Abstract

Measuring optical rotations in materials is a useful tool in many experimental studies. Research may be limited by the ability to measure small rotations due to weak interactions. We propose a novel scheme wherein we use a coupled waveguide and ring resonator to amplify the effects of optical rotation, potentially opening new avenues for investigation. Our proposed device can increase the resulting optical rotation by up to six orders of magnitude.

© 2015 Optical Society of America

1. Introduction

Measuring the rotation of linearly polarized light through a medium provides key data regarding the structure or substance of the material in question. In semiconductor devices, Kerr and Faraday rotation measurements can be used to measure the net spin polarization [1,2]. In chemistry, the isomer chirality ratio of a mixture can be determined from measurements of optical polarization rotation [3]. In materials with weak interactions, polarization rotations may be too small to detect with existing equipment, but can be enhanced in an optical cavity [4,5]. Here we consider the amplification of the rotation of linearly polarized light through a medium when the medium is incorporated in a ring resonator. Waveguides coupled to ring resonators have been fabricated and used for amplification [6], wavelength filtering [7], and lasing [8]. Polarization control in integrated waveguide devices could form the basis for isolators, circulators, modulators and switches, but the current achievable rotation using electrically-generated spin polarization is small [9]. We find that by carefully tuning the coupling of the resonator to a linear waveguide, orders of magnitude increases in the rotation can be achieved.

In Fig. 1, we show a general schematic of the device [7]. Linearly polarized light, a1, is injected into the waveguide. Inside the coupling region, we assume lossless coupling to the resonator, b2, and transmission to the waveguide output, b1. The light in the resonator passes through an interaction region where the polarization is rotated by some angle ϕ. The light decays due to material losses and may gain a phase offset due to the path length of the resonator. The rotated light enters the coupling region as a2, where it is out-coupled to b1 or transmitted through to b2. For a continuous wave input along a1, steady state conditions are reached between the various effects of resonator-waveguide coupling and transmission, optical rotation, decay, and phase offset. By tuning the device, the input signal, a1, can be mostly blocked from the b1 output. Measuring the polarization of light at b1 can lead to detecting much larger rotations than those of passing light once through the interaction region.

 

Fig. 1 Schematic of ring resonator with coupling and interaction regions.

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In this paper, we first consider a simple model of the device to give an intuitive understanding of how the large enhancements in rotation are produced. We then propose a model in which we explicitly study the rotation and coupling from one polarization of light to another. Finally, we consider the relevant device parameters and how the amplification of the rotation in the interaction region can be maximized.

2. Ring resonator with field injection

We consider the simple case in which we have lossless coupling between the waveguide and ring resonator for one polarization of light. Building upon previous work, we assume a unitary scattering matrix can describe the relationship between the fields entering and leaving the coupler [10,11]. Light is coupled from the waveguide to the resonator according to the coupling constant, κ, or transmitted through to the output by the transmission factor, t.

(b1b2)=(tκκt*)=(a1a2)
|t|2+|κ|2=1

As light travels around the ring resonator, three effects take place. Damping due to material loss and phase offset can be described by the parameters α and θ, respectively. The interaction region effectively injects some field, ainj. We model the relationship between a2 and b2 based on these effects.

a2=αeiθb2+ainj

From this equation and our coupling matrix, we determine the output field b1 as a function of the input fields and system parameters.

b1=tαeiθ1t*αeiθa1+κ1t*αeiθainj

Suppose we wish to measure ainj. We consider the conditions necessary to get maximal output at b1 due to ainj. For real coupling constants (κ, t) and resonance conditions (θ = 0) this occurs at critical coupling, α = t. When exactly on resonance, b1 due to a1 goes to zero, and the output of the device is linear with ainj.

b1,crit=ainjκ=ainj1t2

We find that on resonance the greatest output due to the injected field will occur for low coupling parameter κ, corresponding to high α. Thus, a high quality resonator at critical coupling will give the greatest output due to the injected field.

3. Field rotation amplifier

We consider one example of how we could build such an interaction region and exploit it to increase our sensitivity in measuring small rotation angles. We allow vectors a1, a2, b1, and b2 to have components in the x and y directions. Without loss of generality, we send a linearly polarized signal into the coupler with the electric field oriented along the x direction. After coupling to the resonator, the signal enters the interaction region where the material of interest rotates the light toward the y axis by some angle ϕ. We expand our previous coupling matrix to include each polarization and assume there is no change in polarization in the coupling region.

B=(b1xb2xb1yb2y)=(txκx00κx*tx*0000tyκy00κy*κy*)(a1xa2xa1ya2y)=CA

A and B represent the vectors for the input and output of the coupling region, respectively. The matrix C has constraints|tx|2+|κx|2=1 and |ty|2+|κy|2=1 to satisfy lossless coupling. For the interaction region, we have a rotation between the x and y polarizations. We also include the decay term, α and the phase offset, θ. To account for our source, we add a constant term A0.

