focus group on photonics universitatsstr. 27/prg, d-58084 … · 2018. 6. 12. · modes and...
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FOCUS GROUP on PHOTONICSUniversitatsstr. 27/PRG, D-58084 Hagen, Germany
Fundamentals and Theory of Electromagnetics(Allgemeine und Theoretische Elektrotechnik)
Prof. Dr.-Ing. Reinhold Pregla - [email protected]: +49-2331-987-1140, FAX: -353
Students Phone EmailWaldemar Spiller [email protected]
Optical Information Technology(Lehrgebiet Optische Nachrichtentechnik)
Prof. Dr. Jurgen Jahns - [email protected]: +49-2331-987-340, FAX: -352, Email: [email protected]
Staff Phone EmailAndre Edelmann (since 02/2009) 341 [email protected] Heldt (until 08/2009) 1121 [email protected] Dr.-Ing. Stefan Helfert 1144 [email protected]. Hans-W. Knuppertz 1128 [email protected] Lohmann [email protected] Seiler 1122 [email protected] Sortino [email protected] Schaap [email protected] Kral 1120 [email protected]
Optical Microsystems(Juniorprofessur Optische Mikrosysteme)
Jun.-Prof. Dr. Matthias Gruber - [email protected]: +49-2331-987-1131, FAX: -352
Staff Phone EmailMichael Bohling 1195 [email protected] Winkelmann 1195 [email protected]
FD-BPM for Optical Waveguide Structures with Second Order Accuracy
R. Pregla
An improved FD-BPM was developed which is based on the generalized transmission line(GTL) equa-tions (dH/du = −jREE, dE/du = −jRHH) for discretized transverse fields E and H. This BPM isa wide angle algorithm and also full vectorial. The generalized forms take anisotropic material parame-ters into account. In case of a general anisotropy we have a second term on the right sides (for detailssee [3]). Usually, BPM algorithms are based on the wave equations for the potentials or for the fieldcomponents [1] [2]. Because the GTL-equations are of first order, it is possible to distinguish betweenthe forward and backward propagation direction. Therefore, it is not necessary to extract the equationsfor the forward propagating field from the total field equations. We assume that the material parametersare tensors. For each propagation step the parameters in this direction (u) are assumed to be constant.To describe the propagation by an FD-BPM algorithm we replace the first order derivatives on the leftside with central differences. The difference equations have second order accuracy in the middle of twoneigbouring cross-sections Ck and Ck+1 for which we would like to calculate the fields. In case of across-section that is not centered between Ck and Ck+1 a third term must be added to the differenceapproximation. To obtain second order accuracy for the whole FD expression the right sides must becalculated at the same place with second order accuracy. In the Figure is shown how this can be done.Usually only first order approximations are used [4] [5]. A second order function is determined by threepoints. The values of the function between these points can be calculated by the equations below. In
MLDS1060
f
x
f i-1
f if i+1
x i-1 x i x i+1ix l rix
i-1h ih
Quadratic field interpolation
In case of an equidistant discretisation the fol-lowing interpolation scheme for x = xil =xi − hi−1/2 and x = xir = xi + hi/2 (i.e. onpositions left and right from xi) is derived:
f (xil) =3
8fi−1 +
3
4fi −
1
8fi+1
f (xir) =3
4fi +
3
8fi+1 −
1
8fi−1
These expressions have quadratic accuracy.Also non-equidistant discretisation can be used, but with more complicated formulas.
conclusion: We approximate our field functions in the neighbourhood of three points by a function ofsecond order and use formulas for a quadratic field interpolation. By knowing the fields at cross-sectionsCk−1 and Ck (calculated in steps before) we can calculate the field in cross-section Ck+1. Because weuse also the field from previous steps results for the following ones have higher accuracy.
[1] W.P. Huang (ed), Methods for Modeling and Simulation of Guided-Wave Optoelectronic Devices: Part I:Modes and Couplings, II: Waves and Interactions, PIER 10 and 11, resp., in Progress in ElectromagneticResearch, EMW Publishing, Cambridge/USA (1995)
[2] G. Guekos (ed), Photonic Devices for Telecommunications, Springer, Berlin (1998).[3] R. Pregla, Modeling of Optical Waveguide Structures With General Anisitropy in Arbitrary Orthogonal Coo-
dinate Systems, IEEE J. Sel. Topics in Quantum Electron., vol. 8, no. 6, (2002), 1217-1224.[4] R. Pregla, Novel FD-BPM for Optical Waveguide Structures with Isotropic or Anisotropic Material, ECIO’99,
14.-16.04.99, Torino, Italy, 55 - 58.[5] R. Pregla, S. Helfert, Modeling of Optical Devices Including Anisotropic Material by the Method of Lines
and by a Novel FD-BPM, PIERS 2000, Progress in Electromagnetics Research Symposium, Cambridge, Mas-sachusetts, USA, 168.
Analysis of bends and Y-junctions in photonic crystals
W. Spiller, R. Pregla
Photonic crystals play an important rule in designing microwave and optical waveguide circuits. Efficientalgorithms based on the Method of Lines were developed in the past for normal periodic structures likemultilayered waveguides [1]. In this study we analyzed complex periodic structures like a defect wave-guide in photonic crystals. Examples are shown in Fig. 1 a and b) where we have bends and Y-junctionsin photonic crystals that are coupled with each other or with straight waveguides in similar photoniccrystals. The Floquet’s modes of the straight part were obtained in a special way from half of the periodby calculating the short and open circuit impedance matrices. Bends and Y-junctions (propagation inϕ-direction) can be analyzed with the open- and short-circuit matrix parameters of half of the symmetri-cal structures. The MoL is combined with finite differences in conjunction with impedance/admittancetransformation [2] - [3]. Complex structures must be divided into homogeneous parts, whose length isequal to the length of the step used in finite differences. Fig. 2 shows (as example) the field distribution inthe output (part II) for the bend structure and in the straight waveguide (part III) in Fig. 1 a) - determinedwith the described algorithms. The field distributions in the output parts of the Y-junctions are similar tothose of the bend in Fig. 2, but the divided power and the asymmetrical distortions are less pronounced.
