Surface Plasmon Isolator based on Nonreciprocal Coupling. Abstract

Transcription

1 APS/123-QED Surface Plasmon Isolator based on Nonreciprocal Coupling Juan Montoya, Krishnan Parameswaren, Joel Hensley, and Mark Allen Physical Sciences Incorporated 20 New England Business Center, Andover, MA Rajeev Ram Research Laboratory of Electronics, Massachusetts Institute of Technology (Dated: June 23, 2009) Abstract Integrated photonics require optical isolators which achieve low insertion loss and large optical isolation. Here we describe a surface plasmon enhanced optical isolator that is based on nonreciprocal coupling from a dielectric waveguide coupled to a surface plasmon waveguide. The surface plasmon core consists of a magnetic metal which results in a large nonreciprocity, allowing for device lengths on the order of 50 microns. The analysis and modeling presented here indicates greater than 30dB isolation and less than 3dB insertion loss is possible. PACS numbers: ls,42.79.gn,78.66.bz, m Physical Sciences Inc.; Electronic address: montoya@psicorp.com 1

2 I. INTRODUCTION Surface plasmon devices offer several attractive features for photonic integrated circuits, such as strong confinement of optical fields, the ability to simultaneously transport optical and electronic signals, tolerance for sharp bending, and compatibility with existing semiconductor materials and manufacturing processes. Magneto-optic surface plasmon devices exploit the change in the surface plasmon mode effective index with an applied magnetic field. Applications of magneto-optic plasmon devices include optical switching, modulation, and isolation. Here we will focus on an optical isolator application using an InP platform. The analysis however is applicable to other devices and material systems. There is a strong demand for optical isolator devices in integrated photonics. Optical isolators prevent feedback which results in laser source instability, optical amplifier oscillation, and interference in photonic platforms. The demand for optical isolators increases with increasing device density on a chip. Scaling bulk optical isolators to chip dimensions is limited by materials such as YIG which are incompatible with existing photonic material platforms (Si or InP) due to their lattice mismatch [1]. To overcome this limitation, we have developed an optical isolator based on surface plasmon nonreciprocal coupling. While other authors have proposed surface plasmon isolators based on a nonreciprocal phase-shift [2, 3], here we present a design of a surface plasmon isolator based on nonreciprocal coupling [4]. Our concept is illustrated in Fig. 1. An input guided mode at 1550nm propagates through the dielectric waveguide in the forward direction. An adjoining magnetooptic surface plasmon waveguide is designed such that an applied external magnetic field produces a nonreciprocal effective index. In the forward direction, the dielectric waveguide and the surface plasmon waveguide are phase-mismatched and weak coupling occurs resulting in a low insertion loss. In the reverse direction, the two waveguides are matched and coupling into the lossy-surface plasmon mode results in large isolation. This paper is organized into three sections. In section II we present the analysis of the magneto-optic surface plasmon waveguide consisting of a magnetic metal core bounded by air and InP. In section III we will apply the analysis of section II toward an optical isolator application. The conclusions will be discussed in section IV. 2

3 Surface Plasmon Waveguide Magnetic Field Input Output Dielectric Waveguide FIG. 1: Nonreciprocal coupling concept for a surface plasmon optical isolator. In the forward direction weak coupling occurs resulting in low insertion loss. In the reverse direction, the surface plasmon mode and dielectric waveguide mode are index matched and strong coupling occurs resulting in large isolation. 0 h ε 1 Region I ε m, ε xz, ε zx Region II ε 3 Region III M y z x FIG. 2: Geometry used for surface plasmon magneto-optic analysis. Magnetic core (ε m, ε xz, ε zx ) is bounded by material permittivities ε 1, ε 3. II. SURFACE PLASMON MAGNETO-OPTIC ANALYSIS Magneto-optic surface plasmon devices exploit the change in the effective index of the surface plasmon mode in the presence of an externally applied magnetic field. For a nonreciprocal change in the surface plasmon effective index to occur the core or cladding material must be magneto-optic [3]. In this section, we will derive the effective index change in the surface plasmon mode when a magnetic metal is used as the core surrounded by a nonmagnetic upper and lower cladding. We will generalize the approach used in reference [5] for isotropic materials to include the anisotropic permittivity tensor of the magnetic core. A magnetic metal such as iron, cobalt, or nickel may be described by a permittivity tensor 3

