IE84367B1 - A semiconductor laser - Google Patents
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- IE84367B1 IE84367B1 IE2005/0484A IE20050484A IE84367B1 IE 84367 B1 IE84367 B1 IE 84367B1 IE 2005/0484 A IE2005/0484 A IE 2005/0484A IE 20050484 A IE20050484 A IE 20050484A IE 84367 B1 IE84367 B1 IE 84367B1
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Description
A Semiconductor Laser
INTRODUCTION
Field of the Invention
The invention relates to semiconductor lasers, particularly those of the edge-emitting
F abry-Pérot type and to their design and manufacture.
Prior Art Discussion
Semiconductor laser light emitting devices comprise a waveguide, which is formed by
the semiconductor wafer structure in which the laser light is generated. Semiconductor
ridge waveguide Fabry-Pérot (FP) lasers have the advantage of being relatively
straightforward to manufacture but suffer from the drawback that the lasers tend to
operate multimode. A number of different routes have therefore been pursued to achieve
one or more discrete-modes with neighbouring wavelengths suppressed.
It is known that defects or perturbations within the cavity can introduce a modulation,
which can improve the spectral purity of the Fabry-Pérot laser. Following from this
principle, quasi single-moded edge—emitting FP lasers have been demonstrated using a
variety of techniques. Small absorptive sites have been created along the laser cavity
using high energy laser pulses (L. F. diChiaro, J. Lightwave Tech., 9(8) (1991) p. 975).
Side mode suppression (SMS) better than 20 dB was achieved using as few as three such
sites. An undesirable feature of this technique, however, is the large increase in the
device threshold current which can accompany the introduction of the lossy regions.
An alternative technique involves the creation of reflective or scattering sites along the
cavity of the laser by focused ion beam etching (D. A. Kozlowski, J. S. Young, J. M. C.
England and R. G. S. Plumb, IEE Electron. Lett., 31(8) (1995) p. 648). For positioning, a
similar scheme to that of diChiaro was employed; N sites are nominally located at
distances of Lea,/2" , n = l, N, from one of the cleaved facets and where Lw, is cavity
length. In this case SMS as large as 30 dB was achieved with three etch sites and with a
minor increase in the device threshold current.
A coupled cavity laser design has also been proposed (H. Naito, H. Nagai, M. Yuri, K.
Takeoka, M, Kume, K. Harnada and H. Shimizu, J. Appl. Phys., vol. 66, (1989) p. 5726).
Two waveguide cores are connected within the cavity and, owing to the internal
reflection due to the change in the effective index, a modulated effective reflectivity
could be ascribed to one of the facets. While these devices possess desirable features,
multiple growth and etching steps are necessary to form the structure.
Numerical techniques have also been used to design distributions of effective index and
injected current in order to achieve improved spectral purity in edge emitting lasers.
These include the use of genetic breeder algorithms (D. Erni, M. M. Spiihler and J.
Frolich, Opt. Quant. Electron, 30, (1998) p. 287).
A technique which does not require additional processing or regrowth steps involves the
creation of a low density of additional features in the laser ridge waveguide at the
lithographic and etching stages when the ridge itself is formed (B. Corbett and D.
McDonald, IEE Electron. Lett., 3l(25) (1995) p. 2181). These features are typically
made as small as 1 am in length and can have a primarily reflective character. In the case
of ridge waveguide semiconductor lasers emitting near 1.5;zm wavelength, the additional
features resemble slots, which penetrate into the cladding region of the optical waveguide
of the laser.
The invention is directed towards achieving more controlled production of Fabry-Pérot
laser devices so that one or more output modes are accurately achieved.
SUMMARY OF THE INVENTION
According to the invention, there is provided a method for designing an edge-emitting
semiconductor laser device comprising a Fabry-Pérot laser cavity with mirrors for
regenerative feedback for lasing, and at least one feature in the cladding between the
cavity mirrors, each feature causing a local change in refractive index, the method
comprising determining the locations of the features based on a relationship between
feedback in sub—cavities between each feature and a cavity mirror and modulation of the
threshold gain of the Fabry-Pérot modes of the cavity, and wherein the method comprises
the steps of:
setting device Fabry-Pérot reference mode, cavity mirror reflectivities, number of
features, and form of threshold gain modulation required;
providing a feature density function;
sampling the feature density function; and
adjusting feature positions indicated by the sampling to optimise resonant
feedback magnitude.
In one embodiment, the feature density fiinction is provided by multiplying the threshold
modulation amplitude expression by the Fourier transform of the desired threshold gain
modulation function, said feature density function being:
I exp[ 6 Lcavamirjl _ ()2 I exp[— 6 Lcavamir G)
in which,
the gain is distributed uniformly along the length of the cavity,
amir ::-‘log
L lrirzl
are the mirror losses of an unperturbed cavity,
Law is the cavity length,
r1 and F2 are the mirror reflectivities
F(E) is the Fourier transform of the threshold modulation function, and
=77 ‘N2 , "being the position of a feature along the cavity expressed as a
fraction of the total cavity length.
In another embodiment, the Fourier transform has positive and negative components, the
positive and negative components give rise to slot positions located at even integer plus
one half and odd integer plus one half multiples of the values of the quarter wavelength
of light emitted at the selected mode mg with respect to one of the cavity mirrors and
there are multiple modes in the laser spectrum.
In a further embodiment, the feature density function sampling is determined by the total
number of features to be introduced.
In one embodiment, the sampling is performed according to the expression:
- in I exp[xLcavanzir 11“ |Ftdx =1-1/2
cav mir
[Ir] I exp[xL or
in which the normalisation constant A is determined by the number of features to be
introduced, which must be specified in order to sample the feature density function.
In another embodiment, the feature positions are adjusted so that for each feature, a short
sub-cavity on one side has a length which is an odd integer multiple of quarter
wavelengths of the selected mode mo, and the longer sub-cavity on the other side has a
length which is an even integer multiple of quarter wavelengths of the selected mode,
provided that the change is the effective index due to a feature is negative, the mirror
reflectivities are real and positive numbers, and single mode operation is desired.
In a further embodiment, the features are slots in the cladding.
In one embodiment, the slots are in a cladding ridge.
