt coupling power model explanation completely.

Only describe the universal of PMs in current research, summary is arranged, and without any the particular module of modal coupling, particular module may be used for the result of quantitative interpretation experiment.Therefore, need to study the next transmission master mode of a kind of gradually changeable index multimode fiber under space and polarization mode coupling condition.

Summary of the invention

The technical problem to be solved in the present invention is to provide a kind of polarization independent relation based on graded index multimode fiber master mode under space and polarization mode coupling condition and derivation method thereof.

For solving the problems of the technologies described above, technical scheme of the present invention is: a kind of polarization independent relation of graded index multimode fiber master mode, its innovative point is: it is when curvature variance is compared with fiber lengths hour that described polarization independent closes, namely during low coupling mechanism, group delay is its non-coupled value closely, and with fiber lengths linearly, master mode is still produce high degree of polarisation simultaneously, in this mechanism, reproduced the polarization dependence of impulse response, and this polarization dependence is observed in silicon materials multimode optical fiber; When curvature variance and fiber lengths are enough large, i.e. high coupling mechanism, so can reduce the group delay of propagation, and it is proportional with the square root of fiber lengths, depolarizing of master mode simultaneously, in this model, the group delay of model reduces consistent with the propagation group delay observed in plastic material multimode optical fiber MMF.

The present invention also provides a kind of derivation method of polarization independent relation of above-mentioned graded index multimode fiber master mode, and described derivation method concrete steps are as follows:

(1) it is theoretical: assuming that one is launched Electric Field Distribution and polarization, in this electric field, build the multistage model of a space and polarization mode coupled fiber, calculate the propagation operational symbol of optical fiber, then, calculate group delay operational symbol, and obtain its proper vector, these are PMs, when the distribution of given launching site and polarization, calculate the impulse response of optical fiber, prove, it is effective in low coupling mechanism, and the orthogonality of polarization causes minimum and maximum switching process;

The analysis modeling of (2) three modal systems: build a kind of simple three modal systems so that the GDs degree of dependence of regarding fibers curvature and length in low and high coupling mechanism to be described;

(3) the vertical modeling of multimode optical fiber: the numerical evaluation describing actual fiber model, describes PMs characteristic and its group delay in low and high coupling mechanism;

(4) the polarization independent relation of multimode optical fiber master mode is drawn.

Further, when the multistage model of described space and polarization mode coupled fiber is built, first Refractive Index Profile o is determined; Then, draw propagation constant and the field distribution of the normal mode in local, and in bending section, calculate spatial mode coupling coefficient, combine these to obtain finally combine R ⁱand M ⁱcome together to obtain U _total, and calculate GD operational symbol, thus obtain the GDs of Electric Field Distribution and optical fiber PMs.

Further, the construction concrete steps of the multistage model of described space and polarization mode coupled fiber are as follows:

(A) index distribution

The real core of unlimited parabolic refractive index is used to carry out, corresponding with the refractive index of following form

Here n ₀it is the nominal index of refraction of fiber optic hub; For x and y-polarisation, n _0xand n _0ythe background refractive index of the heart in a fiber, and and n ₀difference, and be birefringent half, the refractive index between the real core of Δ parametrization and overlayer, r is outermost radius distance from fiber optic hub to overlayer, and a is real core radius, and α ≈ 2 is power exponents, due to birefringence effect, supposes background refractive index n _0xand n _0ydepend on effect of stress, simultaneously Δ and n ₀have nothing to do with stress, in order to illustrative material dispersion, so adopt Sellmeier equation [18] to calculate n ₀;

Birefringence, is defined as the difference of the refractive index of looking from fiber optic hub x direction and y direction polarization waveform, and hypothesis is caused by curvature [19] stress

Here k represents the curvature of fiber segment, C _s/ k ₀refer to strain-optical coefficient; For single-mode fiber, with δ=1 [20]; In multimode optical fiber, exponential distribution unevenness, real core ellipticity and eccentricity, bending, distortion, inside and outside stress may cause spatial mode to be coupled and birefraction, although these two kinds of effects may not necessarily have consistent initial point in given optical fiber, in order to naive model adopts curvature to generate two kinds of effects, in order to the physics actual value allowing curvature generate these two kinds of models, δ > > 1 must be selected;

(B) ideal state

When Δ < < 1, adopt weak guiding approximation method, the closed loop solution of the ideal state of MMF can be tried to achieve in rectangular coordinate system and cylindrical coordinate system, due to the symmetry that x and y direction is bending, in rectangular coordinate system, so adopt the characteristic modes method of ideal fiber to be easy to find out coupling coefficient, this is orthonormal Hermite – Gaussian function

<E _pq|E _p′q′>＝δ _pp′δ _qq′(4)

Here p and q is the mode numeral in x and y direction, and the maximal value of p and q is determined

And mode radius w is provided (being different from frequency w) by following formula

Total mode number is provided by following formula

Here the factor 2 describes two kinds of polarization states of each ideal space mode, therefore, represents the spatial mode mode along fiber axis each some z, according to ideal state with 2M × 1 composite vector A (z)

Here i _mbe modal index, represent (p, q), for the situation of α=2, propagation constant β _{pq (x, y)}represent

In typical optical fiber, the value of α slowly changes from 2, this makes the closed loop solution being difficult to find ideal state mode, in first-order perturbation is analyzed, suppose that α is not equal to 2, ideal state is constant, propagation constant is only had to change, this is a kind of standard hypothesis of wave equation perturbation analysis, such as, in quantum mechanics, propagation constant calculates with α > 2, but do not consider birefringence, the mode of the background refractive index of polarization is depended on by hypothesis, have modified expression formula, do not revise radial variations refractive index simultaneously, thus obtain propagation constant,

Notice that propagation constant passes through n ₀(x, y), to birefringence sensitive, Gamma function is defined as follows

In formula (10), the non-linear ratio of β and ω, the dispersion of display group delay;

(C) modal coupling system in single hop optical fiber

In order to assess by bending caused modal coupling, adopt coupled mode theoretical, extend modality range according to the normal mode in local, in this approach, at each z point solving wave equations along optical fiber, refractive index is n (x, y, z) here; Suppose that any back scattering wave is not coupled forward-propagating ripple, modal coupling equation is

Here a _pqbe the amplitude of the ripple under mode (p, q), normalization field mode is provided by formula (3); C ' _{pq, p ' q '}from receiving the coupling coefficient that between fiber segment, superposition integral obtains;

C′ _{pq，p′q′}＝0forp+p′＝q+q′.(13)

For normalization modal field, equation is modified, and also should notice that mode is not coupled in same cluster; If x ₀(z) and y ₀z () represents at the fiber optic hub of position z, for α=2, the disturbance of refractive index can be as

In a model, bending is be defined as along x direction, therefore y ₀z ()=0, in order to the validity of perturbation analysis, is provided with 2 Δ x ₀(z) x/a ²< < 1, considers formula (14), formula (13) can be written as

In the model of annular curved, write as

Here the curvature of the k section of being, when the length of every section of curved fiber less than bending radius a lot of time, about equation sets up; The second derivative of formula (16) is substituted in formula (15), obtains

It is effective that equation (17) is propagated for the field in optical fiber usually, and defining normalized one dimension Hermite – Gaussian mode is