4 ε = 0 ε xz 0 ε m 0, (1) 0 ε m ε m ε zx where ε m is the permittivity of the metal in the absence of an externally applied magnetic field, and the off-diagonal terms ε xz,ε xz are proportional to the transverse applied magnetic field M y [6]. The real part of the diagonal elements ε m are negative for excitation below the plasma frequency. For a surface plasmon mode to exist, the magnitude of the real part of the permittivity ε m must be greater than the permittivity in the surrounding medium ε 1, ε 3 as in ε m > ε 1, ε 3 [7]. The geometry we will use to derive the nonreciprocal response is shown in Fig. 2. The surface plasmon mode is bounded by a medium with permittivity ε 1 in region I defined by x < 0, a magnetic metal in region II defined by 0 < x < h, and permittivity ε 3 in region III defined by x > h. We will assume a TM mode field of the form H y = f(x)e ( βz+ωt). The propagation constant is given by β in the z direction and the envelope f(x) takes on the assumed form Ce S 1x Region I f(x) = A cosh(s 2 x) + B sinh(s 2 x) Region II De S 3(x h) Region III (2) for a surface plasmon mode defined with a peak at the interface between the metal-slab and the surrounding medium. Solving for the nonreciprocal response of the surface plasmon mode entails imposing the boundary conditions on the continuity of the tangential electric and magnetic field. The nonreciprocal response is determined by solving for the propogation constant β as a function of the externally applied magnetic field M y. Since reversing the sign of the magnetic field is equivalent to changing the direction of propagation, β(m y ) β( M y ) for a nonreciprocal waveguide. The sign of the magnetic field M y is taken into account in the sign of the off-diagonal elements ɛ xz and ε zx in Eq. 1 which are proportional to the field. For example, to solve for the propagation constant β F in the forward direction we will assume ε xz = jam y where a is the magneto-optic coefficient [3]. In the reverse direction, we will 4

5 assume ε xz = jam y. In the following subsections the boundary conditions are applied to arrive at a dispersion relationship which allows us to solve for β as a function of the material parameters. A. Boundary Condition at the Interface Between Regions I and II In this section we will impose the continuity of the tangential electric and magnetic fields at the interface between regions I and II. We will use a superscript notation as in H y (1), H y (2), H y (3) and E z (1), E z (2), E z (3) regions I,II, and III respectively. to denote the tangential magnetic and electric fields in The first applied boundary condition demands the continuity of the tangential magnetic field at the interface between regions I and II at x = 0. Using the assumed form of Eq. 2 for the magnetic field and setting H y (1) = H y (2) we find A = C. (3) This result is consistent with the assumption that the surface plasmon mode has a peak at the interface and an evanescent decay into the surrounding medium. The tangential electric field component is found by using the curl relationship H (1) = t ε 1E (1). (4) Assuming a harmonic time dependence e jωt for the electric field, we can readily solve Eq. 4 for the tangential electric field component E (1) z in region I. This field component, written in Eq. 5 is used to impose the continuity relation across the interface between regions I and II E (1) z = j ωε 1 x H(1) y = j CS 1 ωε 1. (5) Following the same procedure in region II we once again express the electric field in terms of the magnetic field using the curl relationship as expressed in Eq. 6. Since we are considering a TM magnetic field component H y there will in general be two electric field components E x and E z. H (2) = t ε 2 E (2) (6) 5