In another aspect, the invention provides a method of manufacturing an edge-emitting
semiconductor laser device comprising a Fabry-Perot laser cavity with mirrors for
regenerative feedback for lasing, the method comprising the steps of:
designing the device in any method as defined above; and
fabricating the device with provision of slots in a cavity ridge during lithographic
and etching stages of forming the ridge.
In one embodiment, the device is a multi-mode laser device.
In another aspect, the invention provides an edge-emitting semiconductor laser device
comprising a Fabiy-Pérot laser cavity with mirrors for regenerative feedback for lasing,
and at least one feature in the cladding between the cavity mirrors, said feature or
features being located according to any design method set out above.
DETAILED DESCRIPTION OF THE INVENTION
Brief Description of the Drawings
The invention will be more clearly understood from the following description of some
embodiments thereof given by way of example only with reference to the accompanying
drawings, in which:
Fig. 1(a) is a schematic diagram of a Fabry-Pérot laser device having slots in a
cladding ridge, Fig. l(b) is a flow diagram for design of the device and Fig. 1(c)
is a graphical representation of the step 21 of Fig. l(b);
Fig. 2 is a one dimensional model of the cavity of a laser device;
Fig. 3 is a plot of threshold gain of a homogeneous Fabry-Pérot laser as a function
of cavity mode index m , in which the threshold gain is taken to be constant in
this example;
Fig. 4 is a schematic diagram of another laser structure which is optimally slotted
for mode selection according to the present invention;
Fig. 5(a) is a plot of threshold gain distribution of an unperturbed Fabry-Pérot
laser, in which the variation of the semiconductor gain function, ,1/(/10), with
wavelength is also shown, and Fig. 5(b) shows threshold gain distribution of a
perturbed F abry—Pérot laser where a single mode at m0 is selected;
Fig. 6 is a plot of threshold gain distribution of a perturbed Fabry-Pérot laser
device where a comb of modes at mo fir. na is selected, it being an integer;
Fig. 7 is a plot of threshold gain distribution of a perturbed Fabry-Perot laser
where the losses are reduced at mode mo, with a weaker loss reduction at
mo i na , and with other modes being largely unaffected by the perturbations
introduced;
Fig. 8 is a plot of threshold gain for the laser cavity with sixteen slots described in
Table 1;
Fig. 9 is a plot of below threshold SMSR and peak mode position with
temperature of the device of Fig. 8;
Fig. lO(a) is a plot of an optimum slot density distribution function of a laser
device, in which the inset is a diagram of a laser cavity with a slot pattern
determined according to the invention; and Fig. 10(b) is a plot of the form of the
resultant threshold gain spectrum of this laser;
Fig. 11 is a plot of the lasing spectrum at twice threshold of a single mode laser of
Fig. 10, and the inset is the lasing spectrum at twice threshold of a Fabry—Pérot
laser without slots, for reference purposes;
Fig. 12 is a plot of the form of the threshold gain of modes for a laser cavity
where two Fabry-Perot modes at predetermined wavelengths are selected, and the
inset is a schematic diagram of the laser cavity ridge;
Fig. 13 is a plot of the form of the threshold gain of modes for a laser cavity for
which three Fabry-Pérot modes are selected;
Fig. 14 is a plot of threshold gain for a laser cavity with twenty slots described in
Table 2;
Fig. 15 is a diagram of a multi—section device incorporating two slotted FP
structures designed according to the present invention; and
Fig. 16 is a diagram of a multi-section device in which slotted FP structures are
laterally coupled and each section is independently contacted, such devices
allowing for increased power output in a single mode and for increased
modulation bandwidth.
Detailed Description of the Invention
Referring to Fig. 1(a) a Fabry~Pe'rot (FP) laser device 1 has an n-type substrate 2, an
active region 3, a p-type cladding 4, an insulator 5, and a contact 6. The cladding 4
comprises a ridge 7 having a number of slots 8. The light emitting direction is shown by
the arrow 9.
In the device l, the primary sources of optical feedback are the as-cleaved cavity mirrors.
The structure is grown epitaxially on a substrate. The active region operates under
forward bias to generate light. Confinement layers serve to provide electronic
confinement for the carriers trapped in the active region. The light comes out through the
cavity mirrors. The active region placed within the confinement layer is preferably
formed by any insertion, the energy band of which is narrower than that of the substrate.
Possible active regions include, but are not limited to, a single quantum well or a multi-
laycr system of quantum wells, quantum wires, quantum dots, or any combination
thereof.
The slots 8 cause a partial longitudinal reflection of the light. In the invention the precise
location of the slots is chosen to accurately and predictably achieve a particular selected
mode or modes in the output light.
Device Design Method Overview
Referring to Fig. l(b) a method of designing a laser device such as the device 1 is
illustrated. The invention provides a method to design a slot pattern in a laser device,
both to preferentially select a particular Fabry-Pérot mode as the peak emission
wavelength and also to suppress an arbitrary number of neighbouring F abry-Pérot modes.
The method selects a set of Fabry-Pérot modes in preference to other Fabry-Pérot modes
within the cavity. In this way the method addresses the important problems for
semiconductor lasers of predetermination of the peak lasing wavelength and also stability
of the peak lasing mode with changes in temperature. The method also allows for the
fabrication of multimode devices with increased functionality both as individual devices
and as component parts of more complex multi-section or multi-element devices.
In a step 20 device parameters and properties are set. These include reference FP mode
mo, cavity mirror reflectivities r1 and r;, the number of slots, and form of threshold gain
modulation required. These parameters are set on the basis of:
MJ<<1; M = Lw; and r1= r;
n 211
where
n is the refractive index,
An is the local change in refractive index caused by a slot,
» is the emission wavelength of mode mg,
Lcav is the cavity length, and
r1, 1'; are the cavity mirror reflectivities (ends cleaved and un—treated).
It is to be noted that the mirror reflectivities r1 and r2 are equal, real, and positive. The
data for step 20 is inputted manually, and the remaining steps of the method are
implemented automatically by computer.
In step 21 a slot density function is automatically determined, as set out in more detail
below (particularly Eqn. (4)). This is represented graphically in Fig. l(c), which is a
single mode example for which the parameters are a=20, #0036 and |r,l = r2 .