A=(00000cosϕ0sinϕ00000sinϕ0cosϕ)(αx0000αx0000αy0000αy)eiθB+(a1x0a1y0)=RB+A0

This formulation allows us to analytically solve for the output vector, B, as a function of our input vector, A0, and our system parameters.

B=CA=C(RB+A0)B=(1CR)1CA0

For the case in which the input beam is polarized along the x direction, we set a1x = 1 and a1y = 0.

b1x=αxeiθcosϕty*αxαye2iθ+txty*αyeiθcosϕtxtx*αxeiθcosϕ+ty*αyeiθcosϕtx*ty*αxαye2iθ1
b1y=κx*κyαxeiθsinϕtx*αxeiθcosϕ+ty*αyeiθcosϕtx*ty*αxαye2iθ1

We consider the optimal tx and ty for comparison with the simple case presented before. Since we are now injecting field in the form of a rotation from one polarization to another, we can no longer eliminate b1x entirely. This is due to the fact that light polarized in the y direction that enters the interaction region will rotate to the x direction and then out-couple as b1x. The field rotated from the y to the x polarization is effectively ainj for the x polarization. As we wish to maximaize the polarization rotated from the x to the y direction, we find tx and ty that maximize(b1y/b1x)2.

tx,opt=αxtyαycosϕtyαycosϕ1αx(1+12tyαy+1tyαy1ϕ2)
ty,opt=αyαxtxcosϕαxcosϕtxαy(1+12αx+txαxtxϕ2)

For sufficiently small φ, the optimum values are near critical coupling. However, when tx≈αx, the approximation for ty,opt is no longer valid. However, as we will show later in the paper, device optimization depends more on tuning the coupling parameters for the x polarization than the y.

Near critical coupling, the output from the device would be polarized nearly entirely in the y direction. This would occur for nearly all small values of ϕ, as shown in Fig. 2. For any α, the device has a constant output polarization. The magnitude of b1y would change due to loss, and measuring angles of rotation has the advantage of being independent of the input beam intensity and thus less sensitive to intensity fluctuations. Therefore, our analysis presents the polarization angle of the output rather than the intensity.

 

Fig. 2 Device output polarization as a function of field rotation in the interaction region at critical coupling (t = α).

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Figure 2 shows the output polarization angle of b1 as measured from the x axis. For small ϕ, the output polarization is π/2, or 90°, corresponding to complete conversion from polarization along x to polarization along y. This is a desirable regime for polarization conversion, but it is not ideal if the application of this device is to measure the rotation in the interaction region. For larger ϕ, the rotation in the interaction region is sufficiently large to rotate some non-negligible field from the y polarization back into the original x polarization. The rotated field would be equivalent to an injected x polarization, which would be efficiently out coupled to the waveguide. This leads to a non-constant output polarization for large rotations in the interaction region.

4. Device optimization

An interesting phenomenon occurs when the system is off critical coupling. We define the amplification as the ratio of the polarization angle from the x axis of the output of the device to the polarization change in a single pass through the interaction region, ϕ. In Figs. 3(a) and 3(b), we show the amplification of the polarization for different combinations of parameters α and t. We find that for a large range of interaction region strengths, we have a constant amplification. Constant amplification is ideal for using this device to amplify small rotation signals.

 

Fig. 3 Amplification of the polarization rotation when (a) far off critical coupling and (b) close to critical coupling (δ = α -t).

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In Fig. 3(a), we show the amplification of a device far off critical coupling. A promising feature of this graph is that it demonstrates the constant amplification over many orders of magnitude of rotation. For example, for α = 0.9 and t = 0.85, the polarization amplification is 21.25 from ϕ = 10−2 to 10−6 rad. In Fig. 3(b), we consider the effects very close to critical coupling. For this graph, α = 0.9, although the results are similar for a wide range of α, as will be demonstrated later. Amplification strongly depends on the offset from critical coupling, δ = α-t.

Figure 4 shows the amplification as a function of critical coupling offset for ϕ = 10−6 rad. At low offset, the amplification converges on a constant value. As offset decreases we approach the critical coupling case. In the low δ regime, the b1x output goes to 0 leading to a constant output polarization of π/2 for all b1y. This leads to an amplification of 1.57 × 106, as shown in Fig. 4. Since the amplification is similar for vastly different values of α, we find the amplification is only weakly dependant on the quality of the resonator. Higher α does correspond to higher amplification, as expected. However, a much stronger effect is the offset from critical coupling, δ.

 

Fig. 4 Amplification as a function of offset from critical coupling for ϕ = 10−6 rad.