6 Solving for the electric field in region II as given in Eq. 6 requires using the anisotropic permittivity tensor for the magnetic metal of the form given in Eq. 1. Since the permittivity tensor is Hermitian, the off-diagonal terms are complex conjugates ε xz = ε zx, and are proportional to the magnetic field M y which gives rise to the nonreciprocal response. ε m E x (2) + ε xz E z (2) H(2) y z 0 H (2) y x = jω jβa 0 = jω BS 2 0 ε zx E x (2) + ε m E z (2) ε m E x (2) + ε xz E z (2) 0 ε zx E x (2) + ε m E z (2) Setting ε 2 equal to the permittivity tensor of Eq. 1 results in Eq. 7. The electric field E (2) x may be eliminated allowing one to solve for E (2) z field component H (2) y (7) in terms of the amplitudes of the magnetic in region II. Imposing the continuity relationship of the tangential electric field, namely E z (1) = E z (2), yields the amplitude constant B of the magnetic field in region II B = jβaε zx + [ε 2 m ε xz ε zx ] S 1A ε1. (8) S 2 ε m B. Boundary Condition at the Interface Between Regions II and III Next we apply the boundary conditions at x = h between regions II and III. We begin by evaluating the curl of the magnetic field as expressed in Eq. 6 at x = h. This allows us to solve for the electric field at the interface jβ[a cosh(s 2h) + B sinh(s 2 h)] = (9) AS 2 sinh(s 2 h) + BS 2 cosh(s 2 h) jω ε me x (2) + ε xz E z (2). ε zx E x (2) + ε m E z (2) Eliminating the x component allows us to solve for the tangiential electric field E (2) z at the interface as expressed in Eq. 10. This field component is continuous across the interface in 6

7 region III (E (2) z = E (3) z ). E z (2) = (jβε zxa ε m BS 2 ) cosh(s [jω(ε xz ε zx ε 2 2 h) (10) m)] + (jβε zxb ε m AS 2 ) sinh(s [jω(ε xz ε zx ε 2 2 h) m)] To proceed we must determine the tangential electric field in region III (E z (3) ). This is accomplished by relating the magnetic field to the electric field by the curl relationship. The magnetic field amplitude in region III is found by imposing the continuity boundary condition on the tangiental magnetic field (H y (2) = H y (3) ). From Eq. 2 we observe this results in D = A cosh(s 2 h) + B sinh(s 2 h). (11) The electric field may then be determined by once again applying the curl relation H (3) = ε t 3E (3). Since region III is isotropic the permittivity constant ε 3 is a scalar. The electric field in region III is found to be E (3) z = S 3 jωε 3 [A cosh(s 2 h) + B sinh(s 2 h)]. (12) The fully determined tangential field components result in a dispersion relationship of the surface plasmon modes. The dispersion relationship provides the propagation constant β as a function of the applied magnetic field M y giving rise to ε xz and ε zx. Imposing the continuity of the tangential field components E z (3) = E z (2) in terms of the material and mode decay constants yields a transcendental equation [B + jβε zxb ε m AS 2 ] tanh(s (ε xz ε zx ε 2 m) S 3 2 h) + (13) ε 3 (jβε zx A ε m BS 2 ) + A = 0. [(ε xz ε zx ε 2 m )S 3 ε 3 ] In the limit that (ε xz = 0, ε zx = 0) Eq. 13 reduces to the result given in reference [5] for an isotropic material. This corresponds to the dispersion relationship in the absence of an externally applied magnetic field M y. The surface plasmon mode decay constants S 1, S 2, S 3 may be expressed in terms of the propagation constant by the momentum conservation equations given in Eq. 14. We have solved Eq. 13 using a monte-carlo approach. Our 7

8 results will be presented in the context of an example in section III. S 2 1 = β 2 k 2 0ε 1 S2 2 = β 2 k0ε 2 m (14) S3 2 = β 2 k0ε 2 3 In the next section we will consider an optical isolator application. We will apply the analysis in this section toward the design of a surface plasmon optical isolator. III. OPTICAL ISOLATOR APPLICATION Several authors have exploited a change in the nonreciprocal index to design a nonreciprocal phase shift optical isolator in magneto-optic waveguides [6, 8, 9]. Mostly, magneto-optic materials such as YIG and BIG have been used due to their large magneto-optic response. The large lattice mismatch between these materials and integrated photonic platforms such as Si and InP have precluded their applicability for photonic integrated circuits. Devices which make use of the nonreciprocal index change in surface plasmon waveguides have been proposed in an interferometric application [2, 3]. These devices result in an enhanced nonreciprocal response due to confinement and resonance of the surface plasmon mode in the magneto-optic material. The surface plasmon mode losses in the interferometer geometry create a challenge for low-insertion loss applications. In our approach we design a surface plasmon isolator based on nonreciprocal coupling. In the forward direction, the large index mismatch between the dielectric waveguide and the surface plasmon waveguide prevents efficient coupling resulting in a low insertion loss. In the reverse direction, the nonreciprocal index in the surface plasmon waveguides results in a phase-match condition allowing for strong coupling and isolation. The effective index of the surface plasmon mode in the reverse direction (n eff,sp,rvs ) can be designed to match the effective index of the dielectric waveguide (n effd ) by varying the metal thickness. As an example, we will consider a dielectric waveguide consisting of an InP upper cladding, an In 1 x Ga x As y P 1 y (x=0.290, y=0.628) core, and an InP lower cladding [10]. The surface plasmon waveguide consists of an iron magnetic core on an InP lower cladding with an air upper-cladding. The surface plasmon effective index variation based on the solution of the transcendental equation (Eq. 13) is shown in Fig.3 as a function of the metal thickness. For a 100nm 8

