Abstract

Slot waveguides allow joint confinement of the driving electrical radio frequency field and of the optical waveguide mode in a narrow slot, allowing for highly efficient polymer based interferometers. We show that the optical confinement can be simply explained by a perturbation theoretical approach taking into account the continuity of the electric displacement field. We design phase matched transmission lines and show that their impedance and RF losses can be modeled by an equivalent circuit and linked to slot waveguide properties by a simple set of equations, thus allowing optimization of the device without iterative simulations. We optimize the interferometers for analog optical links and predict record performance metrics (Vπ = 200 mV @ 10 GHz in push-pull configuration) assuming a modest second order nonlinear coefficient (r33 = 50 pm/V) and slot width (100 nm). Using high performance optical polymers (r33 = 150 pm/V), noise figures of state of the art analog optical links can be matched while reducing optical power levels by approximately 30 times. With required optical laser power levels predicted at 50 mW, this could be a game changing improvement by bringing high performance optical analog link power requirements in the reach of laser diodes. A modified transmitter architecture allows shot noise limited performance, while reducing power levels in the slot waveguides and enhancing reliability.

© 2010 OSA

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2010

D. Jin, H. Chen, A. Barklund, J. Mallari, G. Yu, E. Miller, and R. Dinu, “EO polymer modulators reliability study,” Proc. SPIE 7599, 75990H (2010).
[CrossRef]

2009

2008

Z. Shi, J. Luo, S. Huang, X.-H. Zhou, T.-D. Kim, Y.-J. Cheng, B. M. Polishak, T. R. Younkin, B. A. Block, and A. K.-Y. Jen, “Reinforced Site Isolation Leading to Remarkable Thermal Stability and High Electrooptic Activities in Cross-Linked Nonlinear Optical Dendrimers,” Chem. Mater. 20(20), 6372–6377 (2008).
[CrossRef]

T. Baehr-Jones, B. Penkov, J. Huang, P. Sullivan, J. Davies, J. Takayesu, J. Luo, T.-D. Kim, L. Dalton, A. Jen, M. Hochberg, and A. Scherer, “Nonlinear polymer-clad silicon slot waveguide modulator with a half wave voltage of 0.25 V,” Appl. Phys. Lett. 92(16), 163303 (2008).
[CrossRef]

E. I. Ackerman, W. K. Burns, G. E. Betts, J. X. Chen, J. L. Prince, M. D. Regan, H. V. Roussell, and C. H. Cox, “RF-Over-Fiber Links With Very Low Noise Figure,” J. Light. Tech. 26(15), 2441–2448 (2008).
[CrossRef]

2007

J. D. Luo, S. Huang, Y. J. Cheng, T. D. Kim, Z. W. Shi, X. H. Zhou, and A. K.-Y. Jen, “Phenyltetraene-based nonlinear optical chromophores with enhanced chemical stability and electrooptic activity,” Org. Lett. 9(22), 4471–4474 (2007).
[CrossRef] [PubMed]

F. Lucchi, D. Janner, M. Belmonte, S. Balsamo, M. Villa, S. Giurgiola, P. Vergani, and V. Pruneri, “Very low voltage single drive domain inverted LiNbO(3) integrated electro-optic modulator,” Opt. Express 15(17), 10739–10743 (2007).
[CrossRef] [PubMed]

2006

T. E. Darcie and P. F. Driessen, “Class-AB Techniques for High-Dynamic-Range Microwave-Photonic Links,” IEEE Photon. Technol. Lett. 18(8), 929–931 (2006).
[CrossRef]

2005

Q. Xu, B. Schmidt, S. Pradhan, and M. Lipson, “Micrometre-scale silicon electro-optic modulator,” Nature 435(7040), 325–327 (2005).
[CrossRef] [PubMed]

2004

A. Liu, R. Jones, L. Liao, D. Samara-Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, “A high-speed silicon optical modulator based on a metal-oxide-semiconductor capacitor,” Nature 427(6975), 615–618 (2004).
[CrossRef] [PubMed]

V. R. Almeida, Q. Xu, C. A. Barrios, and M. Lipson, “Guiding and confining light in void nanostructure,” Opt. Lett. 29(11), 1209–1211 (2004).
[CrossRef] [PubMed]

2003

J. H. Sinsky, A. Adamiecki, C. A. Burrus, S. Chandrasekhar, J. Leuthold, and O. Wohlgemuth, “A 40-Gb/s Integrated Balanced Optical Front End and RZ-DPSK Performance,” IEEE Photon. Technol. Lett. 15(8), 1135–1137 (2003).
[CrossRef]

2000

Y. Shi, C. Zhang, H. Zhang, J. H. Bechtel, L. R. Dalton, B. H. Robinson, and W. H. Steier, “Low (Sub-1-volt) halfwave voltage polymeric electro-optic modulators achieved by controlling chromophore shape,” Science 288(5463), 119–122 (2000).
[CrossRef] [PubMed]

1999

1998

R. Taylor and S. R. Forrest, “Steering of an optically driven true-time delay phased-array antenna based on a broad-band coherent WDM architecture,” IEEE Photon. Technol. Lett. 10(1), 144–146 (1998).
[CrossRef]

1997

L. T. Nichols, K. J. Williams, and R. D. Esman, “Optimizing the Ultrawide-Band Photonic Link,” IEEE Trans. Microw. Theory Tech. 45(8), 1384–1389 (1997).
[CrossRef]

C. Cox, E. Ackerman, R. Helkey, and G. E. Betts, “Techniques and Performance of Intensity-Modulation Direct-Detection Analog Optical Links,” IEEE Trans. Microw. Theory Tech. 45(8), 1375–1383 (1997).
[CrossRef]

1987

R. A. Soref and B. R. Bennett, “Electrooptical Effects in Silicon,” IEEE J. Quantum Electron. 23(1), 123–129 (1987).
[CrossRef]

Ackerman, E.

C. Cox, E. Ackerman, R. Helkey, and G. E. Betts, “Techniques and Performance of Intensity-Modulation Direct-Detection Analog Optical Links,” IEEE Trans. Microw. Theory Tech. 45(8), 1375–1383 (1997).
[CrossRef]

Ackerman, E. I.

E. I. Ackerman, W. K. Burns, G. E. Betts, J. X. Chen, J. L. Prince, M. D. Regan, H. V. Roussell, and C. H. Cox, “RF-Over-Fiber Links With Very Low Noise Figure,” J. Light. Tech. 26(15), 2441–2448 (2008).
[CrossRef]

Adamiecki, A.