In step 22 the slot density function is sampled, again as set out in more detail below
(particularly Eqn. (21)).
Finally, in step 23 the position of the slots are adjusted to optimize resonant feedback
magnitude, as set out in more detail below with reference to Table I particularly.
Referring to Fig. 2 a model of a Fabry-Perot laser is shown. The cavity is of length Law
and includes 5 slots. The cavity effective index is n and the slot region has effective index
n +An. The cavity is in vacuum with all cavity sections numbered 1' beginning on the
left. The slots are also numbered with index j . The complex transmission and reflection
coefficients of the cavity are 7 and F respectively.
Sub-cavities
In a Fabry-Pérot laser, the cavity mirrors are the sources of regenerative feedback
necessary for lasing oscillation. The addition of a perturbation (in this embodiment a slot)
to the FP cavity forms two sub-cavities between the slot and the cavity mirrors as
illustrated in Fig. 2. The slot perturbs the effective refractive index experienced by an
optical mode propagating in the cavity. The numerical value of this effective index step is
An. Optical modes of the cavity undergo a partial reflection at the boundaries between
the cavity and the slotted region. This partial reflection gives rise to additional feedback
and is the origin of the optical mode selectivity.
In the method of the invention each slot location is selected with consideration of the
sub-cavities on each side between the slot and the cavity mirrors. This consideration is
made irrespective of other slots (other than minor path length corrections), the parameter
values for the sub—cavities for each slot being determined independently.
The method is based on an understanding of how the feedback fiom the slot modulates
the threshold gain of the FP modes. These FP modes are the lasing modes of the device,
and information about light at other wavelengths is unimportant.
The partial reflection provided by a slotted region comprising two parallel interfaces
perpendicular to the laser ridge is maximized. Each of the reflective interfaces provides a
similar amount of optical feedback and choice of the correct slot length then allows the
mode selectivity due to the feedback from the slot to be maximized.
Given that the effective index step associated with the slots is small, the complex
reflection coefficient of the slot, r:, can be approximated by a summation of the two
primary reflections fiom the slot/cavity interfaces, i r... The result is rs =;-I +e3‘9 -(—r,.).
Here 6 = nx/COL: is the phase advance across the slot, n3 is the effective refractive index
of the slotted region, k0 is the free space wavenumber of the cavity mode and L3 is the
length of the slotted region. The reflection coefficient assumes its maximum value of 2r,
provided 26 =(2q+l)7r where q is an integer. This relation implies that
L, = (q +1/2M0 /272:. In order to maximize the partial reflection of a given optical mode
due to the slot, the length of the slotted region must therefore be an odd integer number
of quarter wavelengths of the selected optical mode in question. In the following it is
assumed that the slot lengths (Ls) are as above, but is can be otherwise.
The peak mode is determined by the spacing of the slots with respect to the cavity
mirrors. In general, the two sub-cavities formed by each slot have different lengths. The
change in the threshold gain of the selected mode is maximized in the method. For single
mode design, the selected mode is mg, and the change in threshold gain is maximized for
mg. However, as described below with reference to Fig. 12, where two or more modes are
selected mo may not have the largest change in threshold gain. However, mg is always the
central mode, the selected modes being symmetrical about mo. An important criterion, for
An being negative, is that the long sub—cavity has a length of an integer multiple of a half
wavelength of the selected mode and the short sub-cavity has a length of an odd integer
multiple of a quarter wavelength of this mode. Thus, the length of the long sub—cavity is
such that it is resonant with the mode selected. However, An may alternatively be
positive if the feature is not a slot. If An is positive the roles of the long and short sub-
cavities are reversed.
Fig. 3 illustrates the slight variation of the threshold gain of the optical modes of a
homogeneous Fabry-Perot laser as a function of cavity mode index m . In this example
the threshold gain is taken to be a constant with all modes having equal losses. A
homogeneous FP laser has a grid of allowed modes, with the free-space wavelength of
the mm mode, /lmo , given by
n1/‘two /2n = Law (1)
Here n is the cavity effective index and Law is the cavity length. This relation (equation
1) implies that the condition for resonance is that the cavity length must be equal to an
integer number of the lasing mode half-wavelengths in the cavity. We chose a specific
cavity mode (mode index m = mo) and set the slotted region length in order to maximize
the slot reflectivity for that mode. The first case considered here is the case where the
cavity mirrors are as cleaved, then the change in the threshold gain for that mode will be
maximized provided that one of the sub-cavities formed by the slot has a length equal to
an integral number of the mode half-wavelengths in the cavity. If the slot length is as
described above, then the other sub-cavity will have a length equal to an odd integer
number of the mode quarter-wavelengths in the cavity. In this way the resonant nature of
the feedback due to the slotted region is ensured and the change in the threshold gain of
the selected mode is maximized.
Semiconductor lasers of the invention therefore incorporate slots which are placed on a
discrete set of positions along the laser cavity. Where the cavity mirrors have a coating
applied, or some other means of altering the as—cleaved mirror reflectivity is employed,
the existence of a discrete set of points for the slot positions remains. However, the sub-
cavities formed such that the threshold gain modulation is maximized may in this case no
longer be as described above. The method can accommodate these cases and a suitable
implementation of the method will allow for improved spectral purity and guaranteed
stability of the laser output with temperature in such devices. The general case of
arbitrary facet reflectivity can be described using a complex value for the facet
reflectivity such that r, = | r] |e"”‘ and r2 =| r2 lei”.
For the case where (p, =go2 :0, we now establish the frequency components of the
modulation of the threshold gain due to the introduction of the slots.
The cavity mirrors define the lasing modes of the cavity and the threshold gain, 7/,, with
a modulation period of one cavity mode. We therefore have for the threshold gain
.12.
y! oc cos(2m7r) (2)
Consider the feedback provided by the reflection due to the slots. With respect to the
centre of the slotted region, and as a fraction of the cavity length, the sub-cavities formed
by the slots have lengths n and (l - n) where 77 <1. The feedback due to the slot will
therefore, in general, introduce a modulation of the threshold gain spectrum at two
different frequencies.
