Attempts have been made to significantly increasethe data transfer rate of optical disk equipment[1]. A multibeam optical recording head is usefulfor recording and playback in parallel [2–5], and inthis procedure as many optical spots as possibleshould be aligned at the same interval on a certainsmall plane [6]. To miniaturize such an optical head,we proposed the formation of a multibeam optical re-cording head that uses a pitch conversion component(PCC), which is composed of waveguides with differ-ent bending structures [7–9]. The PCC can alignmul-tiple laser beams close together, while the lasers arekept away from each other, and contribute to the for-mation of a multibeam light source that emits beamsat a narrow pitch.Myriads of waveguides have been developed as op-
tical components of multiplexers, modulators andother optical devices in optical communicationsusing near-infrared light [10]. To use such wave-guides as optical components for optical recordingusing visible light, we have previously reported onthe fabrication of waveguides using poly(perfluoro
butenyl vinyl ether) (PBVE) [11]. Due to the improve-ment of their fabrication process, their transmittancealready achieved 70% for a wavelength of 405nm(blue-violet) , which can convey a high optical inten-sity sufficient for optical recording. In this paper, wereport on a design for a PCC to be applied to opticalrecording using visible light, and discuss a PCC thathas as many waveguides as possible.
2. Fundamental Considerations
A. Requirements
Several trials of optical recording in parallel have al-ready been performed using a semiconductor laserdiode array as a multibeam light source [2–5]. Allthe beams are required to retain a high optical inten-sity, perfect isolation, the same optical properties, thesmall image height that they produce, etc. [7]. To ob-tain such a multibeam, the following requirementsare discussed:
1. A straight waveguide should transmit a beamefficiently (intensity).
2. A PCC should perfectly isolate the respectivebeams optically (isolation).
3. A bent waveguide should transmit a beam asefficiently as a straight waveguide (identity).
4. A PCC should align beams close together(optical view).
Requirements (1) and (3) concern the transmittanceof waveguides. Requirement (2) concerns the forma-tion of waveguides constructed on a PCC. Require-ment (4) concerns the formation of a multibeam.Requirement (1) ensures that a beam through awaveguide constructed on a PCC can retain high op-tical intensity on an optical disk, which contributesto a high-speed recording rate per beam. Require-ment (2) ensures that all the beams through aPCC are prevented from optical crosstalk with eachother, which contributes to their high beam quality,particularly on both the optical disk during recordingand the photodetectors during playback. Require-ment (3) ensures that all the beams through a PCCcan have the same optical intensity, which contri-butes to recording and playback in parallel underthe same optical conditions. Requirement (4) ensuresthat all the beams through a PCC can form opticalspots with the same shape, which contributes toan increase in the number of beams.Table 1 shows our focus in discussing the above re-
quirements. These requirements theoretically con-flict with each other, which makes it too complexto organize them simultaneously. For instance,requirement (1) requires a waveguide with a lowrelative index difference, which does not meet re-quirements (2) and (3).Here, the above requirements take priority in
listed order from the viewpoint of high-speed record-ing. To meet requirements (1) to (4), a PCC should beevaluated, respectively, from the viewpoint of theevaluations listed in Table 1. We design a PCC (para-meters) for the formation of a multibeam optical re-cording head and discuss the number of beams on thebasis of the fabricated waveguides discussed inRefs. [8,9,11]. Prior to the discussion, the structureof a PCC is discussed.
B. Bending Structure
Figure 1 shows a schematic diagram of a PCC com-posed of single-mode waveguides with differentbending structures. Waveguide #0 represents astraight waveguide, and the others represent bentwaveguides composed of two straight sections andone bent section. The waveguide labeled with a lar-ger number is structurally bent more sharply, andwaveguide # − i is positioned symmetrically to wave-guide #i with respect to waveguide #0. Cores ap-
proach each other toward the output, and the pitchis narrowed from p at the input to q at the output.
