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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 17, Iss. 16 — Aug. 3, 2009
  • pp: 13502–13515
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Enhancement of optics-to-THz conversion efficiency by metallic slot waveguides

Zhichao Ruan, Georgios Veronis, Konstantin L. Vodopyanov, Marty M. Fejer, and Shanhui Fan  »View Author Affiliations


Optics Express, Vol. 17, Issue 16, pp. 13502-13515 (2009)
http://dx.doi.org/10.1364/OE.17.013502


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Abstract

A metallic slot waveguide, with a dielectric strip embedded within, is investigated for the purpose of enhancing the optics-to-THz conversion efficiency using the difference-frequency generation (DFG) process. To describe the frequency conversion process in such lossy waveguides, a fully-vectorial coupled-mode theory is developed. Using the coupled-mode theory, we outline the basic theoretical requirements for efficient frequency conversion, which include the needs to achieve large coupling coefficients, phase matching, and low propagation loss for both the optical and THz waves. Following these requirements, a metallic waveguide is designed by considering the trade-off between modal confinement and propagation loss. Our numerical calculation shows that the conversion efficiency in these waveguide structures can be more than one order of magnitude larger than what has been achieved using dielectric waveguides. Based on the distinct impact of the slot width on the optical and THz modal dispersion, we propose a two-step method to realize the phase matching for general pump wavelengths.

© 2009 Optical Society of America

1. Introduction

Many important applications of THz radiation [1

1. M. Tonouchi, “Cutting-edge terahertz technology,” Nature Photonics 1, 97 (2007). [CrossRef]

] demand coherent THz sources. Among all techniques to generate coherent THz radiation, the difference frequency generation (DFG) process is of importance because it offers the advantages of relative compactness, straightforward alignment, and room-temperature working environment. In difference frequency generation, two optical pump beams, with their frequencies separated by a few THz, interact through a χ (2) process to generate a THz beam. In general, the conversion efficiency of DFG is proportional to the intensities of the pump beams [2

2. Y. R. Shen, The principles of nonlinear optics (New York, Wiley-Interscience, 1984).

, 3

3. R. W. Boyd, Nonlinear Optics (Academic Press, 2003).

]. Consequently, in bulk crystals, it is desirable to use a beam with a small radius. On the other hand, an excessively small beam radius can result in strong diffraction effects. As a result, there exists an optimal beam radius. In a bulk GaSe nonlinear crystal, for example, when generating 0.914 THz wave from the mixing of two CO2 lasers near 10 µm, the maximum efficiency of 3.9×10-9 W-1 occurs at an optimal optical beam radius of 10.59 µm [4

4. Y. Jiang and Y. J. Ding, “Efficient terahertz generation from two collinearly propagating CO2 laser pulses,” Appl. Phys. Lett. 91, 091108 (2007). [CrossRef]

].

Fig. 1. (a) Schematic of the metal slot waveguide integrated with an embedded GaAs strip: The background is quartz, the silver film and the GaAs strip have the same thickness h, and the width of the slot and the GaAs strip are w 1 and w 2, respectively. (b) Orientation of the embedded GaAs.

The paper is organized as follows: In Section 2, we outline the basic theoretical requirement for efficient frequency conversion using coupled-mode theory. In Section 3, we present an example of a metallic waveguide structure designed to have a high conversion efficiency when pumped at about 2 µm wavelength. In Section 4, we discuss in detail many of the considerations that lead to the design in Section 3. Finally, in Section 5, we propose a general method for designing such waveguides at different pump wavelengths.

2. Theory

We start by outlining the basic theoretical requirement for efficient frequency conversion. For this purpose, we use a form of coupled-mode theory that is fully vectorial, and is developed specifically for waveguides with substantial loss.