The reflection coefficient of the slot, defined at the center of the slot is
e"4‘°/g.(l~e2”’) =—2ir;. sin6‘. This phase shift of in/2 with respect to the slot/cavity
interfaces implies that the modulation of the threshold gain due to the slot reflection will
be proportional to
cos(277m7r— 77: / 2) + c0s[2(l — 77)m7r+ 21/ 2] = 2cos(m7z') sin(2 Emir). (3)
Here :5: 27 - l / 2 is the position of the center of the slot measured from the cavity centre
as a fraction of the cavity length. Thus the modulation of the threshold gain comprises a
fast modulation at every two cavity modes times a modulation at the frequency equal to
half of the difference between the frequencies of the individual modulation periods due to
each sub-cavity.
Thus semiconductor lasers of the present invention include a slot pattern such that the
frequency of the modulation of the threshold gain due to each slot is incorporated into the
design. This enables the tailoring of the threshold gain distribution in the neighbourhood
of the selected mode, mo, and allows the construction of an effective index pattern that
provides a peak mode that is stable with changing temperature.
Amplitude Selection
The position of the slot relative to the cavity mirrors determines the amplitude of the
modulation of the threshold gain due to the slot. This understanding is also necessary for
the construction of an effective index pattern that provides a peak mode that is stable
with changing temperature.
The change in the threshold gain due to a slot is given by the difference in the amplitude
gain to the left and to the right of the slot. For example, in the case where the gain is
distributed uniformly along the length of the cavity, we have
F2 i eXp[(1 _ 77) Lcavanzir]
Ar, °
A.
cav mir) 1:
Here arm, =—1~—log
C11)‘
| [ are the mirror losses of the unperturbed cavity. The amplitude
Vi’”z
of the modulation of the threshold gain due to a slot is therefore determined by the
reflectivity of each cavity mirror and also by the proximity of the slot to each of the
cavity mirrors.
Thus lasers designed according to the invention include a slot pattern such that the
amplitude of the modulation of the threshold gain due to each slot is known. This
understanding then allows for the choice of a set of slot positions that provides a peak
mode that is stable with changing temperature.
The complete expression for the dependence of the threshold gain modulation due to
each slot in the case where go, = (p 2 = 0 is then given by
A;/, oc ;; sin 6{lr,| exp(77Lmva,m.r) — lrzl exp[(1 — 77)Lma,,u.,_]} x cos(m7z) sin(2 617171‘) (5)
Expression (5) includes components arising from parameters of slot length (sine),
lengths of the sub—cavities (n and (1-11)), amplitude variation (within {}), and frequency
of the output light (In). It implies that there may in general be a position along the
cavity where the modulation of the cavity modes due to the slotted region will
vanish. For example, where the cavity mirrors have equal reflectivities, and the gain
is distributed uniformly along the device, this position coincides with the device centre.
The which determines the modulation strength,
term in expression (5),
-14.
[I3lexp[r7LWam,]—-|r2[exp[(l—77)Lm,ami,], will change in sign as this position is.
traversed. Where the point where the modulation strength vanishes lies between the
cavity mirrors, and where the objective of the placement of the slots is the single mode
operation of the laser, this change in sign requires the introduction of a 72 / 2 phase shift
into the slot pattern. Thus it is appreciated that in this case pairs of slotted regions placed
on either side of this position may be separated by sub-cavities of length equal to an
integral number of cavity half wavelengths of the selected mode. Pairs of slotted regions
placed on the same side of the device with respect to this point are separated by sub-
cavities of lengths equal to an odd integer number of cavity quarter wavelengths of the
selected mode. In the case of an optimal device where go, = goz =0 and where the slot
length is such that the reflection due to the slot is maximized, this property of the slot
pattern appropriate to the single mode operation is then a fundamental property.
A schematic diagram of such an optimized structure is shown in Fig. 4. In this example
the mirror reflectivities 1'] and r2 are taken to be real and positive numbers. It is also
assumed that Ansinél < O , where 6 is the phase advance across the slot, and r1 > 1'2. The
vertical dotted line coincides with the point where the modulation of the threshold gain of
the cavity modes due to a slot vanishes. Sub—cavities formed by the slots and the slots
themselves are quarter wave and half wave in the sense above. In this example we have
taken r, > r2 which results in the point where the modulation strength vanishes moving
toward the left mirror, i.e. the mirror with the larger reflectivity.
The above demonstrates that an understanding of the effect of a slot on the threshold gain
spectrum of the device can be used to tailor the threshold gain spectrum of the device to a
degree such that the spectral purity of the device is improved at a predetermined
wavelength and the stability of the peak mode with changing temperature can be
guaranteed.
We label our selected mode, to have a minimum threshold gain, as mo in the single-
moded case, and in general as mo + Am. The threshold gain modulation can be expressed
in the following form, assuming the positioning of slots for resonant feedback as in Fig.
4:
A}/1 (mo + Am) oc cos(m7r) sin(2 emzz)
= cos(m07r) sin(2 e mozr) cos(Am 7:) cos(2 6 Am 7:) (6)
The threshold gain modulation of the cavity modes defined by their separation Am from
the selected mode can therefore be represented as a cosine series where the frequency of
the modulation is determined by the distance of the slot from the device centre. The
requirement that the slots be placed only on the discrete set of allowed points as
determined by the mirror reflectivities and the gain distribution along the cavity is
necessary for the validity of this representation. In the case considered where
go1:go2 =0, and neglecting the optical path length corrections due to the slots
themselves, these allowed points are defined by the relations sin(2 6}. 17207:) = i1 Where
ej.=nj.——l/2.
The method designs a slot pattern based on the understanding of the effect of a slot on the
threshold gain as described by expression (5) above. Using the above expression, or
similar expressions for the case where gal ¢ goz , an explicit link with the techniques of
Fourier analysis can be made. Thus the method designs a slot pattern (step 21) along the
cavity in order to approximately construct the desired threshold gain modulation.