a) b)
Fig. 1: a) Bends coupled with straight waveguides; b) Y-junctions coupled with straight waveguides
Fig. 2: Field distribution in the output for the bend structure (part II) and in the straight waveguide (part III)
[1] R. Pregla, Analysis of Electromagnetic Fields and Waves, The Method of Lines. Southern Gate, Chichester,England; Baldock, Hertfordshire : John Wiley and Sons, Ltd; Research Studies Press Limited;ISBN 978-0-470-03360-9 (H/B), (2008)
[2] W. Spiller, Analyse von Knicken und Y-Verzweigungen in Defektwellenleitern, FernUniversitat Hagen, (2010)
[3] R. Pregla, Modeling of optical waveguides and devices by combination of the method of lines and finitedifferences of second order accuracy, Optical and Quantum Electronics 38, (2006), 3-17
Self–imaging effect in plasmonic multimode waveguides
A. G. Edelmann, S. F. Helfert, J. Jahns
We analyze the self–imaging effect in plasmonic multimode waveguides. Self–imaging means that awavefield repeats itself after a certain propagation distance (see e.g. [1]). This might have interestingapplications in integrated plasmonic devices. The plasmonic self–imaging effect was the topic of anearlier investigation [2]. However, the considered structure was unbounded in the lateral direction. We gobeyond that case by analyzing plasmonic multimode waveguides where the lateral bounding is essential.We perform numerical simulations using the Method of Lines (MoL) [3] to obtain the field distributions.Firstly we analyze plasmonic waveguides for small metallic layer thickness t = 0.06µm (wavelengthλ = 0.6µm). Here significant deviations in the attenuation and propagation constant between the evenand the odd plasmonic mode sets take place. Figure 1a shows the field distribution for the even modes atthe self–imaging distance Lsi ≈ 15.0µm. In Fig. 1b for the odd modes the field extinguished due to theattenuation before self–imaging occurs.
a)
z [µm]
x [µ
m]
0 10 20 30 400.5
1
1.5
2
2.5
3
3.5
0.2
0.4
0.6
0.8
b)
z [µm]
x [µ
m]
0 10 20 30 400.5
1
1.5
2
2.5
3
3.5
0.2
0.4
0.6
0.8
Fig. 1: Field distribution at the dielectric/metallic interface; a) even- b) odd-modes (t = 0.06µm)
Now, we examine the layer thickness t = 0.1µm. Here the attenuation and propagation constants of theboth mode sets are similar. Fig. 2a shows the respective field distribution (Lsi ≈ 14.5µm). In plasmonicswe can decrease the losses by increasing wavelength. Therefore we analyze the same structure at thewavelength 0.9µm (see Figure 2b). We found that the self–imaging is more pronounced and that thelengths is shorter than in the previous case (Lsi ≈ 9.1µm).
a)
z [µm]
x [µ
m]
0 10 20 30 400.5
1
1.5
2
2.5
3
3.5
0.10.20.30.40.50.6
b)
z [µm]
x [µ
m]
0 10 20 30 400.5
1
1.5
2
2.5
3
3.5
0.2
0.4
0.6
0.8
Fig. 2: Field distribution at the dielectric/metallic interface at a) λ = 0.6µm and b) λ = 0.9µm.
In summary: we observe self–imaging in plasmonic waveguides for the even and odd mode sets at a layerthickness of t = 0.1µm and an operating wavelength of 0.9µm. More details will be published in [4].
[1] L. B. Soldano, E. C. M. Pennings, Optical multi-mode interference devices based on self-imaging: principlesand applications, J. Lightwave Technol. 13, (1995), 615–627.
[2] M. R. Dennis, N. I. Zheludev, F. J. G. de Abajo, The plasmon Talbot effect, Opt. Express 15, (2007),9692–9700.
[3] R. Pregla, Analysis of Electromagnetic Fields and Waves - The Method of Lines, (Wiley & Sons, Chichester,UK, (2008)).
[4] A. G. Edelmann, S. F. Helfert, J. Jahns, Analysis of the self–imaging effect in plasmonic multimode wavegui-des, Appl. Opt. 49 (2010).
Transmission characteristics in plasmonic multimode waveguides
A. G. Edelmann, S. F. Helfert, J. Jahns
Plasmons are electromagnetic waves propagating at optical frequencies along a dielectric metallic inter-face [1,2]. In plasmonic multimode waveguides the field is lateral bounded and interesting self–imagingeffects can be observed [3]. Here we study the transmission characteristics related to this self–imagingeffect. For the analysis the Method of Lines (MoL) [4] was used. Figure 1 shows the plasmonic field pro-pagation at the top of a metallic structure in the plasmonic multimode waveguide. We can clearly see thefield repetition at the self–imaging distance Lsi ≈ 15.8µm. Here, the losses of the metal were neglected.
z [µm]
x [µ
m]
0 5 10 15 20 25 30 35 400.51
1.52
2.53
3.5
0.2
0.4
0.6
0.8
Fig. 1: Field distribution at the top of a metallic structure in the plasmonic multimode waveguide
To analyze the transmission characteristics we determine the correlation between the input field and thefield at the self–imaging distance Lsi. We perform our studies for symmetric (even) and antisymmetric(odd) mode excitation. In Fig. 2 we present the results for the central wavelength λ0 = 0.6µm andλ0 = 0.7µm with ∆λ = 0.2µm. The curves for the central wavelength λ0 = 0.7µm were smoothercompared to those for λ0 = 0.6µm.
0.5 0.55 0.6 0.65 0.7 0.75 0.80.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
wavelength ! [µm]
corre
latio
n F in
/ F Ls
i [a.u
.]
!0 = 0.6 µm (even)
!0 = 0.6 µm (odd)
!0 = 0.7 µm (even)
!0 = 0.7 µm (odd)
Fig. 2: Transmission characteristics for even and odd excitation at the wavelength λ0 = 0.6µm and λ0 = 0.7µm
[1] H. Raether, Surface plasmons on smooth and rough surfaces and on gratings, Springer, Berlin, (1988).
