Electro-Optic beamforming in seamless wireless-to-photonic receiver arrays

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I. INTRODUCTION
The proliferating demand for ultra-high capacity based communication links has been the driving force behind the pursuit of Beyond 5G (B5G) [1]- [3] and 6G [4]- [8] systems.However, high channel-fading hinders long-haul wireless transmission in the B5G and 6G frequency regimes.Wirelessover-optical technology [9]- [16] is one of the most promising candidates for overcoming this challenge.In a typical wirelessover-optical link, high-frequency wireless transmission is confined to short-range cellular areas [17].Wireless-to-optical converters [18]- [21] located within such cells convert the received wireless signals to the optical domain.The resultant optical signals are then transmitted over long-distance optical fiber channels, facilitating high-capacity based long-haul data transmission.
Electro-optic modulators form the backbone of wireless-tooptical converters.Over the last few years, Lithium Niobate (LiNbO 3 ) based wireless-to-optical converters have gained prominence owing to their appreciably high electro-optic conversion efficiencies [22].A thin-film LiNbO 3 electrooptic modulator with an undercut structure was presented recently in [23].In [24], hybrid integration of Silicon photonic components and thin-film LiNbO 3 was demonstrated.A novel thin-film LiNbO 3 platform with a transmission line Niloy Ghosh and Sarang Pendharker are with the Department of Electronics & Electrical Communication Engineering, Indian Institute of Technology Kharagpur, W. Bengal, India.architecture was proposed, designed and simulated in [25].A breakthrough experimental demonstration for overcoming the voltage-bandwidth tradeoff in LiNbO 3 -based modulators using multi-structured electrodes was recently reported in [26].
Since traditional wireless-to-optical converters [27] are composed of electronic RF-downconversion stages, their endto-end latency is high.On the other hand, wireless-to-optical converters that bypass the electronic RF-downconversion stages have lower end-to-end latency, making them even more suitable for B5G and 6G applications.Such systems are referred to as seamless wireless-to-optical converters.Much progress has been made in developing antennacoupled electro-optic modulator-based seamless wireless-tooptical converters.Very recently, a W-band seamless converter array integrated with a nonlinear electro-optic polymer was reported in [28].In [29], a seamless wireless-tooptical conversion technique using ferroelectric crystals with polarization-reversal in a LiNbO 3 substrate was proposed.An extensive investigation of a seamless converter based on multiple-quantum-well structures was presented in [30].The importance of seamless wireless-to-photonic converters in photonic electric-field sensing-based applications has also gained much prominence lately [31]- [33].Extensive research is also being pursued on the innovation of novel plasmonicbased seamless wireless-to-optical converters [34]- [36].
While seamless wireless-to-optical conversion has been extensively explored in recent literature, seamless wirelessto-optical digital information mapping remains unexplored.We have recently proposed a bi-layered model of seamless wireless-to-optical constellation-mapping [37].Nevertheless, the aspect of electro-optic beamforming for seamless constellation mapping has been overlooked.Therefore, it is imperative to investigate electro-optic beamforming to realize seamless digital links in the future.
This paper develops the core theory behind seamless electrooptic beamforming for wireless-to-optical constellationmapping.We show that constellation-mapping in seamless wireless-to-optical converters is intrinsically governed by the direction of wireless reception, even in the case of an isotropic converter.This forms the basis for seamless electro-optic beamforming in the optical-phase domain.In section II, we show that seamless electro-optic beamforming is influenced by (i) direction-dependent electro-optic array-factor of the beamformer and (ii) direction-dependent wireless constellationrotation.In section III, we derive the analytical expression of electro-optic array-factor, which is a measure of electro-optic modulation-depth.We then show that the spatial selectivity of beamformers in terms of constellation-mapping can be enhanced along any particular direction in space by suitably choosing its array periodicity.We highlight the nature of inadvertent secondary maximas of the electro-optic arrayfactor, and suggest a potential solution to get rid of them.In section IV, we highlight that in seamless electro-optic beamforming, an effectively rotated version of the transmitted wireless constellation gets mapped to the optical-domain.The effective constellation-rotation angle is shown to be governed by the direction of wireless reception.We also highlight that constellation-rotation does not lead to degenerate mapping for wireless reception along electro-optic maximas.Section V proposes a seamless beamforming configuration enabling wireless-spatial to optical-wavelength division multiplexing.The novel contributions reported in this paper shall lay the foundation for further investigation and practical realization of seamless wireless-over-optical links in next-generation communication systems.