J. H. Sinsky, A. Adamiecki, C. A. Burrus, S. Chandrasekhar, J. Leuthold, and O. Wohlgemuth, “A 40-Gb/s Integrated Balanced Optical Front End and RZ-DPSK Performance,” IEEE Photon. Technol. Lett. 15(8), 1135–1137 (2003).
[CrossRef]

Almeida, V. R.

Asghari, M.

Baehr-Jones, T.

T. Baehr-Jones, B. Penkov, J. Huang, P. Sullivan, J. Davies, J. Takayesu, J. Luo, T.-D. Kim, L. Dalton, A. Jen, M. Hochberg, and A. Scherer, “Nonlinear polymer-clad silicon slot waveguide modulator with a half wave voltage of 0.25 V,” Appl. Phys. Lett. 92(16), 163303 (2008).
[CrossRef]

Balsamo, S.

Barklund, A.

D. Jin, H. Chen, A. Barklund, J. Mallari, G. Yu, E. Miller, and R. Dinu, “EO polymer modulators reliability study,” Proc. SPIE 7599, 75990H (2010).
[CrossRef]

Barrios, C. A.

Bechtel, J. H.

Y. Shi, C. Zhang, H. Zhang, J. H. Bechtel, L. R. Dalton, B. H. Robinson, and W. H. Steier, “Low (Sub-1-volt) halfwave voltage polymeric electro-optic modulators achieved by controlling chromophore shape,” Science 288(5463), 119–122 (2000).
[CrossRef] [PubMed]

Belmonte, M.

Bennett, B. R.

R. A. Soref and B. R. Bennett, “Electrooptical Effects in Silicon,” IEEE J. Quantum Electron. 23(1), 123–129 (1987).
[CrossRef]

Betts, G. E.

E. I. Ackerman, W. K. Burns, G. E. Betts, J. X. Chen, J. L. Prince, M. D. Regan, H. V. Roussell, and C. H. Cox, “RF-Over-Fiber Links With Very Low Noise Figure,” J. Light. Tech. 26(15), 2441–2448 (2008).
[CrossRef]

C. Cox, E. Ackerman, R. Helkey, and G. E. Betts, “Techniques and Performance of Intensity-Modulation Direct-Detection Analog Optical Links,” IEEE Trans. Microw. Theory Tech. 45(8), 1375–1383 (1997).
[CrossRef]

Block, B. A.

Z. Shi, J. Luo, S. Huang, X.-H. Zhou, T.-D. Kim, Y.-J. Cheng, B. M. Polishak, T. R. Younkin, B. A. Block, and A. K.-Y. Jen, “Reinforced Site Isolation Leading to Remarkable Thermal Stability and High Electrooptic Activities in Cross-Linked Nonlinear Optical Dendrimers,” Chem. Mater. 20(20), 6372–6377 (2008).
[CrossRef]

Burns, W. K.

E. I. Ackerman, W. K. Burns, G. E. Betts, J. X. Chen, J. L. Prince, M. D. Regan, H. V. Roussell, and C. H. Cox, “RF-Over-Fiber Links With Very Low Noise Figure,” J. Light. Tech. 26(15), 2441–2448 (2008).
[CrossRef]

W. K. Burns, M. M. Howerton, R. P. Moeller, R. W. McElhanon, and A. S. Greenblatt, “Low Drive Voltage, Broad-Band LiNbO3 Modulators With and Without Etched Ridges,” J. Lightwave Technol. 17(12), 2551–2555 (1999).
[CrossRef]

Burrus, C. A.

J. H. Sinsky, A. Adamiecki, C. A. Burrus, S. Chandrasekhar, J. Leuthold, and O. Wohlgemuth, “A 40-Gb/s Integrated Balanced Optical Front End and RZ-DPSK Performance,” IEEE Photon. Technol. Lett. 15(8), 1135–1137 (2003).
[CrossRef]

Chandrasekhar, S.

J. H. Sinsky, A. Adamiecki, C. A. Burrus, S. Chandrasekhar, J. Leuthold, and O. Wohlgemuth, “A 40-Gb/s Integrated Balanced Optical Front End and RZ-DPSK Performance,” IEEE Photon. Technol. Lett. 15(8), 1135–1137 (2003).
[CrossRef]

Chen, H.

D. Jin, H. Chen, A. Barklund, J. Mallari, G. Yu, E. Miller, and R. Dinu, “EO polymer modulators reliability study,” Proc. SPIE 7599, 75990H (2010).
[CrossRef]

Chen, J. X.

E. I. Ackerman, W. K. Burns, G. E. Betts, J. X. Chen, J. L. Prince, M. D. Regan, H. V. Roussell, and C. H. Cox, “RF-Over-Fiber Links With Very Low Noise Figure,” J. Light. Tech. 26(15), 2441–2448 (2008).
[CrossRef]

Cheng, Y. J.

J. D. Luo, S. Huang, Y. J. Cheng, T. D. Kim, Z. W. Shi, X. H. Zhou, and A. K.-Y. Jen, “Phenyltetraene-based nonlinear optical chromophores with enhanced chemical stability and electrooptic activity,” Org. Lett. 9(22), 4471–4474 (2007).
[CrossRef] [PubMed]

Cheng, Y.-J.

Z. Shi, J. Luo, S. Huang, X.-H. Zhou, T.-D. Kim, Y.-J. Cheng, B. M. Polishak, T. R. Younkin, B. A. Block, and A. K.-Y. Jen, “Reinforced Site Isolation Leading to Remarkable Thermal Stability and High Electrooptic Activities in Cross-Linked Nonlinear Optical Dendrimers,” Chem. Mater. 20(20), 6372–6377 (2008).
[CrossRef]

Cohen, O.

A. Liu, R. Jones, L. Liao, D. Samara-Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, “A high-speed silicon optical modulator based on a metal-oxide-semiconductor capacitor,” Nature 427(6975), 615–618 (2004).
[CrossRef] [PubMed]

Cox, C.

C. Cox, E. Ackerman, R. Helkey, and G. E. Betts, “Techniques and Performance of Intensity-Modulation Direct-Detection Analog Optical Links,” IEEE Trans. Microw. Theory Tech. 45(8), 1375–1383 (1997).
[CrossRef]

Cox, C. H.

E. I. Ackerman, W. K. Burns, G. E. Betts, J. X. Chen, J. L. Prince, M. D. Regan, H. V. Roussell, and C. H. Cox, “RF-Over-Fiber Links With Very Low Noise Figure,” J. Light. Tech. 26(15), 2441–2448 (2008).
[CrossRef]

Dalton, L.

