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Flat-topped beam output from a double-clad rectangular dielectric waveguide laser with a high-index inner cladding
Optics Communications 282 (2009) 2407–2412
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Flat-topped beam output from a double-clad rectangular dielectric waveguide laser with a high-index inner cladding Hongxiang Kang a,b,*, Haitao Zhang a, Ping Yan a, Dongsheng Wang a, Mali Gong a a b
Center for Photonics and Electronics, State Key Laboratory of Tribology, Department of Precision Instruments, Tsinghua University, Chengfu Road 28#, Haidian, Beijing 100084, China Institute of Radiation Medicine, Academy of Military Medical Sciences, Beijing 100850, China
a r t i c l e
i n f o
Article history: Received 21 October 2008 Received in revised form 24 February 2009 Accepted 1 March 2009
PACS: 42.55.Rz 42.60.Jf 42.79.Gn 42.82.Et
a b s t r a c t A novel method to produce a flat-topped laser beam by using a double-clad rectangular waveguide laser with high-index inner cladding is presented. The waveguide dispersion equation for cosine mode was deduced, the condition for the flattened mode was given out, relative gains for guided modes were calculated numerically and analyzed. Results indicate that a gain advantage for the flattened mode is clear, a flat-topped laser beam can be achieved when the optical confinement factor, the gain intensity and the output coupler are chosen suitably. Ó 2009 Elsevier B.V. All rights reserved.
Keywords: Solid-state lasers Flat-topped beam Rectangular dielectric waveguide Flattened mode
1. Introduction The flat-topped laser beam is needed in many optical systems, for example, laser fusion, optical character recognition (OCR), bioinstrumentation, laser hologram illumination, and material processing such as laser annealing, laser welding, laser cutting, and so on. The conventional methods to obtain flat-topped beams are beam shaping [1–3] or optical cavities with a super Gauss profile reflectivity mirror or a phase gradient mirror [4]. The flattened mode in optical fibers seems to be a promising thing to realize a flat-topped beam output. However, the practical manufacture of optical fibers does not permit the formation of perfect step index profiles, and the bend of the optical fiber is inevitable to induce the mode deformation. These imperfections are no doubt the difficulty of an ideal flat-topped profile in optical fibers [5]. To overcome these difficulties in optical fibers, the thermal-bonded or grown by liquid-phase epitaxy rectangular waveguide seems to be a perfect scheme. The thermal bonding technique involves the assembly by optical contacting of precision finished crystal or glass components and heat treating to increase bonding strength.
Repetition of this process can allow the fabrication of a composite of the given design. If heat treatment does not proceed up to temperatures where inter diffusion between adjacent optically polished surfaces occurs, a wide variety of materials may be bonded without formation of significant stress [6]. Liquid-phase epitaxy (LPE) is a well-known technique for producing high-quality oxide films for laser applications [7], the active layer can be overgrown by overlays to obtain active buried waveguide structures with a fine-controlled refractive-index profile. With the thermal-bonding technology or liquid-phase epitaxy the rectangular waveguide of perfect step index profile can be fabricated. In fact, lots of thermal-bonded or grown by liquid-phase epitaxy single-clad and double-clad planar waveguide lasers have been demonstrated [7–13]. In this paper, we demonstrate how a rectangular waveguide laser outputs a flat-topped beam using the flattened mode in the waveguide. 2. Modes in the double-clad rectangular waveguide with a highindex inner cladding 2.1. The flattened mode
* Corresponding author. Address: Center for Photonics and Electronics, State Key Laboratory of Tribology, Department of Precision Instruments, Tsinghua University, Chengfu Road 28#, Haidian, Beijing 100084, China. Tel.: +86 13381388289. E-mail address: [email protected] (H. Kang). 0030-4018/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2009.03.028
Fig. 1 shows the structure of a double-clad rectangular waveguide with a high-index inner cladding. According to Marcatili’s
H. Kang et al. / Optics Communications 282 (2009) 2407–2412
ay ay+by
y
2408
nx
n1
0 -ay-by -ay
n2 n0
n1
x
ny
y
n2
n0 -ax-bx -ax
0
ax ax+bx
x
Fig. 1. Double-clad rectangular waveguide with a high-index inner cladding.
Fig. 2. Electronic fields of flattened modes in x direction ðn0 ¼ 1:8147; n1 ¼ 1:8247; n2 ¼ 1:8151; ay ¼ 0:01 mm; k ¼ 1:064 lmÞ.
treatment [14], the electronic field can be provided as Ex0 ðx; yÞ ¼ Ex ðxÞEy ðyÞ. Due to the symmetrical characteristic of the rectangular waveguide, using n to replace x or y, the solution En ðnÞ can be expressed as
8 A0n exp½p0 ðn þ an þ bn Þ > > > > > n 6 an bn > > > rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h i > > A 2 > p > > k0n1 t12 k1 þ p20 cos k1 ðn þ an Þ þ k1 bn arctan k10 t1 > > 1 > > > > > an bn 6 n 6 an > > rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > > < A0n t 1 2 k þ p20 cos k1 bn arctan pk10 t 1 k1 2 t 21 1 En ðnÞ ¼ > > > an 6 n 6 an > > > rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h i > > A0n > p0 1 2 2 > t > 2 k1 þ p0 cos k1 ðn an Þ þ k1 bn arctan k1 k1 1 > t > 1 > > > > > an 6 n 6 an þ bn > > > > > A0n exp½p0 ðn an bn Þ > > : n P an þ bn ð4Þ
8 A0n exp½p0n ðn þ an þ bn Þ > > > > > n 6 an bn > > > > > > A1n exp½p1n ðn þ an Þ þ B1n exp½p1n ðn þ an Þ > > > > > an bn 6 n 6 an > > > < A expðp nÞ þ B expðp nÞ 2n 2n 2n 2n En ðnÞ ¼ > > an 6 n 6 an > > 0 0 > > > A1n exp½p1n ðn an Þ þ B1n exp½p1n ðn an Þ > > > > > an 6 n 6 an þ bn > > > > A00n exp½p0n ðn an bn Þ > > > : n P an þ bn
ð1Þ
where
p0n p1n p2n
where
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ k0 n2en n20 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ k0 n2en n21 ¼ ik1n qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ k0 n2en n22 ¼ ik2n
t2 ¼ ð2Þ
1
n¼y
n21 =n22
n¼x
Thus, for the flattened modes the rectangular waveguide must satisfy Eq. (3), and the core widths 2ax ; 2ay are independent. Using the
ney and nex are the effective indexes in y direction and in x direction respectively, and should be n2 for the flattened mode. In y direction, the tangential components of the electronic and magnetic ampli@Ey ðyÞ should be continuous at y ¼ ay and y ¼ tudes Ey ðyÞ and @y ðay þ by Þ. In x direction, the normal components of the dielectric x ðxÞ displacement and magnetic induction intensity n2 ðxÞEx ðxÞ; @E@x should be continuous at x ¼ ax and x ¼ ðax þ bx Þ. Taking these boundary conditions into account, the waveguide parameters for flattened modes should satisfy (See Appendix A):
bn ¼
qn p þ arctan
p0 t k1 1
k1
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where k1 ¼ k0 n21 n22 ;
t1 ¼
1
n¼y
n21 =n20
n¼x
;
;
qn ¼ 0; 1; 2 . . .