Consider the DFG process in a waveguide where two optical beams at frequencies ω 2 and ω 3 mix to produce a THz beam at ω 1=ω 3-ω 2. For each frequency ω j, we assume that the corresponding electromagnetic fields {E j,H j} are in a single mode that propagates along the +z direction, and can be written in the following form:

Ej=Aj(z)(ej,t+ej,z)exp(iβjz)
Hj=Aj(z)(hj,t+hj,z)exp(iβjz).
(1)

dσ·12(ej,t×hj,t)=1,
(2)

cj=dσ·12Re{ej,t×hj,t*}.
(3)

Notice that for lossless waveguides, one can show that both ej,t and hj,t can be taken to be real, and hence cj=1, reproducing the usual normalization condition [16

16. A. Snyder and J. Love, Optical waveguide theory (Kluwer Academic Pub, 1983).

]. Our normalization relation of Eq. (2), however, is more general, and is necessary for the lossy waveguide cases.

Following the derivation in the Appendix, in terms of the mode amplitudes Aj(z), the coupled-mode equations that describe the DFG process in the waveguide are:

A1z=α12A1+iω14κ1A2*A3exp(iΔβz)
A2z=α22A2+iω24κ2A1*A3exp(iΔβz)
A3z=α32A3+iω34κ3A1A2exp(iΔβz),
(4)

where α j is the power propagation loss coefficient for the j-th mode, Δββ 3-β 2-β 1, and the coupling coefficients are

κ1=dσχ=:e2*e3·(e1te1z)
κ2=dσχ=:e1*e3·(e2te2z)
κ3=dσχ=:e1e2·(e3te3z).
(5)

We note that the coupling coefficients have unit of s/(m√W). For a waveguide with length L, pumped by two optical beams with amplitudes A 2(0) and A 3(0) at z=0, the conversion efficiency is: η=P1(L)P2(0)P3(0)=c1c2c3A1(L)A2(0)A3(0)2 where P 1 is the generated THz power and P 2 and P 3 are the pump powers. Under the approximation where the pump is not depleted by the nonlinear conversion process, one can solve Eq. (4) analytically to obtain the conversion efficiency:

η=c1c2c3ω12κ1216exp(α1α2α32L+iΔβL)1α1α2α32+iΔβ2exp(α1L)
(6)

Examining Eq. (6), we note several key factors that are important in order to achieve high conversion efficiency:

(b) Phase matching. The conversion efficiency is typically maximized when the phase-matching condition is satisfied, i.e. Δβ=0. As an illustration, assuming that loss is zero, the efficiency η [Eq. (6)] is then proportional to L 2. Thus, one can achieve high conversion efficiency simply by increasing the length of the waveguide. Since ω 1=ω 3-ω 2, the phase matching condition can equivalently be described as nTHz=ng, where nTHz= 1/ω 1 is the phase index of the THz waves, ng=cβ3β2ω3ω2 is approximately the group index of the optical waves, and c is the speed of light in vacuum.

(c) Reducing propagation losses. Setting Δβ=0 in Eq. (6) and solving /dL=0, we obtain the optimal waveguide length which maximizes the conversion efficiency:

Lmax=2α1α2α3ln(α1α2+α3)
(7)

Thus, the presence of propagating losses limits the length of the waveguides that can be used for conversion purposes, even when phase-matching condition is satisfied.

3. An example of a high-efficiency waveguide device

Based on the theoretical condition presented above, we consider a waveguide geometry as shown in Fig. 1(a). The background material is quartz, and the silver film and the GaAs strip have the same thickness of h=0.3 µm. The width of the slot and the GaAs strip are w 1=4 µm and w 2=0.24 µm respectively.

In choosing the background material, it is desirable that: (a) The background material has low loss in both the optical and THz wavelength; (b) Its index in the THz region is substantially higher than in the optical region. Thus, quartz becomes an interesting choice. Other material can be used as background as well. From a loss perspective alone, an obvious choice of background material might have been air, provided that a long GaAs air bridge can be made (for an example of a GaAs air bridge, see [17

17. P. R. Villeneuve, S. Fan, J. D. Joannopoulos, K. Y. Lim, G. S. Petrich, L. A. Kolodziejski, and R. Reif, “Air-bridge microcavities,” Appl. Phys. Lett. 67, 167 (1995). [CrossRef]

]). In such a case, however, it turns out to be substantially more difficult to design the system for phase matching, while maintaining strong optical confinement in the GaAs region. Quasi-phase matching techniques [18

18. K. L. Vodopyanov, M. M. Fejer, X. Yu, J. S. Harris, Y. S. Lee, W. C. Hurlbut, V. G. Kozlov, D. Bliss, and C. Lynch, “Terahertz-wave generation in quasi-phase-matched GaAs,” Appl. Phys. Lett. 89, 141,119 (2006). [CrossRef]

], therefore, may be needed.