T. Baehr-Jones, B. Penkov, J. Huang, P. Sullivan, J. Davies, J. Takayesu, J. Luo, T.-D. Kim, L. Dalton, A. Jen, M. Hochberg, and A. Scherer, “Nonlinear polymer-clad silicon slot waveguide modulator with a half wave voltage of 0.25 V,” Appl. Phys. Lett. 92(16), 163303 (2008).
[CrossRef]

Dalton, L. R.

Y. Shi, C. Zhang, H. Zhang, J. H. Bechtel, L. R. Dalton, B. H. Robinson, and W. H. Steier, “Low (Sub-1-volt) halfwave voltage polymeric electro-optic modulators achieved by controlling chromophore shape,” Science 288(5463), 119–122 (2000).
[CrossRef] [PubMed]

Darcie, T. E.

T. E. Darcie and P. F. Driessen, “Class-AB Techniques for High-Dynamic-Range Microwave-Photonic Links,” IEEE Photon. Technol. Lett. 18(8), 929–931 (2006).
[CrossRef]

Davies, J.

T. Baehr-Jones, B. Penkov, J. Huang, P. Sullivan, J. Davies, J. Takayesu, J. Luo, T.-D. Kim, L. Dalton, A. Jen, M. Hochberg, and A. Scherer, “Nonlinear polymer-clad silicon slot waveguide modulator with a half wave voltage of 0.25 V,” Appl. Phys. Lett. 92(16), 163303 (2008).
[CrossRef]

Dinu, R.

D. Jin, H. Chen, A. Barklund, J. Mallari, G. Yu, E. Miller, and R. Dinu, “EO polymer modulators reliability study,” Proc. SPIE 7599, 75990H (2010).
[CrossRef]

Dong, P.

Driessen, P. F.

T. E. Darcie and P. F. Driessen, “Class-AB Techniques for High-Dynamic-Range Microwave-Photonic Links,” IEEE Photon. Technol. Lett. 18(8), 929–931 (2006).
[CrossRef]

Esman, R. D.

L. T. Nichols, K. J. Williams, and R. D. Esman, “Optimizing the Ultrawide-Band Photonic Link,” IEEE Trans. Microw. Theory Tech. 45(8), 1384–1389 (1997).
[CrossRef]

Feng, D.

Forrest, S. R.

R. Taylor and S. R. Forrest, “Steering of an optically driven true-time delay phased-array antenna based on a broad-band coherent WDM architecture,” IEEE Photon. Technol. Lett. 10(1), 144–146 (1998).
[CrossRef]

Giurgiola, S.

Greenblatt, A. S.

Helkey, R.

C. Cox, E. Ackerman, R. Helkey, and G. E. Betts, “Techniques and Performance of Intensity-Modulation Direct-Detection Analog Optical Links,” IEEE Trans. Microw. Theory Tech. 45(8), 1375–1383 (1997).
[CrossRef]

Hochberg, M.

T. Baehr-Jones, B. Penkov, J. Huang, P. Sullivan, J. Davies, J. Takayesu, J. Luo, T.-D. Kim, L. Dalton, A. Jen, M. Hochberg, and A. Scherer, “Nonlinear polymer-clad silicon slot waveguide modulator with a half wave voltage of 0.25 V,” Appl. Phys. Lett. 92(16), 163303 (2008).
[CrossRef]

Howerton, M. M.

Huang, J.

T. Baehr-Jones, B. Penkov, J. Huang, P. Sullivan, J. Davies, J. Takayesu, J. Luo, T.-D. Kim, L. Dalton, A. Jen, M. Hochberg, and A. Scherer, “Nonlinear polymer-clad silicon slot waveguide modulator with a half wave voltage of 0.25 V,” Appl. Phys. Lett. 92(16), 163303 (2008).
[CrossRef]

Huang, S.

Z. Shi, J. Luo, S. Huang, X.-H. Zhou, T.-D. Kim, Y.-J. Cheng, B. M. Polishak, T. R. Younkin, B. A. Block, and A. K.-Y. Jen, “Reinforced Site Isolation Leading to Remarkable Thermal Stability and High Electrooptic Activities in Cross-Linked Nonlinear Optical Dendrimers,” Chem. Mater. 20(20), 6372–6377 (2008).
[CrossRef]

J. D. Luo, S. Huang, Y. J. Cheng, T. D. Kim, Z. W. Shi, X. H. Zhou, and A. K.-Y. Jen, “Phenyltetraene-based nonlinear optical chromophores with enhanced chemical stability and electrooptic activity,” Org. Lett. 9(22), 4471–4474 (2007).
[CrossRef] [PubMed]

Janner, D.

Jen, A.

T. Baehr-Jones, B. Penkov, J. Huang, P. Sullivan, J. Davies, J. Takayesu, J. Luo, T.-D. Kim, L. Dalton, A. Jen, M. Hochberg, and A. Scherer, “Nonlinear polymer-clad silicon slot waveguide modulator with a half wave voltage of 0.25 V,” Appl. Phys. Lett. 92(16), 163303 (2008).
[CrossRef]

Jen, A. K.-Y.

Z. Shi, J. Luo, S. Huang, X.-H. Zhou, T.-D. Kim, Y.-J. Cheng, B. M. Polishak, T. R. Younkin, B. A. Block, and A. K.-Y. Jen, “Reinforced Site Isolation Leading to Remarkable Thermal Stability and High Electrooptic Activities in Cross-Linked Nonlinear Optical Dendrimers,” Chem. Mater. 20(20), 6372–6377 (2008).
[CrossRef]

J. D. Luo, S. Huang, Y. J. Cheng, T. D. Kim, Z. W. Shi, X. H. Zhou, and A. K.-Y. Jen, “Phenyltetraene-based nonlinear optical chromophores with enhanced chemical stability and electrooptic activity,” Org. Lett. 9(22), 4471–4474 (2007).
[CrossRef] [PubMed]

Jin, D.

D. Jin, H. Chen, A. Barklund, J. Mallari, G. Yu, E. Miller, and R. Dinu, “EO polymer modulators reliability study,” Proc. SPIE 7599, 75990H (2010).
[CrossRef]

Jones, R.

A. Liu, R. Jones, L. Liao, D. Samara-Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, “A high-speed silicon optical modulator based on a metal-oxide-semiconductor capacitor,” Nature 427(6975), 615–618 (2004).
[CrossRef] [PubMed]

Kim, T. D.

