Light propagation in a resonantly absorbing waveguide array

Light propagation in a resonantly absorbing

waveguide array

Mingneng Feng, Yikun Liu, Yongyao Li, Xiangsheng Xie, and Jianying Zhou*

State Key Laboratory of Optoelectronic Materials and Technologies, Sun Yat-sen University, Guangzhou 510275,

China *[email protected]

Abstract: Light propagation behavior in a resonantly absorbing waveguide

array is analyzed. Both a Lorentzian line shape and an inhomogeneous

broadened absorbing line shape are considered, with their imaginary and

real part of the refractive index determined by a Kramers–Kronig

relationship. The diffracted wave is shown to have the frequency spectra

determined by the material absorption, dispersion as well as the waveguide

structure. An interesting phenomenon is that a spectral hole is produced and

becomes deeper in the diffraction spectrum as the thickness of the

resonantly absorbing waveguide array increases. The experimental

measurements conducted in a waveguide array are found to be in good

agreement with the numerical results.

©2011 Optical Society of America

OCIS codes: (350.5500) Propagation; (230.7370) Waveguides; (050.1940) Diffraction.

References and links

1. D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behavior in linear and nonlinear

waveguide lattices,” Nature 424(6950), 817–823 (2003). 2. Z. G. Chen, H. Martin, E. D. Eugenieva, J. J. Xu, and J. K. Yang, “Formation of discrete solitons in light-induced

photonic lattices,” Opt. Express 13(6), 1816–1826 (2005).

3. A. Szameit, I. L. Garanovich, M. Heinrich, A. Minovich, F. Dreisow, A. A. Sukhorukov, T. Pertsch, D. N. Neshev, S. Nolte, W. Krolikowski, A. Tünnermann, A. Mitchell, and Y. S. Kivshar, “Observation of diffraction-

managed discrete solitons in curved waveguide arrays,” Phys. Rev. A 78(3), 031801 (2008).

4. Y. Y. Li, W. Pang, Y. Z. Chen, Z. Q. Yu, J. Y. Zhou, and H. R. Zhang, “Defect-mediated discrete solitons in optically induced photorefractive lattices,” Phys. Rev. A 80(4), 043824 (2009).

5. T. Schwartz, G. Bartal, S. Fishman, and M. Segev, “Transport and Anderson localization in disordered two-

dimensional photonic lattices,” Nature 446(7131), 52–55 (2007). 6. A. Joushaghani, R. Iyer, J. K. S. Poon, J. S. Aitchison, C. M. de Sterke, J. Wan, and M. M. Dignam, “Quasi-

BLOCH oscillations in curved coupled optical waveguides,” Phys. Rev. Lett. 103(14), 143903 (2009).

7. S. Longhi, “Bloch dynamics of light waves in helical optical waveguide arrays,” Phys. Rev. B 76(19), 195119 (2007).

8. J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of two-dimensional discrete

solitons in optically induced nonlinear photonic lattices,” Nature 422(6928), 147–150 (2003). 9. N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, and M. Segev, “Discrete solitons in

photorefractive optically induced photonic lattices,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 66(4 Pt 2),

046602 (2002). 10. J. D. Liou, C. K. Lee, and K. C. Wu, “Photorefractive crystal-based holographic interferometry system for full-

field wave propagation metrology,” Opt. Express 15(9), 5460–5472 (2007).

11. A. Fratalocchi, G. Assanto, K. A. Brzdakiewicz, and M. A. Karpierz, “Discrete propagation and spatial solitons in nematic liquid crystals,” Opt. Lett. 29(13), 1530–1532 (2004).

12. A. Fratalocchi, G. Assanto, K. A. Brzdakiewicz, and M. A. Karpierz, “Discrete light propagation and self-

trapping in liquid crystals,” Opt. Express 13(6), 1808–1815 (2005). 13. C. Monat, B. Corcoran, M. Ebnali-Heidari, C. Grillet, B. J. Eggleton, T. P. White, L. O’Faolain, and T. F.

Krauss, “Slow light enhancement of nonlinear effects in silicon engineered photonic crystal waveguides,” Opt.

Express 17(4), 2944–2953 (2009). 14. N.-C. Panoiu, B. A. Malomed, and R. M. Osgood, Jr., “Semidiscrete solitons in arrayed waveguide structures

with Kerr nonlinearity,” Phys. Rev. A 78(1), 013801 (2008).

15. D. Vujic and S. John, “Pulse reshaping in photonic crystal waveguides and microcavities with Kerr nonlinearity: critical issues for all-optical switching,” Phys. Rev. A 72(1), 013807 (2005).