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In Fig. 5(a) we see the effects of δ to the off resonant behavior of the device. Here we rescale the polarization amplification relative to the amplification on resonance. As δ decreases, the width of the off-resonance curve decreases significantly. In Fig. 5(b), we plot the full width at half maximum of the relative amplification curve as a function of the critical coupling offset for different values of α. The slope of the curves varies from 1.00 when α = 0.2 to 0.94 when α = 0.999, showing an approximately linear relationship between the width of the amplification curve and the offset from critical coupling. The width of the amplification curve is also dependent on α, with lower α producing greater bandwidths. In practice, the device could be on resonance for one polarization direction while being off resonance for the other due to a polarization dependent refractive index. This would complicate the polarization amplification considered in this paper and is worthy of more investigation. For the proposed application of measuring optical rotation, however, it is possible to design a resonator to have the same resonance position for both polarizations.

 

Fig. 5 (a) Amplification when off resonance depending on offset from critical coupling for α = 0.999 and ϕ = 10−6 rad and (b) The width of the amplitude curve as a function of offset for different α

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Figures 6(a) and 6(b) shows the polarization amplification as a function of the resonator decay parameter, α. Except for the region of high loss (α < 0.05), the amplification varies weakly with α, in very good agreement with the earlier results shown in Fig. 4. In Fig. 6(b), we consider the low loss regime. Again, we see a slowly increasing amplification with α (decreasing with 1-α), but a strong dependence on the offset from critical coupling, δ.

 

Fig. 6 Polarization amplification as a function of (a) α and (b) 1-α for ϕ = 10−6 rad.

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In general, the coupling between the resonator and the waveguide may differ for the two polarizations [12]. Additionally, the decay parameters for the different polarizations can vary based on the device geometry. We now consider the effects of different coupling and decay parameters for each polarization. We first consider a device with a fixed coupling coefficient αy = 0.999 and δy = 10−3. In Fig. 7(a), we show the amplification as a function of the transmission coefficient for the x polarization, tx. The result matches previous studies and our earlier analysis showing that maximal amplification will occur at critical coupling for the x polarization, tx = αx. In Fig. 7(b), we hold the x parameters constant at αx = 0.999 and δx = 10−3. Maximal amplifications occur when ty = αy. We also find that for any ty≠1 we get some polarization amplification. This occurs because we have a field injected into the y polarization in the interaction region. Any ty≠1 will couple that injected field to the waveguide output, creating an output rotation.

 

Fig. 7 Amplification as a function of transmission coefficient for different combinations of αx and αy. Values for constant parameters are (a) αy = 0.999, δy = 10−3 and (b) αx = 0.999, δx = 10−3.

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It may be difficult to build a device in which the coupling constants for the different polarizations can be tuned independently, although this can be realized over a finite range using a tunable coupler based on Mach-Zender interferometry [12]. Based on the results shown in Figs. 7(a) and 7(b), tuning the coupling for the x polarization will have the greatest effect on the amplification.

5. Conclusions

We have demonstrated how integrating a device which rotates the polarization of light with a coupled waveguide and ring resonator can be used to amplify the polarization rotation in a specific interaction region. Amplification factors are strongly dependent on the ability to tune coupling parameters. By varying the device coupling parameters near critical coupling, amplification factors of 104-106 may be achieved. This could potentially allow for studying weak phenomena in materials that were previously unattainable.

Acknowledgments

This material is based upon work supported by the National Science Foundation under Grant No. ECCS-0844908, the Office of Naval Research, the Air Force Office of Scientific Research, and the Defense Threat Reduction Agency, Basic Research Award #HDTRA1-13-1-0013.

References and links

1. Y. K. Kato, R. C. Myers, A. C. Gossard, and D. D. Awschalom, “Observation of the spin hall effect in semiconductors,” Science 306(5703), 1910–1913 (2004). [CrossRef]   [PubMed]  

2. J. M. Kikkawa and D. D. Awschalom, “Resonant spin amplification in n-type GaAs,” Phys. Rev. Lett. 80(19), 4313–4316 (1998). [CrossRef]  

3. H. Kitzerow and C. Bahr, eds., Chirality in Liquid Crystals (Springer 2001).

4. Y. Q. Li, D. W. Steuerman, J. Berezovsky, D. S. Seferos, G. C. Bazan, and D. D. Awschalom, “Cavity enhanced Faraday rotation of semiconductor quantum dots,” Appl. Phys. Lett. 88(19), 193126 (2006). [CrossRef]  

5. R. Giri, S. Cronenberger, M. Vladimirova, D. Scalbert, K. V. Kavokin, M. M. Glazov, M. Nawrocki, A. Lemaître, and J. Bloch, “Giant photoinduced Faraday rotation due to the spin-polarized electron gas in an n-GaAs microcavity,” Phys. Rev. B 85(19), 195313 (2012). [CrossRef]  

6. J. M. Choi, R. K. Lee, and A. Yariv, “Control of critical coupling in a ring resonator-fiber configuration: application to wavelength-selective switching, modulation, amplification, and oscillation,” Opt. Lett. 26(16), 1236–1238 (2001). [CrossRef]   [PubMed]  