J. D. Luo, S. Huang, Y. J. Cheng, T. D. Kim, Z. W. Shi, X. H. Zhou, and A. K.-Y. Jen, “Phenyltetraene-based nonlinear optical chromophores with enhanced chemical stability and electrooptic activity,” Org. Lett. 9(22), 4471–4474 (2007).
[CrossRef] [PubMed]

Kim, T.-D.

T. Baehr-Jones, B. Penkov, J. Huang, P. Sullivan, J. Davies, J. Takayesu, J. Luo, T.-D. Kim, L. Dalton, A. Jen, M. Hochberg, and A. Scherer, “Nonlinear polymer-clad silicon slot waveguide modulator with a half wave voltage of 0.25 V,” Appl. Phys. Lett. 92(16), 163303 (2008).
[CrossRef]

Z. Shi, J. Luo, S. Huang, X.-H. Zhou, T.-D. Kim, Y.-J. Cheng, B. M. Polishak, T. R. Younkin, B. A. Block, and A. K.-Y. Jen, “Reinforced Site Isolation Leading to Remarkable Thermal Stability and High Electrooptic Activities in Cross-Linked Nonlinear Optical Dendrimers,” Chem. Mater. 20(20), 6372–6377 (2008).
[CrossRef]

Krishnamoorthy, A. V.

Kung, C.-C.

Leuthold, J.

J. H. Sinsky, A. Adamiecki, C. A. Burrus, S. Chandrasekhar, J. Leuthold, and O. Wohlgemuth, “A 40-Gb/s Integrated Balanced Optical Front End and RZ-DPSK Performance,” IEEE Photon. Technol. Lett. 15(8), 1135–1137 (2003).
[CrossRef]

Li, G.

Liang, H.

Liao, L.

A. Liu, R. Jones, L. Liao, D. Samara-Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, “A high-speed silicon optical modulator based on a metal-oxide-semiconductor capacitor,” Nature 427(6975), 615–618 (2004).
[CrossRef] [PubMed]

Liao, S.

Lipson, M.

Q. Xu, B. Schmidt, S. Pradhan, and M. Lipson, “Micrometre-scale silicon electro-optic modulator,” Nature 435(7040), 325–327 (2005).
[CrossRef] [PubMed]

V. R. Almeida, Q. Xu, C. A. Barrios, and M. Lipson, “Guiding and confining light in void nanostructure,” Opt. Lett. 29(11), 1209–1211 (2004).
[CrossRef] [PubMed]

Liu, A.

A. Liu, R. Jones, L. Liao, D. Samara-Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, “A high-speed silicon optical modulator based on a metal-oxide-semiconductor capacitor,” Nature 427(6975), 615–618 (2004).
[CrossRef] [PubMed]

Lucchi, F.

Luo, J.

T. Baehr-Jones, B. Penkov, J. Huang, P. Sullivan, J. Davies, J. Takayesu, J. Luo, T.-D. Kim, L. Dalton, A. Jen, M. Hochberg, and A. Scherer, “Nonlinear polymer-clad silicon slot waveguide modulator with a half wave voltage of 0.25 V,” Appl. Phys. Lett. 92(16), 163303 (2008).
[CrossRef]

Z. Shi, J. Luo, S. Huang, X.-H. Zhou, T.-D. Kim, Y.-J. Cheng, B. M. Polishak, T. R. Younkin, B. A. Block, and A. K.-Y. Jen, “Reinforced Site Isolation Leading to Remarkable Thermal Stability and High Electrooptic Activities in Cross-Linked Nonlinear Optical Dendrimers,” Chem. Mater. 20(20), 6372–6377 (2008).
[CrossRef]

Luo, J. D.

J. D. Luo, S. Huang, Y. J. Cheng, T. D. Kim, Z. W. Shi, X. H. Zhou, and A. K.-Y. Jen, “Phenyltetraene-based nonlinear optical chromophores with enhanced chemical stability and electrooptic activity,” Org. Lett. 9(22), 4471–4474 (2007).
[CrossRef] [PubMed]

Mallari, J.

D. Jin, H. Chen, A. Barklund, J. Mallari, G. Yu, E. Miller, and R. Dinu, “EO polymer modulators reliability study,” Proc. SPIE 7599, 75990H (2010).
[CrossRef]

McElhanon, R. W.

Miller, E.

D. Jin, H. Chen, A. Barklund, J. Mallari, G. Yu, E. Miller, and R. Dinu, “EO polymer modulators reliability study,” Proc. SPIE 7599, 75990H (2010).
[CrossRef]

Moeller, R. P.

Nichols, L. T.

L. T. Nichols, K. J. Williams, and R. D. Esman, “Optimizing the Ultrawide-Band Photonic Link,” IEEE Trans. Microw. Theory Tech. 45(8), 1384–1389 (1997).
[CrossRef]

Nicolaescu, R.

A. Liu, R. Jones, L. Liao, D. Samara-Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, “A high-speed silicon optical modulator based on a metal-oxide-semiconductor capacitor,” Nature 427(6975), 615–618 (2004).
[CrossRef] [PubMed]

Paniccia, M.

A. Liu, R. Jones, L. Liao, D. Samara-Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, “A high-speed silicon optical modulator based on a metal-oxide-semiconductor capacitor,” Nature 427(6975), 615–618 (2004).
[CrossRef] [PubMed]

Penkov, B.

T. Baehr-Jones, B. Penkov, J. Huang, P. Sullivan, J. Davies, J. Takayesu, J. Luo, T.-D. Kim, L. Dalton, A. Jen, M. Hochberg, and A. Scherer, “Nonlinear polymer-clad silicon slot waveguide modulator with a half wave voltage of 0.25 V,” Appl. Phys. Lett. 92(16), 163303 (2008).
[CrossRef]

Polishak, B. M.

Z. Shi, J. Luo, S. Huang, X.-H. Zhou, T.-D. Kim, Y.-J. Cheng, B. M. Polishak, T. R. Younkin, B. A. Block, and A. K.-Y. Jen, “Reinforced Site Isolation Leading to Remarkable Thermal Stability and High Electrooptic Activities in Cross-Linked Nonlinear Optical Dendrimers,” Chem. Mater. 20(20), 6372–6377 (2008).
[CrossRef]

Pradhan, S.

Q. Xu, B. Schmidt, S. Pradhan, and M. Lipson, “Micrometre-scale silicon electro-optic modulator,” Nature 435(7040), 325–327 (2005).
[CrossRef] [PubMed]

Prince, J. L.

