## Analytical calculation of nonreciprocal phase shifts and comparison analysis of enhanced magneto-optical waveguides on SOI platform |

Optics Express, Vol. 20, Issue 8, pp. 8256-8269 (2012)

http://dx.doi.org/10.1364/OE.20.008256

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### Abstract

Transfer matrices for one-dimensional (1-D) multi-layered magneto-optical (MO) waveguides are formulated to analytically calculate the nonreciprocal phase shifts (NRPS). The Cauchy contour integration (CCI) method is introduced in detail to calculate the two complex roots of the transcendental equation corresponding to backward and forward waves. By virtue of perturbation theory and the variational principle, we also present the general upper limit of NRPSs in 1-D MO waveguides. These analytical results are applied to compare silicon-on-insulator (SOI) based MO waveguides. First, a three-layered waveguide system with MO medium is briefly examined and discussed to check the validity and efficiency of the above theory. Then we revisited the reported low-index-gap-enhanced NRPSs in MO waveguides and obtained substantially different results. Finally, the potential of common plasmonic waveguides to enhance the nonreciprocal effect is investigated by studying different waveguides composed of Metal, MO medium and dielectrics. Our study shows that the reasonable NRPSs can be optimized to some extent but not as much as claimed in previous publications.

© 2012 OSA

## 1. Introduction

1. Z. F. Yu and S. H. Fan, “Complete optical isolation created by indirect interband photonic transitions,” Nature Photon . **3**, 91–94 (2009). [CrossRef]

2. L. Feng, M. Ayache, J. Q. Huang, Y. L. Xu, M. H. Lu, Y. F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science **333**, 729–733 (2011). [CrossRef] [PubMed]

3. R. L. Espinola, T. Izuhara, M. C. Tsai, R. M. Osgood Jr., and H. Dotsch, “Magneto-optical noreciprocal phase shift in garnet/silicon-on-insulator waveguides,” Opt. Lett. **29**, 941–943 (2004). [CrossRef] [PubMed]

4. L. Bi, J. Hu, P. Jiang, D Kim, G. F. Dionne, L. C. Kimerling, and C. A. Ross, “On-chip optical isolation in monolithically integrated non-reicprocal optical resonators,” Nature Photon. **5**, 758–762 (2011). [CrossRef]

5. N. Bahlmann, M. Lohmeyer, H. Dotsch, and P. Hertel, “Finite-element analysis of nonreciprocal phase shift for TE modes in magnetooptic rib waveguides with a compensation wall,” IEEE J. Quantum Electron. **35**, 250–253 (1999). [CrossRef]

9. J. B. Khurgin, “Optical isolating action in surface plasmon polaritons,” Appl. Phys. Lett. **89**, 251115 (2006). [CrossRef]

10. Y. Shoji and T. Mizumoto, “Ultra-wideband design of waveguide magnetooptical isolator operating in 1.31*μm* and 1.55*μm* band,” Opt. Express **15**, 639–645 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-2-639. [CrossRef] [PubMed]

11. Y. Shoji and T. Mizumoto, “Wideband design of nonreciprocal phase shift magneto-optical isolators using phase adjustment in Mach-Zehnder interferometer,” Appl. Opt. **45**, 7144–7150 (2006). [CrossRef] [PubMed]

12. M. C. Tien, T. Mizumoto, P. Pintus, H. Kromer, and J. Bowers, “Silicon ring resonators with bonded nonreciprocal magneto-optic garnets,” Opt. Express **19**, 11740–11745 (2011), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-2-639. [CrossRef] [PubMed]

13. N. Kono, K. Kakihara, K. Saitoh, and M. Koshiba, “Nonreciprocal microresonators for the miniaturization of optical waveguide isolators,” Opt. Express **15**, 7737–7751 (2007), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-12-7737. [CrossRef] [PubMed]

## 2. Transfer matrices for multi-layered magneto-optical waveguide system

*μm*) to unity [14] and assume the time and

*z*dependent phase factor to be exp [

*j*2

*πν*(

*t*–

*γz*)], where

*ν*is the normalized frequency and

*γ*is the normalized propagation constant (also known as the effective refractive index). The latter can be negative for backward propagating waves. The Faraday’s law and Ampere’s law can be written as the following set of equations and respectively. By dropping the free-space impedance, the three-dimensional field vectors

**E**and

**H**are scaled to the same order of magnitude. Considering the MO effect, the general expression of the relative permittivity tensor