9 Effective Index of SP Mode Magnetic Field Detuning( ) n 3 eff Metal Thickness(nm) Imaginary Index of SP Mode Metal Thickness(nm) Fe M InP Metal Thickness(nm) FIG. 3: Surface plasmon effective index in the reverse direction, detuning parameter, and imaginary coefficient for iron on InP. For a 100nm thickness, the surface plasmon effective index matches the dielectric waveguide effective index in the reverse direction. thickness, the effective index of the surface plasmon mode matches the effective index of the dielectric waveguide mode (n eff,sp,rvs = n effd = 3.25). We define a detuning parameter δβ which results from applying a saturating magnetic field to the surface plasmon magnetic metal. The detuning parameter is proportional to the change in the surface plasmon mode effective index in the forward (n eff,sp,fwd ) and reverse directions (n eff,sp,rvs ) δβ/k 0 = n eff,sp,fwd n eff,sp,rvs. (15) The detuning parameter shown in Fig.3 as a function of metal thickness is determined using the optical properties of iron (n = i)[11]. The off-diagonal elements for iron ε xz, and ε zx are defined by the relation ε xz = 2nθ F /k 0 where θ F is the Faraday rotation and k 0 = 2π/λ. For magnetically saturated iron we use the values from [11] for the Faraday rotation θ F = /cm for a saturated magnetic field. The saturated magnetic field strength in iron is 0.17 Tesla allowing for compact permanent magnets to be used to achieve the desired detuning parameter. The imaginary index of the surface plasmon mode is also illustrated in Fig. 3. The imaginary index illustrates that the surface plasmon mode undergoes rapid attenuation with a 1/e extinction length of 5.7µm. As a result, the power that is transferred from the dielectric waveguide into the surface plasmon waveguide is attenuated ensuring extinction of the power that is removed from the input port. 9

10 A. Insertion Loss and Isolation Determination Using Coupled Mode Theory Coupled mode theory is used to estimate the coupling length required for isolation and for achieving a given insertion loss. To estimate the coupling coefficient the lossless approximation is used for simplicity. The loss is taken into account as a perturbation to the lossless case in section III B and in reference [12]. The surface plasmon and dielectric waveguide modes may be solved using the finite difference full vectorial magnetic field method (FVHFDM) [13, 14]. The mode profiles are illustrated in Fig. 4 for the surface plasmon and dielectric waveguide. The surface plasmon waveguide consists of an InP lower cladding, a 100nm thick iron (Fe) film, and air for the upper cladding. The dielectric waveguide consists of an InP lower cladding, InGaAsP core, and an InP upper cladding. When the two waveguides are brought together as shown in Fig. 1 they share an InP common cladding (ε c ). For the lossless case, the coupling coefficient κ 12 from the dielectric waveguide to the surface plasmon waveguide is equivalent to the coupling coefficient κ 21 from the surface plasmon waveguide to the dielectric waveguide. The coupling coefficient subscripts are subsequently dropped and we define (κ = κ 21 = κ 12 ). The coupling coefficient for the lossless case is given by κ = k2 0 2β 1 (ε1 ε c )U 2 U 1dA U1 2 da where ε 1 is the permittivity of the core of the dielectric waveguide (i.e. InGaAsP), and ε c is the permittivity of the common cladding (i.e. InP). U 2 and U 1 are the unperturbed normalized cross-sectional magnetic field amplitudes of the modes in the surface plasmon waveguide and dielectric waveguide respectively, and β 2 is the real part of the propagation constant of the surface plasmon waveguide. The surface plasmon waveguide length is chosen to allow for full power transfer from the dielectric waveguide in the reverse propogation direction. The coupling length L c required to achieve isolation is inversely proportional to the coupling coefficient κ and is determined by: (16) L c = π 2κ. (17) In principle, the coupling length can be made arbitrarily small by increasing the coupling coefficient κ. The coupling coefficient κ may be controlled by choosing the thickness of the 10