And obtaining superposition integral from formula (17) is

In formula (19), for Hermite – Gaussian mode, see that curvature causes | p-p ' | the coupling of mode between=1 and q-q '=0, although formula (17) is easy to calculate, and can model in use, in order to simplify, the approximate propagation constant being different from formula (9) is further

Employing formula (20), and reaffirming | p-p ' |=1 and q-q '=0, so formula (17) becomes C ' _{pq, p ' q '}=jk ₀n ₀k<E _pq| x|E _{p ' q '}>. (21)

By will the method simplified expression (21) being approximately two straight waveguides and connecting be bent, intersecting angle is Δ θ (abrupt bend), and the coefficient calculated during Δ θ/Δ z → k, this method finds the coupling power from guiding mode to radiation mode;

Formula (19) is substituted into formula (21), modal coupling coefficients can be write as

Coupling coefficient (22) is along bending each definition, therefore they and do not rely on the length of each bending section; They are linearly dependent on curvature k, it is emphasized that because they calculate in Scalar Model, so they are independent of polarization;

(D) intersegmental polarization rotation matrix

Junction between section i and section i+1, fiber axis anglec of rotation θ _i, this rotation represents by unit rotation matrix the effect of electric field polarization

(E) intersegmental mode projection matrix

Junction between section i and section i+1, supposes to turn clockwise about the axle (x, y) of section i at the axle (x ', y ') of section i+1, is written as by mould field type

Use following characteristic:

For formula (28) along the closed loop expression formula of one group of new Hermite – Gaussian mode decomposition such as formula shown in (30), here

k+q-l+m＝2s

p-k+l+n＝2t

s＞k，s＞q-l，s＞m，t＞p-k，t＞l，t＞n.(31)

In M × M rank matrix Ξ, represent coefficient ξ _{mn, pq}, it is the same with two kinds of polarizations for noticing that mode maps, and the mode mapping matrix between the section of obtaining i and section i+1

(F) always operational symbol is propagated

Combine the result from section II-D – F, obtain

(G) group delay operational symbol and master mode

PMs is defined as independent of fundamental frequency, and has the GDs defined, and a transponder pulse in input PM is receive as monopulse in the corresponding PM of output, from propagation operational symbol, obtains GD operational symbol

PMs and corresponding GDs is proper vector and the eigenwert of F respectively, relevant to its time delay respectively, and for harmless optical fiber, U is single, F is Hermitian, and therefore, GDs is real number, and P is single, in ideal fiber, F reduces the diagonal matrix element equal with ideal state GDs in modal coupling optical fiber, feature vector components makes (34) must carry out numerical evaluation usually;

(H) intensity impulse response

Passing through vectorial A _indescribe in modal field mode, if utilizing emitted light signal is in optical fiber, as provided amplitude according to ideal state, the amplitude that can calculate the light signal being coupled to each PMs is

μ _i＝<A _in|P _i>fori＝1，...，2M.(35)

Equation (35) can be considered the superposition integral on electric field, or the dot product of vector according to ideal state, in the latter case, provides the PM amplitude vector μ of 2M × 1; Be transmitted into a pulse of i-th PM with GD τ _ipropagate, for harmless optical fiber, intensity impulse response is the pulse summation that the power that is coupled to PMs is weighed;

Definition intensity impulse response operational symbol is

Intensity impulse response (36) can be written as

h(t)＝<A _in|H|A _in>.(38)

Under matrix form, formula (38) equals

(I) orthogonality of polarization causes minimum and maximum switching process

Emission space modal distribution constant is determined in experiment, intensity impulse response is sensitiveer to the polarization ratio that transmits, in addition, find in the link of the on-off button using direct-detection, cause the polarization nearly orthogonal of minimum and maximum switching process, here the experimental observations of the latter is explained, definition switching process G is the difference between the power of the first spatial mode and the general power of remaining space mode, consider that birefringence is very little, so that first two time delay is corresponding to the lowest-order spatial mode of x and y direction polarization, switching process is written as

If ignore loss, so general power P _tot=Σ _i| μ _i| ²be constant, and maximize G by the mode maximizing first two, described by a secondary objective function

G ₀(μ)＝|μ ₁| ²+|μ ₂| ².(41)

Suppose only with a kind of particular space mode mode A _{m × 1}launch, the polarization of a general ellipse is expressed as

μ can be written as

μ＝P ^HA _in.(43)

Keep spatial mode mode A _{m × 1}constant with general power, the three degree of freedom r of adjustable polarization, with by defining a new variables x

μ can be expressed as

μ＝Qx(45)

Here

Definition with it is obtained by two row before keeping Q and μ respectively, in the first two PMs, and definition for minimum and maximum power ratio

Here last adopts singular value decomposition (SVD) and obtains

So, it is matrix conditional value, will sVD be written as

If select

So obtain the input polarization relevant to maximum odd number value, this causes maximum switching process; On the contrary,

Formula (51) provides the input polarization causing minimum switching process;

Here object be optimize Section 1 and Section 2 PMs power and, when exciting lowest-order spatial mode when trying, and the DGDs between the DGDs and different spaces mode of birefringence induction wants hour; Be very easy to understand promote this analyze with optimize the power launched in any PMs group and, keep total emissive power constant, by suitable definition simultaneously with prove to cause causing orthogonality between minimum and the transmitting polarization of peak power at given group of PMs.

Further, the analysis modeling of described three modal systems is by analyzing a simple system, have studied the correlativity of length in the GDs of regarding fibers curvature and low and high coupling mechanism, in MMF, the minimum number propagating mode in each polarization is 3, two kinds of mode are caused along the bending of a direction, and it is coupled to each other, and allow the third mode not propagate with being coupled separately, therefore, ignore the third spatial mode, in order to simply, suppose that all fiber segment is positioned at x-z plane, so that polarization does not affect, and negligible; Result is two modal systems, and it is mathematically similar to the single mode fiber of PMD, first, in order to low coupling mechanism is described, research single hop has the optical fiber compared with small curve, then, in order to high coupling mechanism is described, study a kind of optical fiber of many sections and statistical segregation parameter.

Further, the analysis modeling concrete steps of described three modal systems are as follows:

(A) DGD of low coupling mechanism

Here, in single hop curved fiber, calculate DGD, definition slowly change envelope A (z) is

A(z)＝exp(-Γz)A(z)(52)

Here Γ is defined by formula (24), and the derivative of formula (52) can be written as

A′(z)＝-Γe ^(-Γz)A(z)+e ^(-Γz)A′(z).(53)

Formula (52) and formula (53) are substituted in formula (23), can be written as

A′(z)＝je ^(Γz)Ce ^(-Γz)A(z).(54)

This is the coupled wave equation slowly converting envelope, and the coupled matrix that definition slowly changes envelope is

C＝e ^(Γz)Ce ^(-Γz).(55)

Definition Δ β=β ₁-β ₂for these two kinds of coupled modes propagation constant between difference, C can be written as

The propagation of envelope A is described by single propogator matrix T, and this is similar with Jones in SMF;

A(z)＝T(z)A(0)(57)

Adopt T, following formula can be used to obtain propogator matrix U

U＝e ^(-Γz)T(z).(59)

Employing formula (59) and T ^hthe situation of T=I, uses formula (34) group delay matrix F to be written as