*ε*reads The off-diagonal elements (i.e. gyrotropy) correspond to the first-order MO effects, where the subscripts denote the coupled field components. The

_{r}*ε*induced by a longitudinal magnetic induction

_{xy}*B*gives rise to the coupling between two transverse electric field components. This is the so-called Faraday rotation effect, which is not covered in this paper. The other two,

_{z}*ε*from the

_{xz}*y*-directed magnetization and

*ε*from

_{yz}*x*-directed produce an NRPS by coupling the longitudinal component

*E*with one transcendental electric field component

_{z}*E*and

_{x}*E*, respectively. Since the element

_{y}*ε*can be treated similar to

_{yz}*ε*, let us assume that

_{xz}*ε*= 0 and

_{xy}*ε*= 0 and only analyze the NRPS induced by the transverse magnetic (TM) mode. For such modes, the longitudinal magnetic field component is missing (

_{yz}*H*= 0) and there are only three non-trivial components

_{z}*H*,

_{y}*E*and

_{x}*E*. Substituting Eq.(3) into Eq.(2) yields and Clearly, the above equation can be inverted as By substituting Eq.(6) into Eq.(4), we get the differential equation for

_{z}*H*component as This equation was first reported in [15

_{y}15. H. Dotsch, P. Hertel, B. Luhrmann, S. Sure, H. P. Winkler, and M. Ye, “Applications of magnetic garnet films in integrated optics,” IEEE Trans. Magn. **28**, 2979–2984 (1992). [CrossRef]

*H*are conserved and no NRPS exists except for a minor modification to the propagation constant. The NRPS must be attributed to the coupling of the two electric fields at the MO material interfaces. By defining

_{y}*κ*Now we focus on the fields in the

*n*-th MO layer. According to Eq.(7), the solutions of the magnetic field

*H*can be separated to a forward wave and a backward wave and take the form of The positive sign in Eq.(8) assures that the

_{y}*A*term is for the forward wave with phase retardation along

_{n}*x*direction. Subsequently, by using

*∂*= −

_{z}*j*2

*πνγ*and substituting Eq.(9) into Eq.(6), the field

*E*can be written as At the interfaces between any two layers, both tangential field components

_{z}*E*and

_{z}*H*must be continuous, so where and the label (

_{y}*n*) should be added to all the local variables on the right hand side except for

*γ*. The origin of

*x*is chosen as the left boundary of the

*n*-th layer. From Eq.(11), we are able to extract the transfer matrix

*S*of the

_{n}*n*-th layer, which relates the weighted indices

*A*and

_{n}*B*with

_{n}*A*

_{n+1}and

*B*

_{n+1}by By simple algebraic manipulation, the transfer matrix can be written as where the transverse phase delay in the

*n*-th layer ϕ

*= 2*

_{n}*πνκ*. The full expressions of

_{n}d_{n}*a*and

*b*describe the jumping process from layer

*n*to layer

*n*+ 1, which can be written as and respectively. If

*b*= 0, the above equation returns to the case of non-magnetized medium. A nonzero

*b*breaks the time-reversal symmetry and gives rise to the NRPS. Let us assume

*A*

_{0}= 1 to normalize all the weighted indices. The reflection coefficient

*R*and transmission coefficient

*T*can be obtained from the cascaded transfer matrix

*S*=

_{T}*S*

_{N}*S*

_{N−1}···

*S*

_{1}by In the two-port network indicated by the above equation, the physical meaning of

*S*

_{T,22}= 0 and to obtain the two roots

*γ*corresponding to backward and forward propagating waves. Subsequently, the NRPSs of unit length are obtained by the difference of the normalized propagation constants Δ

*γ*by

*NRPS*= 2

*πν*Δ

*γ*.

## 3. Cauchy contour integration method

16. J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman, “Planar metal plasmon waveguides: frequency-dependent dispersion, propagation, localization, and loss beyond the free electron model,” Phys. Rev. B **72**, 075405 (2005). [CrossRef]

18. E. Anemogiannis, E. N. Glytsis, and T. K. Gaylord, “Determination of guided and leaky modes in lossless and lossy planar multilayer optical waveguides: reflection pole method and wavevector density method,” J. Lightwave Technol. **17**, 929–941 (1999). [CrossRef]

*D*enclosed by a simple closed curve

*C*. A single-valued function

*f*(

*z*) is analytical in

*D*and has no null point on

*C*. The count and the sum of roots of equation

*f*(

*z*) = 0 in domain

*D*can be calculated by and respectively. Naturally, the

*k*= 1 case is adopted for the root searching. Special attention should be paid in the following issues. First, the function

*f*(

*z*) must be analytical in

*D*, i.e.