11 x ( µm) Fe InP x ( µm) InP 2 InGaAsP InP y ( µm) (a) y ( µm) (b) FIG. 4: (a) Surface plasmon waveguide mode and (b) dielectric waveguide mode computed using the Finite-Difference Full Vectorial Magnetic Field method. common cladding material. Decreasing the distance between the two waveguides results in a stronger interaction. While it is desirable to minimize the device length by increasing the coupling coefficient, the coupling coefficient is constrained to achieve a given insertion loss requirement. The insertion loss is defined in terms of the transmission coefficient between any two points on a device. For the nonreciprocal optical coupling isolator, the insertion loss from the input to the output can be expressed in db as: IL = 10log 10 (1 κ 2 ( β 1 β 2F 2 ) 2 + κ 2 ), (18) where β 2F is the propagation constant of the surface plasmon mode in the forward direction. In evaluating Eq. 18, it is convenient to express the detuning parameter as (δβ = β 1 β 2F ). The 3dB insertion loss occurs at the half power point where the detuning parameter δβ = 2κ. From Fig. 3 we determine that the value of the detuning parameter is δβ = /k 0. Substituting 2κ = δβ in Eq. 17 results in a coupling length of L c = 50µm for a 3dB insertion loss. Designing for a smaller insertion loss requires weaker coupling which results in a longer coupling length. A plot showing the insertion loss in decibels as a function of the detuning parameter is shown in Fig. 5. The source of the detuning is the nonreciprocal change in the propagation constant of the magneto-optic surface plasmon waveguide with an externally applied magnetic field. By increasing the separation between the dielectric waveguide and surface plasmon waveguide, the coupling strength κ decreases and the ratio of δβ/κ increases. This weaker coupling results in a smaller insertion loss. As the detuning parameter is increased, the insertion loss also decreases. This reflects the inefficient coupling between the dielectric waveguide and surface plasmon waveguide as the phase mismatch increases. This insertion 11

12 10 2 Insertion Loss (db) δβ/κ FIG. 5: Insertion loss as a function of the detuning parameter δβ expressed in decibels. loss dependence with phase-matching the real part of the propagation constants is consistent with experimental results using nickel for a surface plasmon polarizer application [15]. An expression for the isolation is found by substituting the reverse propagation constant β 2R in place of β 2F in Eq.18. The surface plasmon mode is designed to match the dielectric waveguide mode (β 1 ) in the reverse direction β 2R = β 1. For perfect phase-matching, complete extinction resulting in perfect isolation is possible. Here we will examine the phase-matching error tolerance to achieve 30dB isolation. The tolerance on the reverse phase-matching to achieve 30dB isolation can be evaluated by expressing the isolation as I 20log 10 ( β 1 β 2R ), where the approximation is valid for a 2κ small phase-mismatch β 1 β 2R 1. For a phase-matching error of β, the surface-plasmon 2κ mode propagation constant in the reverse direction may be expressed as β 2R = β 1 + β as illustrated in Fig.6. For 30dB isolation we find that the surface-plasmon mode reverse propagation constant must be matched to the dielectric waveguide within a fraction of the coupling coefficient ( β =.063κ). For the choice of the coupling coefficient corresponding to a 3dB insertion loss κ = δβ, we can express the phase match error tolerence in terms of 2 the detuning parameter β δβ = 3.16%. B. Accounting for Loss The lossless coupled mode theory analysis was used above to estimate the coupling length, insertion loss, and isolation [16]. Experimental results for surface-plasmon polarizers using lossy metals suggest that phase-matching the real part of the effective index is necessary for coupling from a dielectric waveguide mode into a surface-plasmon mode [15]. This 12