[2] S. A. Maier, Plasmonics: Fundamentals and Applications, Springer, New York, USA, (2007).
[3] A. G. Edelmann, S. F. Helfert, J. Jahns, Analysis of the self–imaging effect in plasmonic multimode wavegui-des, Appl. Opt. 49 (2010).
[4] R. Pregla, Analysis of electromagnetic Fields and Waves: The Method of Lines, (John Wiley & Sons, Ltd.,West Sussex, England, (2008)).
Eigenmodes of symmetric plasmonic waveguides
S. F. Helfert, A. G. Edelmann, J. Jahns
At the interface between metals and dielectric materials so called plasmon waves can propagate [1]. Dueto the short wavelength devices have a potential for future miniaturization. Currently, the high losses inthe metal cause problems for practical utilizations. Here we studied the influence of the height of thewaveguide on propagation characteristics. The examined structure is shown in Fig. 1a. We assume thatthe waveguide is extended to infinity in horizontal direction (i.e. w →∞) making it a 2D problem. Theheight t was varied and the propagation constant was computed. For large heights (t → ∞) the top andbottom sides of the metal are decoupled and the propagation constant can be given analytically as
β/k0 =
√εdεmεd + εm
(see e.g. [2]). If the height t decreases, two modes occur with an even resp. odd symmetry in relation tothe vertical direction. Now, these modes behave completely different as can be seen in Fig. 1a, where weshow the complex propagation constant (β) in the complex plane. The real part of β describes the phase,while its imaginary part gives the losses. As can be seen, the losses increase with decreasing t in case ofthe odd mode. In contrast, the losses for the even mode become smaller and can approach even the valuezero in case of a lossless dielectric medium. However, at the same time also the real part of β becomessmaller. As consequence the field reaches very far into the surrounding dielectric material. Therefore,for practical applications a suitable compromise has to be found. Numerical studies for waveguides withfinite width w showed that the qualitative behavior described for the 2D structure shown above, alsoholds for the 3D case [2] [3].
a)
εdεm
w
t
b)
Re( β / k ) εd1/2
0
Im(β
/ k
) (l
osse
s)0
t - 0(even mode)
t - 0(odd mode)
>
>
t = oo
00
Fig. 1: a) Two-dimensional plasmonic waveguide; b) propagation constant in the complex plane
[1] H. Raether, Surface plasmons, Springer tracts in modern physics 111 , Springer–Verlag, Berlin, Heidelberg,New York, (1988).
[2] P. Berini, Plasmon–polariton waves guided by thin lossy metal films of finite width: Bound modes of symme-tric structures, Phys. Rev. B, 61, (2000) 10484–10503, .
[3] A. G. Edelmann, S. F. Helfert, J. Jahns, Analysis of the self–imaging effect in plasmonic multimode wavegui-des, Appl. Opt., 47, (2010).
Eigenmodes of asymmetric plasmonic waveguides
S. F. Helfert, A. G. Edelmann, J. Jahns
A problem for the practical application of plasmonic waveguide is the coupling of light into these struc-tures. An example is presented in Fig. 1a. We see a plane wave in a dielectric medium (εd1) that hits ametallic structure. To excite a surface plasmon at the metal–dielectric interface the horizontal componentof ~k of the plane wave and ksp (i.e. the wavevector of the surface plasmon wave) must be identical.As can be shown analytically, this condition cannot be fulfilled in symmetric structures (εd1 = εd2). Forεd1 > εd2 the plasmon wave can be excited as shown in Fig. 1a at the side of the dielectric with the lowerpermittivity. Now to gain insight into the effecs it is usefull to determine the eigenmodes. First of all inFig. 1 the eigenmodes for a symmetrical structure εd1 = εd2 are shown. Similar to waveguide couplerswe have even and odd modes (here with respect to the vertical direction). Both of these modes have themaxima of the fields on both interfaces between dielectric and metal. Now, if an asymmetry is introducedthe situation changes. This is shown in Fig. 2a where only a small difference of εd1 occurs. Each of thetwo modes is mainly connected to one interface then. If we increase the permittivity difference the fieldof the mode that is connected to lower permittivity may ”tunnel” through the metal and radiate. As canbe seen, the coupling of light into a plasmon structure requires a careful design.
a)
ε d2
ε d1
ε m
plane wave
ksp
k
t
b) −1 −0.5 0 0.5 1 1.5−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
x [µm]
H [a
.u.]
oddeven
Fig. 1: a) Coupling of a plane wave into a plasmon structure; b) eigenmodes for εd1 = εd2
a)−1 −0.5 0 0.5 1 1.50
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
x [a.u.]
H [a
.u.]
εd1
εd2
b)−2 −1.5 −1 −0.5 0 0.5 1 1.5
−0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
x [µm]
H [a
.u.]
"leaky field"
Fig. 2: Eigenmodes of an asymmetric plasmon waveguide εd1 > εd2: a) small permittivity contrast b) largepermittivity contrast
[1] H. Raether, Surface plasmons, Springer tracts in modern physics 111 , Springer–Verlag, Berlin, Heidelberg,New York, (1988).
[2] P. Berini, Plasmon–polariton waves guided by thin lossy metal films of finite width: Bound modes of asym-metric structures, Phys. Rev. B, 63, (2001) 125417:1-15 .