The perturbation is treated as a separate macroscopic section of the laser cavity where,
according to the transverse structure, we assign a different effective index. Each section
of the laser is assumed to have a square well profile. In the case of a one—dimensional
model of a FP laser cavity of length Law and including a single slot region the complex
transmission of the cavity can be found by considering a matrix product. Since typically,
An/n <<1, where n is the cavity effective index, we can treat the influence of the slot by
only retaining terms to order An/rz in the matrix product. The complex transmission
coefficient of a cavity containing a single defect is then given by
~ : tltz exp(iE 9,.)
1- rlrz exp(2i49,.)
-1
2* 7 2' +.
.{1—gflz,.sin(6J.)r‘exp( '¢’)+r2eXp( mm} (7)
n 1- rlrz exp(2iZ0,.)
In equation 7 above, 6; = kiz ~ L; with kg; = nikoz and L,« is the length of the im section. As
indicated in Fig. 2, the reflectivity of the left mirror is r, and the reflectivity of the right
mirror is r, . The transmission coefficients at these mirrors are ti and tzrespectively. For
the case of a real refractive index distribution, the quantities gt; and gt; are the optical
path lengths from the center of slot j to the left and right facets respectively.
We assume that the effective index step is the same for all the additional slots. For a laser
incorporating s slots (index j) and with the cavity defined between -Lew,/2 and +Lcm/2,
one can show that the change in threshold gain of the mm mode is given by, to first order
in An/n:
A}/t(m) : L
ml
xzjam {ll-,| exp(ej Loam) — |r2l exp(— el. Lami,)}, (8)
where
am) = An / n sin 6}. sin(2¢;i + gal)
Comparison of the above expression (8) with expression (5) confirms the validity of the
method. Expression (8) provides the numerical value of the change in the threshold gain
of each cavity mode m due to the introduction of s slots.
The following are examples of threshold gain distributions to describe how suppression
of cavity modes neighbouring the selected mode at m = mg can be achieved. This
guarantees stability of the peak lasing mode wavelength with temperature provided the
dimensions of the device and the slot lengths and positions are accurately known. A
sufficient number of slots must also be introduced in order that stability of the peak
lasing mode wavelength with temperature can be guaranteed over a given temperature
range of interest. An estimate of the number of slots required can be made using an
expression of the form of equation (5) provided the variation of the gain with
wavelength, the variation of the peak gain with temperature and the index step associated
with the slots are known.
Intensity Spectrum
The peak of the gain spectrum in a semiconductor laser, 7 (/1 0), is generally relatively
flat, varying slowly with m near the peak as illustrated in Fig. 5(a). The position of the
peak, A ,,,a,.(T), also shifts with temperature. This leads to two problems:
(i) Because the gain peak wavelength varies with temperature, the laser peak
emission wavelength will also vary with temperature.
(ii) Because the gain peak is relatively flat, with gain being approximately equal
to loss for many modes, the spectral purity of the laser can be insufficient for
certain applications.
Fig. 5(b) shows how this problem can be overcome if the mirror losses associated with
one mode, mo, are sufficiently reduced compared to the losses associated with all other
modes of wavelength close to that of the selected mode, mo. Because the gain peak is of
finite width, in practice, the mode mo need only be reduced relative to a number of
neighbouring modes, a, on either side of it. This situation is illustrated in Fig. 6, where
the losses are reduced at mode mo, and equally reduced at moi na (n an integer), with
other modes being largely unaffected by the perturbations introduced.
Fig. 7 illustrates another mode loss pattern which can be implemented using the method,
where now the losses are reduced at mode mo, with a weaker loss reduction at mo dz na,
and with other modes being largely unaffected by the perturbations introduced. The
single mode at m : mo now has a lower threshold than all other modes. The difference in
threshold gain between the selected mode and these neighbouring modes is sufficiently
large such that the peak mode is stable over a temperature range (1"mn,Tm). Provided
the difference in mirror losses is larger than a minimum value, stability of the peak lasing
wavelength can be guaranteed with this approach. This minimum value, A;/mm, is
depicted in Fig. 8 and will be determined by the gain spectrum variation with wavelength
and by the temperature range, (T ,,,,~,,, T max), over which we require stability.
If the slots are positioned in order to select a single mode by placing the slots at the
allowed positions according to the scheme above, one can show that the change of the
threshold gain of the (mg + Am)th mode, A}/‘(m + Am) , is proportional to the following
expression in the case where (01 = go 2 = O :
c0sm07rcosAm7rZ}.{r1 exp( e]. L Wam) —r2 exp(— ej L or )}
C cav mir
x sin6'J. sin(2 6] 71107!) cos(2 ej Am/I) . (9)
The method, using expression (9) in the case where go, = (0 2 = 0 , includes the use of
Fourier analysis in order to tailor the threshold gain spectrum to a degree such that the
peak mode wavelength is predetermined and the stability of the device with changing
temperature can be guaranteed.
In order to improve the spectral purity of the FP laser, an example of an ideal functional
form for A;/,(m + Am) would have a maximum at Am = O and would equal zero at all
other integer values of Am as is illustrated in Fig. 5(b). Such a function is sinc Am, i.e.,
sin[7r(m — mo]
(10)
A;/,(m0 + Am) = c ' sinc(m—mO) = c~
71'(m— mo)
where c < 0 is a constant. If the modulus of the constant cis large enough, then, in
principle, the stability of the device with changing temperature can be guaranteed.
This sinc gain modulation can be written as the Fourier transform of the unit rectangle or
top-hat function II ( E) (R. Bracewell, The Fourier transform and its applications,
McGraw-Hill, 1965):
sinc(Am) = J:OH( e) exp[—i27z eAm]d e= cos[27r 5 Am]d e. (1 1)
The method uses the understanding that, in a F abry—Pérot laser, only light at the cavity
mode frequencies indexed by the integer m are of interest and that therefore, to tailor the
threshold gain spectrum of the device, we consider functions defined in the wavenumber
space of integers m which are based on the sine function above with other functions
used in conjunction as appropriate. We now give examples of how more complex
threshold gain distributions are described and approximated according to the present
invention.
In addition, cavity modes over a finite range of frequency are of interest. We therefore
consider the example mirror loss in Fig. 6 and define a periodic distribution of sine
functions with spacing a cavity modes as follows:
A .
p(Am) = lH(%)* s1nc(Am) , (12)
where IH(x) = 2:0 5(x — n) and the symbol * stands for convolution. This function,
p(Am), has a Fourier transform which is proportional to III(ae)-H(e). This Fourier
transform consists of a series of delta functions, centered at the origin, and with equal
spacing a'1 inside the window -1/2 3 e_<_ 1/2.