The focus of our paper is obviously theoretical with the aim to highlight the basic operating principles and considerations for this class of structures. Nevertheless, one could imagine a fabrication process, for the structure shown in Fig. 1, through a combination of wafer bonding [19

19. I. Mehdi, S. C. Martin, R. J. Dengler, R. P. Smith, and P. H. Siegel, “Fabrication and performance of planar Schottky diodes withT-gate-like anodes in 200-GHz subharmonically pumped waveguide mixers,” IEEE Microwave and Guided Wave Lett. 6, 49–51 (1996) [CrossRef]

, 20

20. S. M. Marazita, W. L. Bishop, J. L. Hesler, K. Hui, W. E. Bowen, T. W. Crowe, V. M. W. Inc, and V. A. Charlottesville, “Integrated GaAs Schottky mixers by spin-on-dielectric wafer bonding,” IEEE Trans. Electron Devices 47, 1152–1157 (2000) [CrossRef]

] and micro-manipulation techniques [21

21. P. H. Siegel, R. P. Smith, M. C. Graidis, and S. C. Martin, “2.5-THz GaAs monolithic membrane-diode mixer,” IEEE Trans. Microwave Theory Tech. 47, 596–604 (1999) [CrossRef]

, 22

22. K. Aoki, H. T. Miyazaki, H. Hirayama, K. Inoshita, T. Baba, K. Sakoda, N. Shinya, and Y. Aoyagi, “Microassembly of semiconductor three-dimensional photonic crystals,” Nature Materials 2, 117–121 (2003). [CrossRef] [PubMed]

]. The process consists of three steps: 1) A GaAs-on-quartz wafer is fabricated by the bonding technique, and then a GaAs strip is defined by etching the top GaAs layer; 2) Two metal regions and a void slot are fabricated on another quartz wafer by lithography; 3) The first wafer is mounted face-down on the second by the micro manipulation techniques.

The structure is designed to operate with two optical pump frequencies at 148.5 and 151.5 THz, respectively (the corresponding wavelengths are 2.02 and 1.98 µm). These two optical pumps will mix to generate a wave at 3THz. The refractive indices of all materials involved at these frequencies are summarized in Table 1.

Table 1. Refractive index (n) of materials in the waveguide[23, 24, 25, 26]

table-icon
View This Table

In the nonlinear conversion process, we assume that the two optical pumps have their electric fields predominantly along the y direction, and the THz wave has its electric field along the x direction. In order for these waves to interact, the orientation of GaAs is chosen such that a [011] direction coincides with the y-axis, and a [100] direction coincides with the x-axis [Fig. 1(b)]. As a result, the two optical pump beams can generate a polarization in THz along the x-axis through the d 14 component in the χ̿(2) tensor of GaAs. Here we use d 14=46.1 pm/V [27

27. K. Vodopyanov, “Optical THz-wave generation with periodically-inverted GaAs,” Laser and Photonics Reviews 2, 11 (2008). [CrossRef]

].

Figure 2(a) plots the THz output power versus the waveguide length for different pump powers. The conversion efficiency typically increases as a function of waveguide length, until the loss of the waveguide becomes substantial. At all pump powers, the maximum THz output occurs when the waveguide length is approximately 10 mm, as compared to Lmax=10.4 mm calculated by Eq. (7) under the undepleted-pump approximation. Figure 2(b) shows the THz output power for a 10 mm long waveguide as a function of the pump power. The conversion efficiency is 5.66×10-6 W-1, as compared to η=5.71×10-6 W-1 calculated by Eq. (6). Thus, the full solution of the nonlinear ordinary differential equations of Eq. (4) indicates that the non-depleted pump approximation is generally valid in this system. The conversion efficiency is more than one order of magnitude larger than the highest previously reported for conventional dielectric waveguides [11

11. C. Staus, T. Kuech, and L. McCaughan, “Continuously phase-matched terahertz difference frequency generation in an embedded-waveguide structure supporting only fundamental modes,” Opt. Express 16, 13296–13303 (2008). [CrossRef] [PubMed]

].