E. I. Ackerman, W. K. Burns, G. E. Betts, J. X. Chen, J. L. Prince, M. D. Regan, H. V. Roussell, and C. H. Cox, “RF-Over-Fiber Links With Very Low Noise Figure,” J. Light. Tech. 26(15), 2441–2448 (2008).
[CrossRef]

Pruneri, V.

Qian, W.

Regan, M. D.

E. I. Ackerman, W. K. Burns, G. E. Betts, J. X. Chen, J. L. Prince, M. D. Regan, H. V. Roussell, and C. H. Cox, “RF-Over-Fiber Links With Very Low Noise Figure,” J. Light. Tech. 26(15), 2441–2448 (2008).
[CrossRef]

Robinson, B. H.

Y. Shi, C. Zhang, H. Zhang, J. H. Bechtel, L. R. Dalton, B. H. Robinson, and W. H. Steier, “Low (Sub-1-volt) halfwave voltage polymeric electro-optic modulators achieved by controlling chromophore shape,” Science 288(5463), 119–122 (2000).
[CrossRef] [PubMed]

Roussell, H. V.

E. I. Ackerman, W. K. Burns, G. E. Betts, J. X. Chen, J. L. Prince, M. D. Regan, H. V. Roussell, and C. H. Cox, “RF-Over-Fiber Links With Very Low Noise Figure,” J. Light. Tech. 26(15), 2441–2448 (2008).
[CrossRef]

Rubin, D.

A. Liu, R. Jones, L. Liao, D. Samara-Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, “A high-speed silicon optical modulator based on a metal-oxide-semiconductor capacitor,” Nature 427(6975), 615–618 (2004).
[CrossRef] [PubMed]

Samara-Rubio, D.

A. Liu, R. Jones, L. Liao, D. Samara-Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, “A high-speed silicon optical modulator based on a metal-oxide-semiconductor capacitor,” Nature 427(6975), 615–618 (2004).
[CrossRef] [PubMed]

Scherer, A.

T. Baehr-Jones, B. Penkov, J. Huang, P. Sullivan, J. Davies, J. Takayesu, J. Luo, T.-D. Kim, L. Dalton, A. Jen, M. Hochberg, and A. Scherer, “Nonlinear polymer-clad silicon slot waveguide modulator with a half wave voltage of 0.25 V,” Appl. Phys. Lett. 92(16), 163303 (2008).
[CrossRef]

Schmidt, B.

Q. Xu, B. Schmidt, S. Pradhan, and M. Lipson, “Micrometre-scale silicon electro-optic modulator,” Nature 435(7040), 325–327 (2005).
[CrossRef] [PubMed]

Shafiiha, R.

Shi, Y.

Y. Shi, C. Zhang, H. Zhang, J. H. Bechtel, L. R. Dalton, B. H. Robinson, and W. H. Steier, “Low (Sub-1-volt) halfwave voltage polymeric electro-optic modulators achieved by controlling chromophore shape,” Science 288(5463), 119–122 (2000).
[CrossRef] [PubMed]

Shi, Z.

Z. Shi, J. Luo, S. Huang, X.-H. Zhou, T.-D. Kim, Y.-J. Cheng, B. M. Polishak, T. R. Younkin, B. A. Block, and A. K.-Y. Jen, “Reinforced Site Isolation Leading to Remarkable Thermal Stability and High Electrooptic Activities in Cross-Linked Nonlinear Optical Dendrimers,” Chem. Mater. 20(20), 6372–6377 (2008).
[CrossRef]

Shi, Z. W.

J. D. Luo, S. Huang, Y. J. Cheng, T. D. Kim, Z. W. Shi, X. H. Zhou, and A. K.-Y. Jen, “Phenyltetraene-based nonlinear optical chromophores with enhanced chemical stability and electrooptic activity,” Org. Lett. 9(22), 4471–4474 (2007).
[CrossRef] [PubMed]

Sinsky, J. H.

J. H. Sinsky, A. Adamiecki, C. A. Burrus, S. Chandrasekhar, J. Leuthold, and O. Wohlgemuth, “A 40-Gb/s Integrated Balanced Optical Front End and RZ-DPSK Performance,” IEEE Photon. Technol. Lett. 15(8), 1135–1137 (2003).
[CrossRef]

Soref, R. A.

R. A. Soref and B. R. Bennett, “Electrooptical Effects in Silicon,” IEEE J. Quantum Electron. 23(1), 123–129 (1987).
[CrossRef]

Steier, W. H.

Y. Shi, C. Zhang, H. Zhang, J. H. Bechtel, L. R. Dalton, B. H. Robinson, and W. H. Steier, “Low (Sub-1-volt) halfwave voltage polymeric electro-optic modulators achieved by controlling chromophore shape,” Science 288(5463), 119–122 (2000).
[CrossRef] [PubMed]

Sullivan, P.

T. Baehr-Jones, B. Penkov, J. Huang, P. Sullivan, J. Davies, J. Takayesu, J. Luo, T.-D. Kim, L. Dalton, A. Jen, M. Hochberg, and A. Scherer, “Nonlinear polymer-clad silicon slot waveguide modulator with a half wave voltage of 0.25 V,” Appl. Phys. Lett. 92(16), 163303 (2008).
[CrossRef]

Takayesu, J.

T. Baehr-Jones, B. Penkov, J. Huang, P. Sullivan, J. Davies, J. Takayesu, J. Luo, T.-D. Kim, L. Dalton, A. Jen, M. Hochberg, and A. Scherer, “Nonlinear polymer-clad silicon slot waveguide modulator with a half wave voltage of 0.25 V,” Appl. Phys. Lett. 92(16), 163303 (2008).
[CrossRef]

Taylor, R.

R. Taylor and S. R. Forrest, “Steering of an optically driven true-time delay phased-array antenna based on a broad-band coherent WDM architecture,” IEEE Photon. Technol. Lett. 10(1), 144–146 (1998).
[CrossRef]

Vergani, P.

Villa, M.

Williams, K. J.

L. T. Nichols, K. J. Williams, and R. D. Esman, “Optimizing the Ultrawide-Band Photonic Link,” IEEE Trans. Microw. Theory Tech. 45(8), 1384–1389 (1997).
[CrossRef]

Wohlgemuth, O.

J. H. Sinsky, A. Adamiecki, C. A. Burrus, S. Chandrasekhar, J. Leuthold, and O. Wohlgemuth, “A 40-Gb/s Integrated Balanced Optical Front End and RZ-DPSK Performance,” IEEE Photon. Technol. Lett. 15(8), 1135–1137 (2003).
[CrossRef]

Xu, Q.