7. B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J.-P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15(6), 998–1005 (1997). [CrossRef]  

8. S. L. McCall, A. F. J. Levi, R. E. Slusher, S. J. Pearton, and R. A. Logan, “Wispering-gallery mode microdisk lasers,” Appl. Phys. Lett. 60(3), 289 (1992). [CrossRef]  

9. C. J. Trowbridge, B. M. Norman, J. Stephens, A. C. Gossard, D. D. Awschalom, and V. Sih, “Electron spin polarization-based integrated photonic devices,” Opt. Express 19(16), 14845–14851 (2011). [CrossRef]   [PubMed]  

10. A. Yariv, “Critical coupling and its ccntrol in optical waveguide-ring resonator systems,” IEEE Photon. Technol. Lett. 14(4), 483–485 (2002). [CrossRef]  

11. A. Yariv, “Universal relations for coupling of optical power between microresonators and dielectric waveguides,” Electron. Lett. 36(4), 321 (2000). [CrossRef]  

12. J. K. Doylend, O. Cohen, M. R. Lee, O. Raday, S. Xu, V. Sih, H. Rong, and M. Paniccia, “Tunable ring resonators for silicon Raman laser and amplifier applications,” Proc. SPIE 6896, 68960Q (2008). [CrossRef]  

References

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  1. Y. K. Kato, R. C. Myers, A. C. Gossard, and D. D. Awschalom, “Observation of the spin hall effect in semiconductors,” Science 306(5703), 1910–1913 (2004).
    [Crossref] [PubMed]
  2. J. M. Kikkawa and D. D. Awschalom, “Resonant spin amplification in n-type GaAs,” Phys. Rev. Lett. 80(19), 4313–4316 (1998).
    [Crossref]
  3. H. Kitzerow and C. Bahr, eds., Chirality in Liquid Crystals (Springer 2001).
  4. Y. Q. Li, D. W. Steuerman, J. Berezovsky, D. S. Seferos, G. C. Bazan, and D. D. Awschalom, “Cavity enhanced Faraday rotation of semiconductor quantum dots,” Appl. Phys. Lett. 88(19), 193126 (2006).
    [Crossref]
  5. R. Giri, S. Cronenberger, M. Vladimirova, D. Scalbert, K. V. Kavokin, M. M. Glazov, M. Nawrocki, A. Lemaître, and J. Bloch, “Giant photoinduced Faraday rotation due to the spin-polarized electron gas in an n-GaAs microcavity,” Phys. Rev. B 85(19), 195313 (2012).
    [Crossref]
  6. J. M. Choi, R. K. Lee, and A. Yariv, “Control of critical coupling in a ring resonator-fiber configuration: application to wavelength-selective switching, modulation, amplification, and oscillation,” Opt. Lett. 26(16), 1236–1238 (2001).
    [Crossref] [PubMed]
  7. B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J.-P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15(6), 998–1005 (1997).
    [Crossref]
  8. S. L. McCall, A. F. J. Levi, R. E. Slusher, S. J. Pearton, and R. A. Logan, “Wispering-gallery mode microdisk lasers,” Appl. Phys. Lett. 60(3), 289 (1992).
    [Crossref]
  9. C. J. Trowbridge, B. M. Norman, J. Stephens, A. C. Gossard, D. D. Awschalom, and V. Sih, “Electron spin polarization-based integrated photonic devices,” Opt. Express 19(16), 14845–14851 (2011).
    [Crossref] [PubMed]
  10. A. Yariv, “Critical coupling and its ccntrol in optical waveguide-ring resonator systems,” IEEE Photon. Technol. Lett. 14(4), 483–485 (2002).
    [Crossref]
  11. A. Yariv, “Universal relations for coupling of optical power between microresonators and dielectric waveguides,” Electron. Lett. 36(4), 321 (2000).
    [Crossref]
  12. J. K. Doylend, O. Cohen, M. R. Lee, O. Raday, S. Xu, V. Sih, H. Rong, and M. Paniccia, “Tunable ring resonators for silicon Raman laser and amplifier applications,” Proc. SPIE 6896, 68960Q (2008).
    [Crossref]

2012 (1)

R. Giri, S. Cronenberger, M. Vladimirova, D. Scalbert, K. V. Kavokin, M. M. Glazov, M. Nawrocki, A. Lemaître, and J. Bloch, “Giant photoinduced Faraday rotation due to the spin-polarized electron gas in an n-GaAs microcavity,” Phys. Rev. B 85(19), 195313 (2012).
[Crossref]

2011 (1)

2008 (1)

J. K. Doylend, O. Cohen, M. R. Lee, O. Raday, S. Xu, V. Sih, H. Rong, and M. Paniccia, “Tunable ring resonators for silicon Raman laser and amplifier applications,” Proc. SPIE 6896, 68960Q (2008).
[Crossref]

2006 (1)

