Double spectral sampling transceiver array

Chapter V: Optical phased array transceiver

5.3 Double spectral sampling transceiver array

of the sampling is

𝐸_{𝑓 𝑠}(𝑓_𝑥, 𝑓_𝑦) =

+∞

Õ

𝑚𝑥=−∞

+∞

Õ

𝑚𝑦=−∞

𝐸_𝑓(𝑓_𝑥+ 𝑚_𝑥 𝑋_𝑡

, 𝑓_𝑦+ 𝑚_𝑦 𝑌_𝑡

). (5.5)

Propagation towards the far-field has a low-pass filtering effect on the spectral content [7] such that only spectral components with

𝑓²

𝑥 + 𝑓²

𝑦 ≤ 𝜆², (5.6)

will be present at the far-field. This effect can also be understood by studying equation (2.3) which shows that there are no real valued angles𝜃and𝜙corresponding to the spatial frequencies 𝑓_𝑥 and 𝑓_𝑦 out of this range. Therefore, equation (5.6) specifies the spectral components that form the far-field which is called thevisible range. For 1D arrays,𝑌 and 𝑓_𝑦 can be ignored to simplify the analysis without loss of generality and array pattern is studied for𝜙 =0. Therefore, the visible range for a 1D array reduces to

|𝑓_𝑥| ≤ 𝜆, (5.7)

and

𝑓_𝑥 = 1

𝜆sin𝜃 . (5.8)

Here, for presentation clarity purposes, we do not show the low-pass filtering effect in the derivations while its effect is considered.

Assuming that the spectral components of the continuous excitation 𝐸(𝑥 , 𝑦) are mainly inside the visible range, for sampling periods of𝑋_𝑡 =𝑌_𝑡 =𝜆/2, the spectral replicas will fall outside of the visible range and are far enough to avoid aliasing in the far-field, Fig 5.2(b). Therefore, these replicas (which are artifact of sampling) are filtered out by the low-pass property of free-space propagation and will not affect the radiation pattern. Increasing the element spacings 𝑋_𝑡 and𝑌_𝑡 which translates to reducing the sampling frequency can causes aliasing and more than one replica of the spectral content will appear in the visible range. For instance, for element spacing𝑑 =𝜆, 2 replicas are present in the far-field, Fig. 5.2(c), and for𝑑 =2𝜆, 4 replicas.

While undersampling might leads to aliasing in general, if the aperture excitation is adjusted to form a beam, its Fourier transform is localized in only a fraction of the visible range, and sampling will not cause any spectral overlap or aliasing (Fig. 5.3), and thus the replicated beams are still distinguishable. In other words, the grating lobes which are formed due to aliasing do not overlap even if only two samples

Figure 5.2: (a) Fourier transform of the aperture excitation which is simplified to one dimension (1D array)𝐸_𝑓(𝑓_𝑥, 𝑓_𝑦)(b)

are used in each dimension for the whole aperture. Therefore,undersampling the aperture yields a multi-beam transmitter and each beam covers a fraction of the FOV through beam steering.

Figure 5.3: (a) Undersampling an aperture which is tuned to form a beam does not lead to aliasing.

Since an undersampled transmitter radiates multiple beams, the receiver used to record the reflection needs to be directive to separate the signals coming back from each beam. Therefore, the waveform of the incident wave-front should be captured

by the receiver to extract information about the direction of incidence2. A single photodetector can not be used for this purpose since it does not record any informa- tion about the shape of the incident wave and only measures the incident amplitude.

To record the electric field distribution on the receiver aperture an array of receiving elements can be used to sample the impinging wave-front. Since propagating back from the far-field to the receiver adds another Fourier transformation to the field distribution, the spectrum of the received wave is the same as equation (5.5) due to the Fourier transform duality property. The effect of sampling on the receiver aperture with antennas which are spaced 𝑋_𝑟 and𝑌_𝑟 apart along x and y directions leads to a spectral content of

𝐸_{𝑅𝑥 𝑓 𝑠} =Õ

𝐸_𝑓(𝑓_𝑥+ 𝑚_𝑥 𝑋_𝑡

+ 𝑛_𝑥 𝑋_𝑟

, 𝑓_𝑦 + 𝑚_𝑦

𝑌_𝑡 +

𝑛_𝑦 𝑌_𝑟

), (5.9)

in whichsumis over all the indices. This equation can be modified to 𝐸_{𝑅𝑥 𝑓 𝑠} =Õ