II. SEAMLESS ELECTRO-OPTIC BEAMFORMING
Fig. 1 depicts the model of seamless wireless-to-optical symbol-mapping in an electro-optic beamformer.In this model, an optical carrier E op (t) traveling in the electro-optic beamformer gets phase-modulated by the received digitally modulated wireless signal E w (t) by virtue of Pockel's electrooptic effect [38].Consequently, the digital phase-level b i corresponding to the i th symbol of E w (t) gets encoded within the modulated phase-level θ i of E op (t), resulting in seamless wireless-to-optical symbol-mapping.Furthermore, it can be shown that seamless symbol-mapping is inherently governed by the direction of wireless reception α, as depicted in Fig. 1.This directional-dependence of symbol-mapping lays the foundation for seamless electro-optic beamforming in wireless-over-optical links.
The side-view of an antenna array-based seamless electrooptic beamformer is shown in Fig. 2(a).The beamformer consists of centrally slotted metallic patch antennas of dimension W × L, cascaded at array-periodicity D. Additionally, a single-mode nonlinear optical waveguide made of LiNbO 3 is routed through the array.The individual array elements are appropriately designed to receive y-polarized wireless signals E w (t) of frequency ω w .The free-space phase-constant of E w (t) is k w = ω w /c, where c is the speed of light in air.If E w (t) is incident at an angle α with the z-axis, the freespace vector phase-constant ⃗ k w can be expanded into x-z components as, The i th symbol of a wireless M-PSK modulated signal E w (t) incident on the electro-optic beamformer located at z=0 is expressed as, where |E w | is the electric field-strength of E w (t), and b i is the phase-level corresponding to the i th symbol of E w (t).
When E w (t) is received by the electro-optic beamformer, an electric-field E slot (t) gets induced in the slotted regions of the individual array elements.It was shown in [21] that E slot (t) is the enhanced version of E w (t).
x y We designed a patch antenna array with each element having a 1µm wide centrally located slot.A single-mode LiNbO 3 optical waveguide of dimension 343nm x 183nm was routed through the array.The effective refractive index n op of the optical waveguide was computed to be 1.73.The physical length L of the array elements was chosen to be around 3.57mm to achieve 30GHz wireless reception.The width W of the array elements was selected as 2.9mm in order to optimize wireless-to-optical mapping for 30GHz wireless reception [37].The designed structure was then excited by a y-polarized 30GHz plane-wave E w (t) of strength |E w |=1V/m using full-wave simulation.Consequently, a 30GHz electricfield E slot (t) of strength |E slot |=6.49kV/m was induced in the slots of the individual array elements, as shown in Fig. 2(b).The expression of the instantaneous electric-field E slot n (t) induced in the n th array element when excited by the i th Induction of E slot n (t) causes an instantaneous shift in the effective refractive index of the optical waveguide segment routed through the n th array element.This instantaneous refractive-index shift dn op n (t) [38] is due to Pockel's electrooptic effect in the nonlinear LiNbO 3 optical waveguide, and is governed by following equation.
where r 33 and n op are the Pockel's electro-optic coefficient and the original effective refractive index of LiNbO 3 , respectively.
When a y-polarized optical carrier wavefront E op (t) travels a distance dx through the n th array element, it acquires a differential phase due to electro-optic phase-modulation.This differential phase encapsulates the i th symbol of E w (t), and hence can be denoted as dθ i n (t).
where k op is the phase-constant of E op (t).The net instantaneous phase θ i (t) acquired by E op (t) in an N-element electrooptic beamformer is, where T W is the time elapsed for E op (t) to travel width W of an individual antenna element, T D is the time elapsed for E op (t) to travel distance D between the entry of two consecutive antenna elements, ψ is the electro-optic phasematching factor, and δθ is the electro-optic array-factor.The expressions of ψ and δθ are mentioned in the following equations.
The time-independent offset part of θ i (t) [37] can be expressed as, The electro-optic array-factor δθ is a measure of electro-optic modulation-depth.It is evident from Eq. ( 7) and Eq. ( 8) that δθ is a function of α.Therefore, the array-factor is dependent on the direction of wireless reception.However, it must be pointed out that the antenna gain profile of the individual array elements, which is reflected on the dependence of |E slot | on the wireless reception angle, does not influence δθ.In other words, δθ is intrinsically direction-dependent even if the receiver array of the beamformer is considered to be isotropic.Furthermore, it can also be concluded from Eq. ( 9) that in this mapping scheme, the phase-shifted version of the transmitted wireless symbol b i gets effectively mapped to the optical domain.In other words, the rotated version of the transmitted wireless constellation gets mapped to the optical domain.From Eq.( 10) we can find that the wireless constellationrotation angle γ(α) is also a function of the wireless reception angle.Therefore, electro-optic beamforming is influenced by the array-factor of the beamformer, and effective wireless constellation-rotation, as depicted in Fig. 3.
The influence of direction-dependent electro-optic arrayfactor and wireless constellation-rotation on seamless electrooptic beamforming will be discussed in section III and section IV, respectively.Fig. 3.
Seamless electro-optic beamforming is influenced by directiondependent array-factor and effective wireless constellation-rotation.Before delving deeper into the details of electro-optic beamforming, it is imperative to fundamentally distinguish it from conventional analog beamforming [39].As depicted in Fig. 4(a), conventional analog beamformers operate in the electrical-amplitude domain.The resultant beam pattern in this case represents the directional dependence of the received electrical amplitude V (α).On the other hand, seamless electrooptic beamformers operate in the optical-phase domain, as depicted in Fig. 4(b).In this case, the resultant electro-optic beam pattern indicates the directional profile of the modulated optical phase θ(α).