Fig. 2. (a) THz output power as a function of the waveguide length. Here we assume that the two optical pumps have equal power. (b) THz output power for the 10 mm long waveguide as a function of the pump power.

4. Discussion of the design requirement

In the THz frequency range, the waveguide supports a single quasi-TEM mode with its electric field lines going from one metal region to the other [Fig. 4(a–b)]. Notice that the mode is mostly confined in the slot region. Thus, the mode has a transverse dimension of approximately 8 µm that is much smaller than the 100 µm free space wavelength of a 3 THz wave. Since the THz mode is a quasi-TEM mode, its dispersion relation is strongly influenced by the materials in the slot, which for our geometry is mostly filled with quartz. Our FDFD calculations show a nTHz≃2.14, as compared to the refractive index of 2.13 for quartz at the same wavelength. In the THz wavelength range, there is typically substantial material loss due to phonon-polariton excitations. At 3 THz, the loss coefficient of quartz is approximately 0.6 cm-1. The use of metal induces additional THz loss due to the finite penetration of fields into the metal. The penetration depth is approximately 50 nm. The FDFD calculations indicate that the THz mode has a loss coefficient α 1≃4.59 cm-1. Thus, in this structure the loss at the THz wavelength range is dominated by metal loss.

Fig. 3. Dispersion relation of the slot waveguide shown in Fig. 1. All frequencies are normalized with respect to a length scale of a=1µm. (a) In the optical frequency range 0.3~0.55 (c/a) (i.e. 90~165 THz), the waveguide supports two modes: The electric fields of these two modes are polarized predominantly along the y or x-axis. The dispersion relations of these two modes correspond to the solid and dashed line, respectively. The dotted line is the light line of quartz. (b) Dispersion relation of the slot waveguide in the THz range.

Fig. 4. (a) Power density profile and (b) the real part of Ex of the guided mode at the frequency f=0.01(c/a)=3 THz in Fig. 3(b). (c) Power density profile and (d) the real part of Ey of the second waveguide mode [solid line in Fig. 3(a)] at f=0.495(c/a)=148.5 THz. The white lines give the outline of the waveguide structure.

5. Design procedure for general pump wavelengths

The example presented above is for a specific set of pump wavelengths. Here we point out a design procedure for general pump wavelengths. An important feature, from the discussions above, is that the metal region needs to be placed sufficiently far away from the GaAs strip in order to reduce optical loss. Here, we show that this feature allows near-independent design of optical and THz modes for phase-matching purposes. As a result, we present a two-step process that allows design of slot waveguides for THz conversion process at general pump wavelengths. Because the method is based on the dispersion relation of guided modes, rather than the material dispersion, it can be applied to other material systems.

Fig. 5. Optical propagation loss at ω 2 as a function of slot width. Here the dimensions of the GaAs strip and the thickness of the metal film are fixed at w 2=0.24 µm and h=0.3 µm.
Fig. 6. (a) Coupling coefficient as a function of slot width. (b) Maximum conversion efficiency as a function of slot width. The other geometry parameters are fixed at w 2=0.24 µm and h=0.3 µm, and the pump power is 0.5W. The solid (dashed) line denotes the case without (with) considering the phase mismatching induced by the slot width variation.

Fig. 7. Dispersion relations for two waveguides: (i) a waveguide as in Fig. 1 with slot width w 1=3 µm (solid line); (ii) a GaAs strip waveguide without metal slot structure (circles). In both cases, the dimensions of the GaAs strip are w 2=0.24µm and h=0.3 µm.
Fig. 8. (a) Effective index and (b) propagation loss for 3 THz guided modes as a function of w 1, where the other geometry parameters are fixed at w 2=0.4 µm and h=0.3 µm.

6. Conclusion

This work is supported by AFOSR (Grant No. FA9550-1-04-0437).

