Polarization- Dependent Loss (PDL) Measurement Theory

Transcription

1 Polarization- Dependent Loss PDL) Measureent heory his docuent provides a detailed suary of the theoretical basis of the 4-polarization state ethod for PDL easureent also referred to as the Mueller-tokes ethod). It relates specifically to the swept-wavelength syste W) which eploys this ethod for calculating PDL values fro insertion-loss easureents, and proves the validity of the 4-state ethod for PDL calculations. It is a record of the fundaental equations that are incorporated into the W-PDL software suite. his ethod is described and approved as IEC PDL 4-tate Method Derivation Mueller calculus Mueller calculus is a atheatical tool to deal with polarization that is siilar to the Jones ethod in that the effect of an optical eleent on the polarization of light is treated as a atrix operation. More specifically, the transission of light through an arbitrary device can be expressed as the product of the Mueller atrix of the device under test DU) a 4x4 atrix) and the input tokes vector x4 atrix). he result is another tokes vector. he details of this atheatical operation are reviewed in detail in ref., pg. 83). he transission equations are as follows: 3 sin ω is the intensity of the transitted or output power.,, and 3 relate to the polarization state of the output power. Note that these values are not easured by a standard detector. he first row of the Mueller atrix contains all of the ters that relate to the power transission. he reaining ters are not involved in the deterination of PDL and thus have been oitted. he input tokes vector is in a general for for an elliptically polarized light. is the intensity of the input power, is the angle of the aziuth of the input polarization, and ω is the ellipticity of the input polarization. White Paper 3 []

2 Deterining PDL he PDL of the DU is deterined fro the difference between the iniu and axiu values for the transitted intensity of the variable. o calculate this, we ust solve equation and take the partial transitted derivatives with respect to the polarization variables and ω, and set the resulting equations to, as follows: 3 sinω ω sinω sinω 3 [.] [.a] [.3a] then set.a and.b to : or tan nπ [.b] ω tan 3 [.3b] where n, in order to account for the fact that the aziuth variable has a range fro to π. Equation.b expresses the variable in ters of the Mueller eleents only. It can then be used to eliinate fro equation.3b. o do so, the following anipulation of the first for of.b is required. sin then using sin cos, we get: cos cos sin sin sin ) ) cos cos [] Polarization-Dependent Loss PDL) Measureent heory

3 Equations are then substituted into the denoinator of.3b to yield ω in ters of only the Mueller atrix eleents as follows. ω tan 3 [3a] which siplifies to the following: ω tan 3 [3b] Equations 3b and.b give us the variables ω in ters of only the Mueller eleents. hey can be anipulated to for the trigonoetric expressions in equation for the case of the axiu and iniu transission. he axiu value is the result for the case where n fro equation.b. iilarly n relates to the iniu value. herefore, we require expressions for, and sinω fro equations.b and 3n. in and are already available fro equation. # Fro equation 3b: [4] # Also fro equation 3b: [5] 3 Polarization-Dependent Loss PDL) Measureent heory

4 We can siplify the above expressions by using the substitutions: 3 a hus, the expression for the axiu transission for arbitrary elliptically polarized light becoes: 3 MAX 3 3 [6] where the resulting solution is: 3 MAX a a a [ a that can be siplified to the following: o ax [7] ] o find the iniu transission, we follow the sae path leading to equation 7, but with n. his results in equation.b having the functional for of π. For this expression, we can ake use of the following trigonoetric expressions: cos π cos sin π sin these relationships can be used to solve for equation.3b with n resulting in: ω tan 3 [8] when equation 8 is used instead of equation.3b. Repeating all of the steps leading to the axiu transission equations to 6), we arrive at the condition for iniu transission: 3 MIN 3 3 [9] which, like 7a, siplifies to : in a [7b] 4 Polarization-Dependent Loss PDL) Measureent heory

5 At this point, we can deterine the PDL fro equations 7a and 7b and using the PDL definition: PDL Log in Log ax [] 3 3 Equation illustrates that in order to deterine the PDL of a device, we ust deterine the first row of its Mueller atrix. he four atrix eleents can be deterined by applying four different polarization states to the device. he resulting set of four equations can be solved to find the four unknown Mueller eleents, provided that the 4 input polarization states are linearly independent. he four independent vectors that we eploy are as follows, with their respective tokes vectors: ) horizontal linear polarized light ) vertical linear polarized light 3) 45 linear polarized light 4) left-hand circularly polarized light [] If we use these four polarization states with equation, we get the transission for each state as follows: [] When we algebraically anipulate the above equations and noralize the intensities, we will get the Mueller eleents as a function of the four easurable intensities, which is what we need to finish solving equation and deterine PDL. [3] 5 Polarization-Dependent Loss PDL) Measureent heory

6 Wavelength dependencies Equation 3 gives the correct expressions for the Mueller eleents needed to solve for the PDL by equation when the four input polarization states are exactly as stated in equations. In practice, however, the circularly polarized light is usually elliptically polarized. his is a result of the fact that circularly polarized light is usually ade by an arrangeent of a linear polarizer and a quarter waveplate with its required axis arranged at 45 to the polarizer s axis ref. 4 pg. 94). he quarter waveplate only has its required π/ radians of retardation at a particular wavelength. At other wavelengths, the circular polarizer will produce elliptically polarized light. hus, we will need to apply a correction to equations,, and 3 to account for the wavelength dependencies since the ajor applications P3 eter, W) work at ultiple wavelengths. o begin, we require a tokes vector for the output of a circular polarizer at a wavelength that is different for its center value. he retardation of a quarter waveplate varies with wavelength, to a first-order approxiation, as follows: δ πcw λ for QWP and δ πcw λ for HWP [4] where CW is the center wavelength of the waveplate and λ is the variable wavelength. hen, the Mueller atrix for a linear retarder at an arbitrary angle θ with retardation δ is as follows: [5] where the following abbreviations have been used: Ccosθ, sinθ, µsinδ, βcosδ. For the case of a circular polarizer, the angle θ is 45. With this, we can siplify 5 to: β µ µ β [5b] Fro 5b we can now calculate the tokes vector of the wavelength-dependent elliptically polarized light by inputting the linear polarized light fro the polarizer. he atrix calculation is as follows: β µ µ β β µ cosδ sinδ [6] 6 Polarization-Dependent Loss PDL) Measureent heory

7 hese results can now be used to replace the LHC) fro equation. his leads to the change in the expression for LHC) in equation to becoe: [7] where LHE) is the transitted intensity for the elliptically polarized light. When we algebraically anipulate equation 7 along with the reaining three unchanged equations fro, we once again generate the expressions for the Mueller eleents but now as a function of wavelength. [8] References. IEC Polarized Light in Optics and pectroscopy D. Kiger, J. W. Lewis, C. E. Randall. Acadeic Press pages 75 to 87). 3. yste and Method for easuring Polarization Dependant Loss D. L. Favin, B. M. Nyan, G. M. Wolter, United tates Patent # , Matrix Methods in Optics A. Gerrard, J. M. Burch, Dover Publications, New York, 975 pages 79 to 7). 5. Introduction to Optics F. Pedrotti, L. Pedrotti, Prentice Hall, New Jersey, 993, pages 8 to 95). 7 Polarization-Dependent Loss PDL) Measureent heory

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