Because T is single, so the eigenwert of F is the eigenwert of matrix in parenthesis, as from this derivative seen, for linear fiber, here the eigenwert of F is the GDs value of coupled mode, by provide, in shorter fiber segment, corresponding to low coupling mechanism, A ₁during (z) ≈ 1, can suppose that most of light signal is to propagate in the first mode, and be coupled to the second mode at leisure, make to obtain

Use situation, and only consider single order item in z, can be written as

The derivative of the middle T of consideration formula (58) and formula (62)

Notice that (63) equal T-I/2 ω, and be updated to formula (60), obtain

In order to determine the single order effect of single bending section, obtain GDs by solving following formula

Here the difference between GDs gives the DGD that overall length is the optical fiber of L,

Expression formula (66) display DGD is linear increase along with fiber lengths, and just as the situation of PMD in low coupling mechanism, in addition, for less k, bend and can increase DGD, DGD is and k ²proportional;

(B) DGD of high coupling mechanism

In high coupling mechanism, GDs determined by local optic fibre characteristic; But depend on the cumulative effect of modal coupling on whole optical fiber, the statistical property of GDs is by solving coupled stochastic difference equation to study, for the situation of SMF having PMD, Poole has observed these equations in low and high coupling mechanism, hypothesis is positioned to three mode MMF of x-z plane, in formula (23), coupled modal equations abbreviation is to as follows

As described in chapters and sections above, have a mode not with other two kinds of modal coupling, so analysis in also ignore, the auto-correlation of optical fiber curvature k (z) is defined as follows

R _k(u)＝<k(z)k(z-u)>(68)

Here bracket represents population mean, also defines power spectrum density (PSD) k to be

Solve Random Coupling (67) according to the method for document [6], all square DGD obtained as the function of z is

Parameter h describes overall average rate, and power is changed between mode here, and is defined as

When low coupling limit hz → 0, have

For being in constant bending optical fiber in low coupling mechanism, all square DGD in formula (72) is consistent with the DGD in formula (66), and equation (66) has been extended to and has comprised the bending effect in least significant non-zero rank;

In height coupling limit hz → ∞ considered here, obtain

Equation (73) sees clearly the dependence chance of DGD of regarding fibers statistics to providing, show in high coupling mechanism, the PSD of DGD and curvature is inversely proportional to and changes, just as when SMF [6] PDM, DGD along with length in high coupling mechanism, as square root and change, in order to understand formula (73) further, notice model in because curvature is constant in every segment length, curvature k (z) is by discrete random variable k _i, i=1 ..., N describes, and it is independent identically distributed (i.i.d.), represents k respectively _iexpectation and variance be m _kwith between section i and section i+1, discrete auto-correlation is

In order to find out the PSD of curvature k (z), note k (z), the function class of z is similar to PAM signal, and the function of t uses this analogism, and the PSD obtaining k (z) is

In the problem that these are existing, the first and the second mode are nonsingular here, and such as Δ β ≠ 0, obtaining DGD variance (73) is

Equation (76) is presented in high coupling mechanism, and DGD is and fiber lengths, such as, square root be directly proportional, and with curvature criteria difference σ _kbe inversely proportional to.

Further, the numerical modeling of described multimode optical fiber is, based on the multistage model of the space described in theory and polarization mode coupled fiber, high precision matrix tool in MATLAB is adopted to carry out the numerical modeling of MMF, optical fiber is the MMF of the 50-μm of progressive refractive index silicon materials of real core, and its overall length is L=1000m, and it is NA=0.19 that optical fiber has numerical aperture, and wavelength is the refractive index n of λ=1550nm, fiber optic hub ₀=1.444 is measured when wavelength is λ=1550nm, leaves this wavelength, n ₀be adopt Sellmeier equation to calculate, refractive index is to the derivative dn of frequency ₀/ dw also adopts Sellmeier equation to calculate, but, find model in, waveguide dispersion is larger than material dispersion impact, birefringence scale factor is set to δ=8000, use formula (5) and formula (7), find that 55 spatial modes are propagated in two polarizations, adopt the unlimited real core approximate (2) of α=2.09, select Matching Experiment, in this test, find that lower mode has shorter GDs, and find that DGDs is more much higher than the value predicted when ideal value α=2, except as otherwise noted, optical fiber is divided into 10 ⁴section, namely every segment length is 0.1 meter, and obey independent identically distributed angle θ relative to one section of rotation one above for every section, its probability density function (pdf) is normal distribution, and variance is the curvature of every section is an independent identically distributed stochastic variable k _i, the front of its probability density function normal probability density function, and its variance is because increase, so model is from low coupling mechanism to high coupling mechanism, the Stochastic implementation of the given anglec of rotation and curvature, calculates group delay operational symbol F, carries out diagonal angle abbreviation to obtain PMs and its GDs to F.

Further, described conclusion is that polarization independent closes is when curvature variance is compared with fiber lengths hour, namely during low coupling mechanism, group delay is its non-coupled value closely, and with fiber lengths linearly, simultaneously master mode is still produce high degree of polarisation, in this mechanism, reproduced the polarization dependence of impulse response, and this polarization dependence is observed in silicon materials multimode optical fiber; When curvature variance and fiber lengths are enough large, i.e. high coupling mechanism, so can reduce the group delay of propagation, and it is proportional with the square root of fiber lengths, depolarizing of master mode simultaneously, in this model, the group delay of model reduces consistent with the propagation group delay observed in plastic material multimode optical fiber MMF.

The invention has the advantages that:

When in high speed data rate situation, adopt coherent source time, such as rely on the impulse response of polarization, coupling power model inherently fail to describe multimode optical fiber (MMF) some modal coupling effect; We have developed the field coupling model propagated in a kind of multimode optical fiber at graded index (MMF), be similar to the principal states model of polarization modal dispersion in single-mode fiber; Assuming that one is launched Electric Field Distribution and polarization, our model allows the calculating of fiber pulse response; In order to be coupled into row modeling to space and polarization mode, multimode optical fiber (MMF) is divided into many segments by us, and every section has random curvature and random position angle; This model can only use some parameters to describe, and comprises fiber lengths, hop count and curvature variance; For often kind of Stochastic implementation of multimode optical fiber (MMF), we calculate propogator matrix, master mode (PMs) and corresponding group delay (GDs); When curvature variance hour (low coupling mechanism) compared with fiber lengths, group delay (GDs) is its non-coupled value closely, and with fiber lengths linearly, simultaneously master mode (PMs) is still produce high degree of polarisation; In this mechanism, our model has reproduced the polarization dependence of impulse response, and this polarization dependence is observed in silicon materials multimode optical fiber (MMF); When curvature variance and fiber lengths are enough large (high coupling mechanism), so can reduce the group delay (GDs) of propagation, and proportional with the square root of fiber lengths, master mode (PMs) depolarizing simultaneously; In this model, the group delay of our model reduces consistent with the propagation group delay (GDs) observed in plastic material multimode optical fiber MMF.