Appendix: Derivation of the coupled-mode equations for difference frequency generation in a lossy waveguide under nonlinear interaction

Here we derive the coupled-mode equations to describe nonlinear interaction of co-propagating waves in a waveguide. The derivation is based on the reciprocity theorem, which gives rise to an orthogonality theorem of guided modes. As a starting point, we briefly review these two theorems. Their detailed derivations can be found in Ref. [29

29. P. Bienstman, “Rigorous and efficient modelling of wavelength scale photonic components,” Universiteit Gent Thesis (2001).

].

Reciprocity theorem for waveguides - Consider two guided waves {E i,H i} (i=1, 2) in a z-invariant waveguide, each excited by a current density J i, respectively. From Maxwell’s equations:

×E1=iωμH1
×H1=iωεE1+J1
×E2=iωμH2
×H2=iωεE2+J2,
(8)

and by further assuming that the current densities J i are continuously varying along z, we obtain the reciprocity theorem:

sdσ·z(E1×H2E2×H1)=sdσ(J1·E2J2·E1)
(9)

where S is any slice of the waveguide normal to z. Importantly, this theorem is valid for both lossless and lossy media, i.e. it applies to the case where ε is complex.

Orthogonality theorem of guided modes - We consider a source-free waveguide. Suppose {E i,H i} (i=1, 2) are two guided modes satisfying Eq. (8) with J 1=J 2=0. Splitting each of these two modes into the transverse {e i,t,h i,t} and longitudinal parts {e i,z,h i,z}, we obtain

E1=(e1t+e1z)exp(iq1z)
H1=(h1t+h1z)exp(iq1z)
E2=(e2t+e2z)exp(iq2z)
H2=(h2t+h2z)exp(iq2z)
(10)

where qi are the corresponding propagation constants that in general can be complex. The orthogonality theorem states that, when q 1q 2, the transversal parts of the two modes satisfy the following condition [29

29. P. Bienstman, “Rigorous and efficient modelling of wavelength scale photonic components,” Universiteit Gent Thesis (2001).

]:

sdσ·(e1t×h2t)=0.
(11)

Coupled-mode equations for the nonlinear interaction in waveguides - We now consider a waveguide with a source current density J 1=- P NL 1 that arises from the nonlinear interaction. Such a source generates a guided wave {E 1,H 1} propagating along the +z direction, which can be expanded in terms of all guided modes at the same frequency:

E1=ΣlA˜l(z)(el,t+el,z)exp(iqlz)
H1=ΣlA˜l(z)(hl,t+hl,z)exp(iqlz),
(12)

where {e l,t(z),h l,t(z)} is the transverse (longitudinal) component of each guided mode propagating along +z in the source-free case, and l is the index for each mode. For the L-th mode, applying the mirror symmetry operation of the waveguide about the plane normal to z, one can show that

E2=(eL,teL,z)exp(iqLz)
H2=(hL,t+hL,z)exp(iqLz)
(13)

is also a guided mode that propagates along -z.

By substituting J 1=- P NL 1,J 2=0, and Eqs. (12) and (13) into Eq. (9), we observe that on the left hand of Eq. (9), only the e l,t×h L,t+e L,t×h l,t term is along the z-direction, the other terms are perpendicular to z, and therefore it is the only non-vanishing term once the surface integral along z is performed. By further applying the orthogonality condition [Eq. (11)], we obtain

A˜L(z)z=iω2sdσ(P1(NL)·(eL,teL,z))sdσ·eL,t×hL,texp(iqLz)
(14)

We now specifically consider the DFG process in the present waveguide where two optical waves at frequencies ω 2 and ω 3, mix to produce a THz beam at ω 1=ω 3-ω 2. For each frequency ωj, the guided wave {E j,H j} propagating along +z, has the following form

Ej=A˜j(z)(ej,t+ej,z)exp(iqjz)
Hj=A˜j(z)(hj,t+hj,z)exp(iqjz).
(15)

Aj(z)=A˜j(z)exp(αj2z).
(16)

Also, the THz nonlinear polarization P NL 1=χ̿ : E*2 E 3 at the frequency ω 1 can be written as

P1NL=A˜2*A˜3exp(α3+α22z)exp(i(β3β2)z)χ=:e2*e3
(17)

where χ̿ is the second-order susceptibility tensor. By substituting Eqs. (2), (15), and (17) into Eq. (14), we have

