Abstract

We present an analysis of the diffractional properties of dual-period apodizing gratings. In a previous paper we used these components to obtain single-lateral-mode and dual-longitudinal-mode emission from a broad-area diode laser. We now calculate the diffracted field for a monochromatic beam incident on the grating by using an analytical model. Predictions of the model are compared with experimental measurements made with several dual-period gratings. We also discuss the situation in which a dual-period grating is used as an external coupler of a diode laser in a two-wavelength emission regime.

© 2004 Optical Society of America

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References

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  1. J.-F. Lepage, R. Massudi, G. Anctil, S. Gilbert, M. Piché, N. McCarthy, “Apodizing holographic gratings for the modal control of semiconductor lasers,” Appl. Opt. 36, 4993–4998 (1997).
    [CrossRef] [PubMed]
  2. J.-F. Lepage, N. McCarthy, “Apodizing holographic gratings for dual-wavelength operation of broad-area semiconductor lasers,” Appl. Opt. 37, 8420–8425 (1998).
    [CrossRef]
  3. T. Hidaka, Y. Hatano, “Simultaneous two wave oscillation LD using biperiodic binary grating,” Electron. Lett. 27, 1075–1076 (1991).
    [CrossRef]
  4. A. Othonos, X. Lee, R. M. Measures, “Superimposed multiple Bragg gratings,” Electron. Lett. 30, 1972–1974 (1994).
    [CrossRef]
  5. A. Talneau, J. Charil, A. Ougazzaden, “Superimposed Bragg gratings on semiconductor material,” Electron. Lett. 32, 1884–1885 (1996).
    [CrossRef]
  6. E. A. Sziklas, A. E. Siegman, “Mode calculation in unstable resonators with flowing saturable gain. 2: Fast Fourier transform method,” Appl. Opt. 14, 1874–1889 (1975).
    [CrossRef] [PubMed]

1998

1997

1996

A. Talneau, J. Charil, A. Ougazzaden, “Superimposed Bragg gratings on semiconductor material,” Electron. Lett. 32, 1884–1885 (1996).
[CrossRef]

1994

A. Othonos, X. Lee, R. M. Measures, “Superimposed multiple Bragg gratings,” Electron. Lett. 30, 1972–1974 (1994).
[CrossRef]

1991

T. Hidaka, Y. Hatano, “Simultaneous two wave oscillation LD using biperiodic binary grating,” Electron. Lett. 27, 1075–1076 (1991).
[CrossRef]

1975

Anctil, G.

Charil, J.

A. Talneau, J. Charil, A. Ougazzaden, “Superimposed Bragg gratings on semiconductor material,” Electron. Lett. 32, 1884–1885 (1996).
[CrossRef]

Gilbert, S.

Hatano, Y.

T. Hidaka, Y. Hatano, “Simultaneous two wave oscillation LD using biperiodic binary grating,” Electron. Lett. 27, 1075–1076 (1991).
[CrossRef]

Hidaka, T.

T. Hidaka, Y. Hatano, “Simultaneous two wave oscillation LD using biperiodic binary grating,” Electron. Lett. 27, 1075–1076 (1991).
[CrossRef]

Lee, X.

A. Othonos, X. Lee, R. M. Measures, “Superimposed multiple Bragg gratings,” Electron. Lett. 30, 1972–1974 (1994).
[CrossRef]

Lepage, J.-F.

Massudi, R.

McCarthy, N.

Measures, R. M.

A. Othonos, X. Lee, R. M. Measures, “Superimposed multiple Bragg gratings,” Electron. Lett. 30, 1972–1974 (1994).
[CrossRef]

Othonos, A.

A. Othonos, X. Lee, R. M. Measures, “Superimposed multiple Bragg gratings,” Electron. Lett. 30, 1972–1974 (1994).
[CrossRef]

Ougazzaden, A.

A. Talneau, J. Charil, A. Ougazzaden, “Superimposed Bragg gratings on semiconductor material,” Electron. Lett. 32, 1884–1885 (1996).
[CrossRef]

Piché, M.

Siegman, A. E.

Sziklas, E. A.

Talneau, A.

A. Talneau, J. Charil, A. Ougazzaden, “Superimposed Bragg gratings on semiconductor material,” Electron. Lett. 32, 1884–1885 (1996).
[CrossRef]

Appl. Opt.

Electron. Lett.

T. Hidaka, Y. Hatano, “Simultaneous two wave oscillation LD using biperiodic binary grating,” Electron. Lett. 27, 1075–1076 (1991).
[CrossRef]

A. Othonos, X. Lee, R. M. Measures, “Superimposed multiple Bragg gratings,” Electron. Lett. 30, 1972–1974 (1994).
[CrossRef]

A. Talneau, J. Charil, A. Ougazzaden, “Superimposed Bragg gratings on semiconductor material,” Electron. Lett. 32, 1884–1885 (1996).
[CrossRef]

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Figures (13)

Fig. 1
Fig. 1

Scheme of the diode laser with an external cavity terminated by an apodizing grating (junction is oriented perpendicular to the sheet). AR, antireflection coating; CL, collimating lens; β L , Littrow angle.

Fig. 2
Fig. 2

Geometry used to write the dual-period apodizing holographic gratings. Dashed axes hold for the first exposure (photoresist in the plane z 0 = 0), whereas the solid axes hold for the second exposure (photoresist in the plane z = 0).

Fig. 3
Fig. 3

Example of the modulation profile of a dual-period apodizing grating as a function of the y position (not to scale).