Accompanying drawing explanation

Fig. 1 is the connector that optical fiber is modeled as many bending sections;

Fig. 2 is GDs and accumulation standard deviation sigma _k;

Fig. 3 is for adopting numerical value MMF model to calculate for three modal systems;

Fig. 4 is one the DGD Δ τ of the MMF of 2*55 mode and the ratio of total length L;

Fig. 5 is the intensity impulse response that a long MMF typical case of 1 km realizes;

Fig. 6-9 is intensity sampling and the polarization state of the lowest-order input and output PMs of 1 km optical fiber in low coupling and high coupling mechanism;

Figure 10 is (σ in low coupling mechanism _k=0.95m ^-1) ratio of DOP and fiber lengths;

Figure 11 is at low coupling mechanism (σ _k=1.2m ^-1) ratio of lowest order PMs (numbering 1 – 6) frequency deviation and correlation factor in 1 km MMF in situation;

Embodiment

The invention discloses a kind of polarization independent relation of graded index multimode fiber master mode, it is when curvature variance is compared with fiber lengths hour that this polarization independent closes, namely during low coupling mechanism, group delay is its non-coupled value closely, and with fiber lengths linearly, simultaneously master mode is still produce high degree of polarisation, in this mechanism, reproduced the polarization dependence of impulse response, and this polarization dependence is observed in silicon materials multimode optical fiber; When curvature variance and fiber lengths are enough large, i.e. high coupling mechanism, so can reduce the group delay of propagation, and it is proportional with the square root of fiber lengths, depolarizing of master mode simultaneously, in this model, the group delay of model reduces consistent with the propagation group delay observed in plastic material multimode optical fiber MMF.

The invention also discloses a kind of derivation method of polarization independent relation of above-mentioned graded index multimode fiber master mode, concrete steps are as follows:

(1) theoretical

Optical fiber is modeled as the connector of many bending sections, as shown in Figure 1; Every section is positioned at a phase place, and the phase place of every section rotates gained on the basis of the last period; The curvature of every section causes spatial mode coupling birefringence; The connection of many bending sections causes the spatial mode that polarization is relevant to be coupled; This model may be regarded as the expansion of the multistage model of the PMD of [15] in single-mode fiber; In our model, any mechanism of our non-modal assumption dependent loss;

Mode propagation in such optical fiber is the propogator matrix U by whole optical fiber _totalcarry out complete description; Drawing U _totalduring expression formula, we adopt the normal mode [16] in local to calculate; Therefore, in every section of optical fiber, we calculate according to the ideal model of complete optical fiber, and the plane of coordinate axis and particular segment is in alignment; Propagation in i-th section is by space but is not the impact of polarization modal coupling, and is use propogator matrix represent; Junction between i-th section and the i-th+1 section, local axle anglec of rotation θ _i; The impact of two aspects must be considered here; Polarization mode rotation matrix R ⁱrepresent because axle rotates the polarization coupled produced; In addition, the spatial mode mode defined along previous axle must be expanded along new axle, so that a kind of new ideal state basis to be described; This is by spatial mode mapping matrix M ⁱdescribe; Easily obtain this means that exponent number is unimportant, apply two matrixes here; Therefore,

Here N is total hop count

Assuming that one is launched Electric Field Distribution and polarization, in this electric field, build the multistage model of a space and polarization mode coupled fiber, calculate the propagation operational symbol of optical fiber, then, calculate group delay operational symbol, and obtain its proper vector, these are PMs, when the distribution of given launching site and polarization, calculate the impulse response of optical fiber, prove, it is effective in low coupling mechanism, and the orthogonality of polarization causes minimum and maximum switching process;

When the multistage model of above-mentioned space and polarization mode coupled fiber is built, first determine Refractive Index Profile o; Then, draw propagation constant and the field distribution of the normal mode in local, and in bending section, calculate spatial mode coupling coefficient, combine these to obtain finally combine R ⁱand M ⁱcome together to obtain and calculate GD operational symbol, thus obtain the GDs of Electric Field Distribution and optical fiber PMs;

The construction concrete steps of the multistage model of space and polarization mode coupled fiber are as follows:

(A) index distribution

(B) ideal state

Total mode number is provided by following formula

Notice that propagation constant passes through n _{0 (x, y)}to birefringence sensitive, Gamma function is defined as follows

(C) modal coupling system in single hop optical fiber

Here a _pqbe the amplitude of the ripple under mode (p, q), normalization field mode is provided by formula (3); C _{pq, p ' q '}from receiving the coupling coefficient that between fiber segment, superposition integral obtains;

In the model of annular curved, write as

And obtaining superposition integral from formula (17) is

Employing formula (20), and reaffirming | p-p ' |=1 and q-q '=0, so formula (17) becomes

C _{pq，p′q′}＝jk ₀n ₀k<E _pq|x|E _p′q′>.(21)

(D) intersegmental polarization rotation matrix

(E) intersegmental mode projection matrix

Use following characteristic:

k+q-l+m＝2s

p-k+l+n＝2t

s＞k，s＞q-l，s＞m，t＞p-k，t＞l，t＞n.(31)

In M × M rank matrix Ξ, represent coefficient ξ mn, pq, it is the same with two kinds of polarizations for noticing that mode maps, and the mode mapping matrix between the section of obtaining i and section i+1

(F) always operational symbol is propagated

Combine the result from section II-D – F, obtain

(G) group delay operational symbol and master mode

(H) intensity impulse response

μ _i＝<A _in|P _i>fori＝1，...，2M.(35)

Definition intensity impulse response operational symbol is

Intensity impulse response (36) can be written as

h(t)＝<A _in|H|A _in>.(38)

Under matrix form, formula (38) equals

(I) orthogonality of polarization causes minimum and maximum switching process

G ₀(μ)＝|μ ₁| ²+|μ ₂| ².(41)

μ can be written as

μ＝P ^HA _in.(43)

μ can be expressed as

μ＝Qx(45)

Here

Here last adopts singular value decomposition (SVD) and obtains

So, it is matrix conditional value, will sVD be written as

If select

Formula (51) provides the input polarization causing minimum switching process;

Here object be optimize Section 1 and Section 2 PMs power and, when exciting lowest-order spatial mode when trying, and the DGDs between the DGDs and different spaces mode of birefringence induction wants hour; Be very easy to understand promote this analyze with optimize the power launched in any PMs group and, keep total emissive power constant, by suitable definition simultaneously with prove to cause causing orthogonality between minimum and the transmitting polarization of peak power at given group of PMs;

The analysis modeling of (two) three modal systems

Build a kind of simple three modal systems so that the GDs degree of dependence of regarding fibers curvature and length in low and high coupling mechanism to be described;

The analysis modeling of three modal systems is by analyzing a simple system, have studied the correlativity of length in the GDs of regarding fibers curvature and low and high coupling mechanism, in MMF, the minimum number propagating mode in each polarization is 3, two kinds of mode are caused along the bending of a direction, and it is coupled to each other, and allow the third mode not propagate with being coupled separately, therefore, ignore the third spatial mode, in order to simply, suppose that all fiber segment is positioned at x-z plane, so that polarization does not affect, and negligible; Result is two modal systems, and it is mathematically similar to the single mode fiber of PMD, first, in order to low coupling mechanism is described, research single hop has the optical fiber compared with small curve, then, in order to high coupling mechanism is described, study a kind of optical fiber of many sections and statistical segregation parameter;

The analysis modeling concrete steps of three modal systems are as follows:

(A) DGD of low coupling mechanism

A(z)＝exp(-Γz)A(z)(52)

A′(z)＝-Γe ^(-Γz)A(z)+e ^(-Γz)A′(z).(53)

A′(z)＝je ^(Γz)Ce ^(-Γz)A(z).(54)

C＝e ^(Γz)Ce ^(-Γz).(55)