III. DIRECTION-DEPENDENT ELECTRO-OPTIC ARRAY-FACTOR
It was mentioned earlier that direction-dependent electrooptic array-factor is one of the major factors seamless governing electro-optic beamforming.This section discusses several aspects associated with electro-optic array-factor in detail.

A. Spatially selective electro-optic beamforming
In several practical applications, spatially-selective wirelessto-optical constellation-mapping might play a crucial role.Spatial selectivity of the electro-optic beamformer can be enhanced by maximizing the electro-optic array-factor δθ along the desired direction of interest.The condition for δθ to be maximum can be found out from Eq. ( 8) as, The above equation represents the condition of electro-optic phase-matching.The directions along which δθ is maximum are called the directions of primary maxima.A primary maxima can be oriented along any desired direction by appropriately choosing the array periodicity of the electro-optic beamformer.From Eq. ( 11), the minimum array periodicity D αpm required for orienting the primary maxima of δθ along the direction α pm can be found out to be, Fig. 5(a) shows that a primary maxima can be oriented along α pm =-30 • by choosing the array periodicity of the electrooptic beamformer as D αpm =4.5mm for antenna parameters described in section-II.It can be observed that δθ gets linearly scaled-up with the number of array elements for wireless reception along α pm =-30 • in this case.Consequently, the inter-symbol spacing in the corresponding optical constellation increases in proportion to the number of cascaded array elements as shown in Fig. 5(b).This property can be utilized to mitigate the losses due to free-space wireless fading and increase the noise tolerance of the seamless wireless-to-optical mapping scheme.