Fig. 4
Fig. 4

Angles (ϕ m ) of the diffracted orders for a grating illuminated at an angle β. This case corresponds to a single-period grating, with λ/Λ = 0.5.

Fig. 5
Fig. 5

Representation of the angular spreading ϕ m,n,i of the n = |m| and |m| + 1 terms diffracted in the m = 0 and -1 orders by a dual-period grating. The relative amplitude is used to distinguish between the principal and the secondary terms and is not to scale.

Fig. 6
Fig. 6

Calculated value of the central reflectivity as a function of the normalized central modulation depth for a dual-period grating. The first three n terms are shown: (a) for the -1 order and (b) for the 0 order.

Fig. 7
Fig. 7

Scheme of the setup used to characterize the dual-period apodizing gratings. PD, photodiode.

Fig. 8
Fig. 8

Measured intensity profiles of the diffracted orders for grating D. In (a) and (b) the -1 order is shown for polarization perpendicular and parallel to the lines of the grating, respectively. In (c) and (d) the 0 order is shown for polarization perpendicular and parallel to the lines of the grating, respectively.

Fig. 9
Fig. 9

Measured intensity profiles of the -1 order for gratings presented in Table 1. Polarization of the incident beam is perpendicular to the lines of the grating.

Fig. 10
Fig. 10

Angular distribution of the n, i terms for (a) the m = -1 order and (b) the m = 0 order as a function of the relative difference δΛ between the grating periods. Dots correspond to the measured values for gratings A, B, C, D, and E whereas the curves correspond to the predicted values for n = |m|, |m| + 1, |m| + 2, and |m| + 3; i = {1, 2}.

Fig. 11
Fig. 11

Calculated intensity profiles of the diffracted orders for a dual-period grating with δΛ = 0.00357. In (a) and (b) the -1 order is shown for h 0/λ = 0.1 and h 0/λ = 0.5, respectively. In (c) and (d) the 0 order is shown for h 0/λ = 0.1 and h 0/λ = 0.5, respectively.

Fig. 12
Fig. 12

Intensity profile of the -1 order: (a) measured for grating D with polarization perpendicular to the lines of the grating and (b) calculated with h 0/λ = 0.55 and δΛ = 0.00474.

Fig. 13
Fig. 13

Scheme representing the angular spreading of the principal and secondary (n = |m| + 1) terms of the 0 and -1 orders for a dual-period grating operated in Littrow configuration with two wavelengths (λ1 and λ2) simultaneously. Secondary terms are indicated by a1 = λ1-1,2,1), a2 = λ2-1,2,1), b1 = λ1-1,2,2), b2 = λ2-1,2,2), c1 = λ10,1,1), c2 = λ20,1,1), d1 = λ10,1,2), and d2 = λ20,1,2). Angles and arrow lengths are not to scale.

Tables (1)

Tables Icon

Table 1 Parameters of the Dual-Period Gratings

Equations (19)

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Λ2=Λ1/cos θ,
hx, y=12exp-2x2/wx2exp-2y2/wy2×h01+h01 sin2πΛ1 y+φ01+h02+h02 sin2πΛ2 y+φ02,
Ei=Fx, yexp-j 2πλ y sin β.
Er=m Erm= Gmx, yexp-j 2πλ y sin ϕm,
Er=-Ei exp-jδ,
δ2πλ 2hx, y.
ξx, yπh0λexp-2x2/wx2exp-2y2/wy2.
Er=-Ei exp-j2ξi=12J0ξ+2 n=1 J2nξcos2n 2πΛi y-2j n=1 J2n-1ξsin2n-12πΛi y.
 Gmx, yexp-j 2πλ y sin ϕm=-Fx, yexp-j 2πλ y sin βexp-j2ξJ0ξ2+J0ξn=1-1nJnξexp+j2nπy/Λ1+Jnξexp-j2nπy/Λ1+J0ξn=1-1nJnξexp+j2nπy/Λ2+Jnξexp-j2nπy/Λ2+n=1-1nJnξexp+j2nπy/Λ1+Jnξexp-j2nπy/Λ1×n=1-1nJnξexp+j2nπy/Λ2+Jnξexp-j2nπy/Λ2.
 Gm,n,ix, yexp-j 2πλ y sin ϕm,n,i=-Fx, yexp-j 2πλ y sin βexp-j2ξ×J0ξ2-J0ξJ1ξexp+j2πyΛ1-J0ξJ1ξexp+j2πyΛ2+n=1N-1nJnξ2exp+j2nπyΔΛΛ1Λ2+exp-j2nπyΔΛΛ1Λ2+n=2N-1nJnξJn-1ξ×exp+j2πyΛ1+nΔΛΛ1Λ2+exp+j2πyΛ2-nΔΛΛ1Λ2.
sin ϕm,n,i=sin β+λΛ1Λ2mΛi+-1inΔΛ
sin ϕm,i=sin β+mλ/Λi,
Rm,nx, y=Gm,nx, yFx, y2.
R-1,nx, y=Jnπh0λexp-2x2/wx2×exp-2y2/wy2Jn-1πh0λ×exp-2x2/wx2exp-2y2/wy22,
R0,nx, y=Jnπh0λexp-2x2/wx2×exp-2y2/wy24,
exp-j2ξexp4jπh0λx2wx2+y2wy2expjφ,
expj2πr2/λR,
Rx=wx22h0, Ry=wy22h0
δΛΛ2-Λ1Λ1=1cos θ-1,

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