To achieve the final example mode loss pattern shown in Fig. 7, we define a Gaussian
envelope function g(Am) = exp [— 711 2 (Am)2]. The product of this function with p(Am) is
(g- p)(Am) = g(Am)- sinc(Am —na) (13)
and has Fourier transform proportional to
1"(e)*III(a e)vl'[(e) (14)
where F (e) = exp [-7r E2/T 2]. This is then simply a Gaussian broadening of each of the
delta functions of the previous Fourier transform. The factor 1' detennines the decay of
the envelope and thus the size of the gain modulation at a distance a cavity modes from
the selected mode.
The present invention is based on the understanding that, in order to reproduce any given
threshold gain spectrum, we must correct for the fact that the strength of the gain
modulation due to a slot is determined by its proximity to the laser mirrors. We can then
place a finite number of slots in order to approximately reproduce the distribution of
threshold gain We desire through knowledge of the Fourier transform of the distribution.
The appropriate positions for the placement of the slotted regions will be a discrete set of
points as detemiined by the cavity mirror reflectivities and the peak modes’ quarter
wavelength in the cavity.
Example: Determination of parameters a and Z’
The parameters that primarily determine the variation of the peak lasing mode with
temperature in a FP laser are
- the gain profile as a function of wavelength
- the drift of the gain peak with temperature
- the thermal variation of the cavity length and its effective index
If the temperature range over which we require the peak lasing mode to be stable is
specified, then, using the above parameter set, we can determine a threshold gain
spectrum that will ensure this stability. Our invention allows us to achieve the required
spectrum of mirror losses. Based on the measured gain spectrum, and its temperature
dependence, we can then determine a mode loss pattern to ensure the required stability.
The characteristics of the gain spectrum can be used to determine the choice of the
parameters a and 1'. We assume that the gain curve, ;/(10), has a parabolic variation
about the peak gain position with
(d*o) = ‘b(2~max(T) ‘ d'o)2 +7max (15)
Here 2.,,m\.(T) is the position of the gain peak, ;/ ,,m. is the peak gain value at the given
drive current and b describes how the gain varies with wavelength close to the peak
value. It is appreciated that the parameter b is also in general a function of temperature
but that for the purposes of the present example its dependence on temperature can be
neglected.
.21-
As the operating temperature of the device is varied, the position of the peak gain and the
free space wavelength of each cavity modem will change (amounts A m7 and A me resp).
Typical values of the parameters determining this behaviour are
drift of gain peak: 04 nm K“
thermal variation of index: dn/df: 1.9 x 10*‘ K"
linear expansion coefficient: 4.6 X 106 K4
We consider as an example a device which is required to be temperature stable over a
temperature range of (-20°C, +80°C). Taking room temperature to be 20"C, the device
must be stable over an asymmetric interval —40K 5 A T _<_ +60K. For our example device
we take:
11 : 3.2
Law = 400/011
mo : 1600
The mode spacing at fltmo = 1600 nm is then 1 nm and the gain peak can drift over 40
cavity modes. We therefore place the room temperature gain peak at 1596 nm and set our
fundamental spacing a to be 20 modes. At the extremes of temperature variation we can
show that the separation between the gain peak and our chosen cavity mode will be A my
- A mc ~ 14 modes apart. If we take 17 = 5 X 104 then the difference in gain between our
chosen mode and the peak will be A y ,,,,»,, = 0.1 cm". The parameter r is now be
determined by this difference as follows:
The difference in gain between the chosen mode and the mode at spacing a is
Ame -An = A}/n10.i:1_ g(a)l 2 fA7m. (16)
where A 7 mm is the difference due to gain spectral variation and f > 1 will determine the
SMSR. We therefore have that
./(‘Aymin
M (17)
g(a)S1-
me
We take a = 20 modes, A ;/ ,,,,-,, = 0.1 cm'1 and f = 2. In this example we introduce sixteen
slots with An =-0.02 in which case we estimate A 9/ mo ~ 0.25 cm'1 using expression (5).
We then have 1‘ 3 0.036 in this example.
Determination of slot positions 6}
Case goL =goR =0an_d_|rL l=|rR |;
Here we illustrate how appropriate limits on e can be derived for this case. Here the
mirror reflectivites are equal and the resulting requirement that there exist a half-
wavelength subcavity at the device centre then provides a natural lower limit on e. We
take
em: L]./2=(q+l/2)/two/471]. ~1.41><10'3 (18)
We will also account for the Gaussian broadening beyond e: l/ 2 by setting
eW=l/2+a" (19)
Approximating the sinh x function, which describes the amplitude variation with
position, as x, our normalisation is then
em 1° , x~n/a 2
/IJ. Ex" exp —7r( J dx=s
Em” n=l
(20)
where S is the number of slots. We have not included the broadened feature at e: O in
this example as this is responsible for a primarily d.c. component of Ay . Approximate
slot positions are now determined by the relations
/ 10 _ ‘_ / 2 _ .
Ajepzxlexp [—7r[¥ : 61] }d.X=j—1/2, j=1,2,
m'"n=1
(21)
These slot positions are then be adjusted in order that the quarter wave condition is met.
This requires that the slots be placed on the available discrete set of points defined by the
mirror reflectivities and the wavelength of the selected mode. In the case considered in
this example, the correct positions correspond to the nearest fractions
aJ.=¢;/26,.
of the total optical path length which satisfy the appropriate phase requirement, being
sin(27ra 1.1120) = il in this case. In this example we have placed the first slot on the right
of the device center and subsequent slots on alternate sides as shown in Table 1. The
integer plus one-half values in the third column ensure that sin(27raJ.m0) = :1 according
to whether the slot is placed to the left or to the right of the device center. Further minor
optical path length (OPL) conections resulting from the introduction of the slots
themselves can also be taken into account. These corrections are made by generating the
slot positions using an expression of the form
a‘ = 77} +sJTAn/nfl' (22)
’ l+sAn/71/3
Here 5 and s; are the total number of slots and the number of slots to the left of slot j
respectively, 77 1. is the fraction of the cavity length for fraction of the optical path length
or]. and [f is the slot length as a fiaction of the cavity length. In this example the center of
the device coincides with the point where a phase slip of 21/ 2 must be introduced into
the slot pattern. The resultant threshold gain distribution in the neighbourhood of the
selected mode is shown in Fig. 8.