A waveguide can be structurally bent by either auniform curvature or a gradually changing curvature[12,13]. The latter enables a bent waveguide to besmoothly connected to a straight waveguide by losingits curvature at the interface between them. Then, atthe interface, a beam propagating through the bentwaveguide can be coupled to the straight waveguidewithout a loss in principal, and it is expected to pre-vent nonpropagating light from appearing there.Nonpropagating light usually immediately turnsinto optical noise, and suppressing such light contri-butes to the high quality of a beam propagatingthrough the waveguide, as well as to low opticalcrosstalk toward beams propagating through otherwaveguides constructed on a PCC. This is signifi-cantly important with respect to requirement (2).
As one of such bending structures, a raised-sinefunction shows an appropriate curve [14,15]. The po-sition z is defined as the distance from the interfacebetween the straight and bent sections on the inputside, as shown in Fig. 1. Then, the center axis x of abent waveguide is curved and described as
xðzÞ ¼ A½Bz − sinðBzÞ�; ð1Þ
where A andB represent the coefficients with respectto the lateral shift δ of the waveguide between theinput and output as well as the length ℓ2 of the bent
Table 1. Discussion of Designing PCC
Requirement Parameter Evaluation
(1) Intensity Device length ℓ Transmittance of straight waveguide(2) Isolation Core pitch q Optical coupling(3) Identity Lateral shift δ Transmittance of bent waveguide [Conflict with req. (1)](4) Optical view Number of waveguides 2N þ 1 Image height of optical spots [Conflict with req. (2)]
section. These coefficients of waveguide #i are repre-sented as
Ai ¼p − q2π i; ð2Þ
Bi ¼2 πℓ2
: ð3Þ
The two straight sections play optically importantroles. A laser beam usually differs in shape from abeam propagating through a waveguide, and thestraight section on the input side helps it graduallysettle in the propagation mode of the waveguide. It isreported that a taper with a ridged structure isneeded to be extended to more than 0:5mm sothat a near-infrared beam can be efficiently guidedinto a buried waveguide, which achieves ∼330λ(λ ¼ 1500nm) [16,17]. Here, the straight sectionon the input side is set to be 0:2mm long(ℓ1 ¼ 0:2mm), regardless of the structure of a bentwaveguide, which equals 500λ (λ ¼ 405nm). On theother hand, there is no analytical harm to omit thestraight section on the output side (ℓ3 ¼ 0), becausea raised-sine curve is already selected so that its cur-vature vanishes at the output of the bent section, asdescribed above.
3. Discussions on Design
Table 2 shows the specifications for designing a PCCthat are in accordance with the optical properties ofthe fabricated waveguides discussed in Refs. [8,9,11].Figure 2 shows the propagation mode of a blue-violetbeam (λ¼405nm) that a waveguide with a squarecore allows. The solid line represents a cutoff curve,which is calculated by the point matching method[18]. Below the cutoff curve, the waveguide preventsa propagation of a blue-violet beam in a high-orderpropagation mode, and ensures transmission in onlythe fundamental mode.
A core shape was actually evaluated with a differ-ential interference microscope, and the fluctuation τof the core width is estimated to be approximately0:1 μm, which is not sufficiently small compared withthe core width 2d. Nevertheless, the guiding materi-al, PBVE, can be finely tuned in its refractive index[11]. Here, waveguides are designed just on the cutoffcurve, which includes no margin. Moreover, all theoptical components composed of a multibeam opticalrecording head are assumed to be aligned as pre-cisely as those composed of a conventional opticalhead for a single beam.
Laser arrays for a multibeam optical recordingusing near-infrared light can emit several beamsat pitches of 50 to 100 μm simultaneously [2–5]. Here,a laser array for blue-violet light is applied as an in-put of a PCC, and the core pitch p on the input side isset to 100 μm.
A. Device Length
To retain the optical intensity of a beam, a PCC isrequired to transmit a beam efficiently. Here, theattenuation of a beam is discussed, and a PCC is re-stricted to extend from the viewpoint of the transmit-tance of a straight waveguide.
A guiding material such as SiO2 glass [19] orPBVE [20] already exhibits low intrinsic lossthroughout the visible region (400–800nm). How-ever, single-mode waveguides using such materials
Fig. 2. Cross-sectional structure of a straight waveguide with asquare core of 2d sides (λ ¼ 405nm). The cutoff curve is plotted bythe point matching method [18].