*f*(

*z*) satisfies the Cauchy-Riemann condition. Second, one needs to choose a possible region where the roots may be located at. Third, in each possible region, there may be several roots so that we have to isolate each of them by dividing

*D*into smaller and smaller domains. Instead of the reported Kuhn root-finding algorithm [19], we propose an intuitive and easy-to-implement numerical method to search for the roots. (1) In the complex plane we can plot the figure of

*C*through

*P*

_{0}–

*P*

_{1}–

*P*

_{2}–

*P*

_{0}to enclose the domain

*D*that contains the poles. (2)

*N*is calculated by Eq.(18) to check the number of roots. Let us choose a reasonable numerical tolerance

*ε*(in our case,

*ε*= 10

^{−4}.) (3) If |

*N*| <

*ε*, stop here and recheck the predefined domain

*C*since no root exists in it. (4) If |

*N*− 1| <

*ε*, return the root value evaluated by Eq.(19), else divide the triangle domain

*C*into three child triangles as

*P*

_{0}−

*P*

_{1}−

*P*

_{3}−

*P*

_{0},

*P*

_{1}−

*P*

_{2}−

*P*

_{3}−

*P*

_{1}and

*P*

_{2}−

*P*

_{0}−

*P*

_{3}−

*P*

_{2}. (5) For each child domain, go to step (2) for further calculation. The integration should be conducted clockwise otherwise the sign of the result would be reversed. Triangular domains are recommended because they approach the root faster than other polygons and maintain a similar shape between father and child domains.

*S*

_{T}_{,22}with respect to

*γ*can also be easily derived by a simple linear algebraic manipulation from Eq.(14). All numerical integrations can be easily performed by built-in functions in Matlab. To obtain a continuous and smooth characteristic curve by parameter scanning, for example, the mode dispersion curves, a useful trick is that the triangular domain for the third root can be estimated from the two previous roots.

## 4. The upper limit of NRPS in 1-D waveguides

21. S. Yamamoto and T. Makimoto, “Circuit theory for a class of anisotropic and gyrotropic thin-film optical waveguides and design of nonreciprocal devices for integrated optics,” J. Appl. Phys. **45**, 882–888 (1974). [CrossRef]

*ε*in the permittivity tensor can be treated as a first-order approximation. Δ

_{r}*γ*can be estimated by the expression [6

6. A. F. Popkov, “Nonreciprocal TE-mode phase shift by domain walls in magnetooptic rib waveguides,” Appl. Phys. Lett. **72**, 2508–2510 (1998). [CrossRef]

22. H. Dotsch, N. Bahlmann, O. Zhuromskyy, M. Hammer, L. Wilkens, R. Gerhardt, P. Hertel, and A. F. Popkov, “Applications of magneto-optical waveguides in integrated optics: review,” J. Opt. Soc. Am. B **22**, 240–253 (2005). [CrossRef]

23. M. Fehndrich, A. Josef, L. Wilkens, J. Kleine-Borger, N. Bahlmann, M. Lohmeyer, P. Hertel, and H. Dotsch, “Experimental investigation of the nonreciprocal phase shift of a transverse electric mode in a magnetic-optic rib waveguide,” Appl. Phys. Lett. **74**, 2918–2920 (1999). [CrossRef]

*y*and the principal field as

*E*. We can reduce Eq.(20) to cater to the numerical simulation as Because most of the waveguides we are concerned with have high refractive index contrast, the discontinuity of material index can not be neglected, thus we have to use ∇ ·

_{x}**D**= 0 instead of ∇ ·

**E**= 0 in [23

23. M. Fehndrich, A. Josef, L. Wilkens, J. Kleine-Borger, N. Bahlmann, M. Lohmeyer, P. Hertel, and H. Dotsch, “Experimental investigation of the nonreciprocal phase shift of a transverse electric mode in a magnetic-optic rib waveguide,” Appl. Phys. Lett. **74**, 2918–2920 (1999). [CrossRef]

*H*as [22

_{y}22. H. Dotsch, N. Bahlmann, O. Zhuromskyy, M. Hammer, L. Wilkens, R. Gerhardt, P. Hertel, and A. F. Popkov, “Applications of magneto-optical waveguides in integrated optics: review,” J. Opt. Soc. Am. B **22**, 240–253 (2005). [CrossRef]

*H*is normalized to unity by

_{y}*ε*is favorable to produce the largest NRPS. For a 90°-rotated system, i.e., a system with the non-diagonal permittivity elements of

_{xz}*ε*, one just needs to interchange all the symbols

_{yz}*x*and

*y*in Eq.(22) and Eq.(32).