B. Secondary maximas in electro-optic beamforming
While in the case discussed above, no maximas other than the desired primary maximas are present in the electrooptic array-factor pattern, it may not always be the case in other scenarios.For instance, the array-factor pattern shown in Fig. 6(a) has a secondary maxima along the direction α sm =-47 • , in addition to a primary maxima along α pm =30 • .Similarly, the array-factor pattern shown in Fig. 6(b) has two secondary maximas along the directions α sm =-60 • and α sm =0 • , in addition to a primary maxima oriented along α pm =60 • .In order to understand the reason behind the presence of secondary maximas in the array-factor pattern, the electrooptic phase-matching conditions for primary and secondary maximas must be segregated.By referring to Eq. ( 11) and Eq. ( 12), the electro-optic phase-matching condition for primary maximas can be stated as, If a wireless signal is incident on the same beamformer along the direction α sm such that Eq. ( 14) is satisfied, a secondary maxima is formed along α sm .
The electro-optic phase-matching conditions for primary and secondary maximas have been schematically summarised in Fig. 7.
The relationship between α sm and α pm can be found out by substituting the expression of D αpm from Eq. ( 12) in Eq. ( 14).
It can be interpreted from Eq. ( 15) that all the possible solutions of α sm are governed by α pm .In other words, once the electro-optic beamformer is customized for a particular primary maxima, the orientations of the inadvertent secondary maximas get locked.Consequently, there remains no independent control over the existence of secondary maximas, thereby rendering them inadvertent.
The presence of inadvertent secondary maximas in arrayfactor pattern limits the spatial selectivity of the electro-optic beamformer.However, it is possible to get rid of secondary maximas in certain cases.This will be discussed in the upcoming subsection.

C. Avoiding secondary maximas in electro-optic beamforming
It can be found out from Eq. ( 15) that no solutions of α sm exist when the electro-optic beamformer is customized for a primary maxima α pm < 0. However, when α pm > 0, one or more solutions of α sm may exist.In other words, while secondary maximas may be present when α pm > 0, they are always absent when α pm < 0. We can therefore classify electro-optic beamforming into two independent categories based on whether the electro-optic beamformer is customized for a negative or positive value of α pm .
As per our convention, for an electro-optic beamformer customized for a positive α pm , the x-component of ⃗ k w i.e. k +x w is aligned along ⃗ k op , as shown in Fig. 8(a).This configuration is called the parallel-k mode of electro-optic beamforming.Similarly, for an electro-optic beamformer customized for a negative α pm , the x-component of ⃗ k w i.e. k −x w is aligned opposite to ⃗ k op , as shown in Fig. 8(b).This configuration is called the antiparallel-k mode of electro-optic beamforming.It can be verified from Fig. 8(c) that secondary maximas cease to exist in the antiparallel-k mode of electro-optic beamforming.
We will discuss an interesting application of seamless constellation-mapping based on antiparallel-k mode of beamforming in section V. Prior to that, we will discuss about the impact of direction-dependent wireless constellation-rotation on seamless wireless-to-optical mapping in section IV.