Table 1: Device harmonics and adjusted slot positions: Symmetric case
slot number approx. 6,» OPL fraction >< mo nominal position (pm)
1 +0.0229 1672.5 209.060
2 -0.0347 1489.5 186.184
3 +0.0421 1734.5 216.815
4 0.0487 1443.5 180.429
+0.0561 17805 222.570
6 -0.0674 1383.5 172.924
V 7 +0.088O 1882.5 235.324
8 -0.1023 1273.5 159.171
9 +0.l242 1996.5 249.577
-24.
—O.l509 1117.5 139.669
11 +0.1859 2194.5 274.327
12 -0.2192 899.5 112.420
13 +0.2655 2450.5 306.325
‘ 14 -0.3251 559.5 69.925
+O.3955 2866.5 358.317
16 -0.4782 69.5 8.685
An estimate of the SMSR below threshold and the position of the peak mode as the
temperature is varied are shown in Fig. 9. As expected, no mode hopping is observed and
the SMSR is greater than 90% or l0dB throughout the temperature range. This example
illustrates the potential of the present invention to improve spectral purity and to
guarantee stability with temperature in semiconductor Fabry-Perot lasers.
Experimental data; Single mode case
n order to demonstrate the validity of the present invention, we have designed and
fabricated a single mode laser according to the methods described above. The parameters
specifying this design are as follows:
n = 3.188
Law = 300;1m
m0 = 1236
/imo : 1547.5nm
/*1 = 0.9747
= 0.5292
slot number = 19
This laser is high reflection coated on one end of the cavity, which means that slots are
better placed all on the opposite side of the device center from the high reflection
coating. In this way the amplitude of the modulation of the threshold gain of modes due
to each slot is larger. In order to generate the approximate slot positions, the method uses
equations analogous to equations (20) and (21) but with x”' replaced by
{|r1| exp(xLa) — |r2 | exp(—-xLa)}_1
in the integrand. For this device we used em: 0.0 and em: 0.5. Parameters a and I
were as in the previous example, and together with the facet reflectivities, these
determine the ideal slot density distribution, plotted in Fig. 10 (a). A schematic picture of
the cavity is plotted in the inset of Fig. 10 (a), while the form of the threshold gain of
modes is plotted in Fig. 10 (b).
The side mode suppression at twice threshold of a laser fabricated according to this
design exceeds 40 dB, as shown in Fig. ll. For comparison, an equivalent spectrum of a
plain Fabry-Pérot laser without slots, fabricated on the same bar is shown in the inset of
Fig. 11. This demonstrates that excellent spectral purity with a side mode suppression
ratio exceeding 40db can be achieved at a predetermined wavelength.
It will be noted from Fig. 10(a) that these slots are on the right hand side of the device
center only. This is because slots further from the high reflectivity mirror will provide a
larger modulation of the threshold gain (Eqn. (4)).
Multimode examples
Two mode laser cavity
We wish to select two FP modes with spacing a in preference to all others. In this case
the ideal mirror loss modulation is given by
/2sinc(Am+a /2)+ 1/2sinc(Am—a/2) , (23)
where Am = m—m0, with mo our reference mode as before. This function has Fourier
transform cos(7ras)>< lT(5) . To illustrate how our method allows us to select two modes
we design a laser cavity as in the previous example. In this case, to generate the
approximate slot positions we use an equation, analogous to equation (20), of the form
{|r11exp(xLa) — ]r2lexp(—xLo:)}4lcos(7ra e)|dx = s (24)
The form of the threshold gain of modes is plotted in Fig. 12, while a schematic picture
of the cavity is plotted in the inset of this drawing. Note that because the Fourier
transform of our object spectrum takes on negative values, we must integrate over the
absolute value of the cos function in the above equation. When final slot positions are
calculated, those corresponding to negative or positive Fourier components must be
placed at even or odd integer values plus one half as appropriate.
Three mode laser cavity
We now wish to select three FP modes with spacing a in preference to all others. As in
the single mode case, we define a periodic distribution of sinc functions with spacing a
cavity modes;
p(Am) = 111 sinc(Am) .
The Fourier transfonn is proportional to
lll(a e)-H(e).
We now take the product with an envelope function determined by the difference of two
Gaussian functions
exp(7n',2Am2) ~ A exp(m'fs2Am2).
The Fourier transform of this composite function is then proportional to
[ [e.n/a]t A [
2 exp -71 ——exp —7r
'1 1, S 57,
Again to illustrate how the method allows selection of three modes in this way, a laser
(25)
cavity with parameters is designed. Using the appropriate equation, analogous to
equation (20), and with a = 2 , the form of the threshold gain of modes is plotted in Fig.
13. The central mode has a larger threshold gain. Because of the variation of the material
gain with wavelength in the device, this will then result in a lasing spectrum where the
-27..
optical power in each of the three selected modes is equal provided the material gain
variation is engineered correctly.
The difference of the two Gaussian functions increases the threshold gain at the reference
mode mo. In this way the power in the primary modes can be equal once the peak gain is
We now return to the single mode case in order to illustrate how such a laser cavity can
be designed where (0, ¢ (02. In this general case we write the trigonometric factor,
sin<2¢;i +401) ,
explicitly in terms of the slot positions and the facet phases. Again a given cavity mode
m0 is chosen and we expand as before in terms of m = mo + Am . The mirror loss
modulation can then be expressed in terms of its even and odd components as follows:
sin(2¢;' + go,) = cosm07rcosAm7r x
(26)
{v(ej,m0)c0s271 ej. Am+w(e/.,m0)sin27r G}. Am}.
This expression suggests that in this general case we can seek to approximately
reproduce a given mirror loss spectrum with a finite number of slots once the Fourier
transfonn of the object spectrum and the form of the functions v(eJ.,m0) and w( ej,m0)
are known.