A(z)＝T(z)A(0)(57)

Adopt T, following formula can be used to obtain propogator matrix U

U＝e ^(-Γz)T(z).(59)

Use situation, and only consider single order item in z, can be written as

The derivative of the middle T of consideration formula (58) and formula (62)

Notice that (63) equal T-I/2 ω, and be updated to formula (60), obtain

(B) DGD of high coupling mechanism

R _k(u)＝〈k(z)k(z-u)〉(68)

When low coupling limit hz → 0, have

In height coupling limit hz → ∞ considered here, obtain

(3) numerical modeling of multimode optical fiber

Based on the model described in a first portion, we adopt high precision matrix tool in MATLAB to carry out the numerical modeling [29] of MMF; Optical fiber is the MMF of the 50-μm of progressive refractive index silicon materials of real core, and its overall length is L=1000m; It is NA=0.19 that optical fiber has numerical aperture, and wavelength is λ=1550nm; The refractive index n of fiber optic hub ₀=1.444 is measured when wavelength is λ=1550nm; Leave this wavelength, n ₀adopt Sellmeier equation to calculate; Refractive index is to the derivative dn of frequency ₀/ dw also adopts Sellmeier equation to calculate; But we find in our model, waveguide dispersion is larger than material dispersion impact; Birefringence scale factor is set to δ=8000; Use formula (5) and formula (7), we find that 55 spatial modes are propagated in two polarizations; We adopt the unlimited real core approximate (2) of α=2.09, select Matching Experiment, and in this test, we find that lower mode has shorter GDs, and find that DGDs is more much higher than the value predicted when ideal value α=2; Except as otherwise noted, optical fiber is divided into 10 ⁴section, namely every segment length is 0.1 meter; Obey independent identically distributed angle θ relative to one section of rotation one above for every section, its probability density function (pdf) is normal distribution, and variance is the curvature of every section is an independent identically distributed stochastic variable k _i, the front of its probability density function normal probability density function, and its variance is because increase, so model is from low coupling mechanism to high coupling mechanism; The Stochastic implementation of the given anglec of rotation and curvature, we calculate group delay operational symbol F; We also carry out diagonal angle abbreviation to obtain PMs and its GDs to F;

For the various selections of Model Parameter, Fig. 2 (a) – (c) shows GDs and accumulation standard deviation sigma _k; Fig. 2 (a) shows the GDs that is divided into the optical fiber of N=100 section, every segment length 10 meters; At low σ _ktime, GDs in essence with σ _kindependent, and at very high σ _ktime, GDs disperses; Various parameter value in setting models, this N value too little and can not produce behavior, particularly GDs desired in the statistical study as III-B part in higher σ by the time _kconvergence; When N is increased to 10 by us ⁴time, model produce in essence with N independently result;

For two different values of refractive index power exponent α, Fig. 2 (b) and (c) consider N=10 ⁴section, every segment length is 0.1 meter; In Fig. 2 (b), α=2 here, the scope of minimum and maximum GDs only has 200ps, and this is much less than the result of Germicidal efficacy [11] gained; Therefore, in Fig. 2 (c), index is increased to α=2.09 by us, and it is poor that this produces actual GD;

In Fig. 2 (c), we can distinguish number of mechanisms; At very little σ _ktime, there is less modal coupling, and GDs is nonsingular; At slightly large σ _ktime, corresponding to low coupling, GDs degenerates, and GDs is similar to σ _kquadratic distribution; At higher σ _k(σ _kbelong to interval 1 and 10m-1), corresponding to high coupling mechanism, GDs restrains, and GD difference reduces obviously; This behavior is quantitatively consistent with formula (76); Finally, at very high σ _ktime (about 20m-1), GDs disperses; This non-physical behavior draws due to the hypothesis violated us and do in perturbation analysis;

In Fig. 2 (c), in high coupling mechanism, the convergence of GDs is similar to the single-mode fiber of band PMD, and DGD reduces quite a lot of in high coupling mechanism [6] here; The dependence of GDs to modal coupling is the key feature of a coupling model, but does not occur in coupling power model; In plastics MMFs, observe pulse strenching reduce, be due to modal coupling here, and adopt coupling power model to explain; The minimizing of pulse strenching is not the change due to GDs self; But someone thinks light transmition, and it jumps to another mode from a kind of mode, and all light tended in the time of each mode spending same ratio, so that all light bears almost equal propagation delay; Current work, according to the change of the GDs caused by modal coupling, provides another explanation of the pulse strenching of minimizing; We should note in plastics MMF, and scattering mechanism causes modal coupling to be relevant with the decay increased; By contrast, in our model, curvature causes modal coupling can't cause decay;

In order to study GDs to curvature standard deviations σ _kwith the dependence of fiber lengths L, we define the difference that DGD Δ τ is the GD of two lowest-order PMs;

In figure 3, adopt numerical value MMF model to calculate for three modal systems, we provide DGD Δ τ and curvature standard deviations σ _k; In the theoretical analysis of III part three modal system, for simplicity, we consider all fiber segment being positioned at x-z plane; In numerical model, we loosen this hypothesis, and comprise intersegmental rotation, then obtain result consistent with the analysis of simplification; At Fig. 3 (a) and (b), we consider N=1 section, in this case, and σ _kreduce single curvature k; In Fig. 3 (a), Δ τ seems independent with k, so in Fig. 3 (b), we must eliminate the Δ τ of part independent of k, disclose itself and k ²proportional factor, this is proportional with formula (66); In Fig. 3 (c) – (e), we consider N=100 respectively, 1000,10 ⁴section; In the latter cases, we see that Δ τ is and σ _kinversely proportional, this is also consistent with formula (76);

In the diagram, we show both one and have the DGD Δ τ of the MMF of 2*55 mode and the ratio of total length L; It is constant that segment length remains on 0.1m, as long as length L is increased to 104m from 10 by us, hop count N changes between 100 to 105; We consider Curvature varying σ _kfour kinds of different values; Work as σ _kand (or) N value hour, Δ τ and L is proportional, and this meets (66); For larger σ _kvalue, when N is enough large, Δ τ tend to and proportional; This is consistent with (73) and (76);

Once after the PMs finding MMF specific implementation relevant and their respective GDs, when given launching site distribution, we use the impulse response operational symbol defined in (38) and (39) to carry out calculating strength impulse response; This operational symbol creates a series of pulse; And these pulses by the stress be transmitted in different PMs impact and postponed by their respective GDs; In order to promote and adopt the comparing of band-limited experimental measuring method, we have employed full width at half maximum is the convolution [11] that the Gauss pulse of 50-ps carrys out calculating strength impulse response;

What Fig. 5 (a) and (b) showed is the intensity impulse response that a long MMF typical case of 1 km realizes; The modeling of this intensity impulse response uses N=10 ⁴section and bending change σ _k=1.2m ^-1carry out modeling, to low coupling relevant; Relative to lowest-order ideal space mode, light is launched in Gauss's mode of radius w=8 μm; These parameters are used to qualitative playback experiment result;

In Fig. 5 (a), we suppose n ₀(x, y) depends on the bending stress brought out, and Δ and n ₀independent of stress (see discussion (2)); We find, along with we change transmitting polarization, the quantity being coupled into the light in high-order PMs also can change; This is qualitative consistent with experimental observation [11] [10]; Use (50) and (51), we have found two cross polarizations of the minimum and maximum switching process caused; In these two polarizations, we observe in lowest-order PM, and the ratio of peak power and minimum power is ζ=1.31; As far as we know, be first research that the polarization dependence of MMF impulse response is made explanations herein;