Y. Q. Li, D. W. Steuerman, J. Berezovsky, D. S. Seferos, G. C. Bazan, and D. D. Awschalom, “Cavity enhanced Faraday rotation of semiconductor quantum dots,” Appl. Phys. Lett. 88(19), 193126 (2006).
[Crossref]

2004 (1)

Y. K. Kato, R. C. Myers, A. C. Gossard, and D. D. Awschalom, “Observation of the spin hall effect in semiconductors,” Science 306(5703), 1910–1913 (2004).
[Crossref] [PubMed]

2002 (1)

A. Yariv, “Critical coupling and its ccntrol in optical waveguide-ring resonator systems,” IEEE Photon. Technol. Lett. 14(4), 483–485 (2002).
[Crossref]

2001 (1)

2000 (1)

A. Yariv, “Universal relations for coupling of optical power between microresonators and dielectric waveguides,” Electron. Lett. 36(4), 321 (2000).
[Crossref]

1998 (1)

J. M. Kikkawa and D. D. Awschalom, “Resonant spin amplification in n-type GaAs,” Phys. Rev. Lett. 80(19), 4313–4316 (1998).
[Crossref]

1997 (1)

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J.-P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15(6), 998–1005 (1997).
[Crossref]

1992 (1)

S. L. McCall, A. F. J. Levi, R. E. Slusher, S. J. Pearton, and R. A. Logan, “Wispering-gallery mode microdisk lasers,” Appl. Phys. Lett. 60(3), 289 (1992).
[Crossref]

Awschalom, D. D.

C. J. Trowbridge, B. M. Norman, J. Stephens, A. C. Gossard, D. D. Awschalom, and V. Sih, “Electron spin polarization-based integrated photonic devices,” Opt. Express 19(16), 14845–14851 (2011).
[Crossref] [PubMed]

Y. Q. Li, D. W. Steuerman, J. Berezovsky, D. S. Seferos, G. C. Bazan, and D. D. Awschalom, “Cavity enhanced Faraday rotation of semiconductor quantum dots,” Appl. Phys. Lett. 88(19), 193126 (2006).
[Crossref]

Y. K. Kato, R. C. Myers, A. C. Gossard, and D. D. Awschalom, “Observation of the spin hall effect in semiconductors,” Science 306(5703), 1910–1913 (2004).
[Crossref] [PubMed]

J. M. Kikkawa and D. D. Awschalom, “Resonant spin amplification in n-type GaAs,” Phys. Rev. Lett. 80(19), 4313–4316 (1998).
[Crossref]

Bazan, G. C.

Y. Q. Li, D. W. Steuerman, J. Berezovsky, D. S. Seferos, G. C. Bazan, and D. D. Awschalom, “Cavity enhanced Faraday rotation of semiconductor quantum dots,” Appl. Phys. Lett. 88(19), 193126 (2006).
[Crossref]

Berezovsky, J.

Y. Q. Li, D. W. Steuerman, J. Berezovsky, D. S. Seferos, G. C. Bazan, and D. D. Awschalom, “Cavity enhanced Faraday rotation of semiconductor quantum dots,” Appl. Phys. Lett. 88(19), 193126 (2006).
[Crossref]

Bloch, J.

R. Giri, S. Cronenberger, M. Vladimirova, D. Scalbert, K. V. Kavokin, M. M. Glazov, M. Nawrocki, A. Lemaître, and J. Bloch, “Giant photoinduced Faraday rotation due to the spin-polarized electron gas in an n-GaAs microcavity,” Phys. Rev. B 85(19), 195313 (2012).
[Crossref]

Choi, J. M.

Chu, S. T.

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J.-P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15(6), 998–1005 (1997).
[Crossref]

Cohen, O.

J. K. Doylend, O. Cohen, M. R. Lee, O. Raday, S. Xu, V. Sih, H. Rong, and M. Paniccia, “Tunable ring resonators for silicon Raman laser and amplifier applications,” Proc. SPIE 6896, 68960Q (2008).
[Crossref]

Cronenberger, S.

R. Giri, S. Cronenberger, M. Vladimirova, D. Scalbert, K. V. Kavokin, M. M. Glazov, M. Nawrocki, A. Lemaître, and J. Bloch, “Giant photoinduced Faraday rotation due to the spin-polarized electron gas in an n-GaAs microcavity,” Phys. Rev. B 85(19), 195313 (2012).
[Crossref]

Doylend, J. K.

J. K. Doylend, O. Cohen, M. R. Lee, O. Raday, S. Xu, V. Sih, H. Rong, and M. Paniccia, “Tunable ring resonators for silicon Raman laser and amplifier applications,” Proc. SPIE 6896, 68960Q (2008).
[Crossref]

Foresi, J.

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J.-P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15(6), 998–1005 (1997).
[Crossref]

Giri, R.