𝐸_𝑓(𝑓_𝑥+ 𝑚_𝑥𝑋_𝑟 +𝑛_𝑥𝑋_𝑡 𝑋_𝑡𝑟

, 𝑓_𝑦+

𝑚_𝑦𝑌_𝑟+𝑛_𝑦𝑌_𝑡 𝑌_𝑡𝑟

), (5.10) 𝑋_𝑡𝑟 = 𝑋_𝑡𝑋_𝑟 and 𝑌_𝑡𝑟 =𝑌_𝑡𝑌_𝑟. (5.11) Therefore, the undersampling on the receiver aperture adds extra aliasing compo- nents to the spectrum. If the width of the beam is small enough such that all the shifted replicas of𝐸_𝑓(𝑓_𝑥, 𝑓_𝑦)are separated which translates to

(𝑚_𝑥−𝑚

0

𝑥)𝑋_𝑟 + (𝑛_𝑥−𝑛

0

𝑥)𝑋_𝑡 𝑋_𝑡𝑟

≥ 𝐵𝑊_𝑥 𝑖 𝑓 (𝑚_𝑥, 𝑛_𝑥) ≠ (𝑚

0

𝑥, 𝑛

0

𝑥), (5.12) and

(𝑚_𝑦−𝑚

0

𝑦)𝑌_𝑟 + (𝑛_𝑦−𝑛

0

𝑦)𝑌_𝑡 𝑌_𝑡𝑟

≥ 𝐵𝑊_𝑦 𝑖 𝑓 (𝑚_𝑦, 𝑛_𝑦) ≠ (𝑚

0

𝑦, 𝑛

0

𝑦), (5.13) there is no overlap between the replicated beams and the information can be ex- tracted. If equation (5.12) is not met, the overlap between the replicas will create large side lobes. Moreover, this equation does not considered the amplification of side lobes created by one aperture with a grating lobe of the other. Therefore, to achieve the optimum values of𝑋_{𝑡 ,𝑟} and𝑌_{𝑡 ,𝑟} for minimum side lobe, we can perform an optimization over these parameters. For a 1D OPA transceiver, only 𝑋_𝑡 and 𝑋_𝑟 are relevant which are the sampling periods (element spacing) on the transmitter and receiver, respectively. Since these are only two free variables, it is also possible to plot the maximum side lobe level of the full transceiver chain versus 𝑋_𝑡 and 𝑋_𝑟.

2Lenses perform wave-front processing in many optical systems.

Figure 5.4 shows the plot for the maximum side lobe level of a transceiver with an 8-element array on the transmitter side and an 8-element array on the receiver side achieving the minimum of−17.85 dB for𝑋_𝑡 =2𝜆and 𝑋_𝑟 =2.75𝜆. Figure 5.5 shows the same plot for a 16-element arrays on both transmitter and receiver sides achieving minimum side lobe level of −24.3 dB for 𝑋_𝑡 = 2.9𝜆 and 𝑋_𝑟 = 2𝜆3. In these plots, a minimum of 2𝜆 element spacing is assumed considering the typical size of the nano-photonic antennas.

Figure 5.4: Side lobe level of the transceiver with an 8-element array transmitter and an 8-element array receiver with minimum side lobe level of−17.85 dB . It should be noted that in equation (5.10) an infinite aperture is assumed for the receiver. To understand the effect of the limited aperture, the receiver can be mod- eled as an OPA receiver. Since the receiver element spacing is larger than 𝜆/2, its array pattern has several grating lobes. However, for a large receiver aperture, if the receiver pattern is steered across the FOV, maximum of one receiver beam overlaps with one transmitter beam at each steering angle, Fig. 5.7. Therefore, the signal reflected from each illuminated point can be captured by the receiver when the other points are filtered by the array pattern and steering the beam collects all the information. However, the finite aperture size (and thus beamwidth) of the transmit- ter and receiver leads to an averaging effect for the neighboring points of the main beam direction. To consider this effect and also achieve a faster optimization/sweep

3If element spacings are limited to only integer multiples of 𝜆/2, then the non-overlapping condition is met only if the integer coefficients are co-prime with respect to each other, leading to a co-prime sampling technique [132], [133].

Figure 5.5: Side lobe level of the transceiver with an 16-element array transmitter and an 16-element array receiver with minimum side lobe level of−24.3 dB .

algorithm, the patterns of the transmitter and the receiver can be multiplied to get the full transceiver pattern.

Figure 5.6 shows the transceiver design with 2-element arrays on the transmitter and receiver apertures. The beam patterns of the two aperture are wide and have a considerable overlap that leads to large transceiver pattern side lobes. Increasing the number of elements, Fig. 5.7, reduces the beamwidth and thus higher isolation between information of the beams. Since the element spacing is defined by the maximum side lobe level, to further reduce the overlap and account for angle tuning inaccuracies, larger element count can be used and the optimization should be repeated. Since efficient power splitters are in the form of binary trees, the practical element counts are integer powers of 2.