#140568 - $15.00 USD Received 4 Jan 2011; revised 2 Mar 2011; accepted 25 Mar 2011; published 31 Mar 2011(C) 2011 OSA 11 April 2011 / Vol. 19, No. 8 / OPTICS EXPRESS 7222

16. M. Blaauboer, B. A. Malomed, and G. Kurizki, “Spatiotemporally localized multidimensional solitons in self-

induced transparency media,” Phys. Rev. Lett. 84(9), 1906–1909 (2000). 17. J. P. Prineas, J. Y. Zhou, J. Kuhl, H. M. Gibbs, G. Khitrova, S. W. Koch, and A. Knorr, “Ultrafast ac Stark effect

switching of the active photonic band gap from Bragg-periodic semiconductor quantum wells,” Appl. Phys. Lett.

81(23), 4332–4334 (2002). 18. O. Toader, S. John, and K. Busch, “Optical trapping, field enhancement and laser cooling in photonic crystals,”

Opt. Express 8(3), 217–222 (2001).

19. J. Y. Zhou, H. G. Shao, J. Zhao, X. Yu, and K. S. Wong, “Storage and release of femtosecond laser pulses in a resonant photonic crystal,” Opt. Lett. 30(12), 1560–1562 (2005).

20. I. V. Mel’nikov and J. S. Aitchison, “Gap soliton memory in a resonant photonic crystal,” Appl. Phys. Lett.

87(20), 201111 (2005). 21. R. Khomeriki and J. Leon, “Driving light pulses with light in two-level media,” Phys. Rev. Lett. 99(18), 183601

(2007).

22. J. T. Li and J. Y. Zhou, “Nonlinear optical frequency conversion with stopped short light pulses,” Opt. Express 14(7), 2811–2816 (2006).

23. A. Kozhekin and G. Kurizki, “Self-induced transparency in Bragg reflectors: gap solitons near absorption

resonances,” Phys. Rev. Lett. 74(25), 5020–5023 (1995). 24. A. E. Kozhekin, G. Kurizki, and B. Malomed, “Standing and moving gap solitons in resonantly absorbing

gratings,” Phys. Rev. Lett. 81(17), 3647–3650 (1998).

25. M. Blaauboer, G. Kurizki, and B. A. Malomed, “Spatiotemporally localized solitons in resonantly absorbing Bragg reflectors,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 62(1 Pt A 1 Pt A), R57–R59

(2000).

26. J. Zhu, J. Y. Zhou, and J. Cheng, “Moving and stationary spatial-temporal solitons in a resonantly absorbing Bragg reflector,” Opt. Express 13(18), 7133–7138 (2005).

27. Y. Y. Li, B. A. Malomed, M. N. Feng, and J. Y. Zhou, “Array and checkerboard optical waveguides controlled

by the electromagnetically induced transparency,” Phys. Rev. A 82(6), 063813 (2010). 28. J. T. Li, B. Liang, Y. K. Liu, P. Q. Zhang, J. Y. Zhou, S. O. Klimonsky, A. S. Slesarev, Y. D. Tretyakov, L.

O’Faolain, and T. F. Krauss, “Photonic crystal formed by the imaginary part of the refractive index,” Adv. Mater. (Deerfield Beach Fla.) 22(24), 2676–2679 (2010).

29. M. Campbell, D. N. Sharp, M. T. Harrison, R. G. Denning, and A. J. Turberfield, “Fabrication of photonic

crystals for the visible spectrum by holographic lithography,” Nature 404(6773), 53–56 (2000). 30. L. Wu, Y. Zhong, C. T. Chan, K. S. Wong, and G. P. Wang, “Fabrication of large area two- and three-

dimensional polymer photonic crystals using single refracting prism holographic lithography,” Appl. Phys. Lett.

86(24), 241102 (2005). 31. Y. V. Miklyaev, D. C. Meisel, A. Blanco, G. von Freymann, K. Busch, W. Koch, C. Enkrich, M. Deubel, and M.

Wegener, “Three-dimensional face-centered-cubic photonic crystal templates by laser holography: fabrication,

optical characterization, and band-structure calculations,” Appl. Phys. Lett. 82(8), 1284–1286 (2003).

32. O. Peleg, G. Bartal, B. Freedman, O. Manela, M. Segev, and D. N. Christodoulides, “Conical diffraction and gap

solitons in honeycomb photonic lattices,” Phys. Rev. Lett. 98(10), 103901 (2007).