*γ*. By removing the ∫

*dy*in Eq.(32), the formula reduces to the one-dimensional case and we have

*γ*

^{2}with respect to small variations of the eigen-field [24]. For the transverse electric modes, we have the following relation With the aid of the H

*ö*lder inequality, one may get It is also well-known that the guided mode of any asymmetric waveguide has a cutoff condition for the case where

*γ*=

*n*

_{2}, where

*n*

_{2}is the higher refractive index of the cladding layers at the waveguide sides. Therefore, we have an upper limit of |Δ

*γ*| expressed as where

*ε*is the maximum dielectric constant in the waveguide system. This equation is very instructive to predict the greatest NRPS value one can obtain no matter how to optimize the waveguide structure. Also indicated is that larger NRPS may be expected in high-contrast MO waveguides.

_{m}## 5. Magneto-optical waveguides on SOI platform

_{3}N

_{4}, SiO

_{2}and Ce:YIG are assumed to be 3.477, 2.0, 1.444 and 2.22, respectively. The values of Faraday rotation of Ce:YIG films was reported experimentally to be as high as −4500°cm

^{−1}in the wavelength of interest, which corresponds to a non-diagonal element of 0.0086 in the permittivity tensor. For the comparison analysis with prior publications, we conservatively assume

*ε*= 0.005 in the whole text. Actually, all the calculated NRPSs can be scaled to

_{xz}*ε*for any realistic MO material.

_{xz}### 5.1. Multi-layered MO waveguides

15. H. Dotsch, P. Hertel, B. Luhrmann, S. Sure, H. P. Winkler, and M. Ye, “Applications of magnetic garnet films in integrated optics,” IEEE Trans. Magn. **28**, 2979–2984 (1992). [CrossRef]

3. R. L. Espinola, T. Izuhara, M. C. Tsai, R. M. Osgood Jr., and H. Dotsch, “Magneto-optical noreciprocal phase shift in garnet/silicon-on-insulator waveguides,” Opt. Lett. **29**, 941–943 (2004). [CrossRef] [PubMed]

*b*

_{01}and

*b*

_{21}as small quantities, one can relate Δ

*γ*to

*γ*

_{0}by where

*γ*

_{0}is the normalized propagation constant of the correspondent reciprocal waveguide with

*b*

_{01}=

*b*

_{21}= 0 in Eq. (29).

*γ*is equal to the larger refractive index

*n*

_{2}of the two claddings, which can be written as

_{2}/MO/SiO

_{2}structures because of the cancelation. For a symmetric three-layered structure, we have

*b*

_{01}= −

*a*

_{01}

*b*

_{12}to simplify Eq. (28) so that all the linear terms for

*γ*are missing in the nominator and denominator of Eq. (28). As a consequence, NRPS vanishes.

*γ*| <13.7rad/mm for the SiO

_{2}/MO/Air structure. These results agree well with many reasonable results reported in previous publications [3

3. R. L. Espinola, T. Izuhara, M. C. Tsai, R. M. Osgood Jr., and H. Dotsch, “Magneto-optical noreciprocal phase shift in garnet/silicon-on-insulator waveguides,” Opt. Lett. **29**, 941–943 (2004). [CrossRef] [PubMed]

10. Y. Shoji and T. Mizumoto, “Ultra-wideband design of waveguide magnetooptical isolator operating in 1.31*μm* and 1.55*μm* band,” Opt. Express **15**, 639–645 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-2-639. [CrossRef] [PubMed]

11. Y. Shoji and T. Mizumoto, “Wideband design of nonreciprocal phase shift magneto-optical isolators using phase adjustment in Mach-Zehnder interferometer,” Appl. Opt. **45**, 7144–7150 (2006). [CrossRef] [PubMed]

15. H. Dotsch, P. Hertel, B. Luhrmann, S. Sure, H. P. Winkler, and M. Ye, “Applications of magnetic garnet films in integrated optics,” IEEE Trans. Magn. **28**, 2979–2984 (1992). [CrossRef]