Plasmonic multimode waveguide with transversely structured core
A. G. Edelmann, S. F. Helfert, J. Jahns
The propagation of plasmonic fields is well-known and described in [1, 2]. Here we suggest the use oftransversely structured plasmonic waveguides where the metallic core is not homogenous, but rathersubdivided by well-defined stripes of different metals. In that case, the propagation constants and theattenuation will be different in the different sections. As example we analyze the plasmonic waveguidestructures shown in Fig. 1. The “conventional” plasmonic waveguide has an unstructured silver core thatis specified by the permittivity εm, the layer thickness tm and the layer width wm. The metallic core issurrounded by a dielectric medium of the permittivity εd. In the “structured” plasmonic waveguide weuse two different metals and introduce a gold inlay of width win and permittivity εm2 (εm1 6= εm2) in thecenter of the silver core. For our numerical studies we use the Method of Lines (MoL) [3].
wm
tm
‘’conventional’’
win
‘’structured‘’
Fig. 1: “Conventional” plasmonic waveguide consist of silver core and “structured” with a gold inlay in the center
In Fig. 2a the field distribution of the first five eigenmodes of the “structured” plasmonic waveguideare shown and compared to the “conventional” ones (dotted line). We can clearly see that the fields forthe eigenmodes m = 1, 3, 5 are more affected by the inlay than the eigenmodes m = 2, 4. Figure 2bshows the respective deviations of the attenuation ∆αm = αstructm − αconvm (top) and the propagationconstant∆βm = βstructm−βconvm (below). Here we found higher attenuation and propagation constantsfor the odd number modes (m = 1, 3, ...) than for the even number modes (m = 2, 4, ...). This behaviorcan be explained with an effective index model and considering that waveguides with gold have higherattenuation and a large propagation constant in comparison to silver.
a) b)
1 2 3 4 5 6 7 80
2
4
6x 10−3
!"
[µm
]−1
1 2 3 4 5 6 7 80
0.005
0.01
0.015
0.02
eigenmode
!#
[µm
]−1
Fig. 2: Absolute value of the magnetic field component for m = 1− 5 eigenmodes of structured (continuous) andunstructured (dotted) core. b) Deviations of the attenuation constant ∆α (top) and propagation constant ∆β(below) for the m = 1− 8 eigenmodes.
Finally we want to advise that a combination of different metals may state a further degree of freedom inengineering of plasmonic devices and may offer new potentials for several applications.
[1] H. Raether, Surface plasmons on smooth and rough surfaces and on gratings, Springer, Berlin, (1988).
[2] S. A. Maier, Plasmonics: Fundamentals and Applications, Springer, New York, USA, (2007).
[3] R. Pregla, Analysis of electromagnetic Fields and Waves: The Method of Lines, (John Wiley & Sons, Ltd.,West Sussex, England, (2008)).
Influence of the dielectric medium in plasmonic waveguides
A. G. Edelmann, S. F. Helfert, J. Jahns
The propagation of plasmons in waveguide structures strongly depends on the geometrical and materialparameters [1]. Here we examine the influence of the dielectric medium in plasmonic waveguides byapplying numerical calculations. Figure 1a shows the cross-section of the plasmonic waveguide understudy. The metallic core of width w and thickness t consists of silver with the permittivity εm. Thesurrounded dielectric medium has the permittivity εd. We obtain the permittivity εm by applying theDrude-model with the parameters taken from [2]. In Fig. 1b we plotted εm as function of the wavelengthλ. As known, both real and imaginary part of εm increase with increasing wavelength λ.
a)
!d w
t!m
x
y
z
b)
200 400 600 800 1000 1200 1400 1600 1800−150
−125
−100
−75
−50
−25
0
wavelength [nm]
! m
imaginary part
real part
Fig. 1: a) Plasmonic waveguide under study and b) frequency dependency of εm
Now, for our analysis we use the Method of Lines (MoL) [3] as numerical algorithm. We have determinedthe attenuation α and propagation constant β of first plasmonic odd mode as function of the frequency,see Fig. 2. For the dielectric medium we used the values εd = 4, 6, 8. We see that the attenuation αand propagation constants β are shifted to higher wavelengths for higher εd. The decrease of α and β athigher wavelength is well-known in plasmonics. More details will be published in [4].
a)
0.4 0.6 0.8 1 1.2 1.4 1.60
0.05
0.1
0.15
0.2
0.25
0.3
! [µm]
" /
k 0
#d = 4#d = 6#d = 8
b)
0.4 0.6 0.8 1 1.2 1.4 1.61
1.5
2
2.5
3
3.5
4
4.5
5
5.5
! [µm]
" /k
0
#d = 4#d = 6#d = 8
Fig. 2: a) Attenuation and b) propagation constant of the first fundamental plasmonic odd mode as function of thefrequency
[1] P. Berini, Plasmon–polariton waves guided by thin lossy metal films of finite width: Bound modes of symme-tric structures, Phys. Rev. B 61, (2000) 10484.
[2] M. Besbes et al., Numerical analysis of a slit-groove diffraction problem, J. Eur. Opt. Soc. 2, (2007) 07022.
[3] R. Pregla, Analysis of Electromagnetic Fields and Waves - The Method of Lines, (Wiley & Sons, Chichester,UK, (2008)).
[4] A. G. Edelmann, S. F. Helfert, J. Jahns, Analysis of the self–imaging effect in plasmonic multimode wavegui-des, Appl. Opt. 49, (2010).
Modelling the electrostatic fields in dielectric corners with finite differences
S. F. Helfert
The simulation of electromagnetic fields in waveguide structures can only be done in exceptional caseswith analytic expressions. Usually numerical algorithms are required. One of the biggest challenge is thetreatment of dielectric corners, because the fields can have poles there. While analytic expressions existfor the static case see e.g. [1] [2] , no exact expressions have been developed for waveguide structuresthough studies have been done since decades (see e.g. [3]).The goal of our future work is to model the corners with finite differences in waveguide structures withfinite differences (FD). However, as test we began with computing the electric fields in corners for thestatic case. Here the results can be compared with the exact analytic ones in order to get a estimation ofthe suitability of simple FD-expressions.Fig. 1a shows the examined structure. A dielectric material with permittivity ε2 is embedded in a sur-rounding one (permittivity ε1). For ε1 < ε2 a pole of the electric fields should occur, whereas in caseε1 > ε2 the electric fields in the corner should be zero [1].Results are shown in Fig. 1b-c and Fig. 1. The computed field distribution in a cross–section throughthe corner is presented with the discretization distance as parameter. It can be seen that the fields in thecorner has a sharp peak which indicates the presence of pole for ε1 > ε2. In the second case a dip occursi.e. the field drops to zero. By decreasing the discretization distance the effects are more pronounced.Hence the results for the static case agree (at least qualitatively) with the analytic predicted ones.
a)
ε r1
ε r2
x y
z
b) 8 9 10 11 12 131.5
2
2.5
3
3.5
4
4.5
5x 10
−3
x [a.u.]