Q. Xu, B. Schmidt, S. Pradhan, and M. Lipson, “Micrometre-scale silicon electro-optic modulator,” Nature 435(7040), 325–327 (2005).
[CrossRef] [PubMed]

V. R. Almeida, Q. Xu, C. A. Barrios, and M. Lipson, “Guiding and confining light in void nanostructure,” Opt. Lett. 29(11), 1209–1211 (2004).
[CrossRef] [PubMed]

Younkin, T. R.

Z. Shi, J. Luo, S. Huang, X.-H. Zhou, T.-D. Kim, Y.-J. Cheng, B. M. Polishak, T. R. Younkin, B. A. Block, and A. K.-Y. Jen, “Reinforced Site Isolation Leading to Remarkable Thermal Stability and High Electrooptic Activities in Cross-Linked Nonlinear Optical Dendrimers,” Chem. Mater. 20(20), 6372–6377 (2008).
[CrossRef]

Yu, G.

D. Jin, H. Chen, A. Barklund, J. Mallari, G. Yu, E. Miller, and R. Dinu, “EO polymer modulators reliability study,” Proc. SPIE 7599, 75990H (2010).
[CrossRef]

Zhang, C.

Y. Shi, C. Zhang, H. Zhang, J. H. Bechtel, L. R. Dalton, B. H. Robinson, and W. H. Steier, “Low (Sub-1-volt) halfwave voltage polymeric electro-optic modulators achieved by controlling chromophore shape,” Science 288(5463), 119–122 (2000).
[CrossRef] [PubMed]

Zhang, H.

Y. Shi, C. Zhang, H. Zhang, J. H. Bechtel, L. R. Dalton, B. H. Robinson, and W. H. Steier, “Low (Sub-1-volt) halfwave voltage polymeric electro-optic modulators achieved by controlling chromophore shape,” Science 288(5463), 119–122 (2000).
[CrossRef] [PubMed]

Zheng, D.

Zheng, X.

Zhou, X. H.

J. D. Luo, S. Huang, Y. J. Cheng, T. D. Kim, Z. W. Shi, X. H. Zhou, and A. K.-Y. Jen, “Phenyltetraene-based nonlinear optical chromophores with enhanced chemical stability and electrooptic activity,” Org. Lett. 9(22), 4471–4474 (2007).
[CrossRef] [PubMed]

Zhou, X.-H.

Z. Shi, J. Luo, S. Huang, X.-H. Zhou, T.-D. Kim, Y.-J. Cheng, B. M. Polishak, T. R. Younkin, B. A. Block, and A. K.-Y. Jen, “Reinforced Site Isolation Leading to Remarkable Thermal Stability and High Electrooptic Activities in Cross-Linked Nonlinear Optical Dendrimers,” Chem. Mater. 20(20), 6372–6377 (2008).
[CrossRef]

Appl. Phys. Lett.

T. Baehr-Jones, B. Penkov, J. Huang, P. Sullivan, J. Davies, J. Takayesu, J. Luo, T.-D. Kim, L. Dalton, A. Jen, M. Hochberg, and A. Scherer, “Nonlinear polymer-clad silicon slot waveguide modulator with a half wave voltage of 0.25 V,” Appl. Phys. Lett. 92(16), 163303 (2008).
[CrossRef]

Chem. Mater.

Z. Shi, J. Luo, S. Huang, X.-H. Zhou, T.-D. Kim, Y.-J. Cheng, B. M. Polishak, T. R. Younkin, B. A. Block, and A. K.-Y. Jen, “Reinforced Site Isolation Leading to Remarkable Thermal Stability and High Electrooptic Activities in Cross-Linked Nonlinear Optical Dendrimers,” Chem. Mater. 20(20), 6372–6377 (2008).
[CrossRef]

IEEE J. Quantum Electron.

R. A. Soref and B. R. Bennett, “Electrooptical Effects in Silicon,” IEEE J. Quantum Electron. 23(1), 123–129 (1987).
[CrossRef]

IEEE Photon. Technol. Lett.

R. Taylor and S. R. Forrest, “Steering of an optically driven true-time delay phased-array antenna based on a broad-band coherent WDM architecture,” IEEE Photon. Technol. Lett. 10(1), 144–146 (1998).
[CrossRef]

T. E. Darcie and P. F. Driessen, “Class-AB Techniques for High-Dynamic-Range Microwave-Photonic Links,” IEEE Photon. Technol. Lett. 18(8), 929–931 (2006).
[CrossRef]

J. H. Sinsky, A. Adamiecki, C. A. Burrus, S. Chandrasekhar, J. Leuthold, and O. Wohlgemuth, “A 40-Gb/s Integrated Balanced Optical Front End and RZ-DPSK Performance,” IEEE Photon. Technol. Lett. 15(8), 1135–1137 (2003).
[CrossRef]

IEEE Trans. Microw. Theory Tech.

L. T. Nichols, K. J. Williams, and R. D. Esman, “Optimizing the Ultrawide-Band Photonic Link,” IEEE Trans. Microw. Theory Tech. 45(8), 1384–1389 (1997).
[CrossRef]

C. Cox, E. Ackerman, R. Helkey, and G. E. Betts, “Techniques and Performance of Intensity-Modulation Direct-Detection Analog Optical Links,” IEEE Trans. Microw. Theory Tech. 45(8), 1375–1383 (1997).
[CrossRef]

J. Light. Tech.

E. I. Ackerman, W. K. Burns, G. E. Betts, J. X. Chen, J. L. Prince, M. D. Regan, H. V. Roussell, and C. H. Cox, “RF-Over-Fiber Links With Very Low Noise Figure,” J. Light. Tech. 26(15), 2441–2448 (2008).
[CrossRef]

J. Lightwave Technol.

Nature

A. Liu, R. Jones, L. Liao, D. Samara-Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, “A high-speed silicon optical modulator based on a metal-oxide-semiconductor capacitor,” Nature 427(6975), 615–618 (2004).
[CrossRef] [PubMed]

Q. Xu, B. Schmidt, S. Pradhan, and M. Lipson, “Micrometre-scale silicon electro-optic modulator,” Nature 435(7040), 325–327 (2005).
[CrossRef] [PubMed]

Opt. Express

Opt. Lett.

Org. Lett.