IV. DIRECTION-DEPENDENT EFFECTIVE WIRELESS CONSTELLATION-ROTATION
It was mentioned earlier that in addition to directiondependent electro-optic array-factor, wireless constellationrotation plays a crucial role in seamless electro-optic beamforming.From Eq. ( 7), Eq. ( 10) and Eq. ( 12) , the expression of the wireless constellation-rotation angle γ(α) in an Nelement electro-optic beamformer customized for a primary maxima along α pm can be found out to be, From the above equation, it is evident that the wireless constellation-rotation angle is a function of the wireless reception angle.In [37], the challenge of degenerate mapping in seamless digital wireless-to-optical conversion was reported.In order to overcome this challenge, a modified M-PSK modulation scheme with all the symbols confined in the upper half of the wireless constellation was proposed.However, this solution may not ensure non-degenerate mapping in direction-dependent seamless mapping because of wireless constellation-rotation.Due to wireless constellation-rotation, not all wireless symbols will effectively remain confined in the upper half of the constellation, thereby increasing the risk of optical symbol degeneracy.Fig. 9(a) shows the original wireless constellation of a modified M-PSK wireless signal, in which two consecutive symbols i & j differ by the phase ∆b ij .From the geometry of this constellation, it can be found that degenerate mapping occurs when the effective wireless constellation-rotation angle γ(α) is, where m is any integer except 2π/∆b ij , 4π/∆b ij , 6π/∆b ij , and so on.The angle of reception α=α d for which the above equation is satisfied can be found by substituting the expression of γ(α) from Eq. ( 16) into Eq.( 17).
where m is any integer except (2π/∆b ij ), (4π/∆b ij ), (6π/∆b ij ), and so on.For the modified 4-PSK constellation shown in Fig. 9(b) with ∆b ij =π/3, the above-mentioned equation becomes, From Eq. ( 19), it can be found that the number of solutions of α d increases as N is increased.This can be verified from Fig. 9(c), which shows the variation of the net optical phase θ i with the angle of reception of 30GHz wireless Modified 4-PSK signal on the electro-optic beamformer optimized for reception along -30 • .It can be observed that the number of inter-symbol intersections increase when N =6, as compared to when N =4.The angle of reception α d corresponding to each inter-symbol intersection denotes the angle of reception for which the effective wireless constellation-rotation results in optical symbol degeneracy.
However, it must be pointed out from Fig. 9(c) that optical symbol degeneracy does not occur for wireless reception along the primary maxima.The reason behind this can be explained using Eq. ( 16), according to which γ(α = α pm ) is equal to integral multiples of π.This implies that for wireless reception along the primary maxima, the wireless constellation with circular geometry rotates by integral multiples of π.Consequently, the effective constellation-rotation does not cause optical symbol degeneracy.Therefore, even though the number of degeneracy spots increase and the electro-optic beam becomes narrower with the increase in the number of array elements, each wireless symbol received along the maxima gets mapped to unique optical symbol.Therefore, wireless constellation-rotation does not cause optical symbol degeneracy for reception along maximas.
In the upcoming section, we will propose a seamless electrooptic beamforming configuration that enables wireless-spatial to optical-wavelength division multiplexing.

V. SEAMLESS WIRELESS-SPATIAL TO OPTICAL-WAVELENGTH DIVISION MULTIPLEXING
The proposed seamless wireless-spatial to opticalwavelength division multiplexing scheme has been schematically represented in Fig. 10(a).In this scheme, wireless symbols incident along two different directions (α pm1 & α pm2 =-α pm1 ) are independently mapped to the optical domain in an electro-optic beamformer whose array periodicity is appropriately chosen in order to operate it in antiparallel-k mode.In this scheme, two independent wireless symbols are made to ride upon two counter-propagating optical carriers of slightly different wavelengths (λ 1 & λ 2 ), which are then multiplexed over a single optical-fiber channel.The overall directional electro-optic array-factor pattern of this configuration is shown in Fig. 10(b).
The working of this configuration can be elaborated using Fig. 11.A Laser Diode LD-1, generates an optical carrier E op1 (t) of wavelength λ 1 , propagating along the +xdirection.The optical signal E op1 (t) is then circulated from port-1 to port-2 of circulator-1, after which it enters the electro-optic beamformer block.We consider the electro-optic beamformer to be simultaneously excited by two wireless α pm1 signals E w1 (t) and E w2 (t), received along α pm1 and α pm2 , respectively (where α pm2 =-α pm1 ).Therefore, E op1 (t) forms an antiparallel-k pair with E w1 (t), and a parallel-k pair with E w2 (t).Since the array periodicity of the electro-optic beamformer is chosen to be D αpm1 , it essentially operates in the antiparallel-k mode from the point of view of E op1 (t).Consequently, the wireless symbol d i of E w1 (t) received along α pm1 gets encoded within the optical signal E op1 (t) as θ i (d i , α pm1 ), while it remains blind to E w2 (t).Another optical carrier E op2 (t) of wavelength λ 2 propagating along the -x-direction is generated by Laser Diode LD-2.The optical signal E op2 (t) enters the electro-optic beamformer after being circulated from port-1 to port-2 of circulator-2.In this case, E op2 (t) forms an antiparallel-k pair with E w2 (t), and a parallel-k pair with E w1 (t).Since, α pm2 =α pm1 , it can be deduced that D αpm1 =D −αpm2 .Therefore, the electro-optic beamformer operates in antiparallel-k mode from the point of view of E op2 (t).Consequently, the wireless symbol b i of E w2 (t) received along α pm2 gets encoded within E op2 (t) as θ i (b i , α pm2 ), while it remains blind to E w1 (t) received along α pm1 .An optical coupler multiplexes the phase-modulated versions of E op1 (t) and E op2 (t) into a single optical fiber channel, thereby facilitating wireless-spatial to optical-wavelength division multiplexing, as shown in Fig. 11.