### 5.2. Gap-assisted MO waveguides

7. R. Y. Chen, G. M. Jiang, Y. L. Hao, J. Y. Yang, M. H. Wang, and X. Q. Jiang, “Enhancement of nonreciprocal phase shift by using nanoscale air gap,” Opt. Lett. **35**, 1335–1337 (2010). [CrossRef] [PubMed]

*ε*= 0.0086) in a 2-D Ce:YIG/Gap/Si/SiO

_{xz}_{2}waveguide. If this effect is efficient, the air gap can be readily replaced by another LRI material that can be controlled very accurately in realistic fabrication, e.g. SiO

_{2}film, to enhance the MO effect. However, our results show that

*NO*enhancement can be made by such a sided slot.

_{2}structure, which is a four-layered problem. Our method avoids the integration of a discontinuous function in perturbation theory and excludes uncertainty in 2-D cross-section simulation arising from integrals being taken at infinitesimal distances from lines of discontinuity [7

7. R. Y. Chen, G. M. Jiang, Y. L. Hao, J. Y. Yang, M. H. Wang, and X. Q. Jiang, “Enhancement of nonreciprocal phase shift by using nanoscale air gap,” Opt. Lett. **35**, 1335–1337 (2010). [CrossRef] [PubMed]

22. H. Dotsch, N. Bahlmann, O. Zhuromskyy, M. Hammer, L. Wilkens, R. Gerhardt, P. Hertel, and A. F. Popkov, “Applications of magneto-optical waveguides in integrated optics: review,” J. Opt. Soc. Am. B **22**, 240–253 (2005). [CrossRef]

*μm*to 1

*μm*, and from 1nm to 100nm, respectively. We can see that the NRPS decreases monotonically with increasing gap width, without any finite gap width specially good for enhancement.

*d*are shown in Fig. 3(b). For step-like dielectric waveguides, the double integral in Eq. (32) is converted into a sum of single integrals along material boundaries

*x*by [22

_{h}**22**, 240–253 (2005). [CrossRef]

*H*|

_{y}^{2}at the Ce:YIG/Gap boundary decreases as well. The NRPS value is optimal at

*d*= 0 corresponding to no LRI gap and decreases monotonically as

*d*decreases. Hence, the high

*E*in the LRI slot is just an illusion to enhance NRPS and the presence of the LRI gap does not result in NRPS enhancement.

_{x}### 5.3. Slotted waveguide with a compensation wall

*y*and −

*y*direction, respectively. As shown in Fig. 4, for

*x*= 0, the NRPS is zero with no MO material. Increasing the MO layer thickness will increase the the NRPS to a maximum value of 6rad/mm. The NRPSs will then decrease and stablize at a value of 5rad/mm for increasing the MO layer thickness. This tendency can be easily explained by the separation of the two silicon layers. If they are far away from each other, the slot effect is missing and the waveguide system is actually two isolated asymmetric waveguides, as shown in Fig. 5(a). The dotted line in Fig. 4 is for the realistic 2-D silicon/MO(+)/MO(−)/silicon waveguide with air cladding and oxide substrate. The NRPS monotonically increases with the increasing slot width instead of any enhancement due to slot effect and finally tend towards the same limit of the 1-D case due to separation of the two silicon rectangular waveguides, as shown in Fig 5(b–c). This result is totally opposite to that in [8

8. W. F. Zhang, J. W. Mu, W. P. Huang, and W. Zhao, “Enhancement of nonreciprocal phase shift by magneto-optical slot waveguide with a compensation wall,” Appl. Phys. Lett. **98**, 171109 (2011). [CrossRef]

*x*= 0) and the two sided MO/Si walls (

5. N. Bahlmann, M. Lohmeyer, H. Dotsch, and P. Hertel, “Finite-element analysis of nonreciprocal phase shift for TE modes in magnetooptic rib waveguides with a compensation wall,” IEEE J. Quantum Electron. **35**, 250–253 (1999). [CrossRef]

8. W. F. Zhang, J. W. Mu, W. P. Huang, and W. Zhao, “Enhancement of nonreciprocal phase shift by magneto-optical slot waveguide with a compensation wall,” Appl. Phys. Lett. **98**, 171109 (2011). [CrossRef]