J. D. Luo, S. Huang, Y. J. Cheng, T. D. Kim, Z. W. Shi, X. H. Zhou, and A. K.-Y. Jen, “Phenyltetraene-based nonlinear optical chromophores with enhanced chemical stability and electrooptic activity,” Org. Lett. 9(22), 4471–4474 (2007).
[CrossRef] [PubMed]

Proc. SPIE

D. Jin, H. Chen, A. Barklund, J. Mallari, G. Yu, E. Miller, and R. Dinu, “EO polymer modulators reliability study,” Proc. SPIE 7599, 75990H (2010).
[CrossRef]

Science

Y. Shi, C. Zhang, H. Zhang, J. H. Bechtel, L. R. Dalton, B. H. Robinson, and W. H. Steier, “Low (Sub-1-volt) halfwave voltage polymeric electro-optic modulators achieved by controlling chromophore shape,” Science 288(5463), 119–122 (2000).
[CrossRef] [PubMed]

Other

PhotonicSystems, part number PSI-3600-MOD-D1.

T. Pinguet, V. Sadagopan, A. Mekis, B. Analui, D. Kucharski, and S. Gloeckner, “A 1550 nm, 10 Gbps optical modulator with integrated driver in 130 nm CMOS”, Proc. IEEE conf. on Group IV Photonics, 1–3 (2007).

R. Ding, T. Baehr-Jones, Y. Liu, R. Bojko, J. Witzens, S. Huang, J. Luo, S. Benight, P. Sullivan, J.-M. Fedeli, M. Fournier, L. Dalton, A. Jen, M. Hochberg, “Demonstration of a low VπL modulator with GHz bandwidth based on electro-optic polymer-clad silicon slot waveguides,” submitted for publication.

J. Witzens, G. Masini, S. Sahni, B. Analui, and C. Gunn, “10 Gbits/s transceiver on silicon”, Proc. SPIE 6996, 699610 1–10 (2008).

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Figures (24)

Fig. 5
Fig. 5

Optimum ridge width as a function of slot width.

Fig. 1
Fig. 1

Overview of the device geometry (not to scale). Gray areas represent aluminum, green areas silicon, blue silicon dioxide and orange the electro-optic polymer. The green shading indicates the implant concentrations. The inset shows the whole structure with both arms of the MZI operated in push-pull operation (arrows indicate the relative orientation of the E-field during operation).

Fig. 18
Fig. 18

Schematic of an optical analog link. SWGs are represented in green and regular waveguides (on-chip) or fibers (off-chip) in orange.

Fig. 2
Fig. 2

SEM micrograph of a SWG with a 120 nm slot fabricated with optical lithography at BAE Systems. The slot was purposely defined off center, it was found that this increases the single mode frequency region.

Fig. 3
Fig. 3

(a) Optical Ex field distribution in a.u. for a transverse electric (x-polarized) mode and (b) RF Ex field distribution. The waveguide geometry is described in Table 1. The very high E-field regions at the corners are artifacts of the finite-elements mode solver.

Fig. 4
Fig. 4

(a) optical field overlap with the slot region as a function of slot width, and (b) overlap multiplied by the Ex RF field strength for an applied bias of 1V. wg_rh = 200 nm and wg_ch = 50 nm. wg_rw is adjusted to maintain optimum overlap (Fig. 5). The inset shows that the FOM in (b) converges to a finite value.

Fig. 6
Fig. 6

Optical Ex field across the waveguide cross-section shown in the inset. (a) is the extreme case of a 2 nm slot and (b) a typical realistic device with a 100 nm slot.

Fig. 7
Fig. 7

(a) Comparison of the optical E-fields across SWGs with 2nm and 100 nm slots. Both modes have been normalized to carry equal flux. (b) Comparison between the simulated FOM (blue line) and the semi-analytical model (stars).

Fig. 8
Fig. 8

φ as a function of slot width as extracted from mode profiles (black line) and as predicted with Eq. (6).

Fig. 9
Fig. 9

Overlap of the optical field with the core silicon, cladding silicon, and decay length of the optical field inside the cladding silicon. The SWG ridge width, wg_rw, is intrinsically varied with wg_sw in order to maintain optimum slot overlap. wg_rh = 200 nm and wg_ch = 50 nm.

Fig. 10
Fig. 10

SWG capacitance as a function of slot width. The continuous line is simulated (PISCES), while the dashed line corresponds to the parallel plate approximation.

Fig. 11
Fig. 11

TL effective index and propagation losses as a function of design parameters at a frequency of 10 GHz. Each parameter is varied around the geometry given by Table 1. The propagation losses shown here include excess losses from the high and moderate resistivity silicon regions. These curves were obtained by iterative simulations with a finite-elements mode solver.

Fig. 12
Fig. 12

(a) Equivalent circuit of the loaded TL and (b) simplified equivalent circuit. Elements in red correspond to the SWG. In (c) the series RC model for the SWG is transformed into a parallel model to allow using the standard telegraph line equations.

Fig. 13
Fig. 13

(a) RF H-field distribution and (b) RF E-field distribution. The scale in (a) is 0 to 2e4 A/m. The scale in (b) is 0 to 1.15e8 V/m. The mode is normalized to carry 1W of power. Solutions are based on fully vectorial solutions of Maxwell’s equations at 10 GHz.

Fig. 14
Fig. 14

(a) RF loss volume density (b) Jz current distribution (c) Jx current distribution and (d) rescaled Jz current distribution. The scale in (a) is 0 to 2.2e15 W/m3, the scale in (b) 0 to 6e10 A/m2, the scale in (c) 0 to 2.5e9 A/m2, and the scale in (d) 0 to 1e8 A/m2. The mode is normalized to carry 1W of power.

Fig. 15
Fig. 15

(a) E-field in V/m and (b) H-field in A/m for an SWG driven as a lumped element at 10 GHz, i.e., with a homogeneous AC voltage applied along the entire length.

Fig. 16
Fig. 16

Real parts of the TL impedance and effective index. Black dots correspond to simulation data for the loaded TL. The black curves correspond to the fits described in the text. As a reference, the blue curves show the simulation results for the unloaded TL (silicon conductance uniformly set to zero). The red curves correspond to the analytical models described in the text, with independently evaluated values for SWG and TL characteristics applied to the formulas (“independently simulated” column of Table 3).

Fig. 17
Fig. 17

TL losses as a function of frequency (continuous black curve). The dashed curve shows the transmission losses when the aluminum is replaced by a perfect metal, and corresponds to the excess losses from the silicon. The blue curve represents the losses from the unloaded aluminum TL. The green curve corresponds to the analytical model for the excess losses and should be compared to the dashed curve. (b) is a detailed view of the lower frequency portion of (a).