In Fig. 5 (b), we consider that same MMF realizes, but hypothesis n ₀(x, y), Δ and n ₀all depend on stress; Impulse response depends on polarization hardly, and we have observed ζ=1.01; Which illustrate, Δ and n ₀the model of hypothesis to us not relying on stress is very important; Therefore, in all calculating of next display, we do this hypothesis;

In figs. 6-9, we show both intensity sampling and the polarization state of the lowest-order input and output PMs of 1 km optical fiber in low coupling and high coupling mechanism; In each figure, form pairing with the PMs being close to identical GDs, and sort by GD increase; In order to obtain intensity sampling, but our overlap has the territory of different modulus identical polarization, calculates the intensity of each coincidence, then calculate two polarizations and; We are according to Stokes parameter display polarization state [30]; In order to calculate i ^ththese values of PM, we have made the product between field component into dot product between PM amplitude vecotr; S ₁, S ₂and S ₃be standardized as general power S ₀, three standardized points are then shown as the point on spherome surface; Radius imply that higher degree of polarization (DOP) close to 1, and less radius means lower degree of polarization; Compared by Fig. 6 and Fig. 7, Fig. 8 and Fig. 9 compares, and we observe when all, and an input PM has different intensity samplings and different polarization states with relative output PM;

Fig. 6 and 7 describes a 1 km MMF in low coupling mechanism, and its bending change is σ _k=0.95m ^-1; More symmetrical than low level PMs of the intensity sampling of low level ideal state; Have among substantially identical PMs at often pair, but both have substantially identical intensity sampling different cross polarization; The DOP of detailed inspection display lowest order PMs1 and 2 is closely 1, but PMs3, the DOPs of 4,5 and 6 displays are less than 1 slightly;

The MMF of what Fig. 8 and 9 described is 1 km in high coupling mechanism, its bending change is σ _k=4.2m ^-1; Even the PMs of lowest order also represents the coincidence of several ideal state, therefore their intensity sampling is complicated and asymmetric; Have in the PMs of substantially identical GDs at often pair, both have visibly different intensity sampling and polarization state; PMs is partial polarization, and the DOPs of its display is significantly less than 1, this imply that coupling important between spatial degrees of freedom and polarization degree of freedom; Fiber lengths and bending change are on the impact display of DOP in Fig. 10; PMs sorts by GDs growth; 1 – 2,3 – 4 and 5 – 6 defines the right of basic degeneration;

Figure 10 (a) and (b) show (σ in low coupling mechanism _k=0.95m ^-1) ratio of DOP and fiber lengths; Start to show some at the fiber lengths local PMs that is hundreds of rice to depolarize, in the process increased at fiber lengths along with high-order PMs, go attenuation ratio low order PMs fast; Figure 10 (c) and (d) show (σ in high coupling mechanism _k=4.2m ^-1) ratio of DOP and fiber lengths; The local PMs being short to 10 meters at fiber lengths just starts display and goes decay; And along with the increase of fiber lengths, correlativity obvious, not consistent between mode number and DOP;

By definition, PMs has the field type [14] independent of fundamental frequency; People can think that these field types are substantially constant in some frequency ranges; We are referred to as " coherence bandwidth "; At the SMF with PMD in particular cases, this coherence bandwidth is called as " bandwidth of Principal State of Polarization " [31]; Reference section [11] demonstrates the electric field that usage space photomodulator (SLM) controls to be transmitted in MMF, with this priority activation low order PMs, reduces the impact of modal dispersion; We find, in 11 km silicon materials MMF of the low Model coupling of display, one group of SLM just can compensate modal dispersion when bandwidth is 600GHz; This imply that PMs may reach the degree of tens to hundreds of GHz in such optical fiber;

In order to estimate the coherence bandwidth in our model, we given optical frequency f=w/ (2 π) and displacement frequency f+Δ f on calculate a PMs and correlation factor; This factor is the size of normalized inner product between PMs in these two frequencies; Coherence bandwidth is defined as the value of Δ f and result in correlation factor and be reduced to 0.8. Figure 11 (a) and (b) and show at low coupling mechanism (σ by randomly _k=1.2m ^-1) ratio of lowest order PMs (numbering 1 – 6) frequency deviation and correlation factor in 1 km MMF in situation; In Figure 11 (a), we consider an index, power exponent α=2.00; As shown in Fig. 2 (b), GD that this index provides propagate than experimental observation arrive low five times; Use this value of α, the coherence bandwidth of this model prediction PMs is about 300GHz; In Figure 11 (b), we think α=2.09, and we consider α=2.09; This numerical value is display in Fig. 2 (c), is used for the more real GD of generation one to propagate; Use this value of α, the coherence bandwidth of model prediction lowest order PMs (1 – 2) is close to 10GHz, and high-order PMs (3 – 4 and 5 – 6) has the coherence bandwidth diminished gradually; Therefore, our model assessment coherence bandwidth is at least less than an order of magnitude desired by experiment; At present, cause the reason of this inconsistency also indefinite;

(4) conclusion

It is when curvature variance is compared with fiber lengths hour that polarization independent closes, namely during low coupling mechanism, group delay is its non-coupled value closely, and with fiber lengths linearly, master mode is still produce high degree of polarisation simultaneously, in this mechanism, reproduced the polarization dependence of impulse response, and this polarization dependence is observed in silicon materials multimode optical fiber; When curvature variance and fiber lengths are enough large, i.e. high coupling mechanism, so can reduce the group delay of propagation, and it is proportional with the square root of fiber lengths, depolarizing of master mode simultaneously, in this model, the group delay of model reduces consistent with the propagation group delay observed in plastic material multimode optical fiber MMF.

More than show and describe ultimate principle of the present invention and principal character and advantage of the present invention.The technician of the industry should understand; the present invention is not restricted to the described embodiments; what describe in above-described embodiment and instructions just illustrates principle of the present invention; without departing from the spirit and scope of the present invention; the present invention also has various changes and modifications, and these changes and improvements all fall in the claimed scope of the invention.Application claims protection domain is defined by appending claims and equivalent thereof.

Claims

1. a polarization independent relation for multimode optical fiber master mode, is characterized in that: described polarization independent closes and is

When curvature variance is compared with fiber lengths hour, namely during low coupling mechanism, group delay is its non-coupled value closely, and with fiber lengths linearly, master mode is still produce high degree of polarisation simultaneously, in this mechanism, reproduced the polarization dependence of impulse response, and this polarization dependence is observed in silicon materials multimode optical fiber;

When curvature variance and fiber lengths are enough large, i.e. high coupling mechanism, so can reduce the group delay of propagation, and it is proportional with the square root of fiber lengths, depolarizing of master mode simultaneously, in this model, the group delay of model reduces consistent with the propagation group delay observed in plastic material multimode optical fiber MMF.