Table 2. Specifications of PCC
Parameter Value
Wavelength λ 405nmCore size 2d On cutoff curveFluctuation of core width τ 0:1 μmCore pitch (input) p 100 μm
may result in an intense increase in loss at the visibleregion, because they have small cores with somedefects produced during fabrication. Figure 3 showsthe extrinsic factors that may enable a waveguide toattenuate a laser beam, and these can be classifiedinto two major factors [21]. Propagation loss is in-duced when a beam is propagating through a wave-guide, which is considered to be caused by animperfect fabrication of waveguides. This loss in-creases in proportion to the device length of thewaveguide. On the other hand, connection loss is in-duced on only the input and output faces of awaveguide and is caused by an optical mismatch ofan interface between a PCC and an optical system.Here, connection loss appears constant, regardlessof the length of a waveguide.It is difficult to exactly formulate the propagation
loss of an actual buried waveguide because it takes acomplicated shape at a cross section. Here it is esti-mated on the basis of the electromagnetic analysis ofan optical fiber or a slab waveguide. Then, the pro-pagation loss σ dB=m of a buried waveguide can beexpressed as the following function regarding across-sectional structure, Δ and 2a (¼ 2d − τ):
σðΔÞ ¼ EΔ2
aþ F
1Δþ C: ð4Þ
The first term shows mode conversion loss [22], andthe second term shows microbending loss [23]. Thethird term shows absorption loss. E and F representcoefficients that strongly depend on the fabrication ofa waveguide. As Δ increases, mode conversion ap-pears enhanced and σ becomes more strongly depen-dent on it, whereas microbending disappears. On theother hand, as Δ decreases, microbending appearsenhanced, although mode conversion disappears.Thus, the coefficients E and F show a barometer ofσ of an actual waveguide, and Δ should be designedappropriately according to the optical properties of
fabricated waveguides. Generally, σ is minimizedwhen Δ takes a small value.
The coefficients E, F, and C can be determined byevaluating the propagation properties of more thanthree types of straight waveguides that are designedto differ in only their cross-sectional structure,Δ and2a. For instance, regarding the fabricated wave-guides discussed in Refs. [8,9,11], E, F, and C are es-timated to be 3.6, 0.0125, and 35, respectively.Figure 4 shows σ, and σ becomes ∼55dB=m forΔ ¼ 0:25% and drops to its minimum 43dB=mfor Δ ≈ 0:13%.
It is difficult to formulate connection loss exactly,because both reflection and coupling strongly dependon a surface condition of a waveguide and an opticalmatching between a PCC and an input optical sys-tem. Here, the connection loss is estimated as fol-lows. A single-mode optical fiber was placed facinga waveguide fabricated in Refs. [8,9,11], and a beamwas directly guided into the waveguide through theoptical fiber. It was confirmed that the optical fiberproduced an almost circular Gaussian beam andthat, on average, ∼86% of a blue-violet beam wasguided into the waveguide, which includes reflectionat the input and output faces. Here, the connectionloss of a waveguide is set to 14%.
To retain the optical intensity of a beam, wave-guides should retain the transmittance of a beam.Here, a PCC is permitted to extend to the length ℓuntil straight waveguide #0 itself retains the effi-cient utilization η of a beam, which excludes connec-tion loss:
ℓ ≤−10 · logðηÞ
σðΔÞ : ð5Þ
All the fabricated straight waveguides discussed inRefs. [8,9,11] can transmit over 70% of a blue-violetbeam, which takes a connection loss of 14% into ac-count. Then, straight waveguide #0 itself is required
Fig. 4. Propagation loss of a blue-violet beam propagatingthrough a straight waveguide designed on a cutoff curve.
Fig. 5. Device length ℓ of a PCC when a straight waveguide #0transmits over 70% of a blue-violet beam.
to retain η ≤ 0:7=ð1–0:14Þ ¼ 0:81% excluding connec-tion loss.Figure 5 shows the device length ℓ when a straight
waveguide on a cutoff curve is designed to transmit70% of a blue-violet beam (η ¼ 0:81%). Then, ℓ takesa maximum of 19:0mm for Δ ≈ 0:13%.