VI. CONCLUSION
In this paper, it has been shown that constellation-mapping in an electro-optic beamformer is dependent on the wireless reception angle through a nonlinear relationship.We showed that seamless electro-optic beamforming is governed by direction-dependent electro-optic array-factor, and wireless constellation-rotation.We showed that the spatial-selectivity of an electro-optic beamformer can be enhanced by suitably choosing its array periodicity.The challenge of degenerate constellation-mapping due to direction-dependent effective wireless constellation-rotation has been pointed out.However, we have also pointed out that wireless constellation-rotation does not cause optical symbol degeneracy for wireless reception along electro-optic maximas.We have also proposed a novel electro-optic beamforming configuration that enables spatial-to-wavelength division multiplexing.Unlike conventional analog beamformers, the seamless electro-optic beamformer presented in this paper does not consist of an electronic RF-downconversion chain, thereby ensuring low latency.The theoretical contributions and the design guidelines mentioned in this paper will be crucial for further investigation and practical realization of seamless wireless-over-optical links in the next-generation communication systems.

Fig. 2 .
Fig. 2. (a) Side-view of a 2-element slotted patch antenna array-based seamless electro-optic beamformer.(b) A 30GHz electric-field E slot (t) of strength |E slot |=6.49kV/m is induced in the 1µm wide centrally located slot of a 2.9mm x 3.57mm patch antenna on receiving a y-polarized 30GHz wireless signal Ew(t) of field-strength |Ew|=1V/m.

Fig. 4 .
Fig. 4. Schematic representation of (a) Conventional analog beamforming performed in the electrical-amplitude domain, and (b) Seamless electro-optic beamforming performed in the optical-phase domain.

Fig. 6 .
Fig. 6.(a) A secondary maxima is obtained in the electro-optic array-factor pattern along αsm=-47 • when the electro-optic beamformer is customized for a primary maxima along αpm=30 • by setting the array-periodicity as Dα pm =8.1mm for 30GHz reception.(b) Two secondary maximas are obtained in the electro-optic array-factor pattern along αsm=-60 • and αsm= 0 • , when the electro-optic beamformer is customized for a primary maxima along αpm=60 • by setting the array-periodicity as Dα pm =11.58mm for 30GHz reception.

Fig. 10 .
Fig. 10.(a) Schematic representation of dual antiparallel-k mode of electrooptic beamforming for wireless-spatial to optical-wavelength division multiplexing.(b) The overall electro-optic array-factor pattern for dual antiparallelk mode of electro-optic beamforming.

Fig. 11 .
Fig. 11.Proposed setup for dual antiparallel-k mode for wireless-spatial to optical-wavelength division multiplexing.