To illustrate the application of the method in this asymmetric case, it will suffice to chose
a given set of parameters rl and r2 and to desciibe how the appropriate slot pattern is
designed in order to select a single cavity resonance as the peak lasing mode of the
device. An instructive example to treat is where the cavity has a high reflection metallic
coating at one facet while the other is as cleaved. In this case we have go, = 77, (02 = 0 and
then:
v=s1n27z 5}. 7710511171’ 5]. +cos27r (51. mo coszrej, (27)
w=cos27r e/. mosinrzej. —sin27z' ej mocoszzej. (28)
Since ‘C0577 e i> |sin7r e i for he i
the mirror loss modulation through v while keeping the odd component to a minimum,
we must place the slots such that Icos(27z 5]. m0)|=l for .5 1<1/ 4 and such that
lsin(27r <5]. m0)I = 1 for [E [<1/4. The higher reflection at the left facet means that the
mirror loss modulation is larger for slot placed on the right of the device centre where
e> 0. Thus in this case 72' / 4phase shift is present in the slot pattern at three-quarters of
the device length.
For the example device we consider the same laser as before but with facet reflectivities
given by r1 = 0.95e"’ and r2 =0.524. The values of T and a are as for the previous
example and here we introduce 20 slots on the right of the device centre. To generate the
approximate slot positions in this case, we first integrate the product of the Fourier
transform of our object spectrum with the inverse of the modulation amplitude functions,
which are
)]cos(7r e) ,
)—lr2|exp(— EL 0:
Cav mir
[|r]lexp(eL a
Cav ml‘
for 0
r2 | exp(- e LmV0cm,.,_)] sin(72' e) ,
cav mir
[|r1lexp(eL at )-
for 1/ 4
over the first interval (0
(1/4
expressions of the type used in the first example.
The resultant slot positions are given in Table 2. Note how the first 12 slots are placed
such that cos(27r 5}. mo) :1 and the final 8 slots are placed such that sin(27r ej m0)=1 .
Table 2: Device harmonics and adjusted slot positions, Asymmetric case
slot number approx. Ej OPL fraction >< mo nominal position (pm)
+0.0067 1622 202.678
2 +0.0294 1694 211.682
3 +0.0481 1754 219.186
4 +0.0644 1806 225.691
+0.09l0 1892 236.444
6 +0.l070 1942 242.699
7 +0.1339 2028 253.452
8 +0.1521 2086 260.707
L 9 +0.1749 2160 269.960
+0.1977 2232 278.964
11 +0.2156 2290 286.219
12 +0.2422 2376 296.972
13 +O.2581 24265 303.289
14 +O.287O 2518.5 314.792
+0.3071 2582.5 322.797
16 +0.3407 26905 336.299
17 +O.3660 2770.5 346.299
18 +0.4020 28865 360.804
19 +O.4422 3014.5 376.806
+0.4814 3140.5 392.557
The resultant threshold gain distribution in the neighbourhood of the selected mode is
shown in Fig 14. Again, excellent mode selectivity is achieved.
.30.
It will be appreciated that the invention provides a method to improve the spectral purity
of a Fabry-Perot semiconductor laser at a predetennined wavelength. The method is
based on an understanding of the role of each additional feature in predetermining the
peak lasing wavelength. The method achieves temperature stability, minimises the mirror
losses associated with selected modes, and specifies the losses associated with a range of
neighbouring modes.
Although we are primarily concerned with the optimisation of semiconductor lasers
emitting near 1.3 - 1.5izm for the telecoms market, our method is valid for any device
where the cavity mirrors are the primary source of regenerative feedback for lasing. The
additional features can in principle take any fonn that will provide the internal reflection
and thus a modulation of the threshold gain of the cavity modes.
The method allows the efficient design and manufacture of semiconductor lasers with
improved spectral purity and temperature stability, and at a predetermined wavelength
but at a fraction of the cost of creating a DFB or a DBR laser. Embodiments of the
present invention will also include coupled cavity devices and multi—contact devices
where the distinct cavities or sections of the device have a slot pattern designed using the
methods described.
A schematic diagram of a multi-section device where two slotted FP lasers are connected
longitudinally is shown in Fig. 15. In this example the device includes a phase section
and a mirror segment. Each section is independently contacted. In such a device the peak
lasing mode wavelength can be dynamically tuned through the Vernier effect. The basis
for this functionality is the difference in the peak mode spacing a between the two
slotted FP sections and the variation of the wavelength of these peak modes with injected
carried density. The device can also incorporate further sections such as an electro-
absorption modulator or an amplifying section. An advantage of the method in this case
is much reduced fabrication cost as compared to devices based on, for example, sampled
grating DBR lasers.
A schematic of a device where four slotted FP lasers are coupled laterally is shown in
Fig. 16. Each section can be independently contacted. In such a device the individual FP
modes are coupled across the device. Such devices allow for increased power output and
larger modulation bandwidths. The advantage of the invention in this case is improved
spectral purity as compared to devices based on plain FP lasers.
In a manufacturing method to implement the design, the slots are preferably formed at
the ridge lithographic and etching stages. The invention is particularly advantageous for
designing and manufacturing tailored multi—mode F P edge-emitting lasers inexpensively.
The invention is not limited to the embodiments described but may be varied in
construction and detail. For example, the invention may be applied to any edge—emitting
laser of the FP type. These include lasers in which the optical gain is provided by inter-
band or intra-band electronic transitions. Examples are quantum cascade lasers or surface
plasmon enhanced quantum cascade lasers.
The cladding may alternatively be of a metal.
In the embodiments of the invention described above the features to alter the refractive
index are slots. However, different refractive index altering features may be used, such as
a projection in the cladding (added matter, rather than missing matter as in the case of a
slot) or a discontinuity in the cladding material. Indeed any feature which causes a
discrete local change in effective refractive index in the transverse direction could be
employed.
Claims (1)
1- |r,|exp[xL 11'1lF(x)[dx = 1' —
Priority Applications (1)
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IEIRELAND16/07/20042004/0484 | |||
IE20050253 | 2005-04-26 | ||
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