B. Core Pitch
One possible cause of degradation of the optical se-paration of beams is that evanescent light invadesother waveguides constructed on a PCC through acladding region, whereas a beam propagates througha waveguide. Such optical coupling is observed moreclearly as multiple single-mode waveguides ap-proach each other. Here, a core pitch is discussedfrom the viewpoint of such optical coupling.When two core films with the refractive index n2e
are stacked in a cladding region at intervals D apartfrom each other, this composes two slab waveguidesrunning in parallel. Then, a coupling coefficient χ canbe expressed as [24],
χ ¼ κx02γx022βek02n2e
2Δeð1þ aγx0Þexp½−γx0ðD − 2aÞ�; ð6Þ
where κx0 and γx0 represent the wave numbers of abeam on the transverse plane of the waveguide,and k0 represents the wave number of a beam travel-ing in free space.Provided that a beam is not electromagnetically
deformed while propagating through a bent wave-guide, two beams propagating through bent wave-guides #i − 1 and #iþ 1 equivalently couple withbent waveguide #i. Thus, it is focused consideringthat a beam propagating through bent waveguide#1 optically couples with straight waveguide #0.Although the interval D between waveguides #0and #1 is designed to vary with the position z, onlythe transverse electromagnetic components of abeam propagating through bent waveguide #1 contri-bute to the optical coupling with the straight wave-guide #0. Consequently, the coupling amount Q isgiven by
Q ¼ 2sin2
�Zℓ2
0χðzÞdz
�; ð7Þ
where the coefficient “2” indicates that bent wave-guide #i possesses two adjacent waveguides (#i − 1and #iþ 1), except for bent waveguides #Nand # −N.Recording media can be removed from optical disk
equipment at any time, andmisplacement in a centerplace when a recording medium is attached inducesreadout noise. For an optical disk for a single beam,such noise is required to be reduced to <−45dB to-ward a beam reflected at an optical disk [25]. Thus,optical interference among multibeams should besuppressed sufficiently below the readout noise.Here, cores are allowed to approach each other untilthe optical coupling Q reaches −60dB.
Figure 6 shows the core pitch q at the output of aPCC when the device length ℓ is designed as in Fig. 5.A buried waveguide is transformed into a model of aslab waveguide by the effective index method [26]. qis required to become wider as Δ decreases, becausea waveguide weakens the effect of confining a beaminside a core.
C. Lateral Shift
A bent waveguide more intensely radiates a beam asit is structurally bent with a smaller radius. How-ever, it is difficult to exactly formulate the radiationof a beam due to the bending structure of a buriedwaveguide with some defects, because it assumes acomplicated shape at a cross section. Here, radiationis estimated due to the bending structure of a slabwaveguide, and a PCC is restricted in bending radiusto transmit beams as efficiently as a straightwaveguide.
When a slab waveguide is structurally curved witha uniform radius R, a beam is estimated to be ra-diated according to an exponential function concern-ing R, and the argument on the exponential termalmost determines the amount of radiation [27].On the other hand, an actual buried waveguide ap-pears to have rough sidewalls, which induces moreintense radiation of a beam. Thus, it can be analyti-cally considered that this only increases the coeffi-cient of the exponential term while the exponentialterm itself remains intact. Thus, the argumentshould be designed to exceed a certain criterion ε[28]:
ΛðRÞ½1 −H2ðRÞ�1:53
≥ ε: ð8Þ
Here Λ and H represent the following functions con-cerning R, and the propagation parameters of a slabwaveguide are defined in Eq. (6):
Fig. 6. Core pitch q at output of a PCC. A blue-violet beam of < −
60dB optically couples with a straight waveguide #0, while propa-gating through bent the waveguides #1.