5. N. Bahlmann, M. Lohmeyer, H. Dotsch, and P. Hertel, “Finite-element analysis of nonreciprocal phase shift for TE modes in magnetooptic rib waveguides with a compensation wall,” IEEE J. Quantum Electron. **35**, 250–253 (1999). [CrossRef]

### 5.4. NRPS in plasmonic-enhanced MO waveguides

25. P. Berini and I. D. Leon, “Surface plasmon-polariton amplifiers and lasers,” Nature Photon. **6**, 16–24 (2012). [CrossRef]

9. J. B. Khurgin, “Optical isolating action in surface plasmon polaritons,” Appl. Phys. Lett. **89**, 251115 (2006). [CrossRef]

*γ*and (7) short-range plasmon polariton with lower

*γ*. For the reason of symmetry, a compensation wall is set for the case (6–7) to obtain the largest NRPS.

27. I. Avrutsky, R. Soref, and W. Buchwald, “Sub-wavelength plasmonic modes in a conductor-gap-dielectric system with a nanoscale gap,” Opt. Express **18**, 348–363 (2009), http://www.opticsinfobase.org/abstract.cfm?URI=oe-18-1-348 [CrossRef]

*d*= 0 for the cases (2,4), the MO layer vanishes;

*L*goes to infinity and nonreciprocal effect is missing. Conversely, when

_{π}*d*= 0 for the cases (1,3) and

*d*→ ∞ for the cases (5–7), all the structures return to the simplest bilayer surface plasmon waveguide Cu/MO discussed in [9

9. J. B. Khurgin, “Optical isolating action in surface plasmon polaritons,” Appl. Phys. Lett. **89**, 251115 (2006). [CrossRef]

*ε*is metal permittivity. Under the first-order approximation,

_{m}*f*(

*γ*) ≈

*f*(

*γ*) +

_{spp}*f*′(

*γ*)(

_{spp}*γ*–

*γ*) and

_{spp}*μm*,

*ε*≈ −68 + 10

_{Cu}*i*(whereas

*ε*≈ −87 + 8.7

_{Ag}*i*); the

*L*is 618

_{π}*μm*(705

*μm*) and the

*L*

_{1dB}is 2.15

*μm*(4.07

*μm*), corresponding to an insertion loss of 288dB (173dB). In theory the push-pull scheme can reduce the insertion loss by half. For the Cu/MO(+)/MO(−)/Cu structure, the NRPS is very stable when

*d*> 20

*nm*and rapidly increase when the MO layer thickness goes down. This is because the field intensity contrast at

*x*= 0 and

*L*and

_{π}*L*

_{1dB}, which is critical to the feasibility of combining plasmonics and MO medium. The data in Fig.6 (a) and (b) are plotted differently in magnitude by 3 order. As calculated for the bilayer Cu/MO case, an insertion loss at the scale of 100dB is almost meaningless for practical use in that even a dielectric waveguide can achieve the comparable

*L*without any metallic loss. Just considering enhancement, the case (7) may be appealing because an

_{π}*L*value of 100

_{π}*μm*can be obtained when

*d*≈ 10

*nm*, which is very challenging for dielectric MO waveguide. However, the loss problem is very serious due to the relatively fast decrease of

*L*

_{1dB}. To conclude, the loss problem seriously constrains the use of plasmonics for enhancement of NRPS.

## 6. Conclusions

## Acknowledgments

## References and links

1. | Z. F. Yu and S. H. Fan, “Complete optical isolation created by indirect interband photonic transitions,” Nature Photon . |

2. | L. Feng, M. Ayache, J. Q. Huang, Y. L. Xu, M. H. Lu, Y. F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science |

3. | R. L. Espinola, T. Izuhara, M. C. Tsai, R. M. Osgood Jr., and H. Dotsch, “Magneto-optical noreciprocal phase shift in garnet/silicon-on-insulator waveguides,” Opt. Lett. |

4. | L. Bi, J. Hu, P. Jiang, D Kim, G. F. Dionne, L. C. Kimerling, and C. A. Ross, “On-chip optical isolation in monolithically integrated non-reicprocal optical resonators,” Nature Photon. |

5. | N. Bahlmann, M. Lohmeyer, H. Dotsch, and P. Hertel, “Finite-element analysis of nonreciprocal phase shift for TE modes in magnetooptic rib waveguides with a compensation wall,” IEEE J. Quantum Electron. |

6. | A. F. Popkov, “Nonreciprocal TE-mode phase shift by domain walls in magnetooptic rib waveguides,” Appl. Phys. Lett. |