ð3Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p0 ¼ k0 n22 n20 ,
qn is the inner clad ordinal number:
The electronic field equation for flattened modes can be written as (See Appendix A):
Fig. 3. Intensity distribution of the flat-topped beam ðn0 ¼ 1:8147; n1 ¼ 1:8247; n2 ¼ 1:8151; ax ¼ ay ¼ 0:01 mm; k ¼ 1:064 lm; qx ¼ qy ¼ 0Þ.
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equation Ex0 ðx; yÞ ¼ Ex ðxÞEy ðyÞ the electronic field can be obtained. Fig. 2 shows electronic fields of flattened modes in y direction, it is clear that an idea flat-topped beam can be obtained when qy ¼ 0. Fig. 3 shows the intensity distribution of the flattened mode in the x–y plane. 2.2. Cosine modes When the mode effective index ne satisfies n0 < ne < n2 , the guided mode amplitude should be a cosine function and the waveguide dispersion equation can be deduced as (See Appendix B)
k1n 2k2n an þ 2 arctan t 2 tanðk1n bn hn Þ ¼ mn p; k2n where hn ¼ arctan
p0n t k1n 1
mn ¼ 0; 1; 2; . . .
; mn is the mode ordinal number.
ð5Þ
The electronic field equation for cosine modes can be written as (See Appendix B):
Grel ¼
3. Design for flat-topped beam outputting
dðx; y; zÞ/p ðx; y; zÞ R R R dðx;y;zÞ/ðx;y;zÞ
½1 þ S/ðx; y; zÞ
1þS/ðx;y;zÞ
dV
dV
ð7Þ
where dðx; y; zÞ is the normalized dopant profile, S gives a measure of the degree of saturation caused by the intracavity laser power. In a rectangular waveguide shown in Fig. 1, dðx; y; zÞ is a constant in the core and zero outside the core. On the assumption that the flattened mode is the lasing mode, /ðx; y; zÞ is a constant in the core. Thus, Eq. (7) can now be reduced as:
Grel
RRR /p ðx; y; zÞdV ¼ R R Rcore /ðx; y; zÞdV core
ð8Þ
It should be noted that, from Eq. (8), as a direct result of the flattened lasing mode and the uniform pumping distribution, the spatial distribution profile of the inversion remains constant for various levels of saturation and the mode relative gain Grel is independent of the saturation degree S.
8 A0n exp½p0n ðn þ an þ bn Þ > > > rffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > > p2 > > A0n 12 þ 20n cosðk1n n þ k1n an þ k1n bn hn Þ > t1 > k1n > > > rffiffiffiffiffiffiffiffiffiffiffiffiffiffirffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > 2 > p > k2 2 > > A0n t12 þ 20n t22 cos2 ðk1n bn hn Þ þ 1n sin ðk1n bn hn Þ cosðk2n n þ aÞ < k1n k22n 1 En ðnÞ ¼ > ðcosðk1n bn hn Þ P 0Þ mn p=2 > > a ¼ > > > p þ m p =2 ðcosðk n 1n bn hn Þ < 0Þ > > rffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > 2 > > > A0n 12 þ p20n cosðk1n n þ k1n an þ k1n bn hn þ mn pÞ > > t1 k1n > > > : A0n exp½p0n ðn an bn Þ
Using the equation Ex0 ðx; yÞ ¼ Ex ðxÞEy ðyÞ the electronic field for cosine modes can be obtained.
Z Z Z
n 6 an bn an bn 6 n 6 an
ð6Þ an 6 n 6 an an 6 n 6 an þ bn n P an þ bn
Applying Eq. (8) and the electronic field distribution, relative gains for each of the waveguide modes were calculated. As a calculational sample, the outer cladding refractive index n0 ¼ 1:8147 and the core refractive index n2 ¼ 1:8151 are corresponding to
3.1. The optical confinement factor Fig. 4 shows the optical confinement factor curves for the flattened mode with respect to half core width a (provided a ¼ a1 ¼ a2 for simplification and symmetry), the energy ratio in the flat top increases as half core width a increasing. Generally, for a rectangular beam, the spot size should contain about 90% of the whole energy [15]. In our examples shown in Fig. 4, when the optical confinement factor amounts to 0.9 the half core width a is 44 lm, 57 lm, 64 lm, and 82 lm corresponding to the inner cladding index n1 ¼ 1:8154; 1:8157; 1:816; 1:82, respectively. It is seen that the optical confinement factor increases when the inner cladding index increases. It is obvious that the rectangular waveguide laser would output a flat-topped beam when only the flattened mode works. However, the flattened mode may not always be the exclusive mode in the rectangular waveguide. The output laser beam is determined by the result of mode competition. 3.2. The relative gain, mode competition The gain for a particular mode, /p ðx; y; zÞ ¼ E2p ðx; y; zÞ, relative to the lasing mode, /ðx; y; zÞ ¼ E2 ðx; y; zÞ, can be expressed as [16]:
Fig. 4. The optical confinement factor curves for the flattened mode with respect to half core width a ðn0 ¼ 1:8147; n2 ¼ 1:8151; k ¼ 1:064 lm; qx ¼ qy ¼ 0; a ¼ ax ¼ ay Þ.
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Fig. 5. Relative gain for the Eymn modes (only the cases of m ¼ n are given out for simplification, curve m for mode Eymm ) ðn0 ¼ 1:8147; n1 ¼ 1:8157; n2 ¼ 1:8151; k ¼ 1:064 lm; q1 ¼ q2 ¼ 0; a ¼ a1 ¼ a2 Þ.
Fig. 7. The maximal relative gain curves in different inner cladding indexes for cosine modes with respect to half core width a ðn0 ¼ 1:8147; n2 ¼ 1:8151; k ¼ 1:064 lm; qx ¼ qy ¼ 0; a ¼ ax ¼ ay Þ.
