Optical isolation based on space-time engineered asymmetric photonic band gaps

Nonreciprocal electromagnetic devices play a crucial role in modern microwave and optical technologies. Conventional methods for realizing such systems are incompatible with integrated circuits. With recent advances in integrated photonics, the need for efficient on-chip magnetless nonreciprocal devices has become more pressing than ever. This paper leverages space-time engineered asymmetric photonic band gaps to generate optical isolation. It shows that a properly designed space-time modulated slab is highly reflective/transparent for opposite directions of propagation. The corresponding design is magnetless, accommodates low modulation frequencies, and can achieve very high isolation levels. An experimental proof of concept at microwave frequencies is provided.

Figure 1

Principle of nonreciprocal Bragg reflection based on asymmetric photonic band gaps. The red and blue colors represent forward and backward propagation, respectively. The dashed curves correspond to the dispersion curves of the input and output medium while the solid and dotted curves represent the real ($\beta$) and imaginary ($\alpha$) parts of the wave number, respectively, of the (central) photonic band gap medium, assuming the harmonic time dependence $e^{j\omega t}$. The horizontal line corresponds to the excitation frequency, ${\omega }_0$. (a) In a reciprocal system, the band gap is symmetric with respect to positive and negative directions. Red/blue dots correspond to evanescent waves for excitation from the left/right. (b) In a nonreciprocal system, the band gap is tilted with a given slope. Red/blue dots correspond to evanescent and propagating waves, respectively, for excitation from the left/right. In one direction, the wave is totally reflected; in the opposite direction, it is fully transmitted.

Figure 2

Dispersion diagram for an infinite space-time modulated medium with permittivity given by Eq. (4) and parameters ${\epsilon }_r=12.25, M=0.5, {\omega }_{\text{m}}/{\omega }_0=0.13$, and $k_{\text{m}}/k_0=-2.27$. The dotted lines represent the limit of vanishingly small modulation depth $M\to 0$. The red vector, $(k_{\text{m}}a,{\omega }_{\text{m}}a/c)$, corresponds to the space-time modulation wave number and frequency, where $a=2\pi /k_{\text{m}}$ is the spatial period and $c$ is the velocity of light. The dashed lines represent the Brillouin zone. The highlighted green region represents an oblique band gap. The yellow window corresponds to the asymmetric band gap structure in Fig. 1.

Figure 3

Scattering from a space-time modulated slab. A finite part of a material with permittivity ${\epsilon }_r$ is spatiotemporally modulated with the space-time varying permittivity Eq. (1) or Eq. (4). The structure responds differently when excited from the left side or the right side, and is therefore nonreciprocal. Top arrows represent forward excitation, where the structure is excited from the left. Bottom arrows represent backward excitation, where the structure is excited from the right.

Figure 4

Bloch-Floquet modes excited in the space-time modulated slab of Fig. 3, with the same parameters as in Fig. 2, by an incident wave with frequency ${\omega }_0$. The horizontal dashed line represents ${\omega }_0$ and the dots represent the corresponding excited modes. The red dots represent the modes with a positive group velocity while the blue dots represent the modes with a negative group velocity. The dashed lines delimit by the (oblique) Brillouin zone. The green lines represent the dispersion curves of the incident medium. (a) General representation. (b) Modes or space-time harmonics transferred into the Brillouin zone.

Figure 5

Magnitude of the modes excited in the space-time modulated slab of Fig. 4 with $L=200\lambda$. The normalized excitation frequency is ${\omega }_0=0.259c/a$. (a) Slab excited from the left. The gap mode [red index 0 in Fig. 4] is excited dominantly. All the other forward propagating modes (red) and backward propagating modes (blue) are very weakly excited. (b) Slab excited from the right. The dominantly excited mode is a propagating mode [blue index 0 in Fig. 4].

Figure 6

Reflection and transmission (outside the modulated slab) for the space-time modulated slab with the same parameters as in Fig. 5. (a) Slab excited from the left. The dominantly excited evanescent gap mode decays exponentially and conveys no power to the transmitted region. All of the power is reflected. The reflected wave is blue-shifted. (b) Slab excited from the right. The dominantly excited propagating mode transfers all its energy to the other end. Almost all the power is transmitted at the fundamental frequency (${\omega }_0$).

Figure 7

Dispersion diagram and geometrical parameters in the limit $M\to 0$ for explaining the Doppler shift of the reflected wave in Fig. 6. The solid lines represent the dispersion curves of the background medium and $c$ represents the velocity of light in this medium. The green bands correspond to the infinitesimal gaps.