Ex [a
.u.]
hx=h
0hx=h
0/2
hx=h
0/4
c) 8 9 10 11 12 131.4
1.6
1.8
2
2.2
2.4
2.6
2.8
3
3.2x 10
−3
x [a.u.]
Ez [a
.u.]
hx=h
0hx=h
0/2
hx=h
0/4
Fig. 1: a) Dielectric structure, electric field through the corner for ε1 > ε2; b) Ex c) Ez
a) −4 −3.5 −3 −2.5 −20.008
0.01
0.012
0.014
0.016
0.018
0.02
x [a.u.]
Ex [a
.u.]
hx=h
0hx=h
0/2
hx=h
0/4
b) −3.5 −3.4 −3.3 −3.2 −3.1 −3 −2.9 −2.80.016
0.018
0.02
0.022
0.024
0.026
0.028
0.03
0.032
0.034
x [a.u.]
Ez [a
.u.]
hx=h
0hx=h
0/2
hx=h
0/4
Fig. 2: a) Electric field through the corner for ε1 < ε2; a) Ex, b) Ez
[1] R. E. Collin, Field Theory of Guided Waves, Series of Electromagnetic Waves. IEEE press, New York, 2edition, (1991) 26.
[2] J. Meixner, The Behavior of Electromagnetic Fields at Edges, IEEE Trans. Antennas. Propagation, AP-20,(1972) 442–446.
[3] J. Bach Andersen, V. V. Solodukhov, Field behavior near a Dielectric Wedge, IEEE Trans. Antennas. Propa-gation, AP-26, (1978) 598–602.
Step approximation of oblique boundaries to compute band structures of photonic crystals
S. F. Helfert
We want to determine the band structures of photonic crystals with hexagonal elementary cell. Obliquecoordinates had been successfully applied for this purpose (see e.g. [1] [2]). However, when we use themethod of lines [3] or finite differences as numerical algorithms we obtain matrices whose size is twiceas large as those appearing when Cartesian coordinates are used. If rectangular cells are used instead, thearea is twice as large as that of the elementary cell leading to non-unique values of the bands.Here, we want to combine these approaches, i.e. use oblique boundaries and perform the calculationswith Cartesian coordinates. For this purpose, we introduce a step approximation (see Fig. 1a). Now, a stepapproximation of waveguide structures is commonly used (e.g. for modeling a tilted waveguide) and onemight wonder about the novelty. As one can see in Fig. 1a) the boundaries of the computational windowitself is modeled with a staircase approximation. This is in contrast to the analysis of tilted waveguides,where the step approximation is only done for the inner structure, whereas the outer boundaries remainhomogeneous.By looking at the point labeled k in Fig. 1a we see the difficulties when we model the computationalwindow with stairs. As known, the wave equation contains derivatives with respect to the vertical (herez) coordinate. If we use e.g. finite differences, we need the points k − 1 and k + 1 for an approximationof these derivatives. If we apply the method of lines [3] as alternative numerical method we divide thestructure into homogenous regions. (The two lowest ones are labeledA andB in Fig. 1a). As can be seen,for both cases we must consider the area outside the computational window. Now, for the computationof band structures in PhCs, we know that the fields are periodic in horizontal (x resp. v) direction. Thispermits us to transform the fields from the right boundary to left one.A second problem is indicated in Fig. 1a as well. The PhC is periodic in v and u direction. However,when using Cartesian coordinates, we relate the point ”1” and ”2”, whereas we know from the obliquecoordinate system that point ”1” and point ”3”, have to be connected. Mathematically, we solve thisproblem by introducing a ”virtual layer” at the top boundary, which causes a ”shift” of the fields torelate the correct fields at the top and bottom boundary with each other. Numerical results obtained withoblique coordinates and with a step approximation (Cartesian coordinates) are shown in Fig. 1b. A verygood agreement can be recognized.
a)
k
k+1
k-11
2 3
boundary ofboundary ofelementary cellelementary cell
step approximationstep approximation of elementary of elementary cell cell
a
a
discretization linesdiscretization lines
x , vy
zu
A
B
b)
0.00
0.05
0.10
0.15
0.20
0.25
0.30
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0.40
a / w
avel
engt
h
M K Γ Γ
oblique coord. Cartesian coord.
Fig. 1: a) Elementary cell of a hexagonal photonic crystal and step approximation; b) determined band structure
[1] S. F. Helfert, Applying oblique coordinates to the method of lines, PIERS, 61 (2006) 271–278 .
[2] A. G. Edelmann, S. F. Helfert, Three-dimensional analysis of hexagonal structured photonic crystals usingoblique coordinates, Opt. Quantum Electron., (2010).
[3] R. Pregla, Analysis of Electromagnetic Fields and Waves - The Method of Lines, Wiley & Sons, Chichester,UK, (2008).
Generation of orbital angular momentum in femtosecond laser pulses by means of spiral phaseelements
A. T. N. Richter1, R. Grunwald1, J. Jahns(1 Max-Born-Institute, Berlin)
Higher order Bessel-Gauss beams and Laguerre-Gauss beams are well known to carry a certain orbitalangular momentum of `~ per photon[1]. Optical vortices in highly intense femtosecond laser pulses areexpected to lead towards a variety of specific new applications like momentum selective spectroscopy,nonlinear excitation and material interaction as well as ultrafast quantum information processing. Ex-periments were performed, where diffractive spiral phase elements were used to generate Bessel-Gaussand Laguerre-Gauss beams with topological charges of ` = 1 and ` = 2. The propagation behaviourand spatio-spectral distribution of a Ti:sapphire laser oscillator beam was investigated under cw andmode-locked conditions. Experimental results were compared with data from analytical and numericalmodeling[2,3].