YAG and Nd: YAG, respectively. Fig. 5 shows the results of the numerical calculation for the Eymn modes (only the cases of m ¼ n are given out for simplification). The number of cosine modes and relative gain Grel for each of the cosine modes increases as half core width a increasing. Fig. 6 shows curves of the optical confinement factor and the maximal relative gain for all the possible cosine modes with respect to half core width a. For the optical confinement factor 0.9, the maximal relative gain for cosine modes is about 0.9. There is a gain advantage for the flattened mode, implying that the flattened mode will lase first. Although this advantage is not very great, the lasing flattened mode will decrease the uniform gain intensity well-proportioned that a great competition advantage for the flattened mode can be anticipated in practice. A greater discriminability can be achieved by reducing the core widths in the cost of decreasing the optical confinement factor. If the optical confinement factor, the gain intensity and the output coupler are chosen suitably, a flat-topped beam is undoubtedly to be achieved in such a rectangular waveguide laser.
3.3. The influence from the inner cladding index
Fig. 6. The optical confinement factor for the flattened mode and the maximal relative gain for cosine modes with respect to half core width a ðn0 ¼ 1:8147; n1 ¼ 1:8157; n2 ¼ 1:8151; k ¼ 1:064 lm; qx ¼ qy ¼ 0; a ¼ ax ¼ ay Þ.
Fig. 7 shows the maximal relative gain curves in different inner cladding indexes for cosine modes with respect to half core width a. The maximal relative gain for cosine modes reduces when the inner cladding index n1 becomes small except for very small half core width a. However, as shown in Fig. 4, the optical confinement factor reduces also when the inner cladding index n1 becomes small. It is infeasible to reduce the inner cladding index for greater discriminability. 4. The implementation for the flat-topped beam In fact, it is difficult for Eq. (3) to be satisfied ideally, the practical thickness b is only made to approach the required thickness b0 in some accuracy as the waveguide being fabricated. Thus, the case that the practical thickness b deviates from the required thickness b0 should be discussed. For simplification, we only take the case in one direction into account.
Fig. 8. The optical confinement factors for the possible modes with respect to the inner cladding thickness b ðn0 ¼ 1:8147; n1 ¼ 1:8157; n2 ¼ 1:8151; k ¼ 1:064 lm; q ¼ 0; ay ¼ 0:03 mmÞ:
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According to the conditions that the boundaries, we have
@En ðnÞ @n
should be continuous at
A0n p0 ¼ A1n p1 expðp1 bn Þ B1n p1 expðp1 bn Þ
ðA1Þ
A1n ¼ B1n
ðA2Þ
A01n ¼ B01n
ðA3Þ
A00n p0 ¼ A01n p1 expðp1 bn Þ þ B01n p1 expðp1 bn Þ
ðA4Þ
According to the conditions that Ey ðyÞ should be continuous at the boundaries in y direction and n2 ðxÞEx ðxÞ should be continuous at the boundaries in x direction, we have
1 A0n ¼ A1n expðp1 bn Þ þ B1n expðp1 bn Þ t1 t2 ðA1n þ B1n Þ ¼ A2n
ðA7Þ
1 0 A ¼ A01n expðp1 bn Þ þ B01n expðp1 bn Þ t 1 0n
ðA8Þ
where
Fig. 8 shows the optical confinement factors for the possible modes with respect to the inner cladding thickness b. The flattened mode is an example of the cosine mode m ¼ 0 as b ¼ b0 , the flattened mode does not exist and the cosine mode m ¼ 0 would have a gain advantage in the case of b < b0 . Thus the cosine mode m ¼ 0 would become the lasing mode. The flattened mode and the cosine mode m ¼ 0 do not exist, and a flat-topped beam can not be obtained in the case of b > b0 . Fig. 9 shows the electronic field distributions for the cosine mode m ¼ 0 when b ¼ 0:95b0 ; 0:96b0 ; 0:97b0 ; 0:98b0 ; 0:99b0 . The practical thickness b is closer to the required thickness b0 , the field distribution is flatter. Given the relative errors of 1%, 5%, 10%, and 15%, calculated numerically using the parameters in Fig. 9, the relative intensity fluctuations in the core are 0.08%, 1.8%, 7.4%, and 17.5% respectively. Therefore, its relative intensity fluctuation is small enough that the cosine mode m ¼ 0 can be regarded as an approximate flattened mode in the condition that the relative error of b is small enough, and the laser can be regarded as a flat-topped beam.
From Eqs. (A2), (A3), (A6) and (A7), we have
In this paper, we present a novel method to produce a flattopped laser beam by using a double-clad rectangular waveguide laser with high-index inner cladding. The flattened mode and cosine modes in the double-clad rectangular waveguide were reasoned out, and their relative gains were calculated numerically and analyzed. A gain advantage for the flattened mode is clear and a flat-topped laser beam can be achieved when the optical confinement factor, the gain intensity and the output coupler are chosen suitably. A thermal-bonded or grown by liquid-phase epitaxy rectangular waveguide laser has the advantage of perfect step index profiles, with no mode deformation from the bend and no need for beam shaping or optical cavities with a super Gauss profile reflectivity mirror or a phase gradient mirror, we believe it is a promising scheme for the flat-topped laser beam. Appendix A For the flattened mode, the effective q refractive ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi index nn should be n2 , from Eq. (1) we have p0n ¼ k0 n22 n20 ¼ p0 ; p1n ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k0
n22
n21
¼ ik1 , and p2n ¼ 0.
ðA6Þ
t2 ðA01n þ B01n Þ ¼ A2n
Fig. 9. The field distributions for the cosine mode m=0 with respect to the practical thickness b (The humps from the top down corresponding to b ¼ 0:95b0 ; 0:96b0 ; 0:97b0 ; 0:98b0 ; 0:99b0 . n0 ¼ 1:8147; n1 ¼ 1:8157; n2 ¼ 1:8151; k ¼ 1:064 lm; q ¼ 0; ay ¼ 0:03 mm).
5. Conclusion
ðA5Þ
t1 ¼
1
n¼y
n21 =n20
n¼x
;
t2 ¼
1
n¼y
n21 =n22
n¼x
A1n ¼ B1n ¼ A01n ¼ B01n ¼ A2n =ð2t 2 Þ
ðA9Þ
From Eqs. (A1) and (A5), we have
A0n 1 p0 þ expðp1 bn Þ 2 t 1 p1 A0n 1 p0 ¼ expðp1 bn Þ ¼ A1n 2 t 1 p1
A1n ¼
ðA10Þ
B1n
ðA11Þ
Substitute Eqs. (A10) and (A11) into Eq. (A9), we have
bn ¼
qn p þ arctan k1
p0 t k1 1
; qn ¼ 0; 1; 2 . . .
