Optical structure for imparting a distributed phase shift to an optical signal, electro-optic modulator incorporating such structure for reduced size, low signal loss, and high extinction ratio, and related system and method

1. An optical structure, comprising:

a core having first and second ends, the core including:

a first sidewall between the first and second ends, the first sidewall with a first grating profile defined by a first periodic grating structure, the first grating profile having a first phase shift distributed between the first and second ends; and

a second sidewall opposite the first sidewall, the second sidewall between the first and second ends, the second sidewall with a second grating profile defined by a second periodic grating structure, the second grating profile having a second phase shift distributed between the first and second ends of the core; and

a cladding disposed around the core;

wherein the first grating profile is apodized such that the first periodic grating structure has a non-uniform amplitude between the first and second ends.

2. The optical structure of claim 1 wherein:

the core includes lithium niobate; and

the cladding includes silicon dioxide.

3. The optical structure of claim 1 wherein the first grating profile includes a sinusoidal grating profile.

4. The optical structure of claim 1 wherein the first phase shift is 180° of a spatial wavelength of the first grating profile.

5. The optical structure of claim 1 wherein the first grating profile has one half of the first phase shift distributed between the first end of the core and a midpoint between the first and second ends of the core, and has another half of the first phase shift distributed between the midpoint and the second end of the core.

6. The optical structure of claim 1 wherein the first phase shift is distributed linearly between the first and second ends of the core.

7. The optical structure of claim 1 wherein the second grating profile is apodized such that the second periodic grating structure has a non-uniform amplitude between the first and second ends of the core.

8. The optical structure of claim 1 wherein the first and second periodic grating structures are configured to produce a pi phase shift in an optical signal propagating in the optical structure, such that the optical structure is configured to be a Bragg resonator.

9. The optical structure of claim 1 wherein:

a peak-to-peak amplitude of the first grating profile decreases from the first end of the core to an abrupt-phase-change cross section of the core; and

the peak-to-peak amplitude of the first grating profile increases from the abrupt- phase-change cross section to the second end of the core.

10. A electro-optic modulator, comprising:

an optical structure, including:

a core having first and second ends, the core comprising:

a first sidewall between the first and second ends, the first sidewall with a first grating profile defined by a first periodic grating structure, the first grating profile having a first phase shift distributed between the first and second ends, and

a second sidewall opposite the first sidewall, the second sidewall between the first and second ends, the second sidewall with a second grating profile defined by a second periodic grating structure, the second grating profile having a second phase shift distributed between the first and second ends; and

a cladding disposed around the core;

a first electrode disposed adjacent to the first sidewall of the core; and

a second electrode disposed adjacent to the second sidewall of the core, the second sidewall being opposite the first sidewall;

wherein at least one of the first and second grating profiles is apodized such that at least one of the first and second periodic grating structures has a non-uniform amplitude between the first and second ends of the core.

11. The electro-optic modulator of claim 10 wherein the first and second periodic grating structures are configured to produce a pi phase shift in an optical signal propagating in the optical structure, such that the optical structure is configured to be a Bragg resonator.

12. The electro-optic modulator of claim 11 wherein:

the second grating profile is approximately the same as the first grating profile;

the second phase shift is approximately the same as the first phase shift; and

the second phase shift is distributed between the first and second ends in a manner similar to a manner in which the first phase shift is distributed between the first and second ends.

13. The electro-optic modulator of claim 11 wherein first and second phase shifts are each distributed approximately uniformly between the respective first and second ends.

14. The electro-optic modulator of claim 10 wherein:

the optical structure is configured to resonate at an optical frequency in response to a first voltage between the first and second electrodes; and

the optical structure is configured to attenuate an optical signal at the optical frequency in response to a second voltage between the first and second electrodes.

15. The electro-optic modulator of claim 10 , wherein the core comprises:

a first reflector disposed at the first end and configured to impart the first phase shift to an optical signal propagating in the optical structure; and

a second reflector disposed at the second end and configured to impart the second phase shift to the optical signal;

wherein a combination of the first reflector and the second reflector is configured to resonate at an optical wavelength in response to a sum of the phase shifts equaling 360° of the optical wavelength.

16. The electro-optic modulator of claim 10 , wherein the core comprises:

a first Bragg reflector disposed at the first end and configured to impart, to an optical signal propagating in the optical structure, a phase shift of approximately 180° of a spatial wavelength of the first and second grating profiles; and

a second Bragg reflector disposed at the second end and configured to impart, to the optical signal, a phase shift of approximately 180° of the spatial wavelength of the first and second grating profiles.

17. The electro-optic modulator of claim 10 wherein:

a peak-to-peak amplitude of at least one of the first and second grating profiles decreases from the first end of the core to an abrupt-phase-change cross section of the core; and

the peak-to-peak amplitude of the at least one of the first and second grating profiles increases from the abrupt-phase-change cross section to the second end of the core.