Lecture 9/1

Diode laser emission

y

x

z

Diode laser emission has oblong cross-section.

Y-axis with large divergence angle is called fast axis

X-axis with smaller divergence angle is called slow axis

Lecture 9/2

Laser emission: beam divergence and astigmatism

Beam emitted from a small facet is equivalent to the beam emitted by an imaginary point source P. When dX is larger than dY the distance between PXand PY can be nonzero. This phenomenon is called astigmatism, and the distance ∆between PX and PY is the numerical description of astigmatism.

from 1 to 200µm

less than 1µmLASER OUTPUT FACET

Top View ΘX

dX

ΘY

dY

∆

Side View

PX PY

y

x

Lecture 9/3

Field emitted from the laserNEAR FIELD - E(x,y)

At distances larger than λd 2

i

FAR FIELD – EF(ΘX,ΘY)

xy

z

ΘYΘXSpatial distribution

of the optical field across laser mirror

Angle distribution of the optical field far from the laser mirror

R

Laser emission: far and near field emission patterns

Near field pattern uniquely determines far field pattern

Lecture 9/4

Laser near field

Size of the laser mirror (at least in y direction) is comparable with wavelength of light and wave optic analysis techniques must be used.

Main result of this analysis is that laser near field pattern can take shapes only from certain discrete set of special shapes called modes. No other shapes of near field pattern are allowed.

This effect is very similar in nature to quantization of the energy states of electron in atom.

Each laser mode is characterized by certain number – effective refractive index (neff). This number determines mode shape and its group velocity.

Number of possible (allowed) near field shapes (modes) is given by laser wavelength, refractive indexes of all layers and actual sizes of layers.

xy

z

neff_1

neff_2

neff_3

Lecture 9/5

Nature of discreteness of the possible near field shapesConsider light propagating in between of two perfect mirrors (metallic with 100% reflection)

λ

d ~ λ

yz

Because of total reflection from the mirror the energy transfer will occur only in direction z and in direction y standing wave must be formed.

Standing wave condition for perfect

mirrors is: 1,2,3...m m,2λ

d y =⋅=

We used λy because only y-projection of the light wave vector matters in y-direction:

Resulting optical field will look like: z_m

z_my_m

y_m2z_m

2y_m

2

λ2πk and

λ2πk ,kkk ==+=

( ) ( ) ( )( )

( )( )( )⎩

⎨⎧

+=⋅+=⋅

∝

⋅=

⋅−⋅⋅⋅=

number odd1m ,yksinnumbereven 1m ,ykcos

yE

nλ2πk where

,zktωjexpyEzy,E

y_m

y_mm

eff_m0

z_m

z_mm

∗ λ0 –wavelength in vacuum

Em(y)

m = 1 neff_1

m = 2 neff_2

m = 3 neff_3

m2dλy_m =

Lecture 9/6

Physical meaning of the effective refractive index

* Mode effective refractive index determines group velocity for a given mode

( ) ( )

( ) ( )

( ) 22

0m

2

0

m0

my_m

eff_m0

m0

mz_m

eff_mn-n

λ2πθsin-1n

λ2π

θcosnλ2πθcoskk

nλ2πθsinn

λ2πθsinkk

⋅=⋅⋅=

=⋅⋅=⋅=

⋅=⋅⋅=⋅=

nθm

nλ2πk

0⋅=y

z

In semiconductor lasers no metallic mirrors are available (huge loss would have appeared). The waveguide is formed by stack of semiconductors with different refractive indexes. Optical field can propagate in semiconductors as opposed to metals. As a result, we will not have the luxury of zero optical field on the physical surface of the mirrors.

m = 1 neff_1

m = 2 neff_2

m = 3 neff_3

n1

n3

n2

Standing wave in y-direction will be formed again. Optical field will not be fully confined in region with refractive index n2 but will have exponentially decaying tails in regions n1 and n3. Effective refractive index can be understood as being weighted average between n1, n2 and n3 as soon as field is present in all these regions.

ZigZag waves model can still be applied with some care and θm can be assigned to each mode

Lecture 9/7

Possible range of the effective refractive index and maximum number of supported modes.