7. | R. Y. Chen, G. M. Jiang, Y. L. Hao, J. Y. Yang, M. H. Wang, and X. Q. Jiang, “Enhancement of nonreciprocal phase shift by using nanoscale air gap,” Opt. Lett. |

8. | W. F. Zhang, J. W. Mu, W. P. Huang, and W. Zhao, “Enhancement of nonreciprocal phase shift by magneto-optical slot waveguide with a compensation wall,” Appl. Phys. Lett. |

9. | J. B. Khurgin, “Optical isolating action in surface plasmon polaritons,” Appl. Phys. Lett. |

10. | Y. Shoji and T. Mizumoto, “Ultra-wideband design of waveguide magnetooptical isolator operating in 1.31 |

11. | Y. Shoji and T. Mizumoto, “Wideband design of nonreciprocal phase shift magneto-optical isolators using phase adjustment in Mach-Zehnder interferometer,” Appl. Opt. |

12. | M. C. Tien, T. Mizumoto, P. Pintus, H. Kromer, and J. Bowers, “Silicon ring resonators with bonded nonreciprocal magneto-optic garnets,” Opt. Express |

13. | N. Kono, K. Kakihara, K. Saitoh, and M. Koshiba, “Nonreciprocal microresonators for the miniaturization of optical waveguide isolators,” Opt. Express |

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15. | H. Dotsch, P. Hertel, B. Luhrmann, S. Sure, H. P. Winkler, and M. Ye, “Applications of magnetic garnet films in integrated optics,” IEEE Trans. Magn. |

16. | J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman, “Planar metal plasmon waveguides: frequency-dependent dispersion, propagation, localization, and loss beyond the free electron model,” Phys. Rev. B |

17. | J. A. Nelder and R. Mead, “A simplex method for function minimization,” Comput. J. |

18. | E. Anemogiannis, E. N. Glytsis, and T. K. Gaylord, “Determination of guided and leaky modes in lossless and lossy planar multilayer optical waveguides: reflection pole method and wavevector density method,” J. Lightwave Technol. |

19. | Z. K. Wang, “An implementation of Kuhn’s rootfinding algorithm for polynomials and related discussion,” J. Numer. Methods Comput. Appl. |

20. | Y. L. Long, X. L. Wen, and C. F. Xie, “An implementation of a root-finding algorithm for transcendental functions in a complex plane,” J. Numer. Methods Comput. Appl. |

21. | S. Yamamoto and T. Makimoto, “Circuit theory for a class of anisotropic and gyrotropic thin-film optical waveguides and design of nonreciprocal devices for integrated optics,” J. Appl. Phys. |

22. | H. Dotsch, N. Bahlmann, O. Zhuromskyy, M. Hammer, L. Wilkens, R. Gerhardt, P. Hertel, and A. F. Popkov, “Applications of magneto-optical waveguides in integrated optics: review,” J. Opt. Soc. Am. B |

23. | M. Fehndrich, A. Josef, L. Wilkens, J. Kleine-Borger, N. Bahlmann, M. Lohmeyer, P. Hertel, and H. Dotsch, “Experimental investigation of the nonreciprocal phase shift of a transverse electric mode in a magnetic-optic rib waveguide,” Appl. Phys. Lett. |

24. | M. J. Adams, |

25. | P. Berini and I. D. Leon, “Surface plasmon-polariton amplifiers and lasers,” Nature Photon. |

26. | E. D. Palik, |

27. | I. Avrutsky, R. Soref, and W. Buchwald, “Sub-wavelength plasmonic modes in a conductor-gap-dielectric system with a nanoscale gap,” Opt. Express |

**OCIS Codes**

(130.2790) Integrated optics : Guided waves

(130.3120) Integrated optics : Integrated optics devices

(230.3810) Optical devices : Magneto-optic systems

(230.4170) Optical devices : Multilayers

(240.6680) Optics at surfaces : Surface plasmons

**ToC Category:**

Integrated Optics

**History**

Original Manuscript: January 4, 2012

Revised Manuscript: January 26, 2012

Manuscript Accepted: January 26, 2012

Published: March 26, 2012

**Citation**

Haifeng Zhou, Jingyee Chee, Junfeng Song, and Guoqiang Lo, "Analytical calculation of nonreciprocal phase shifts and comparison analysis of enhanced magneto-optical waveguides on SOI platform," Opt. Express **20**, 8256-8269 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-8-8256

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