A˜1z=iω14κ1A˜2*A˜3exp(α3+α22z+α12z)exp(iΔβz).
(18)

Here Δβ=β 3-β 2-β 1, and κ 1 is the coupling coefficient

κ1=Sdσχ=:e2*e3·(e1te1z).
(19)

By substituting Eq. (16) into Eq. (18), we finally obtain the coupled-mode equation for the amplitude of the THz wave

A1z=α12A1+iω14κ1A2*A3exp(iΔβz)
(20)

The coupled mode equations for the optical wave can be derived similarly.

References and links

1.

M. Tonouchi, “Cutting-edge terahertz technology,” Nature Photonics 1, 97 (2007). [CrossRef]

2.

Y. R. Shen, The principles of nonlinear optics (New York, Wiley-Interscience, 1984).

3.

R. W. Boyd, Nonlinear Optics (Academic Press, 2003).

4.

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11.

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A. Snyder and J. Love, Optical waveguide theory (Kluwer Academic Pub, 1983).

17.

P. R. Villeneuve, S. Fan, J. D. Joannopoulos, K. Y. Lim, G. S. Petrich, L. A. Kolodziejski, and R. Reif, “Air-bridge microcavities,” Appl. Phys. Lett. 67, 167 (1995). [CrossRef]

18.

K. L. Vodopyanov, M. M. Fejer, X. Yu, J. S. Harris, Y. S. Lee, W. C. Hurlbut, V. G. Kozlov, D. Bliss, and C. Lynch, “Terahertz-wave generation in quasi-phase-matched GaAs,” Appl. Phys. Lett. 89, 141,119 (2006). [CrossRef]

19.

I. Mehdi, S. C. Martin, R. J. Dengler, R. P. Smith, and P. H. Siegel, “Fabrication and performance of planar Schottky diodes withT-gate-like anodes in 200-GHz subharmonically pumped waveguide mixers,” IEEE Microwave and Guided Wave Lett. 6, 49–51 (1996) [CrossRef]

20.

S. M. Marazita, W. L. Bishop, J. L. Hesler, K. Hui, W. E. Bowen, T. W. Crowe, V. M. W. Inc, and V. A. Charlottesville, “Integrated GaAs Schottky mixers by spin-on-dielectric wafer bonding,” IEEE Trans. Electron Devices 47, 1152–1157 (2000) [CrossRef]

21.

P. H. Siegel, R. P. Smith, M. C. Graidis, and S. C. Martin, “2.5-THz GaAs monolithic membrane-diode mixer,” IEEE Trans. Microwave Theory Tech. 47, 596–604 (1999) [CrossRef]

22.

K. Aoki, H. T. Miyazaki, H. Hirayama, K. Inoshita, T. Baba, K. Sakoda, N. Shinya, and Y. Aoyagi, “Microassembly of semiconductor three-dimensional photonic crystals,” Nature Materials 2, 117–121 (2003). [CrossRef] [PubMed]

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W. C. Hurlbut, Y. S. Lee, K. L. Vodopyanov, P. S. Kuo, and M. M. Fejer, “Multiphoton absorption and nonlinear refraction of GaAs in the mid-infrared,” Opt. Lett. 32, 668–670 (2007). [CrossRef] [PubMed]

29.

P. Bienstman, “Rigorous and efficient modelling of wavelength scale photonic components,” Universiteit Gent Thesis (2001).

OCIS Codes
(190.4390) Nonlinear optics : Nonlinear optics, integrated optics
(230.7370) Optical devices : Waveguides

ToC Category:
Nonlinear Optics

History
Original Manuscript: May 5, 2009
Revised Manuscript: June 16, 2009
Manuscript Accepted: June 18, 2009
Published: July 22, 2009

Citation
Zhichao Ruan, Georgios Veronis, Konstantin L. Vodopyanov, Marty M. Fejer, and Shanhui Fan, "Enhancement of optics-to-THz conversion efficiency by metallic slot waveguides," Opt. Express 17, 13502-13515 (2009)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-16-13502


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References

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