Figure 8

Explanation of the frequency upshift in the reflected field in terms of intraband photonic transitions. The dashed lines correspond to the dispersion curves of the background medium. The arrows represent the momentum end energy provided by the space-time modulated region, $\pm (k_{\text{m}},{\omega }_{\text{m}})$. (a) Left excitation: the forward propagating mode gradually transforms into a backward propagating mode with frequency ${\omega }_0+{\omega }_{\text{m}}$ and is reflected. (b) Right excitation: the propagating mode passes through the space-time section, as $(k_0\pm k_{\text{m}},{\omega }_0\pm {\omega }_{\text{m}})$ does not correspond to any waveguide mode.

Figure 9

Electric field pattern and frequency spectrum of the gap mode for excitation from the left. (a) Electric field pattern for the evanescent gap mode, corresponding to the red index 0 in Fig. 4. (b) Frequency harmonics of the evanescent gap mode plotted in (a). This mode has only two significant harmonics, at frequencies ${\omega }_0$ and ${\omega }_0+{\omega }_{\text{m}}$.

Figure 10

Normalized impedance for the different harmonics of the gap mode plotted in Fig. 9. Each point represents the normalized impedance corresponding to the ratio $E_n/H_n$ for the frequency harmonic ${\omega }_n={\omega }_0+n{\omega }_{\text{m}}$. The harmonic at frequency ${\omega }_0$ has a positive impedance (forward propagating), and is fully matched to the incident region. The harmonic at frequency ${\omega }_0+{\omega }_{\text{m}}$ has a negative impedance (backward propagating), and is fully matched to the incident region.

Figure 11

Transformation of an incident forward propagating wave to an upshifted backward propagating wave. An incident field with frequency ${\omega }_0$ impinges on the slab from the left side. The only two significant harmonics of the gap mode, at frequencies ${\omega }_0$ and ${\omega }_0+{\omega }_{\text{m}}$, are represented at the right side, where ${\omega }_{\text{m}}$ is the modulation frequency. The incident wave is fully matched to the evanescent harmonic at frequency ${\omega }_0$, therefore it excites this harmonic without back-reflection. The excited evanescent harmonic at frequency ${\omega }_0$ decays exponentially inside the slab, and its energy is transferred to the backward propagating harmonic at frequency ${\omega }_0+{\omega }_{\text{m}}$, whose energy increases exponentially as it reaches the slab interface. The backward propagating harmonic is fully matched to the incident region. It excites the reflected wave at frequency ${\omega }_0+{\omega }_{\text{m}}$ without any back-reflection inside the slab.

Figure 12

Isolation versus modulated slab length for different modulation depths. For weaker modulations, longer space-time sections are required to achieve a specific amount of isolation. For each modulation depth, the isolation saturates at a specific level due to coupling to the undesired propagating modes, which act as a noise floor. For weaker modulations, coupling to undesired modes is weaker and higher isolation levels are achievable.

Figure 13

Isolation versus frequency for different modulation depths and slab lengths corresponding to the knee points in Fig. 12. The bandwidth is directly proportional to the modulation depth.

Figure 14

Operation of the space-time modulated slab as a reflection-type mixer. When the structure is excited from the left at the down-tilted forward band gap, the wave is fully reflected and blue-shifted. When the structure is excited from the right at the up-tilted backward band gap, the wave is fully reflected and red-shifted.

Figure 15

Experimental realization of the space-time varying system in the form of a space-time varying artificial microstrip transmission line. (a) Schematic of the system, with distributed-capacitance varactors modulated by a radio wave emulating Eq. (4). (b) Photograph of the fabricated structure. The varactors are BB833 from Infineon Technologies, with capacitance ratio $C_{\text{max}}/C_{\text{min}}=12$. The structure is $L=8$ in long and is excited at ${\omega }_0=2\pi \times 2.5$ GHz. The substrate is RT6010 from Rogers with permittivity 10.2, thickness $h=100$ mil, and $tan\delta =0.0023$.

Figure 16

Dispersion diagram corresponding to Eq. (16) with parameters ${\omega }_{\text{m}}=2\pi \times 0.675$ GHz, $k_{\text{m}}=415.79$ rad/m, and $M=C_{\text{m}}/C_{\text{av}}=0.15$ in the structure of Fig. 15. The yellow window corresponds to the asymmetric band gap structure in Fig. 1. The red and blue dots represent the dominantly excited mode for forward and backward excitations, respectively.

Figure 17

Experimental (solid lines) vs theoretical (dashed lines) results for the isolator in Fig. 15 for the same parameters as in Fig. 16. (a) Forward excitation: the wave is almost fully reflected at the blue-shifted frequency ${\omega }_0+{\omega }_{\text{m}}=2\pi \times 3.175$ GHz with a transmission level less than $-10$ dB. (b) Backward excitation: the backward incident wave is fully transmitted at ${\omega }_0=2\pi \times 2.5$ GHz. For clarity, the theoretical results are shifted by $0.1{\omega }_{\text{m}}$.