2. a derivation method for the polarization independent relation of multimode optical fiber master mode according to claim 1, is characterized in that: described derivation method concrete steps are as follows:

It is theoretical: assuming that one is launched Electric Field Distribution and polarization, in this electric field, build the multistage model of a space and polarization mode coupled fiber, calculate the propagation operational symbol of optical fiber, then, calculate group delay operational symbol, and obtain its proper vector, these are PMs, when the distribution of given launching site and polarization, calculate the impulse response of optical fiber, prove, it is effective in low coupling mechanism, and the orthogonality of polarization causes minimum and maximum switching process;

The analysis modeling of three modal systems: build a kind of simple three modal systems so that the GDs degree of dependence of regarding fibers curvature and length in low and high coupling mechanism to be described;

The vertical modeling of multimode optical fiber: the numerical evaluation describing actual fiber model, describes PMs characteristic and its group delay in low and high coupling mechanism;

Draw the polarization independent relation of multimode optical fiber master mode.

3. the transmission master mode of gradually changeable index multimode fiber according to claim 2, is characterized in that: when the multistage model of described space and polarization mode coupled fiber is built, first determine Refractive Index Profile o; Then, draw propagation constant and the field distribution of the normal mode in local, and in bending section, calculate spatial mode coupling coefficient, combine these to obtain ; Finally combine with come together to obtain , and calculate GD operational symbol, thus obtain the GDs of Electric Field Distribution and optical fiber PMs.

4. the transmission master mode of gradually changeable index multimode fiber according to claim 3, is characterized in that: the construction concrete steps of the multistage model of described space and polarization mode coupled fiber are as follows:

(A) index distribution

Here it is the nominal index of refraction of fiber optic hub; For x and y-polarisation, with the background refractive index of the heart in a fiber, and with difference, and be birefringent half, refractive index between the real core of parametrization and overlayer, r is outermost radius distance from fiber optic hub to overlayer, real core radius, be power exponent, due to birefringence effect, suppose background refractive index with depend on effect of stress, simultaneously with have nothing to do with stress, in order to illustrative material dispersion, so adopt Sellmeier equation [18] to calculate ;

Here represent the curvature of fiber segment, refer to strain-optical coefficient; For single-mode fiber, with [20]; In multimode optical fiber, exponential distribution unevenness, real core ellipticity and eccentricity, bending, distortion, inside and outside stress may cause spatial mode to be coupled and birefraction, although these two kinds of effects may not necessarily have consistent initial point in given optical fiber, in order to naive model adopts curvature to generate two kinds of effects, in order to the physics actual value allowing curvature generate these two kinds of models, must select ;

(B) ideal state

? time, adopt weak guiding approximation method, the closed loop solution of the ideal state of MMF can be tried to achieve in rectangular coordinate system and cylindrical coordinate system, due to the symmetry that x and y direction is bending, in rectangular coordinate system, so adopt the characteristic modes method of ideal fiber to be easy to find out coupling coefficient, this is orthonormal Hermite – Gaussian function

And mode radius provided by following formula and (be different from frequency )

Total mode number is provided by following formula

Here the factor 2 describes two kinds of polarization states of each ideal space mode, therefore, uses composite vector represent the spatial mode mode along fiber axis each some z, according to ideal state

Here be modal index, represent , for situation, propagation constant represent

In typical optical fiber, value slowly change from 2, this makes the closed loop solution being difficult to find ideal state mode, first-order perturbation analyze in, suppose be not equal to 2, ideal state is constant, only has propagation constant to change, and this is a kind of standard hypothesis of wave equation perturbation analysis, such as, in quantum mechanics, propagation constant be with calculate, but do not consider birefringence, depended on the mode of the background refractive index of polarization by hypothesis, have modified expression formula, do not revise radial variations refractive index simultaneously, thus obtain propagation constant,

Notice that propagation constant is passed through to birefringence sensitive, Gamma function is defined as follows

In formula (10), with non-linear ratio, the dispersion of display group delay;

(C) modal coupling system in single hop optical fiber

In order to assess by bending caused modal coupling, adopt coupled mode theoretical, extend modality range according to the normal mode in local, in this approach, each along optical fiber point solving wave equations, refractive index is here ; Suppose that any back scattering wave is not coupled forward-propagating ripple, modal coupling equation is

Here in mode under the amplitude of ripple, normalization field mode is provided by formula (3); from receiving the coupling coefficient that between fiber segment, superposition integral obtains;

For normalization modal field, equation is modified, and also should notice that mode is not coupled in same cluster; If with represent in position fiber optic hub, for , the disturbance of refractive index can be as

In a model, bending is be defined as along x direction, therefore , in order to the validity of perturbation analysis, be provided with , consider formula (14), formula (13) can be written as

In the model of annular curved, write as

Here the curvature of the section of being, when the length of every section of curved fiber less than bending radius a lot of time, about equation sets up; The second derivative of formula (16) is substituted in formula (15), obtains

And obtaining superposition integral from formula (17) is

In formula (19), for Hermite – Gaussian mode, see that curvature causes with between the coupling of mode, although formula (17) is easy to calculate, and can model in use, in order to simplify, the approximate propagation constant being different from formula (9) is further

Employing formula (20), and reaffirm with , so formula (17) becomes

By bending the method simplified expression (21) being approximately two straight waveguides and connecting, intersecting angle will be (abrupt bend), and calculate time coefficient, this method find from guide mode to radiation mode coupling power;

Coupling coefficient (22) is along bending each definition, therefore they and do not rely on the length of each bending section; They are linearly dependent on curvature , it is emphasized that because they calculate in Scalar Model, so they are independent of polarization;

(D) intersegmental polarization rotation matrix

Junction between section i and section i+1, the fiber axis anglec of rotation , this rotation represents by unit rotation matrix the effect of electric field polarization

(E) intersegmental mode projection matrix

Junction between section i and section i+1, supposes the axle at section i+1 about the axle of section i turn clockwise, mould field type is written as

Use following characteristic:

At one rank matrix in, represent coefficient , it is the same with two kinds of polarizations for noticing that mode maps, and the mode mapping matrix between the section of obtaining i and section i+1

；

(F) always operational symbol is propagated

Combine the result from section II-D – F, obtain

；

(G) group delay operational symbol and master mode

PMs and corresponding GDs is respectively proper vector and eigenwert, relevant to its time delay respectively, for harmless optical fiber, U is single, and F is Hermitian, and therefore, GDs is real number, be single, in ideal fiber, F reduces the diagonal matrix element equal with ideal state GDs, , in modal coupling optical fiber, feature vector components makes (34) must carry out numerical evaluation usually;

(H) intensity impulse response

Passing through vector describe in modal field mode, if utilizing emitted light signal is in optical fiber, as provided amplitude according to ideal state, the amplitude that can calculate the light signal being coupled to each PMs is

Equation (35) can be considered the superposition integral on electric field, or the dot product of vector according to ideal state, in the latter case, provides one pM amplitude vector ; Be transmitted into a pulse of i-th PM with GD propagate, for harmless optical fiber, intensity impulse response is the pulse summation that the power that is coupled to PMs is weighed;

Definition intensity impulse response operational symbol is

Intensity impulse response (36) can be written as

Under matrix form, formula (38) equals

；

(I) orthogonality of polarization causes minimum and maximum switching process

Emission space modal distribution constant is determined in experiment, intensity impulse response is sensitiveer to the polarization ratio that transmits, in addition, find in the link of the on-off button using direct-detection, cause the polarization nearly orthogonal of minimum and maximum switching process, here the experimental observations of the latter is explained, definition switching process for the difference between the power of the first spatial mode and the general power of remaining space mode, consider that birefringence is very little, so that first two time delay is corresponding to the lowest-order spatial mode of x and y direction polarization, switching process is written as