Fig. 1: Synthetic interference fringes measured with a white-light interferometer, SE1 with magnification 10x.
Fig. 2: Averaged rotation cuts (pulsed).
The measurements showed angular dispersion only in terms of a blurring of the fringes in the diffractionpatterns. The clean dark centres of the donut modes were always very well observable. By using spiraldiffractive phase elements for the generation of optical vortices in femtosecond laser pulses no angulardispersion compensation is necessary. The beam diameters of donut modes created by this method areonly slightly larger (a few %) in pulsed mode than they are in cw mode.
[1] L. Allen, S. Barnett, M. Padget, Orbital Angular momentum, Intitute of Physics Publishing Bristol and Phil-adelphia, (2003).
[2] A. T. N. Richter, Generation of orbital angular momentum in femtosecond laser pulses by means of spiralphase elements, Master Thesis FernUniversity Hagen, (2009).
[3] A. T. N. Richter, M. Bock, J. Jahns, R. Grunwald, Orbital angular momentum experiments with broadbandfew cycle pulses, SPIE Phot. West, San Francisco, 23.-28.01.2010, 7613-07.
Experimental and Numerical Study of Multiple Quantum Well Electroabsorption Modulator
T.M. Knyazyan1, H.V. Baghdasaryan1, J. Jahns, H. Knuppertz(1 State Engineering University of Armenia, Yerevan)
Electro-absorption (EA) modulator is a key component of contemporary fiber-optics communication net-works and high-speed free space optical interconnects due to high-speed operation, improved modulationefficiency and high contrast [1, 2]. Experimental and theoretical study of the contrast of one-dimensionalarray of 2x64 surface-normal EA modulators consisting of multiple quantum-wells (MQWs) sandwichedbetween top and bottom distributed Bragg reflectors (DBRs) is performed. A radiation from the tunablelaser is directed on the modulator chip, which is driven by a corresponding reverse voltage by supplyingit with two voltage levels switching the modulator pixels from a reflection to an absorption mode. Theresults of the measurements of the modulator contrast versus the wavelength and the supplied reversevoltage are depicted in Fig. 1a, b. As it is seen the modulator is wavelength sensitive and operates withinwavelength range of 850 854 nm. The modulator contrast depends also on the supplied voltage and thehighest contrast is observed for the reverse voltage of 9.5 V.
Theoretical analysis is carried out through numerical simulation by the method of single expression(MSE) [3]. The optical structure of EA modulator analysed theoretically is presented in Fig. 2.
a) b) c)
Fig. 1: a) Dielectric structure, electric field through the corner for ε1 > ε2; b) Ex c) Ez
Supplying a reverse voltage to a modulator causes a change of permittivity of QWs. By decreasing thecontrast of the modulator is increased (Fig. 3a) and then it falls (Fig. 3b). This is in a good qualitativeaccordance with the results of the measurements (Fig. 1b). It is also seen that the modulator is wave-length selective, having high contrast at specific operating wavelength of 852 nm. This fact is also in agood qualitative agreement with the results of the experimental analysis (Fig. 1a). The analysed EA mo-dulator is a wavelength sensitive device and should be driven by a certain reverse voltage for the efficientoperation.
[1] Q. Wang, S. Junique, D. Agren, S. Almqvist, B. Noharet, Arrays of vertical-cavity electroabsorption modula-tors for parallel signal processing, Opt. Express 13, (2005) 3323-3300.
[2] M. Jarczynski, Th. Seiler, J. Jahns, Integrated three-dimensional optical multilayer using free-space optics,Appl. Opt. 45, (2006) 6335-6341.
[3] H.V. Baghdasaryan, T.M. Knyazyan, Problem of plane EM wave self-action in multilayer structure: an exactsolution, Opt. and Quant. Electron., 31, No. 9/10, (1999) 1059-1072.
Fabrication of a micro-retroreflector by LIGA
J. Jahns, Th. Seiler, J. Mohr1, M. Borner1
(1Karlsruhe Institute of Technology)
The fabrication of microoptical elements with prismatic shape is, in general, a difficult task. It requiresa continuous linear profile for implementing the phase function and a sharp edge. The difficulty lies inachieving both simultaneously: the fabrication of a smooth surface requires a low-pass characteristic ofthe fabrication process (including both, lithography and pattern transfer into a substrate), but the sharpedge requires a high-pass characteristic. Unavoidable phenomena like surface tension, for example, ea-sily lead to a rounding of, both, the continuous profile and the edge. So far, the fundamental approachesto fabricate prismatic micro-structures are based on surface-micromachining (Fig. 1):
(1) anisotropic etching in silicon [1] yields excellent surface and edge quality, however, it is limitedto certain angles only as given by the crystal structure of the silicon; (2) gray-scale lithography [2] requi-res suitable gray-scale masks, good results can be achieved, however, the sag of the elements is limited;(3) ultaprecision micromachining [3] has been developed to generate free-form surfaces, however, thefeature size is limited to a minimum value.
Here, we report on the fabrication of a specific prismatic element, a 1D micro-retroreflector. Such anelement is of interest, for example, to realize a tapped delay line filter for ultrashort optical pulses [4].The requirements for the realization are a) a pitch on the order of 100 µm, b) to achieve a wedge an-gle of 90◦ as precisely as possible, and c) phase error from pitch to pitch of less than /10. None of theapproaches mentioned above can satisfy all requirements. Therefore, the fabrication was done here byprocessing the device from the edge of the structure rather than from the surface (Fig. 2). The advan-tage lies in the fact, that one can make use of the high precision of lithographic masks to generate theprismatic profile, in particular, the 90◦ angle. Virtually no phase error occurs across the device makingit suitable for interferometric experiments. To achieve large enough depth in direction of the grooves X-ray lithography and the LIGA process were used [5]. The device was fabricated at Karlsruhe Insitute ofTechnology as a Au-coated PMMA structure (Fig. 2) with a pitch of 100 µm and a depth in the directionof the grooves of 1 mm.
a)
variable exposure
photoresist b)
2D mask
X-ray illumination
photoresist c)
[1] W. Menz, J. Mohr, Mikrosystemtechnik fur Ingenieure, VCH-Verlag,(1997).
