VCSEL Industry. Babu Dayal Padullaparthi. Читать онлайн. Newlib. NEWLIB.NET

Автор: Babu Dayal Padullaparthi
Издательство: John Wiley & Sons Limited
Серия:
Жанр произведения: Техническая литература
Год издания: 0
isbn: 9781119782216
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rel="nofollow" href="#fb3_img_img_5cf673db-495d-513f-91ed-c68b1d789a92.png" alt="r 1 equals StartRoot upper R 1 EndRoot normal e Superscript minus j phi 1"/> r 2 equals StartRoot upper R 2 EndRoot normal e Superscript minus j phi 2

       g: gain coefficient

        α : loss coefficient

       β: propagation constant (=w/c = 2 πf /c)

       ω: angular frequency

      Consider that the light wave in the resonator travels in the z direction from z = 0 and is reflected by the reflector r2 at z = L; then it goes backward by the length of L and returns to the starting point z = 0. If the electric field is sustainable, we should have:

      By comparing the imaginary and real parts, we have:

      (1.3)g Subscript t h Baseline equals alpha StartFraction upper L Over d EndFraction plus StartFraction 1 Over 2 d EndFraction ln left-parenthesis StartFraction 1 Over upper R 1 upper R 2 EndFraction right-parenthesis period

      The first term is absorption by the medium, and in GaAs, absorption by the free carrier has a magnitude of about 10 cm−1. In the second term, the reflectance of a reflector made by cleaving the surface of a semiconductor is

      (1.4)upper R 1 equals upper R 2 equals upper R equals left-brace left-parenthesis n minus 1 right-parenthesis slash left-parenthesis n plus 1 right-parenthesis right-brace squared left-parenthesis n colon upper E q u i v a l e n t r e f r a c t i v e i n d e x o f s e m i c o n d u c t o r right-parenthesis

      Therefore, in the case of a GaAs edge‐emitting laser (n = 3.5) with L = d = 300 μm, it is about 39 cm−1. To oscillate, a threshold gain of 10 + 39 = 49 cm−1 or more is required.

      The electric field Eout of the output light, with E0: field at the end of cavity, is given by:

      (1.5)upper E Subscript o u t Baseline equals StartRoot 1 minus upper R 2 EndRoot upper E 0 period

      1.1.5.2 Resonant Wavelength

      (1.6)2 left-parenthesis 2 pi n slash lamda right-parenthesis upper L plus phi 1 left-parenthesis lamda right-parenthesis plus phi 2 left-parenthesis lamda right-parenthesis equals 2 pi normal q left-parenthesis normal q colon i n t e g e r right-parenthesis period

      When the phase shift is 0 and the reflector is at the fixed end, for example, the standing wave is as shown in Figure 1.8b. The total length L is an integer q times the half wavelength λ /(2n) in the medium:

      (1.7)left-parenthesis lamda slash 2 n right-parenthesis q equals upper L period

      Now, in a laser cavity with a resonator length L longer than the wavelength, waves of many wavelengths with slightly different lengths can resonate. These modes are called longitudinal modes. On the other hand, the modes in the perpendicular direction are called the transverse modes.

      Considering a normal semiconductor laser, if λ is 1.3 μm, n = 3.5, and L = 3 μm, then q = 16.

      Therefore, even if q differs by 1, the resonance wavelength changes only slightly as Δλ. With |Δ λ | ≪̸ λ in mind, if λ λ 0 + Δλ , qq + 1, then we obtain,

      (1.8)StartFraction upper Delta lamda Over lamda 0 EndFraction equals minus 2 StartFraction lamda 0 Over n Subscript e f f Baseline upper L EndFraction period

      This |∆ λ | is called free spectral range (FSR) and is inversely proportional to cavity length, L.

      Here, neff is the effective index considering the dispersion of the medium and is given by the following expression:

      (1.9)n Subscript e f f Baseline equals n left-brace 1 minus left-parenthesis lamda 0 slash n right-parenthesis left-parenthesis partial-differential n slash partial-differential lamda right-parenthesis vertical-bar Subscript lamda equals lamda 0 Baseline right-brace period

      Since ∂n/∂ λ < 0 in ordinary semiconductors, neff is usually larger than n. In the above example, neff = 4.0 and |∆ λ | = 70 nm.

      1.1.5.3 Cavity Formation

      In Table 1.1 we have touched on DFB and DBR structures for single‐mode operation of edge‐emitting lasers [20–23]. In both cases, we utilize a pair of Bragg mirrors having an electric field reflectivity expressed by r equals StartRoot upper R EndRoot exp left-parenthesis minus j phi right-parenthesis that sandwich some space