If ignore loss, so general power be constant, and maximize by the mode maximizing first two , described by a secondary objective function

Suppose only in a kind of particular space mode mode launch, the polarization of a general ellipse is expressed as

Can be by be written as

Keep spatial mode mode constant with general power, the three degree of freedom of adjustable polarization , with , by defining a new variables x

Can be by be expressed as

Here

Definition with , it is by keeping respectively with front two row obtain, in the first two PMs, definition for minimum and maximum power ratio

Here last adopts singular value decomposition (SVD) and obtains

So, it is matrix conditional value, will sVD be written as

If select

Formula (51) provides the input polarization causing minimum switching process;

Here object be optimize Section 1 and Section 2 PMs power and, when exciting lowest-order spatial mode when trying, and the DGDs between the DGDs and different spaces mode of birefringence induction wants hour; Be very easy to understand promote this analyze with optimize the power launched in any PMs group and, keep total emissive power constant, by suitable definition simultaneously with , prove to cause to cause orthogonality between minimum and the transmitting polarization of peak power at given group of PMs.

5. the transmission master mode of gradually changeable index multimode fiber according to claim 2, it is characterized in that: the analysis modeling of described three modal systems is by analyzing a simple system, have studied the correlativity of length in the GDs of regarding fibers curvature and low and high coupling mechanism, in MMF, the minimum number propagating mode in each polarization is 3, two kinds of mode are caused along the bending of a direction, and it is coupled to each other, and allow the third mode not propagate with being coupled separately, therefore, ignore the third spatial mode, in order to simply, suppose that all fiber segment is positioned at plane, so that polarization does not affect, and negligible, result is two modal systems, and it is mathematically similar to the single mode fiber of PMD, first, in order to low coupling mechanism is described, research single hop has the optical fiber compared with small curve, then, in order to high coupling mechanism is described, study a kind of optical fiber of many sections and statistical segregation parameter.

6. the transmission master mode of gradually changeable index multimode fiber according to claim 5, is characterized in that: the analysis modeling concrete steps of described three modal systems are as follows:

the DGD of low coupling mechanism

Here, in single hop curved fiber, calculate DGD, definition slowly changes envelope for

Here defined by formula (24), the derivative of formula (52) can be written as

Definition for these two kinds of coupled modes propagation constant between difference, can be by be written as

Adopt T, following formula can be used to obtain propogator matrix U

Employing formula (59) and situation, use formula (34) group delay matrix F is written as

Because T is single, so the eigenwert of F is the eigenwert of matrix in parenthesis, as from this derivative seen, for linear fiber, here , the eigenwert of F is the GDs value of coupled mode, by provide, in shorter fiber segment, corresponding to low coupling mechanism, time, can suppose that most of light signal is to propagate in the first mode, and be coupled to the second mode at leisure, make to obtain

Use situation, and only consider single order item in z, can be written as

The derivative of the middle T of consideration formula (58) and formula (62)

Notice that (63) equal , and be updated to formula (60), obtain

Expression formula (66) display DGD is linear increase along with fiber lengths, just as the situation of PMD in low coupling mechanism, in addition, for less , bending can increase DGD, DGD be with proportional;

(B) DGD of high coupling mechanism

In high coupling mechanism, GDs determined by local optic fibre characteristic; But depend on the cumulative effect of modal coupling on whole optical fiber, the statistical property of GDs is by solving coupled stochastic difference equation to study, for the situation of SMF having PMD, Poole has observed these equations in low and high coupling mechanism, is positioned at for hypothesis three mode MMF of plane, in formula (23), coupled modal equations abbreviation is to as follows

As described in chapters and sections above, have a mode not with other two kinds of modal coupling, so analysis in also ignore, optical fiber curvature auto-correlation be defined as follows

Here bracket represents population mean, also defines power spectrum density (PSD) for

Solve Random Coupling (67) according to the method for document [6], obtain conduct the all square DGD of function be

Parameter describe overall average rate, power is changed between mode here, and is defined as

In the low coupling limit time, have

The height coupling limit considered here in, obtain

Equation (73) sees clearly the dependence chance of DGD of regarding fibers statistics to providing, show in high coupling mechanism, the PSD of DGD and curvature is inversely proportional to and changes, just as when SMF [6] PDM, DGD along with length in high coupling mechanism, as square root and change, in order to understand formula (73) further, notice model in because curvature is constant in every segment length, curvature by discrete random variable describe, it is independent identically distributed (i.i.d.), represents respectively expectation and variance be with , between section i and section i+1, discrete auto-correlation is

In order to find out curvature pSD, note , the function class of z is similar to PAM signal, and the function of t uses this analogism, obtains pSD be

In the problem that these are existing, the first and the second mode are nonsingular here, such as , obtaining DGD variance (73) is

Equation (76) is presented in high coupling mechanism, and DGD is and fiber lengths, such as, square root be directly proportional, and poor with curvature criteria be inversely proportional to.

7. the transmission master mode of gradually changeable index multimode fiber according to claim 5, it is characterized in that: the numerical modeling of described multimode optical fiber is, based on the multistage model of the space described in theory and polarization mode coupled fiber, adopt high precision matrix tool in MATLAB to carry out the numerical modeling of MMF, optical fiber is the MMF of the progressive refractive index silicon materials of real core, its overall length is , optical fiber has numerical aperture and is , and wavelength is , the refractive index of fiber optic hub be at wavelength time measured, leave this wavelength, be adopt Sellmeier equation to calculate, refractive index is to the derivative of frequency also adopt Sellmeier equation to calculate, but, find model in, waveguide dispersion is larger than material dispersion impact, and birefringence scale factor is set to , use formula (5) and formula (7), find that 55 spatial modes are propagated in two polarizations, adopt unlimited real core approximate (2), select Matching Experiment, in this test, find that lower mode has shorter GDs, and find that DGDs is than in ideal value time the value predicted much higher, except as otherwise noted, optical fiber is divided into section, namely every segment length is 0.1 meter, obeys independent identically distributed angle for every section relative to one section of rotation one above , its probability density function (pdf) is normal distribution, and variance is , the curvature of every section is an independent identically distributed stochastic variable , the front of its probability density function normal probability density function, and its variance is , because increase, so model is from low coupling mechanism to high coupling mechanism, the Stochastic implementation of the given anglec of rotation and curvature, calculates group delay operational symbol F, carries out diagonal angle abbreviation to obtain PMs and its GDs to F.

8. the transmission master mode of gradually changeable index multimode fiber according to claim 5, it is characterized in that: described conclusion is that polarization independent closes is when curvature variance is compared with fiber lengths hour, namely during low coupling mechanism, group delay is its non-coupled value closely, and with fiber lengths linearly, master mode is still produce high degree of polarisation simultaneously, in this mechanism, reproduced the polarization dependence of impulse response, and this polarization dependence is observed in silicon materials multimode optical fiber; When curvature variance and fiber lengths are enough large, i.e. high coupling mechanism, so can reduce the group delay of propagation, and it is proportional with the square root of fiber lengths, depolarizing of master mode simultaneously, in this model, the group delay of model reduces consistent with the propagation group delay observed in plastic material multimode optical fiber MMF.