ðA12Þ
Eq. (A12) is the condition should be satisfied for the flattened mode. Substitute Eq. (A12) into Eq. (A10), we have
A2n A1n ¼ B1n ¼ 2t 2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A0n 1 2 p0 2 cos k ¼ k þ p b arctan t 1 n 1 0 2k1 t 21 1 k1 Substitute Eq. (A13) and p0n ¼ k0
ðA13Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n22 n20 ¼ p0 ; p1n ¼ k0 n22 n21 ¼
ik1 ; p2n ¼ 0 into Eq. (1), En ðnÞ for the flattened mode can be obtained as Eq. (4). Appendix B According to the conditions that Ey ðyÞ should be continuous at the boundaries in y direction and n2 ðxÞEx ðxÞ should be continuous at the boundaries in y direction, from Eq. (1) we have
1 A0n ¼ A1n expðp1n bn Þ þ B1n expðp1n bn Þ t1 t2 ðA1n þ B1n Þ ¼ A2n expðp2n an Þ þ B2n expðp2n an Þ t2 ðA01n
þ
B01n Þ
¼ A2n expðp2n an Þ þ B2n expðp2n an Þ
1 0 A ¼ A01n expðp1n bn Þ þ B01n expðp1n bn Þ t 1 0n According to the conditions that boundaries, from Eq. (1) we have
@En ðnÞ @n
ðB1Þ ðB2Þ ðB3Þ ðB4Þ
should be continuous at the
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H. Kang et al. / Optics Communications 282 (2009) 2407–2412
A0n p0n ¼ A1n p1n expðp1n bn Þ B1n p1n expðp1n bn Þ
ðB5Þ
ðA1n B1n Þp1n ¼ A2n p2n expðp2n an Þ B2n p2n expðp2n an Þ
ðB6Þ
ðA01n þ B01n Þp1n ¼ A2n p2n expðp2n an Þ B2n p2n expðp2n an Þ
ðB7Þ
A00n p0n
ðB8Þ
¼
A01n p1n
expðp1n bn Þ þ
B01n p1n
expðp1n bn Þ
B1n
A0n 1 p0n expðp1n bn Þ þ 2 t1 p1n A0n 1 p0n ¼ expðp1n bn Þ 2 t 1 p1n
ðB9Þ
1 p1n A2n ¼ ðA1n B1n Þ þ t2 ðA1n þ B1n Þ expðp2n an Þ 2 p2n p 1 B2n ¼ 1n ðA1n B1n Þ þ t2 ðA1n þ B1n Þ expðp2n an Þ 2 p2n
B01n
ðB11Þ ðB12Þ
1 p0n expðp1n bn Þ þ t1 p1n A0 1 p0n expðp1n bn Þ ¼ 0n 2 t 1 p1n
ðB15Þ
B2n
ðB16Þ
[1] [2] [3] [4] [5] [6]
[8]
ðA1n B1n Þ þ t 2 ðA1n þ B1n Þ p1n p2n ðA1n B1n Þ þ t2 ðA1n þ B1n Þ
p1n B01n A01n þ t2 A01n þ B01n expð4p2n an Þ p2n ¼
p
p1n B01n A01n þ t2 A01n þ B01n
ðB19Þ
rffiffiffiffiffiffiffiffiffiffiffiffiffirffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p2 k2 A 2 A2n ¼ B2n ¼ 20n 1 þ k20n cos2 ðk1n bn hn Þ þ k1n 2 sin ðk1n bn hn Þ expðiaÞ 1n 2n ðcosðk1n bn hn Þ P 0Þ mn p=2 a¼ p þ mnp =2 ðcosðk1n bn hn Þ < 0Þ ðB20Þ
[7]
From Eqs. (B11), (B12), (B13) and (B8), we have p1n p2n
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u p2 A0n u t 1 þ 0n exp½iðk1n bn hn Þ ¼ 2 2 t 1 k21n
References
ðB14Þ
A2n ¼
A1n ¼
B1n
ðB13Þ
From Eqs. (B4) and (B8), we have
1 p1n 0 B1n A01n þ t 2 A01n þ B01n expðp2n an Þ 2 p2n p
1 ¼ 1n B01n A01n þ t 2 ðA01n þ B01n Þ expðp2n an Þ 2 p2n
.
Substitute Eqs. (B19) and (B20) into Eq. (1), En ðnÞ for the cosine mode can be obtained as Eq. (6).
2
p0n t k1n 1
Substitute Eq. (B19) into Eqs. (B11) and (B12), we have
From Eqs. (B3) and (B7), we have
A01n ¼
where hn ¼ arctan
ðB10Þ
From Eqs. (B2) and (B6), we have
A00n
ðB18Þ
A1n ¼
mn ¼ 0; 1; 2;
From Eqs. (B9) and (B10), we have
From Eqs. (B1) and (B5), we have
k1n 2k2n an þ 2 arctan t 2 tanðk1n bn hn Þ ¼ mn p; k2n
[9] [10] [11]
ðB17Þ
2n
Substitute Eqs. (B9), (B10), (B13), and (B14) into Eq. (B17), the waveguide dispersion equation for the cosine mode can be obtained as
[12] [13] [14] [15] [16]
Ximing Den, Xiangchun Liang, Zezun Chen, et al., Appl. Opt. 25 (1986) 377. Nieholas C. Roberts, Appl. Opt. 28 (1989) 31. Steven K. Dew, Robert R. Parsons, Appl. Opt. 31 (1992) 3416. Baida Lv, Laser Optics: Beam Characterization and Transformation, Resonator Technology and Physics, third ed., Beijing, 2003. Jay W. Dawson, Raymond Beach, Igor Jovanovic, Benoit Wattellier, Zhi Liao, Stephen A. Payne, C.P.J. Barty, in: Proceedings of SPIE, 2004, p. 132. C.T.A. Brown, C.L. Bonner, T.J. Warburton, D.P. Shepherd, A.C. Tropper, D.C. Hanna, H.E. Meissner, Appl. Phys. Lett. 71 (1997) 1139. Yaroslav E. Romanyuk, Camelia N. Borca, Markus Pollnau, Simon Rivier, Valentin Petrov, Uwe Griebne, Opt. Lett. 31 (2006) 53. S. Rivier, X. Mateos, V. Petrov, U. Griebner, Y.E. Romanyuk, C.N. Borca, F. Gardillou, M. Pollnau, Opt. Exp. 15 (2007) 5885. J. Wang, J.I. Mackenzie, D.P. Shepherd, in: Conference on Lasers and ElectroOptics, CLEO, 2005, p. 1694. M. Gong, H. Zhang, H. Kang, D. Wang, L. Huang, P. Yan, Q. Liu, Laser Phys. Lett. 5 (2008) 518. D.P. Shepherd, S.J. Hettrick, C. Li, J.I. Mackenzie, R.J. Beach, S.C. Mitchell, H.E. Meissner, J. Phys. D: Appl. Phys. 34 (2001) 2420. J.I. Mackenzie, IEEE J. Selected Topics Quantum Electron. 13 (2007) 626. M. Pollnau, Y.E. Romanyuk, Academie des Sciences, Comptes Rendus, Physique 8 (2007) 123. E.A.J. Marcatili, Bell Syst. Technol. J. 48 (1969) 2071. Walter Koechner, Solid-state Laser Engineering, fifth ed., Springer, 1999. Tajamal Bhutta, Jacob I. Mackenzie, David p. Shepherd, Raymond J. Beach, J. Opt. Soc. Am. B 19 (2002) 1539.
Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Flat-topped beam output from a double-clad rectangular dielectric waveguide laser with a high-index inner cladding Hongxiang Kang a,b,*, Haitao Zhang a, Ping Yan a, Dongsheng Wang a, Mali Gong a a b
Center for Photonics and Electronics, State Key Laboratory of Tribology, Department of Precision Instruments, Tsinghua University, Chengfu Road 28#, Haidian, Beijing 100084, China Institute of Radiation Medicine, Academy of Military Medical Sciences, Beijing 100850, China
a r t i c l e
i n f o
Article history: Received 21 October 2008 Received in revised form 24 February 2009 Accepted 1 March 2009
PACS: 42.55.Rz 42.60.Jf 42.79.Gn 42.82.Et
a b s t r a c t A novel method to produce a flat-topped laser beam by using a double-clad rectangular waveguide laser with high-index inner cladding is presented. The waveguide dispersion equation for cosine mode was deduced, the condition for the flattened mode was given out, relative gains for guided modes were calculated numerically and analyzed. Results indicate that a gain advantage for the flattened mode is clear, a flat-topped laser beam can be achieved when the optical confinement factor, the gain intensity and the output coupler are chosen suitably. Ó 2009 Elsevier B.V. All rights reserved.
Keywords: Solid-state lasers Flat-topped beam Rectangular dielectric waveguide Flattened mode
1. Introduction The flat-topped laser beam is needed in many optical systems, for example, laser fusion, optical character recognition (OCR), bioinstrumentation, laser hologram illumination, and material processing such as laser annealing, laser welding, laser cutting, and so on. The conventional methods to obtain flat-topped beams are beam shaping [1–3] or optical cavities with a super Gauss profile reflectivity mirror or a phase gradient mirror [4]. The flattened mode in optical fibers seems to be a promising thing to realize a flat-topped beam output. However, the practical manufacture of optical fibers does not permit the formation of perfect step index profiles, and the bend of the optical fiber is inevitable to induce the mode deformation. These imperfections are no doubt the difficulty of an ideal flat-topped profile in optical fibers [5]. To overcome these difficulties in optical fibers, the thermal-bonded or grown by liquid-phase epitaxy rectangular waveguide seems to be a perfect scheme. The thermal bonding technique involves the assembly by optical contacting of precision finished crystal or glass components and heat treating to increase bonding strength.
Repetition of this process can allow the fabrication of a composite of the given design. If heat treatment does not proceed up to temperatures where inter diffusion between adjacent optically polished surfaces occurs, a wide variety of materials may be bonded without formation of significant stress [6]. Liquid-phase epitaxy (LPE) is a well-known technique for producing high-quality oxide films for laser applications [7], the active layer can be overgrown by overlays to obtain active buried waveguide structures with a fine-controlled refractive-index profile. With the thermal-bonding technology or liquid-phase epitaxy the rectangular waveguide of perfect step index profile can be fabricated. In fact, lots of thermal-bonded or grown by liquid-phase epitaxy single-clad and double-clad planar waveguide lasers have been demonstrated [7–13]. In this paper, we demonstrate how a rectangular waveguide laser outputs a flat-topped beam using the flattened mode in the waveguide. 2. Modes in the double-clad rectangular waveguide with a highindex inner cladding 2.1. The flattened mode
* Corresponding author. Address: Center for Photonics and Electronics, State Key Laboratory of Tribology, Department of Precision Instruments, Tsinghua University, Chengfu Road 28#, Haidian, Beijing 100084, China. Tel.: +86 13381388289. E-mail address: [email protected] (H. Kang). 0030-4018/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2009.03.028
Fig. 1 shows the structure of a double-clad rectangular waveguide with a high-index inner cladding. According to Marcatili’s
H. Kang et al. / Optics Communications 282 (2009) 2407–2412
ay ay+by
y
2408
nx
n1
0 -ay-by -ay
n2 n0
n1
x
ny
y
n2
n0 -ax-bx -ax
0
ax ax+bx
x
Fig. 1. Double-clad rectangular waveguide with a high-index inner cladding.
Fig. 2. Electronic fields of flattened modes in x direction ðn0 ¼ 1:8147; n1 ¼ 1:8247; n2 ¼ 1:8151; ay ¼ 0:01 mm; k ¼ 1:064 lmÞ.