[2] Ch. Gimkiewicz, D. Hagedorn, J. Jahns, E.-B. Kley, F. Thoma, Appl. Opt. 38, (1999) 2986-2990.
[3] Y. Takeuchi, M. Murota, T. Kawai, K. Sawada, CIRP Ann. 52, (2003) 41-44.
[4] A. Sabatyan, J. Jahns, J. Eur. Opt. Soc. - Rapid Publ. 1, (2006) 06022.
[5] J. Mohr, Polymer Optics and Optical MEMS, in: V. Saile, U. Wallrabe, O. Tabata, J. G. Korvink, LIGA andits applications, Wiley-VCH, (2009).
Talbot band experiment with micro-retroreflector as tapped delay line for ultrashort opticalpulses
Th. Seiler, J. Jahns
In [1], the use of a retroreflector array (RA) for implementing an optical tapped delay line was suggestedand demonstrated. The experiments at that time were done with an RA that had been fabricated by ul-traprecision micromachining in aluminum. Although the surface quality of the element was very good,phase errors due to fabrication tolerances led to a visible reduction of the contrast of the diffraction pat-terns. Recently, a micro-RA was fabricated by using the LIGA process which does not suffer from suchphase errors [2]. With this element, the earlier experiments were repeated.
The experimental setup is shown in Fig. 1. A collimated beam from a broadband source illuminates theRA and is reflected back into the same direction. By means of a beamsplitter cube, the reflected signal issent to a grating interferometer. One of the two first diffraction orders is modulated. This is in analogy tothe classical Talbot band experiment which was extended to microstructured optical elements in ref. 3. Inour experiment very sharp peaks across the spectrum occur as a result of multibeam interference. Fig. 2shows the continuous spectrum of the white-light source (top) and the modulated spectrum in the outputplane (bottom). The theory for calculating the intensity distribution across the spectrum is given in refs.1 and 3. The sharpness of the peaks are a demonstration of the high quality of the micro-retroreflectorarray. Furthermore, the experiment demonstrates that such a device can, in principle, be used to build asa tapped delay line filter for ultrashort optical pulses similar to known devices like the VIPA [4].
tilted micro-retroreflector
array
spectrum
grating
point source
Fig. 1: Optical setup for analyzing theperformance of the retroreflector.
Fig. 2: Top: spectrum from a white-lightsource used for illumination. Bottom:modulated spectrum exhibiting sharppeaks.
[1] A. Sabatyan, J. Jahns, J. Eur. Opt. Soc. - Rapid Publ. 1, (2006) 06022.
[2] J. Jahns, Th. Seiler, J. Mohr, M. Borner, SPIE Phot. Europe, (2010) 7716-89.
[3] A. W. Lohmann, Optical Information Processing, 2nd ed., Universitatsverlag Ilmenau (2006).
[4] M. Shirasaki, Opt. Lett. 21, (1996) 366.
Spatio-temporal properties of a micro-retroreflector array
J. Jahns
The filtering and shaping of short optical pulses with pulse durations in the ps/fs-range is a task thatgains increasingly significance as applications emerge in all areas of information optics. These includedata storage, optical interconnection and information processing. Consequently, there exists an interest inthe handling and manipulation of the information content of signals represented by short optical pulses.The conventional approach for filtering of fs-pulses uses far-field diffraction at a grating by using staticor dynamic masks in the spectral plane [1,2]. A number of additional techniques has been suggested.One of them is the so-called ”virtual phased-array”(VIPA) [3]. It is based on the use of a Lummer-Gehrke interferometer which, in turn, has some common features with a Fabry-PErot interferometer. Theinteresting aspect of the VIPA is that it achieves a large spectral dispersion, about a factor of 10-20 largerthan usually achieved with diffraction gratings. This advantage can be used for stationary wavefields.However, practical drawbacks occur for short pulses due to material dispersion and spatial aberrations.Recently, we suggested the use of a retroreflector array (RA) as a tapped delay line [4-6]. Its mainfesatures are, that it is a reflective device (thus avoiding material dispersion), that it allows filteringdirectly in the temporal domain with time delays of O(10 fs - 1000 fs) and that it can be tuned (Fig.1). Here, we discuss its spatio-temporal properties. These can, for example, be represented in terms ofthe dependency of the diffraction angle sinα as a function of the wavelength λ. The relationship can bederived from the interference condition, which reads:
This is represented graphically in Fig. 2. The different lines represent different values of m (diffractionorder). Their slope (dispersion) is determined by the rotation tilt γ of the RA. The bundle of lines emergesfrom a point on the ordinate (for λ→ 0) given by sinα 0 = 2 sinγcosγ. This dispersion relationship isvery similar to the one that can be derived for the VIPA, which is due to the fact that both are multi-beaminterferometers. However, the physics is slightly different. The main advantage of the RA-interferometeris that it is purely reflective, hence not material dispersion occurs.
a)W
axis of rotation
!
Wsin! 2Wsin!
pulse front!
b)
Fig. 1: Left: Retroreflector structure: W is the period.Right: Time delays occur for the beamlets reflected fromneighbouring facets when the RA is rotated by an angle γ.
sin!
"
!(sin")=(#/W)cos$
sin!0
Fig. 2: Graphical representation of the dispersionrelationship. α is the direction of observation, λ isthe wavelength.
[1] B. Colombeau, M. Vampouille, C. Froehly, Opt. Comm. 19 (2008) 201-204.
[2] A. M. Weiner, J. P. Heritage, E. M. Kirschner, J. Opt. Soc. Am. B 5 (1988) 1563-1572.
[3] M. Shirasaki, Opt. Lett. 21 (1996) 366-368
[4] A. Sabatyan, J. Jahns, J. Eur. Opt. Soc. - RP 1 (2006) 06022.
[5] R. Grunwald, M. Bock, J. Jahns, CLEO/QELS (2008) CTu3.