R. Giri, S. Cronenberger, M. Vladimirova, D. Scalbert, K. V. Kavokin, M. M. Glazov, M. Nawrocki, A. Lemaître, and J. Bloch, “Giant photoinduced Faraday rotation due to the spin-polarized electron gas in an n-GaAs microcavity,” Phys. Rev. B 85(19), 195313 (2012).
[Crossref]

Glazov, M. M.

R. Giri, S. Cronenberger, M. Vladimirova, D. Scalbert, K. V. Kavokin, M. M. Glazov, M. Nawrocki, A. Lemaître, and J. Bloch, “Giant photoinduced Faraday rotation due to the spin-polarized electron gas in an n-GaAs microcavity,” Phys. Rev. B 85(19), 195313 (2012).
[Crossref]

Gossard, A. C.

C. J. Trowbridge, B. M. Norman, J. Stephens, A. C. Gossard, D. D. Awschalom, and V. Sih, “Electron spin polarization-based integrated photonic devices,” Opt. Express 19(16), 14845–14851 (2011).
[Crossref] [PubMed]

Y. K. Kato, R. C. Myers, A. C. Gossard, and D. D. Awschalom, “Observation of the spin hall effect in semiconductors,” Science 306(5703), 1910–1913 (2004).
[Crossref] [PubMed]

Haus, H. A.

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J.-P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15(6), 998–1005 (1997).
[Crossref]

Kato, Y. K.

Y. K. Kato, R. C. Myers, A. C. Gossard, and D. D. Awschalom, “Observation of the spin hall effect in semiconductors,” Science 306(5703), 1910–1913 (2004).
[Crossref] [PubMed]

Kavokin, K. V.

R. Giri, S. Cronenberger, M. Vladimirova, D. Scalbert, K. V. Kavokin, M. M. Glazov, M. Nawrocki, A. Lemaître, and J. Bloch, “Giant photoinduced Faraday rotation due to the spin-polarized electron gas in an n-GaAs microcavity,” Phys. Rev. B 85(19), 195313 (2012).
[Crossref]

Kikkawa, J. M.

J. M. Kikkawa and D. D. Awschalom, “Resonant spin amplification in n-type GaAs,” Phys. Rev. Lett. 80(19), 4313–4316 (1998).
[Crossref]

Laine, J.-P.

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J.-P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15(6), 998–1005 (1997).
[Crossref]

Lee, M. R.

J. K. Doylend, O. Cohen, M. R. Lee, O. Raday, S. Xu, V. Sih, H. Rong, and M. Paniccia, “Tunable ring resonators for silicon Raman laser and amplifier applications,” Proc. SPIE 6896, 68960Q (2008).
[Crossref]

Lee, R. K.

Lemaître, A.

R. Giri, S. Cronenberger, M. Vladimirova, D. Scalbert, K. V. Kavokin, M. M. Glazov, M. Nawrocki, A. Lemaître, and J. Bloch, “Giant photoinduced Faraday rotation due to the spin-polarized electron gas in an n-GaAs microcavity,” Phys. Rev. B 85(19), 195313 (2012).
[Crossref]

Levi, A. F. J.

S. L. McCall, A. F. J. Levi, R. E. Slusher, S. J. Pearton, and R. A. Logan, “Wispering-gallery mode microdisk lasers,” Appl. Phys. Lett. 60(3), 289 (1992).
[Crossref]

Li, Y. Q.

Y. Q. Li, D. W. Steuerman, J. Berezovsky, D. S. Seferos, G. C. Bazan, and D. D. Awschalom, “Cavity enhanced Faraday rotation of semiconductor quantum dots,” Appl. Phys. Lett. 88(19), 193126 (2006).
[Crossref]

Little, B. E.

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J.-P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15(6), 998–1005 (1997).
[Crossref]

Logan, R. A.

S. L. McCall, A. F. J. Levi, R. E. Slusher, S. J. Pearton, and R. A. Logan, “Wispering-gallery mode microdisk lasers,” Appl. Phys. Lett. 60(3), 289 (1992).
[Crossref]

McCall, S. L.

S. L. McCall, A. F. J. Levi, R. E. Slusher, S. J. Pearton, and R. A. Logan, “Wispering-gallery mode microdisk lasers,” Appl. Phys. Lett. 60(3), 289 (1992).
[Crossref]

Myers, R. C.

Y. K. Kato, R. C. Myers, A. C. Gossard, and D. D. Awschalom, “Observation of the spin hall effect in semiconductors,” Science 306(5703), 1910–1913 (2004).
[Crossref] [PubMed]

Nawrocki, M.

R. Giri, S. Cronenberger, M. Vladimirova, D. Scalbert, K. V. Kavokin, M. M. Glazov, M. Nawrocki, A. Lemaître, and J. Bloch, “Giant photoinduced Faraday rotation due to the spin-polarized electron gas in an n-GaAs microcavity,” Phys. Rev. B 85(19), 195313 (2012).
[Crossref]

Norman, B. M.