The radiation of a blue-violet beam was directlymeasured using the fabricated bent waveguidescurved by constant radii, which are discussed inRefs. [8,9,11]. Here, the minimum radius Rmin isevaluated as the radius at which a bent waveguidedegrades its transmittance by 0:1dB (≈1:0%) com-pared with a straight waveguide, which is the samedefinition as the cutoff radius of an optical fiber.Here, the criterion ε is set to 18.When waveguides are structured by a raised-sine
curve according to Eqs. (1)–(3), the curvature radiusR is geometrically given by
RðzÞ always takes a minimum Rmin at Bz ≈ π=2, re-gardless of the coefficient B. Then, a lateral shiftof a bent waveguide between an input and an outputtakes a maximum, δmax, and is expressed as
δmax ≈ðℓ − ℓ1Þ22πRmin
: ð12Þ
Figure 7 shows δmax when the device length ℓ is de-signed as in Fig. 5. The curve of δmax is shifted towardthe positive direction of Δ compared with that of ℓ,because Rmin decreases with Δ. δmax takes a maxi-mum of 2:75mm for Δ ≈ 0:24%.
D. Number of Beams
Figure 8 shows the basic configuration of a focusingsystem. Optical spots formed on an optical disk aremore distorted due to the wavefront aberration ofa focusing system, particularly an objective lens,as a beam takes a more oblique direction in a focus-ing system [6]. Here, the number of beams is dis-cussed taking account of all the requirements aswell as an optical view of an objective lens.
A single-mode waveguide provides a beam with aprofile similar to a Gaussian distribution. Accordingto Kirchhoff ’s diffraction theory, such a beam di-verges in the air at the angle θ (1=e half angle) ina Fraunhofer field, expressed as
θ ¼ tan�1
� λ2πω0
�: ð13Þ
ω0 represents a 1=e half size of a beam propagatingthrough a waveguide (beam waist), and here ω0 is es-timated by the point matching method [18].
When beams through a PCC are expanded to aneffective aperture of an objective lens, a collimatorlens is designed to have the numerical aperture
Fig. 7. Lateral shift δmax between the input and the output of abent waveguide.
Fig. 8. (Color online) Basic configuration of a focusing system.
Fig. 9. (Color online) Number of blue-violet beams. A multibeamoptical recording head is designed to consist of a PCC using PBVE.
NAc of sin θ. Then, a focusing system reduces the lat-eral image of the beams at the magnificationα ¼ sin θ=NAo, where NAo represents the numericalapertures of the objective lens.Blu-ray Disc equipment employs an objective lens
with a numerical aperture NAo of 0.85, and the lensperforms wavefront aberration of <λ=20 inside asmall circle with a radius r of 23 μm for a blue-violetbeam [29]. Figure 9 shows the number of beams thatan optical recording head is allowed to have. Thedotted line represents the number of optical spotswithin the optical view of a focusing system, whichis projected on the output of a PCC (2r=αq). On theother hand, the dashed line represents the numberof waveguides that can be constructed on a PCC(δmax=ðp − qÞ). Single-mode waveguides using PBVEare desired to have a relative index difference Δ of∼0:23%, and the PCC should be designed accordingto Table 3.The solid line in Fig. 9 represents the smaller of
the numbers of optical spots or waveguides. Theodd number that does not exceed the solid line indi-cates the number of blue-violet beams that a multi-beam optical recording head is allowed to have,which is restricted by the optical view of a focusingsystem. As a result, the recording head can have 51blue-violet beams (N ¼ 25) for 0:17% < Δ < 0:32%.
4. Conclusions
We describe a design for a PCC from the viewpoint ofits application to multibeam optical recording. Wediscussed four fundamental requirements, intensity,isolation, identity, and optical view; note that theserequirements conflict theoretically. Bent waveguideswere structured using a raised-sine curve, and theserequirements were considered by their priority inlisted order. According to the fabrication of wave-guides using PBVE, a PCC can have over 50 wave-guides for λ ¼ 405nm, and contributes to theformation of amultibeam optical recording head withnumerous beams.It is difficult to exactly study a beam for propaga-
tion through buried waveguides, particularly asingle-mode waveguide for visible light, because theyhave a small structure and a complicated shape.Thus, their optical properties strongly depend onthe fabrication process of the waveguide. As dis-cussed in this paper, a PCC can be designed on thebasis of the optical performance of the actual wave-guides under such a fabrication condition.
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