[6] J. Jahns, CLEO/Pacific Rim (2009) WJ3-2.
ZEMAX-simulations of the polarization properties of a 1x4 PIFSO signal distribution system
M. Bohling, J. Jahns
Earlier, we presented a PIFSO-System (planar integrated free-space optics) for 1x4 signal distribution(Fig. 1). It was made of plastic and used prismatic arrays in reflection for beam-splitting and beam-redirection. Initially, the system was demonstrated experimentally (Fig. 2). Here, we analyze the pola-rization properties by simulation. In particular, we make use of the possibility to carry out polarizationcalculations with ZEMAX®. Fig. 1 shows the raytracing set-up in ZEMAX®. The raytracing calculati-on starts at position 1 with TM- and TE- polarized (TM: transversal magnetic, TE: transversal electric)light, respectively. We assume normal incidence of the incoming rays. The raytracing calculation ends atthe output detector positions 21. During the light propagation inside the PIFSO-System the light bundleexperiences dielectric as well as metallic layers indicated here by some position. In Fig. 2 the relative
Fig. 1: Raytracing set-up in ZEMAX® with input and output detectors. Raytracing start is at position 1 (sourceHeNe-laser), raytracing end is at position 21 (Detector OUT). Some positions in the PIFSO system are indicatedby numbers: 2: Detector IN; 3: input left subsystem; 6 and 17: metallic layer on microprism.
Fig. 2: Simulated rel. optical power as function of the position in the signal distributor and on the detectors,respectively. TM and TE indicates the polarization direction of the ray bundle at the entrance.
optical power vs. in the PIFSO-system for TM- and TE-polarization is shown. Till the first metallic layer(position 6) there is no difference in the polarization behavior. Between Position 6 and position 17 thecurve shapes for both polarization directions are similar but the optical power is different. From position17 the TM- and TE-curves overlap again. And finally at position 21 (detector OUT) the same opticaloutput power was calculated. In other words, for normal incidence, the system is invariant against theinput polarization direction.
[1] M. Bohling, J. Jahns, Optische Signalverteilung mit ultraprazisionsgefertigten mikrooptischen Systemen,ORT2006, Workshop Universitat Siegen, Oktober (2007), 62-66
[2] M. Bohling, Ultraprazisionsgefertigte Mikrooptiken fur ps/fs-Puls-Anwendungen, Masterthesis (2007).
Mixed media modelling and simulation of VCSEL based optical chip level interconnections
U. Lohmann, H. Knuppertz, J. Jahns
We demonstrate a MATLAB based simulation system and model to analyse optical interconnections inthe gigabit range in view of the performance and power consumption. The system enables a combinationbetween optical and electrical simulation functions. Furthermore same results are shown in terms ofminimization of power consumption of electro-optical links.Optoelectronic chip to chip connections in the gigabit range of up to 10 GB/s become more and moreinteresting for communication applications. The demonstrated simulation model and system enables ananalysis with focus on electrical and optical performance to support the system design regarding theperformance and power consumption.The used simulation system is based on MATLAB libraries for communication and signal analysis toimplement functions like “bit error rate calculation“ and “eye diagram scope“. With the CADENCEPSPICE libraries a combination between intrinsic laser functions and electrical functions of the com-ponents (VCSEL driver) was realized. With these separable functions of single components a closed
Fig. 1: Function modules of the modeland the sequence of signal conversion(current to opt. power)
Fig. 2: Eye diagram of a simulated 10GB/s data stream
transmitter and receiver signal conversion sequence from input voltage over VCSEL injection current tooptical power and back to photocurrent could be realized based on known dependences [1][2][3] (Fig.1).With several designed VCSEL-driver arrangements a total power consumption down to 5 mW/GB/s(BER defined at 10-9) could be achieved. Effects like the necessary electrical matching and the influenceof reflexions on the micro strip lines were shown in eye diagrams (Fig. 2).Due to the influences of the electrical line theory at wavelengths in the few cm-ranges, the dependency ofcomponents and system design increases for higher data rates. Thus, for further system design, a realisticsimulation combining optical and electrical aspects is essential.
[1] G. Sialm, VCSEL Modeling and CMOS Transmitters up to 40 GB/s for High-Desnsity Optical Links,Hartung-Gorre Verlag, (2007).
[2] G. Zimmer, Hochfrequenztechnik, lineare Modelle, Springer Verlag, (2000).
Statistical BER analysis of simulated optical interconnections
U. Lohmann, H. Knuppertz, J. Jahns
Measuring the bit error rate (BER) of optoelectronic connections in range of up to 10 Gb/s requires costintensive equipment like bit-error testers. The demonstrated simulation analysis, based on MATLAB andPSPICE libraries with the focus on statistical BER calculation, is a helpful and inexpensive alternativefor the system design. Here, we use it to analyze the relationship between BER and minimized powerconsumption for a short-distance optical link. This investigation was performed in connection with astudy on optical interconnection for the European Space Agency (Fig. 1).
Fig. 1: 3D density function of a simulated interconnection (10 Gb/s pseudorandom digital data stream)
With the density function approximation of the signal levels of the detected signal a relationship betweenthe fitted error function of the levels and the BER is given based on known dependences (Fig. 2). TheFigure 2 shows the cross section of the 3D density function at t = 0 and the fitted error function of thesignal levels “logical high“ and “logical low“.
Fig. 2: Fitted density function of signal levels (photocurrent in mA) and resulting optimal decision level gammaplus bit error rate (for example: Q = 6.9, gamma = 0.156, BER = 1.8e-9)
This statistical method allows calculating either the best decision level gamma and the resulting BER forseveral form factors Q of digital signal levels or s min values for a given BER (for example: BER min =1e-9) and given µvalues of signal levels.
[1] J. Jahns, H. Knuppertz, D. Fey, ESA Study 22188: State of the Art Optical Interchip Interconnections andPhotonic PCBs, (2009).
[2] G. Agrawal, Fiber Optic Communication Systems, Wiley-Interscience Verlag, (1997).