33. O. B. Treidel, O. Peleg, and M. Segev, “Symmetry breaking in honeycomb photonic lattices,” Opt. Lett. 33, 2251–2253 (2009).

34. B. Freedman, G. Bartal, M. Segev, R. Lifshitz, D. N. Christodoulides, and J. W. Fleischer, “Wave and defect

dynamics in nonlinear photonic quasicrystals,” Nature 440(7088), 1166–1169 (2006). 35. R. Driben, B. A. Malomed, A. Gubeskys, and J. Zyss, “Cubic-quintic solitons in the checkerboard potential,”

Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 76(6 Pt 2), 066604 (2007).

36. R. Driben and B. A. Malomed, “Stabilization of two-dimensional solitons and vortices against supercritical collapse by lattice potentials,” Eur. Phys. J. D 50(3), 317–323 (2008).

37. F. Fedele, J. K. Yang, and Z. G. Chen, “Defect modes in one-dimensional photonic lattices,” Opt. Lett. 30(12),

1506–1508 (2005). 38. X. S. Wang, Z. G. Chen, and P. G. Kevrekidis, “Observation of discrete solitons and soliton rotation in optically

induced periodic ring lattices,” Phys. Rev. Lett. 96(8), 083904 (2006).

39. J. W. Fleischer, G. Bartal, O. Cohen, O. Manela, M. Segev, J. Hudock, and D. N. Christodoulides, “Observation of vortex-ring “discrete” solitons in 2D photonic lattices,” Phys. Rev. Lett. 92(12), 123904 (2004).

40. A. Argyros, I. M. Bassett, M. A. van Eijkelenborg, M. C. J. Large, J. Zagari, N. A. P. Nicorovici, R. C.

McPhedran, and C. M. de Sterke, “Ring structures in microstructured polymer optical fibres,” Opt. Express

9(13), 813–820 (2001).

1. Introduction

The propagation of light in a waveguide array is a topic of great current interest in optics.

Novel phenomena such as discrete solitons [1–4], Anderson localization [5] and Quasi-Bloch

Oscillations [6,7] are observed as optical fields propagating in a waveguide array.

Traditional waveguide arrays, such as those made of photorefractive crystals [8–10] or

liquid crystals [11,12], etc., are created in passive materials. While the nonlinearity of the


waveguide can be fairly large, the response to the input field is in general slow. On the other

hand, waveguide with Kerr nonlinearity [13–15] provides a much faster nonlinear response,

but a fairly intense laser is required to induce the nonlinear effect.

Being different from passive waveguide arrays, a resonantly absorbing waveguide array

(RAWA) consisting of periodically doped active material is proposed and demonstrated in

this work. The absorptive material, such as atoms with a near-resonant transition frequency,

has a complex dielectric constant. Hence it has characteristics of a large nonlinear effect, fast

response speed and a possible low working threshold. In addition, its wavelength-dependent

refractive index always offers abundant dispersion properties, which brings in more freedom

to control the light field. Therefore, a RAWA may be tuned more easily than conventional

waveguides and offer additional management of the waveguiding and diffraction

characteristics. This has applications in optical devices, such as bullet generation [16], optical

switching [17], optical storage [18–21], and nonlinear optical frequency conversion [22]. It is

necessary to point out that, an active discrete system named resonantly absorbing Bragg

reflector (RABR) was well studied in detail [23–26]. The difference of our system from the

RABR is that the light is guided along the waveguide, rather than perpendicular to it. Very

recently, an active discrete system in a medium periodically doped by resonant multi-level

atoms, which is akin to our system, was theoretically studied [27].

Here, a RAWA is proposed and experimentally demonstrated. We firstly analyze the

propagation behavior of the light field in the one-dimensional (1-D) RAWA, we then derive

the diffracted light field behavior. Finally, we describe an experimental two-dimensional (2-

D) RAWA consisted of periodically arrayed of SU-8 polymer and SU-8 doped with

Rhodamine B (RhB). We found that the numerical simulations have a good agreement with

the experimental results for the diffraction spectrum and light field distribution.

2. Numerical simulation and analysis

The structure of RAWA is shown in Fig. 1. Such a system is described (in the paraxial limit)

by a Schrödinger-like equation:

2

0

1( , ) ( ) ( , ) ( , ).