Fig. 19
Fig. 19

Performance metrics of the optimized MZI. Insertion losses are relatively insensitive to slot width and are plotted as a function of optimization frequency for a 100 nm slot. The TL impedance is relatively insensitive to optimization frequency and is plotted as a function of slot width for a 10 GHz frequency. The insertion losses only take into account implanted SWG losses. Fiber to chip and ridge to SWG coupling losses have to be accounted for separately. Data points correspond to devices optimized for the operation frequency at which they are reported (in particular, Vπ is reported at speed). The reported bandwidth is the optical bandwidth of the entire MZI (i.e., S21 = 0.5 or equivalently Vπ(f) = 2Vπ(DC)), including effects of both the SWG bandwidth limitation and the frequency dependent TL losses. The blue, green and red curves correspond to devices optimized for and operated at 10 GHz, 20 GHz and 50 GHz.

Fig. 20
Fig. 20

Optimized device geometries and internal device characteristics as a function of optimized frequency, assuming a 100 nm slot width. In (a), imp_c1 is given by the red curve and imp_c2 by the black curve. Optimization was run assuming donor type impurities. In plots (c) and (d), blue corresponds to SWG characteristics and red to TL characteristics. In (c), the red curve is the effective length of an infinite TL (this gives an idea of the limit on the MZI length if the insertion losses where heavily neglected relative to drive voltage).

Fig. 21
Fig. 21

TL propagation losses (red) and SWG propagation losses (blue) as a function of the lowest density implant concentration when varying around the optimum (dashed line) at 10 GHz.

Fig. 22
Fig. 22

(a) Single ended analog link NF as a function of slot width, assuming r33 = 150 pm/V and a polymer optical power density equivalent to launching 15 mW into each of the two SWG branches of the MZI when the slot is equal to 100 nm. (b) Single ended analog link NF as a function of optical power launched into each of the two SWG branches of the MZI assuming r33 = 150 pm/V and a 100 nm slot. (c) Dual fiber analog link NF as a function of optical power launched into each of the two SWG branches of the MZI assuming r33 = 150 pm/V and a 100 nm slot. The complementary outputs of the MZI are assumed to be sent into a balanced receiver. In all three plots the blue, green and red curves correspond to devices optimized for and operated at 10 GHz, 20 GHz and 50 GHz. In (b) and (c) the dashed curves correspond to the NF assuming a perfect, noiseless receiver.

Fig. 23
Fig. 23

Balanced receiver analog link with homodyne signal amplification at the Tx. In the lower schematic, SWGs are represented in green, while conventional waveguides or fibers are represented in orange. The laser power is split with an asymmetric tap to send Psig into the SWG and Pamp into the other branch. Biasing of the modulator at the 3 dB point is intrinsically assumed (control system or waveguide delay not shown).

Fig. 24
Fig. 24

NF of a balanced link with homodyne amplification Tx (a) with Pamp = 20 mW and (b) Pamp = 100 mW. Blue, green and red curves are for devices optimized for and operated at 10 GHz, 20 GHz and 50 GHz. (c) is the NF as a function of Pamp for Psig = 3 mW and for a device optimized for and operated at 20 GHz. The dashed curves correspond to the NF assuming a perfect, noiseless receiver.

Tables (4)

Tables Icon

Table 1 Summery of device parameters.

Tables Icon

Table 2 SWG characteristics for different ridge and cladding thicknesses.

Tables Icon

Table 3 Summary of fitted parameters.

Tables Icon

Table 4 Optical power inside the SWGs and required laser power for various link architectures. The required laser power for the balanced link utilizing the Lithium Niobate modulator was derived assuming a total modulator insertion loss of 9 dB and a Vπ of 1.35V at 18 GHz [10]. Numbers in parenthesis correspond to values obtained for an ideal, noiseless receiver. Coupling losses for bringing Pamp onto the chip are assumed to be 3 dB, since no on chip mode conversion is required. Unless otherwise specified, r33 is assumed to be 150 pm/V. Maximum electrical input power is evaluated assuming the peak-to-peak input voltage should be kept below Vπ/10 in order to maintain sufficient link linearity.

Equations (18)

Equations on this page are rendered with MathJax. Learn more.

Ov =  slot n | E x | 2 dxdy Z 0 Re ( E × H * ) dxdy
Ov wg_sw  =  x=0 n | E x | 2 dy Z 0 Re ( E × H * ) dx dy  =  n silicon 3 n polymer 3 x=0 n silicon | E ¯ x | 2 dy Z 0 Re ( E ¯ × H ¯ * ) dx dy
α  =  n eff 2 + ( n Si 2 - n slab 2 ) - n polymer 2 λ
β  = n eff λ
k =  n slab 2 - n eff 2 λ
φ  = arctan ( α k tanh ( α wg_sw 2 ) n 2 Si n 2 polymer )
× H =  ε E t   H y z  =  ε E x t    E x  =  2πn λ H y ε ω  =  Z 0 n H y                            H x z  =  ε E y t    E y  =  2πn λ H x ε ω  =  Z 0 n H x
× H =  ε E t     H z y  =  ε E x t    E x  =  i l y H z ε ω                           H z x  =  ε E y t    E y  =  i l x H z ε ω
Z =  1 2 L C+G/i ω  =  1 2 L TL C TL +C SWG 1 1+i f BW                              =  1 2 1 ( C TL L TL ) + ( C SWG L TL ) 1 1+i f BW
n eff  = c 0 L ( C+G/i ω )  = c 0 C TL L TL +C SWG L TL 1 1+i f BW
α  =  10 log ( 10 ) Re ( i ω C SWG 1+i f BW ) ( 2Z )  =  10 log ( 10 ) ω 2 C 2 SWG R SWG 1+ ( f BW ) 2 ( 2Z )
G =  ( g TIA Rsp π V π P in IL 2 ) 2 Z in Z out  =  ( π V π g TIA I av ) 2 Z in Z out
N ex  =  ( 2qI av B ) g 2 TIA Z out + ( I av 2 10 RIN/10 B ) g 2 TIA Z out
NF = 10log10 ( 1+ N ex GN in )  = 10log10 ( 1+ V π 2 Z in 1 π 2 k B T ( 4q P in IL Rsp +10 RIN/10 ) )
V π  =  λ 4L eff ( Ov n 3 polymer r 33 2wg_sw ) -1 1+ ( f/BW ) 2
L eff  =  0 L e α 2 l dl
L opt = 2 α log ( 1+ α β )
P Shot GN in     P amp 2 ( 2Z in ) ( P sig P amp π ( 2V π ) ) 2  =  V π 2 Z in π 2 P sig  =  2 V π 2 Z in π 2 P in

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