treatment [14], the electronic field can be provided as Ex0 ðx; yÞ ¼ Ex ðxÞEy ðyÞ. Due to the symmetrical characteristic of the rectangular waveguide, using n to replace x or y, the solution En ðnÞ can be expressed as
8 A0n exp½p0 ðn þ an þ bn Þ > > > > > n 6 an bn > > > rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h i > > A 2 > p > > k0n1 t12 k1 þ p20 cos k1 ðn þ an Þ þ k1 bn arctan k10 t1 > > 1 > > > > > an bn 6 n 6 an > > rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > > < A0n t 1 2 k þ p20 cos k1 bn arctan pk10 t 1 k1 2 t 21 1 En ðnÞ ¼ > > > an 6 n 6 an > > > rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h i > > A0n > p0 1 2 2 > t > 2 k1 þ p0 cos k1 ðn an Þ þ k1 bn arctan k1 k1 1 > t > 1 > > > > > an 6 n 6 an þ bn > > > > > A0n exp½p0 ðn an bn Þ > > : n P an þ bn ð4Þ
8 A0n exp½p0n ðn þ an þ bn Þ > > > > > n 6 an bn > > > > > > A1n exp½p1n ðn þ an Þ þ B1n exp½p1n ðn þ an Þ > > > > > an bn 6 n 6 an > > > < A expðp nÞ þ B expðp nÞ 2n 2n 2n 2n En ðnÞ ¼ > > an 6 n 6 an > > 0 0 > > > A1n exp½p1n ðn an Þ þ B1n exp½p1n ðn an Þ > > > > > an 6 n 6 an þ bn > > > > A00n exp½p0n ðn an bn Þ > > > : n P an þ bn
ð1Þ
where
p0n p1n p2n
where
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ k0 n2en n20 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ k0 n2en n21 ¼ ik1n qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ k0 n2en n22 ¼ ik2n
t2 ¼ ð2Þ
1
n¼y
n21 =n22
n¼x
Thus, for the flattened modes the rectangular waveguide must satisfy Eq. (3), and the core widths 2ax ; 2ay are independent. Using the
ney and nex are the effective indexes in y direction and in x direction respectively, and should be n2 for the flattened mode. In y direction, the tangential components of the electronic and magnetic ampli@Ey ðyÞ should be continuous at y ¼ ay and y ¼ tudes Ey ðyÞ and @y ðay þ by Þ. In x direction, the normal components of the dielectric x ðxÞ displacement and magnetic induction intensity n2 ðxÞEx ðxÞ; @E@x should be continuous at x ¼ ax and x ¼ ðax þ bx Þ. Taking these boundary conditions into account, the waveguide parameters for flattened modes should satisfy (See Appendix A):
bn ¼
qn p þ arctan
p0 t k1 1
k1
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where k1 ¼ k0 n21 n22 ;
t1 ¼
1
n¼y
n21 =n20
n¼x
;
;
qn ¼ 0; 1; 2 . . .
ð3Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p0 ¼ k0 n22 n20 ,
qn is the inner clad ordinal number:
The electronic field equation for flattened modes can be written as (See Appendix A):
Fig. 3. Intensity distribution of the flat-topped beam ðn0 ¼ 1:8147; n1 ¼ 1:8247; n2 ¼ 1:8151; ax ¼ ay ¼ 0:01 mm; k ¼ 1:064 lm; qx ¼ qy ¼ 0Þ.
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H. Kang et al. / Optics Communications 282 (2009) 2407–2412
equation Ex0 ðx; yÞ ¼ Ex ðxÞEy ðyÞ the electronic field can be obtained. Fig. 2 shows electronic fields of flattened modes in y direction, it is clear that an idea flat-topped beam can be obtained when qy ¼ 0. Fig. 3 shows the intensity distribution of the flattened mode in the x–y plane. 2.2. Cosine modes When the mode effective index ne satisfies n0 < ne < n2 , the guided mode amplitude should be a cosine function and the waveguide dispersion equation can be deduced as (See Appendix B)
k1n 2k2n an þ 2 arctan t 2 tanðk1n bn hn Þ ¼ mn p; k2n where hn ¼ arctan
p0n t k1n 1
mn ¼ 0; 1; 2; . . .
; mn is the mode ordinal number.
ð5Þ
The electronic field equation for cosine modes can be written as (See Appendix B):
Grel ¼
3. Design for flat-topped beam outputting
dðx; y; zÞ/p ðx; y; zÞ R R R dðx;y;zÞ/ðx;y;zÞ
½1 þ S/ðx; y; zÞ
1þS/ðx;y;zÞ
dV
dV
ð7Þ
where dðx; y; zÞ is the normalized dopant profile, S gives a measure of the degree of saturation caused by the intracavity laser power. In a rectangular waveguide shown in Fig. 1, dðx; y; zÞ is a constant in the core and zero outside the core. On the assumption that the flattened mode is the lasing mode, /ðx; y; zÞ is a constant in the core. Thus, Eq. (7) can now be reduced as:
Grel
RRR /p ðx; y; zÞdV ¼ R R Rcore /ðx; y; zÞdV core
ð8Þ
It should be noted that, from Eq. (8), as a direct result of the flattened lasing mode and the uniform pumping distribution, the spatial distribution profile of the inversion remains constant for various levels of saturation and the mode relative gain Grel is independent of the saturation degree S.
8 A0n exp½p0n ðn þ an þ bn Þ > > > rffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > > p2 > > A0n 12 þ 20n cosðk1n n þ k1n an þ k1n bn hn Þ > t1 > k1n > > > rffiffiffiffiffiffiffiffiffiffiffiffiffiffirffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > 2 > p > k2 2 > > A0n t12 þ 20n t22 cos2 ðk1n bn hn Þ þ 1n sin ðk1n bn hn Þ cosðk2n n þ aÞ < k1n k22n 1 En ðnÞ ¼ > ðcosðk1n bn hn Þ P 0Þ mn p=2 > > a ¼ > > > p þ m p =2 ðcosðk n 1n bn hn Þ < 0Þ > > rffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > 2 > > > A0n 12 þ p20n cosðk1n n þ k1n an þ k1n bn hn þ mn pÞ > > t1 k1n > > > : A0n exp½p0n ðn an bn Þ
Using the equation Ex0 ðx; yÞ ¼ Ex ðxÞEy ðyÞ the electronic field for cosine modes can be obtained.
Z Z Z
n 6 an bn an bn 6 n 6 an
ð6Þ an 6 n 6 an an 6 n 6 an þ bn n P an þ bn
Applying Eq. (8) and the electronic field distribution, relative gains for each of the waveguide modes were calculated. As a calculational sample, the outer cladding refractive index n0 ¼ 1:8147 and the core refractive index n2 ¼ 1:8151 are corresponding to
3.1. The optical confinement factor Fig. 4 shows the optical confinement factor curves for the flattened mode with respect to half core width a (provided a ¼ a1 ¼ a2 for simplification and symmetry), the energy ratio in the flat top increases as half core width a increasing. Generally, for a rectangular beam, the spot size should contain about 90% of the whole energy [15]. In our examples shown in Fig. 4, when the optical confinement factor amounts to 0.9 the half core width a is 44 lm, 57 lm, 64 lm, and 82 lm corresponding to the inner cladding index n1 ¼ 1:8154; 1:8157; 1:816; 1:82, respectively. It is seen that the optical confinement factor increases when the inner cladding index increases. It is obvious that the rectangular waveguide laser would output a flat-topped beam when only the flattened mode works. However, the flattened mode may not always be the exclusive mode in the rectangular waveguide. The output laser beam is determined by the result of mode competition. 3.2. The relative gain, mode competition The gain for a particular mode, /p ðx; y; zÞ ¼ E2p ðx; y; zÞ, relative to the lasing mode, /ðx; y; zÞ ¼ E2 ðx; y; zÞ, can be expressed as [16]:
Fig. 4. The optical confinement factor curves for the flattened mode with respect to half core width a ðn0 ¼ 1:8147; n2 ¼ 1:8151; k ¼ 1:064 lm; qx ¼ qy ¼ 0; a ¼ ax ¼ ay Þ.