2 2

E ii E x y k n V x y E x y

z k

(1)

Here, in Eq. (1), 2 = (2 /x

2 +

2 /y

2) and k = k0n, where n = 1.62 is the refractive index

of the background, k0 is the vacuum wave vector of E(x, y). α is the absorption coefficient of

the active medium, Δn is the corresponding refractive index change obtained by the Kramers–

Kronig (K-K) relationship. V(x, y) is the periodic modulation function, as shown in Fig. 1

where the black areas are described by V(x, y) = 1 while the white areas by V(x, y) = 0.

First, we consider the 1-D RAWA (Fig. 1(a)) whose absorbing coefficient is assumed to

have a Lorentzian line shape, i.e.

2 2

0

,( )

A

B

(2)

with λ0 the absorption center, A /B2 the maximum value of the absorbing coefficient and 2B

the line width. Based on the K-K relationship, the refractive index change can be calculated

by

'

'

2 ' 2

1 ( )( ) .

2 1 ( / )n d

(3)

In the numerical simulation, the parameters in Eq. (2) are assumed to be λ0 = 564 nm, A = 1.3

× 105

um, and B = 5 × 103

um. Both α and the corresponding Δn vary strongly in the vicinity

of the resonance (Fig. 2(a)). The refractive index of the active medium on the shorter


wavelength side of the resonance is lower than the background medium, while on the longer

wavelength side the refractive index is higher. In the 1-D system, 2 = 2 /x

2, E(x, y) =

E(x), and V(x, y) = V(x). As shown in Fig. 1(a), d = 2.86 um is the lattice constant, d1 = 0.5d is

the width of the active medium layer.

Fig. 1. The schematic diagram of (a) 1-D and (b) 2-D square-lattice RAWA. The black areas

are the active media, and the white areas are the background media.

Fig. 2. The α (solid line) and Δn (dashed line) of the active medium vary with the wavelength

for the (a) theoretical line shape (b) experimental line shape.

Based on Eq. (1), the propagation behaviors of the light field with different wavelengths in

the RAWA are simulated. The incident light is described as a Gaussian function: E = E0 exp(-

x2/(4d

2)), and the wavelengths are chosen as λ1 = 554 nm and λ2 = 574 nm. The former

wavelength is experienced a negative value of Δn while the latter is experienced a positive

value of Δn (Fig. 2(a)). As shown in Fig. 3(a), the background media act as the guiding layer

for the incident wavelength at 554 nm. However, the situation is reversed when the incident

wavelength is 574 nm (Fig. 3(b)). Under this circumstance, the active media act as the guiding

layer because the light field prefers to propagate in the layer with a higher refractive index.

Therefore, the waveguiding property of the system can be tuned by the light frequency. Due to

the absorption of the active medium, the light field with the wavelength of λ2 = 574 nm (Fig.

3(d)) chooses the active media as the guiding layer, which decays more seriously than the

light at λ1 = 554 nm (Fig. 3(c)), which propagates mainly in the background media of the

structure.


Fig. 3. The propagation behavior of light in the 1-D RAWA. (a) The light propagates in the

background media (i.e. in the areas where V(x) = 0) for the incident wavelength at 554 nm. (b) The light propagates in the active media (i.e. in the areas where V(x) = 1) for the incident

wavelength at 574 nm. The corresponding transmissions are shown in (c) and (d) for the

incident wavelengths at 554 nm and 574 nm, respectively.

According to the diffraction theory, the diffraction field distribution is the Fourier

transform of the light field distribution at the exit of the array. The first order diffraction

efficiency η is defined as the ratio of the first order diffraction intensity to the incident light

intensity. Figure 4(a) shows the numerical simulation of the diffraction spectra of the RAWA

at different thickness.

Fig. 4. (a) The diffraction spectra of the 1-D RAWA. The different color correspond to the

different thickness of the RAWA with 5 um (black), 10 um (red) and 15 um (green). (b) The η

as a function of the thickness of the 1-D RAWA for the incident wavelengths at 554 nm (black), 564 nm (red) and 574 nm (green), respectively.

An interesting phenomenon is that a diffraction spectral hole is produced at the absorption

center at the thickness greater than 10 um. The reason of the spectral hole is that η is

determined by α and Δn. In Fig. 4(b), we show η with a variation of the thickness for three

chosen wavelengths: 564 nm, which is located exactly at the absorption peak, 554 nm and 574


nm, at the two sides of the absorption peak, which experience a negative value or a positive

value of Δn, respectively. For incident light of 564 nm, the diffraction is caused solely by α,

while for the lights of 554 nm and 574 nm, their diffractions are contributed both by α and Δn.