Paniccia, M.

J. K. Doylend, O. Cohen, M. R. Lee, O. Raday, S. Xu, V. Sih, H. Rong, and M. Paniccia, “Tunable ring resonators for silicon Raman laser and amplifier applications,” Proc. SPIE 6896, 68960Q (2008).
[Crossref]

Pearton, S. J.

S. L. McCall, A. F. J. Levi, R. E. Slusher, S. J. Pearton, and R. A. Logan, “Wispering-gallery mode microdisk lasers,” Appl. Phys. Lett. 60(3), 289 (1992).
[Crossref]

Raday, O.

J. K. Doylend, O. Cohen, M. R. Lee, O. Raday, S. Xu, V. Sih, H. Rong, and M. Paniccia, “Tunable ring resonators for silicon Raman laser and amplifier applications,” Proc. SPIE 6896, 68960Q (2008).
[Crossref]

Rong, H.

J. K. Doylend, O. Cohen, M. R. Lee, O. Raday, S. Xu, V. Sih, H. Rong, and M. Paniccia, “Tunable ring resonators for silicon Raman laser and amplifier applications,” Proc. SPIE 6896, 68960Q (2008).
[Crossref]

Scalbert, D.

R. Giri, S. Cronenberger, M. Vladimirova, D. Scalbert, K. V. Kavokin, M. M. Glazov, M. Nawrocki, A. Lemaître, and J. Bloch, “Giant photoinduced Faraday rotation due to the spin-polarized electron gas in an n-GaAs microcavity,” Phys. Rev. B 85(19), 195313 (2012).
[Crossref]

Seferos, D. S.

Y. Q. Li, D. W. Steuerman, J. Berezovsky, D. S. Seferos, G. C. Bazan, and D. D. Awschalom, “Cavity enhanced Faraday rotation of semiconductor quantum dots,” Appl. Phys. Lett. 88(19), 193126 (2006).
[Crossref]

Sih, V.

C. J. Trowbridge, B. M. Norman, J. Stephens, A. C. Gossard, D. D. Awschalom, and V. Sih, “Electron spin polarization-based integrated photonic devices,” Opt. Express 19(16), 14845–14851 (2011).
[Crossref] [PubMed]

J. K. Doylend, O. Cohen, M. R. Lee, O. Raday, S. Xu, V. Sih, H. Rong, and M. Paniccia, “Tunable ring resonators for silicon Raman laser and amplifier applications,” Proc. SPIE 6896, 68960Q (2008).
[Crossref]

Slusher, R. E.

S. L. McCall, A. F. J. Levi, R. E. Slusher, S. J. Pearton, and R. A. Logan, “Wispering-gallery mode microdisk lasers,” Appl. Phys. Lett. 60(3), 289 (1992).
[Crossref]

Stephens, J.

Steuerman, D. W.

Y. Q. Li, D. W. Steuerman, J. Berezovsky, D. S. Seferos, G. C. Bazan, and D. D. Awschalom, “Cavity enhanced Faraday rotation of semiconductor quantum dots,” Appl. Phys. Lett. 88(19), 193126 (2006).
[Crossref]

Trowbridge, C. J.

Vladimirova, M.

R. Giri, S. Cronenberger, M. Vladimirova, D. Scalbert, K. V. Kavokin, M. M. Glazov, M. Nawrocki, A. Lemaître, and J. Bloch, “Giant photoinduced Faraday rotation due to the spin-polarized electron gas in an n-GaAs microcavity,” Phys. Rev. B 85(19), 195313 (2012).
[Crossref]

Xu, S.

J. K. Doylend, O. Cohen, M. R. Lee, O. Raday, S. Xu, V. Sih, H. Rong, and M. Paniccia, “Tunable ring resonators for silicon Raman laser and amplifier applications,” Proc. SPIE 6896, 68960Q (2008).
[Crossref]

Yariv, A.

A. Yariv, “Critical coupling and its ccntrol in optical waveguide-ring resonator systems,” IEEE Photon. Technol. Lett. 14(4), 483–485 (2002).
[Crossref]

J. M. Choi, R. K. Lee, and A. Yariv, “Control of critical coupling in a ring resonator-fiber configuration: application to wavelength-selective switching, modulation, amplification, and oscillation,” Opt. Lett. 26(16), 1236–1238 (2001).
[Crossref] [PubMed]

A. Yariv, “Universal relations for coupling of optical power between microresonators and dielectric waveguides,” Electron. Lett. 36(4), 321 (2000).
[Crossref]

Appl. Phys. Lett. (2)

Y. Q. Li, D. W. Steuerman, J. Berezovsky, D. S. Seferos, G. C. Bazan, and D. D. Awschalom, “Cavity enhanced Faraday rotation of semiconductor quantum dots,” Appl. Phys. Lett. 88(19), 193126 (2006).
[Crossref]