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H. Kang et al. / Optics Communications 282 (2009) 2407–2412
Fig. 5. Relative gain for the Eymn modes (only the cases of m ¼ n are given out for simplification, curve m for mode Eymm ) ðn0 ¼ 1:8147; n1 ¼ 1:8157; n2 ¼ 1:8151; k ¼ 1:064 lm; q1 ¼ q2 ¼ 0; a ¼ a1 ¼ a2 Þ.
Fig. 7. The maximal relative gain curves in different inner cladding indexes for cosine modes with respect to half core width a ðn0 ¼ 1:8147; n2 ¼ 1:8151; k ¼ 1:064 lm; qx ¼ qy ¼ 0; a ¼ ax ¼ ay Þ.
YAG and Nd: YAG, respectively. Fig. 5 shows the results of the numerical calculation for the Eymn modes (only the cases of m ¼ n are given out for simplification). The number of cosine modes and relative gain Grel for each of the cosine modes increases as half core width a increasing. Fig. 6 shows curves of the optical confinement factor and the maximal relative gain for all the possible cosine modes with respect to half core width a. For the optical confinement factor 0.9, the maximal relative gain for cosine modes is about 0.9. There is a gain advantage for the flattened mode, implying that the flattened mode will lase first. Although this advantage is not very great, the lasing flattened mode will decrease the uniform gain intensity well-proportioned that a great competition advantage for the flattened mode can be anticipated in practice. A greater discriminability can be achieved by reducing the core widths in the cost of decreasing the optical confinement factor. If the optical confinement factor, the gain intensity and the output coupler are chosen suitably, a flat-topped beam is undoubtedly to be achieved in such a rectangular waveguide laser.
3.3. The influence from the inner cladding index
Fig. 6. The optical confinement factor for the flattened mode and the maximal relative gain for cosine modes with respect to half core width a ðn0 ¼ 1:8147; n1 ¼ 1:8157; n2 ¼ 1:8151; k ¼ 1:064 lm; qx ¼ qy ¼ 0; a ¼ ax ¼ ay Þ.
Fig. 7 shows the maximal relative gain curves in different inner cladding indexes for cosine modes with respect to half core width a. The maximal relative gain for cosine modes reduces when the inner cladding index n1 becomes small except for very small half core width a. However, as shown in Fig. 4, the optical confinement factor reduces also when the inner cladding index n1 becomes small. It is infeasible to reduce the inner cladding index for greater discriminability. 4. The implementation for the flat-topped beam In fact, it is difficult for Eq. (3) to be satisfied ideally, the practical thickness b is only made to approach the required thickness b0 in some accuracy as the waveguide being fabricated. Thus, the case that the practical thickness b deviates from the required thickness b0 should be discussed. For simplification, we only take the case in one direction into account.
Fig. 8. The optical confinement factors for the possible modes with respect to the inner cladding thickness b ðn0 ¼ 1:8147; n1 ¼ 1:8157; n2 ¼ 1:8151; k ¼ 1:064 lm; q ¼ 0; ay ¼ 0:03 mmÞ:
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H. Kang et al. / Optics Communications 282 (2009) 2407–2412
According to the conditions that the boundaries, we have
@En ðnÞ @n
should be continuous at
A0n p0 ¼ A1n p1 expðp1 bn Þ B1n p1 expðp1 bn Þ
ðA1Þ
A1n ¼ B1n
ðA2Þ
A01n ¼ B01n
ðA3Þ
A00n p0 ¼ A01n p1 expðp1 bn Þ þ B01n p1 expðp1 bn Þ
ðA4Þ
According to the conditions that Ey ðyÞ should be continuous at the boundaries in y direction and n2 ðxÞEx ðxÞ should be continuous at the boundaries in x direction, we have
1 A0n ¼ A1n expðp1 bn Þ þ B1n expðp1 bn Þ t1 t2 ðA1n þ B1n Þ ¼ A2n
ðA7Þ
1 0 A ¼ A01n expðp1 bn Þ þ B01n expðp1 bn Þ t 1 0n
ðA8Þ
where
Fig. 8 shows the optical confinement factors for the possible modes with respect to the inner cladding thickness b. The flattened mode is an example of the cosine mode m ¼ 0 as b ¼ b0 , the flattened mode does not exist and the cosine mode m ¼ 0 would have a gain advantage in the case of b < b0 . Thus the cosine mode m ¼ 0 would become the lasing mode. The flattened mode and the cosine mode m ¼ 0 do not exist, and a flat-topped beam can not be obtained in the case of b > b0 . Fig. 9 shows the electronic field distributions for the cosine mode m ¼ 0 when b ¼ 0:95b0 ; 0:96b0 ; 0:97b0 ; 0:98b0 ; 0:99b0 . The practical thickness b is closer to the required thickness b0 , the field distribution is flatter. Given the relative errors of 1%, 5%, 10%, and 15%, calculated numerically using the parameters in Fig. 9, the relative intensity fluctuations in the core are 0.08%, 1.8%, 7.4%, and 17.5% respectively. Therefore, its relative intensity fluctuation is small enough that the cosine mode m ¼ 0 can be regarded as an approximate flattened mode in the condition that the relative error of b is small enough, and the laser can be regarded as a flat-topped beam.
From Eqs. (A2), (A3), (A6) and (A7), we have
In this paper, we present a novel method to produce a flattopped laser beam by using a double-clad rectangular waveguide laser with high-index inner cladding. The flattened mode and cosine modes in the double-clad rectangular waveguide were reasoned out, and their relative gains were calculated numerically and analyzed. A gain advantage for the flattened mode is clear and a flat-topped laser beam can be achieved when the optical confinement factor, the gain intensity and the output coupler are chosen suitably. A thermal-bonded or grown by liquid-phase epitaxy rectangular waveguide laser has the advantage of perfect step index profiles, with no mode deformation from the bend and no need for beam shaping or optical cavities with a super Gauss profile reflectivity mirror or a phase gradient mirror, we believe it is a promising scheme for the flat-topped laser beam. Appendix A For the flattened mode, the effective q refractive ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi index nn should be n2 , from Eq. (1) we have p0n ¼ k0 n22 n20 ¼ p0 ; p1n ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k0
n22
n21
¼ ik1 , and p2n ¼ 0.