13 FIG. 6: Surface plasmon mode phase matching error β. Surface Plasmon propagation constant β 2F,β 2R in the forward and reverse direction. Dielectric waveguide propagation constant β 1. When β = 0, the surface plasmon mode matches the dielectric waveguide mode in the reverse propagation direction. experimental observation is consistent with the lossless coupled mode theory analysis. In general, the complex permittivity of a metal (ε r jε i ) results in an effective index for the surface-plasmon mode that is complex n eff = n eff,sp jn i. The real part of the effective index is used to determine the phase-matching dependence of the coupling. To verify the coupling length between a dielectric waveguide and a lossy surface-plasmon mode, the loss is treated as a perturbation to the lossless solution. Fig. 7 shows the results for increasing the loss of the surface-plasmon mode. The simulations were performed using a commercial beam propagation method solver produced by RSOFT Design Group (Ossining, NY). The lossless case is determined by using an ideal permittivity ε m that results in the same real effective index as the lossy surface-plasmon mode. In Fig. 7(a), the lossless case is shown where power is continually transferred between the dielectric waveguide and the surface-plasmon waveguide. A coupling length of 50 µm results for the lossless case. The energy oscillates between the dielectric waveguide and the surface-plasmon waveguide. In Fig. 7(b)-(d), adding loss to the metal results in attenuation as power is transferred into the lossy surface plasmon mode. This attenuated power is lost and does not couple back into the dielectric waveguide. The coupling length is fixed in both the lossless and lossy case while the number of damped oscillations decreases with increasing loss. This result is consistent with the lossy waveguide coupling analysis found in the literature [12]. We argue by extension from this perturbation analysis that Eq. 16 provides a valid estimate of the coupling length 13

14 Field Power Field Power Z(µm) Z(µm) (a) Y(µm) (b) Y(µm) Z(µm) Z(µm) (c) Y(µm) (d) Y(µm) FIG. 7: Power transfer from the dielectric waveguide to the surface plasmon mode (a) lossless case n eff,sp = 3.25, increasing loss (b) n eff,sp = i,(c) n eff,sp = i, (d) n eff,sp = i. between the dielectric waveguide and the lossy surface-plasmon waveguide. In practice, the separation between the surface-plasmon waveguide and the dielectric waveguide may be optimized to achieve the desired insertion loss and isolation. IV. CONCLUSIONS We have designed a surface plasmon isolator based on nonreciprocal coupling between a dielectric waveguide and a surface plasmon waveguide. Our unique geometry which uses a magnetic metal as the core of a surface plasmon waveguide offers several advantages. The confinement in the magnetic metal results in an enhanced nonreciprocal response resulting in device lengths on the order of 50 µm for an insertion loss of 3dB. Moreover, the power that is transferred to the surface plasmon waveguide is attenuated insuring isolation. These devices are favorable for photonic integrated circuits as they are compatible with semiconductor manufacturing. 14

15 Acknowledgments This material is based upon work supported by DARPA and the US Army Aviation and Missile Command under Contract Number W31P4Q-08-C Any opinions, findings and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of DARPA and the US Army Aviation and Missile Command. This article has been approved for public release. 15

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17 V. FIGURE CAPTIONS Figure (1) Nonreciprocal coupling concept for a surface plasmon optical isolator. In the forward direction weak coupling occurs resulting in low insertion loss. In the reverse direction, the surface plasmon mode and dielectric waveguide mode are index matched and strong coupling occurs resulting in large isolation. Figure (2) Geometry used for surface plasmon magneto-optic analysis. Magnetic core (ε m, ε xz, ε zx ) is bounded by material permittivities ε 1, ε 3. Figure (3) Surface plasmon effective index in the reverse direction, detuning parameter, and imaginary coefficient for iron on InP. For a 100nm thickness, the surface plasmon effective index matches the dielectric waveguide effective index in the reverse direction. Figure (4) (a) Surface plasmon waveguide mode and (b) dielectric waveguide mode computed using the Finite-Difference Full Vectorial Magnetic Field method. Figure (5) Insertion loss as a function of the detuning parameter δβ expressed in decibels. Figure (6) Surface plasmon mode phase matching error β. Surface plasmon propagation constant β 2F,β 2R in the forward and reverse direction. Dielectric waveguide propagation constant β 1. When β = 0, the surface plasmon mode matches the dielectric waveguide mode in the reverse propagation direction. Figure (7) Power transfer from the dielectric waveguide to the surface plasmon mode (a) lossless case n eff,sp = 3.25, increasing loss (b) n eff,sp = i,(c) n eff,sp = i, (d) n eff,sp = i. 17