S. L. McCall, A. F. J. Levi, R. E. Slusher, S. J. Pearton, and R. A. Logan, “Wispering-gallery mode microdisk lasers,” Appl. Phys. Lett. 60(3), 289 (1992).
[Crossref]

Electron. Lett. (1)

A. Yariv, “Universal relations for coupling of optical power between microresonators and dielectric waveguides,” Electron. Lett. 36(4), 321 (2000).
[Crossref]

IEEE Photon. Technol. Lett. (1)

A. Yariv, “Critical coupling and its ccntrol in optical waveguide-ring resonator systems,” IEEE Photon. Technol. Lett. 14(4), 483–485 (2002).
[Crossref]

J. Lightwave Technol. (1)

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J.-P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15(6), 998–1005 (1997).
[Crossref]

Opt. Express (1)

Opt. Lett. (1)

Phys. Rev. B (1)

R. Giri, S. Cronenberger, M. Vladimirova, D. Scalbert, K. V. Kavokin, M. M. Glazov, M. Nawrocki, A. Lemaître, and J. Bloch, “Giant photoinduced Faraday rotation due to the spin-polarized electron gas in an n-GaAs microcavity,” Phys. Rev. B 85(19), 195313 (2012).
[Crossref]

Phys. Rev. Lett. (1)

J. M. Kikkawa and D. D. Awschalom, “Resonant spin amplification in n-type GaAs,” Phys. Rev. Lett. 80(19), 4313–4316 (1998).
[Crossref]

Proc. SPIE (1)

J. K. Doylend, O. Cohen, M. R. Lee, O. Raday, S. Xu, V. Sih, H. Rong, and M. Paniccia, “Tunable ring resonators for silicon Raman laser and amplifier applications,” Proc. SPIE 6896, 68960Q (2008).
[Crossref]

Science (1)

Y. K. Kato, R. C. Myers, A. C. Gossard, and D. D. Awschalom, “Observation of the spin hall effect in semiconductors,” Science 306(5703), 1910–1913 (2004).
[Crossref] [PubMed]

Other (1)

H. Kitzerow and C. Bahr, eds., Chirality in Liquid Crystals (Springer 2001).

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Figures (7)

Fig. 1
Fig. 1 Schematic of ring resonator with coupling and interaction regions.
Fig. 2
Fig. 2 Device output polarization as a function of field rotation in the interaction region at critical coupling (t = α).
Fig. 3
Fig. 3 Amplification of the polarization rotation when (a) far off critical coupling and (b) close to critical coupling (δ = α -t).
Fig. 4
Fig. 4 Amplification as a function of offset from critical coupling for ϕ = 10−6 rad.
Fig. 5
Fig. 5 (a) Amplification when off resonance depending on offset from critical coupling for α = 0.999 and ϕ = 10−6 rad and (b) The width of the amplitude curve as a function of offset for different α
Fig. 6
Fig. 6 Polarization amplification as a function of (a) α and (b) 1-α for ϕ = 10−6 rad.
Fig. 7
Fig. 7 Amplification as a function of transmission coefficient for different combinations of αx and αy. Values for constant parameters are (a) αy = 0.999, δy = 10−3 and (b) αx = 0.999, δx = 10−3.

Equations (12)

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( b 1 b 2 )=( t κ κ t * )=( a 1 a 2 )
| t | 2 + | κ | 2 =1
a 2 =α e iθ b 2 + a inj
b 1 = tα e iθ 1 t * α e iθ a 1 + κ 1 t * α e iθ a inj
b 1,crit = a inj κ = a inj 1 t 2
B=( b 1x b 2x b 1y b 2y )=( t x κ x 0 0 κ x * t x * 0 0 0 0 t y κ y 0 0 κ y * κ y * )( a 1x a 2x a 1y a 2y )=CA
A=( 0 0 0 0 0 cosϕ 0 sinϕ 0 0 0 0 0 sinϕ 0 cosϕ )( α x 0 0 0 0 α x 0 0 0 0 α y 0 0 0 0 α y ) e iθ B+( a 1x 0 a 1y 0 )=RB+ A 0
B=CA=C( RB+ A 0 ) B= ( 1CR ) 1 C A 0
b 1x = α x e iθ cosϕ t y * α x α y e 2iθ + t x t y * α y e iθ cosϕ t x t x * α x e iθ cosϕ+ t y * α y e iθ cosϕ t x * t y * α x α y e 2iθ 1
b 1y = κ x * κ y α x e iθ sinϕ t x * α x e iθ cosϕ+ t y * α y e iθ cosϕ t x * t y * α x α y e 2iθ 1
t x,opt = α x t y α y cosϕ t y α y cosϕ1 α x ( 1+ 1 2 t y α y +1 t y α y 1 ϕ 2 )
t y,opt = α y α x t x cosϕ α x cosϕ t x α y ( 1+ 1 2 α x + t x α x t x ϕ 2 )

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