ðA6Þ
t2 ðA01n þ B01n Þ ¼ A2n
Fig. 9. The field distributions for the cosine mode m=0 with respect to the practical thickness b (The humps from the top down corresponding to b ¼ 0:95b0 ; 0:96b0 ; 0:97b0 ; 0:98b0 ; 0:99b0 . n0 ¼ 1:8147; n1 ¼ 1:8157; n2 ¼ 1:8151; k ¼ 1:064 lm; q ¼ 0; ay ¼ 0:03 mm).
5. Conclusion
ðA5Þ
t1 ¼
1
n¼y
n21 =n20
n¼x
;
t2 ¼
1
n¼y
n21 =n22
n¼x
A1n ¼ B1n ¼ A01n ¼ B01n ¼ A2n =ð2t 2 Þ
ðA9Þ
From Eqs. (A1) and (A5), we have
A0n 1 p0 þ expðp1 bn Þ 2 t 1 p1 A0n 1 p0 ¼ expðp1 bn Þ ¼ A1n 2 t 1 p1
A1n ¼
ðA10Þ
B1n
ðA11Þ
Substitute Eqs. (A10) and (A11) into Eq. (A9), we have
bn ¼
qn p þ arctan k1
p0 t k1 1
; qn ¼ 0; 1; 2 . . .
ðA12Þ
Eq. (A12) is the condition should be satisfied for the flattened mode. Substitute Eq. (A12) into Eq. (A10), we have
A2n A1n ¼ B1n ¼ 2t 2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A0n 1 2 p0 2 cos k ¼ k þ p b arctan t 1 n 1 0 2k1 t 21 1 k1 Substitute Eq. (A13) and p0n ¼ k0
ðA13Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n22 n20 ¼ p0 ; p1n ¼ k0 n22 n21 ¼
ik1 ; p2n ¼ 0 into Eq. (1), En ðnÞ for the flattened mode can be obtained as Eq. (4). Appendix B According to the conditions that Ey ðyÞ should be continuous at the boundaries in y direction and n2 ðxÞEx ðxÞ should be continuous at the boundaries in y direction, from Eq. (1) we have
1 A0n ¼ A1n expðp1n bn Þ þ B1n expðp1n bn Þ t1 t2 ðA1n þ B1n Þ ¼ A2n expðp2n an Þ þ B2n expðp2n an Þ t2 ðA01n
þ
B01n Þ
¼ A2n expðp2n an Þ þ B2n expðp2n an Þ
1 0 A ¼ A01n expðp1n bn Þ þ B01n expðp1n bn Þ t 1 0n According to the conditions that boundaries, from Eq. (1) we have
@En ðnÞ @n
ðB1Þ ðB2Þ ðB3Þ ðB4Þ
should be continuous at the
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H. Kang et al. / Optics Communications 282 (2009) 2407–2412
A0n p0n ¼ A1n p1n expðp1n bn Þ B1n p1n expðp1n bn Þ
ðB5Þ
ðA1n B1n Þp1n ¼ A2n p2n expðp2n an Þ B2n p2n expðp2n an Þ
ðB6Þ
ðA01n þ B01n Þp1n ¼ A2n p2n expðp2n an Þ B2n p2n expðp2n an Þ
ðB7Þ
A00n p0n
ðB8Þ
¼
A01n p1n
expðp1n bn Þ þ
B01n p1n
expðp1n bn Þ
B1n
A0n 1 p0n expðp1n bn Þ þ 2 t1 p1n A0n 1 p0n ¼ expðp1n bn Þ 2 t 1 p1n
ðB9Þ
1 p1n A2n ¼ ðA1n B1n Þ þ t2 ðA1n þ B1n Þ expðp2n an Þ 2 p2n p 1 B2n ¼ 1n ðA1n B1n Þ þ t2 ðA1n þ B1n Þ expðp2n an Þ 2 p2n
B01n
ðB11Þ ðB12Þ
1 p0n expðp1n bn Þ þ t1 p1n A0 1 p0n expðp1n bn Þ ¼ 0n 2 t 1 p1n
ðB15Þ
B2n
ðB16Þ
[1] [2] [3] [4] [5] [6]
[8]
ðA1n B1n Þ þ t 2 ðA1n þ B1n Þ p1n p2n ðA1n B1n Þ þ t2 ðA1n þ B1n Þ
p1n B01n A01n þ t2 A01n þ B01n expð4p2n an Þ p2n ¼
p
p1n B01n A01n þ t2 A01n þ B01n
ðB19Þ
rffiffiffiffiffiffiffiffiffiffiffiffiffirffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p2 k2 A 2 A2n ¼ B2n ¼ 20n 1 þ k20n cos2 ðk1n bn hn Þ þ k1n 2 sin ðk1n bn hn Þ expðiaÞ 1n 2n ðcosðk1n bn hn Þ P 0Þ mn p=2 a¼ p þ mnp =2 ðcosðk1n bn hn Þ < 0Þ ðB20Þ
[7]
From Eqs. (B11), (B12), (B13) and (B8), we have p1n p2n
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u p2 A0n u t 1 þ 0n exp½iðk1n bn hn Þ ¼ 2 2 t 1 k21n
References
ðB14Þ
A2n ¼
A1n ¼
B1n
ðB13Þ
From Eqs. (B4) and (B8), we have
1 p1n 0 B1n A01n þ t 2 A01n þ B01n expðp2n an Þ 2 p2n p
1 ¼ 1n B01n A01n þ t 2 ðA01n þ B01n Þ expðp2n an Þ 2 p2n
.
Substitute Eqs. (B19) and (B20) into Eq. (1), En ðnÞ for the cosine mode can be obtained as Eq. (6).
2
p0n t k1n 1
Substitute Eq. (B19) into Eqs. (B11) and (B12), we have
From Eqs. (B3) and (B7), we have
A01n ¼
where hn ¼ arctan
ðB10Þ
From Eqs. (B2) and (B6), we have
A00n
ðB18Þ
A1n ¼
mn ¼ 0; 1; 2;
From Eqs. (B9) and (B10), we have
From Eqs. (B1) and (B5), we have
k1n 2k2n an þ 2 arctan t 2 tanðk1n bn hn Þ ¼ mn p; k2n
[9] [10] [11]
ðB17Þ
2n
Substitute Eqs. (B9), (B10), (B13), and (B14) into Eq. (B17), the waveguide dispersion equation for the cosine mode can be obtained as
[12] [13] [14] [15] [16]
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