As the thickness of the RAWA increases, the η first increases and then decreases. The

optimal thickness to maximize η can be obtained. There is a peak in all the three curves for

different wavelengths (Fig. 4(b)). The magnitude and position of the peak are determined by

the value of α and Δn. Figure 4(b) shows that the peak for 564 nm appears in a shorter

distance (about 8 um) and the maximum value is smaller than the maximum efficiencies of the

light field at 554 nm and 574 nm respectively. Therefore, after a certain distance (about 10

um), a spectral hole occurs and becomes deeper as the thickness is increased.

The numerical simulation about the propagation and diffraction based on Eq. (1) can be

extended to the inhomogeneous broadened medium as long as the active material obeys the

K–K relationship. In the next section, the experimental measurements compared with the

numerical results on the diffraction spectrum and light field distribution in the inhomogeneous

broadened medium will be discussed.

3. Experiment compared with theory

Imaginary-part photonic crystals (IPPCs) [28] described in this work is periodically arrayed of

SU-8 polymer and SU-8 doped with RhB. In this section, our 2-D RAWA was fabricated akin

to IPPCs with the multi-beam holographic lithography [29–31]. The α and Δn of RhB with a

variation of the wavelength are shown in Fig. 2(b). Though the line shape of α and Δn are

different form the Lorentzian case, they still obey the K-K relationship. The structure diagram

of this 2-D square lattice RAWA is shown in Fig. 1(b), with the lattice constant d = 2.86 um.

According to the experiment result of the IPPCs [28], the shape of the active layer is assumed

to be circular with radius at d1 = 0.33d.

For the diffraction spectrum measurement, a Xenon lamp is used as a light source. An

Ocean Optic USB2000 + spectrometer is used to measure the diffraction spectrum, as shown

in Fig. 5. The thickness of the 2-D RAWA is 5 um. Figure 6 shows the experimental curve of

η. The corresponding numerical result from Eq. (1) is shown in the same figure, which shows

a striking agreement with the experimental curve.

Fig. 5. The experimental setup for the diffraction spectrum measurement of a 2-D RAWA.


Fig. 6. Numerical result (dashed line) and experimental result (solid line) for the η of the 2-D RAWA.

The measurement of pattern distribution in the end-facet is shown in Fig. 7. In the

experiment, a confocal laser scanning microscope (CLSM) combining with a tunable laser

source (Opium Auto 100, Radiants Light S.L.) is used to obtain the end-facet pattern

distribution of the light field at λ1 = 547 nm and λ2 = 581 nm. The CLSM (AlphaSNOM,

WITec GmbH) is working in the transmission mode with both the input point (incident laser

focus) and the output point (light collection point) adjustable respectively. It shows that the

experimental results coincide well with the numerical simulations.

Fig. 7. The distributions of the light field with the wavelength (a) λ1 = 547 nm and (b) λ2 = 581

nm in the end-facet for a 2-D experimental RAWA with the sample thickness at 5 um. The

insets in each figure are numerical results.


4. Conclusion

The propagation behavior and diffraction characteristic of the RAWA are analyzed. For the

incident light with the wavelength in the vicinity of the absorption peak, the system will

choose corresponding higher refractive index layer as the guiding layer. A diffraction spectral

hole is observed due to the competition of the contribution to the diffraction by real and

imaginary parts of the refractive index. Furthermore, for a 2-D RAWA, it is shown that the

numerical simulations agree well with the experimental results for the measurement of

diffraction spectrum and light field distribution. As a result, the numerical simulation can be

used to guide the design of a RAWA to achieve useful functionality.

There are potential applications for a RAWA. For example, by exploiting saturable

absorption of the RhB, an all-optical switch can be realized at a much lower pump power,

while the nonlinear optical index is considerably greater than that of non-resonant Kerr effect.

Hence thin optical sample, low pump threshold, room temperature operation and high contrast

ratio for optical modulation are made possible with the RAWA. Furthermore, discrete solitons

[1–4] can be generated at a low pump power with the RAWA structure. Many complex

structures, such as honeycomb lattices [32,33], quasicrystals [34], checkerboard lattices

[35,36], defect lattices [37], ring lattices [38–40], etc., can also be designed and fabricated

with the theoretical simulation and experimental realization developed in this work.

Acknowledgments

The authors thank Prof. B. A. Malomed, Dr. J. T. Li, Y. F. Guan, and M. D. Zhang for useful

discussion. This work is supported by the National Key Basic Research Special Foundation

(G2010CB923204) and by the Chinese National Natural Science Foundation (10934011).


Documents

Light propagation in a resonantly absorbing waveguide array