1. Perfect mirror case: zero field on the boundary between n-region and mirrors.

nθm

nλ2πk

0⋅=y

z

2nλ

dmor n λ2πm

dπ hence ,k k

1,2...m m,dπm

2d2π

λ2πk ,kkn

λ2πk

00y_m

y_my_m

2z_m

2y_m

2

0

2

≤≤⋅≤

=⋅=⋅==+=⎟⎟⎠

⎞⎜⎜⎝

⎛=

Maximum number of modes is equal to number of half waves that can be squeezed in d. Minimum neff is zero and corresponds to extreme case when k = ky and no energy transfer occurs in z-direction – pure standing wave. Maximum neff = n, this corresponds to pure traveling wave when no mirrors present. When d < λ0/2n this waveguide supports no modes.

Effective refractive index can be found from:

k

kz_m

ky_m

2022

eff_m

2

eff_m0

22z_m

2y_m

2

0

2

dm

2λnn

nλ2πm

dπkkn

λ2πk

⎟⎠⎞

⎜⎝⎛ ⋅−=

⎟⎟⎠

⎞⎜⎜⎝

⎛⋅+⎟

⎠⎞

⎜⎝⎛ ⋅=+=⎟⎟

⎠

⎞⎜⎜⎝

⎛⋅=

Lecture 9/8

Possible range of the effective refractive index and maximum number of supported modes.

2. Semiconductor laser: nonzero field in all layers n1, n2 and n3.

In symmetric waveguide single lobe mode always exists and is called fundamental mode. Next mode exists if maximums of the field are inside n2.

Number of supported modes m is given by: ( )221

20

nn1

2nλ

d1m −⋅+≤

Important correction to perfect mirror case , see appendix for exact solution

Fundamental mode – always exists in symmetric waveguide

n1

n3 = n1

n2

fundamental mode always supported

about λ0/2n2

The mode is guided if all maximums of the standing wave in y-direction are inside the confining region n2. For the first mode (see figure) to exist waveguide width d should be larger than half wave in n2 media to have standing wave maximums inside. For the second – larger than two half waves, for the third – three, etc.

first mode

For simplicity consider symmetric waveguide n1 = n3. In symmetric dielectric waveguide neff can change between n1 and n2 because optical field penetrate into all regions. Actual value of neff_m can be found numerically

d

* before we plotted absolute value of the electric filed. Here we plot electric field with phase to assist the eye to recognize half wave distance between maximums.

Lecture 9/9

Single mode lasers.Symmetric dielectric waveguide always supports at least one mode called fundamental mode.

( )2

0221 2n

λnn1d <−⋅For the waveguide not to support any other higher order modes the condition should be satisfied:

Single spatial mode operation of semiconductor laser produces current independent single lobe far field pattern. Single spatial mode operation lasers have the highest brightness.

*Asymmetric dielectric waveguide does not necessary supports this mode

Brightness B by definition is:ΩS

PB ν

⋅= , where S – emitting area, Ω – solid angle

into which the power Pν is emitted

Near field is spatial distribution of optical field across laser mirror, i.e. S. Far field is angle distribution of the optical field far from laser mirror, i.e. Ω. When laser emits single spatial mode S and Ω are related as Fourier transform pair and:

2ν2

λPBmax maximum is brightness and ,λΩS ==⋅ High brightness is desired in

almost all applications.

Single spatial mode operation is ideal operation condition of semiconductor laser. Its advantages have to be sacrificed when high output power level is required.

Usually, semiconductor laser are single mode in y-direction (transverse) and can have many spatial modes in x-direction (lateral).

Lecture 9/10

Transverse-electric (TE) and Transverse-magnetic (TM) modes.y

z x

E HTE TM

light wave electric field component oscillates in the plane of active layer

light wave magnetic field component oscillates in the plane of active layer

yz

Electric and magnetic components of the light wave in isotropic media are orthogonal and are related by: z

E

H377EnE

µµεεH

0

0 ⋅=⋅⋅⋅

=

EH

TE TM

TE modes have lower mirror loss than TM modes and laser emission is usually TE polarized. In QW lasers mode gain for TE and TM polarization is also different.

n n

mirror mirror

H

E

TE and TM polarized light have different reflection coefficients at both waveguide boundaries and laser mirrors.

Lecture 9/11

Gain-guided WeaklyIndex-guided

StronglyIndex-guided

Lateral optical confinement (X-direction)

Oxide-Stripe Ridge CMBH

Current ConfinementGain modifies Imε

I(x) leads to Imε(x)Multimode

Im index step ~ 0.001*Dependent on pumping level*Antiguiding

Current ConfinementOptical Confinement

Reε(x) & Imε(x)Single mode

Refractive index step < 0.01

*Antiguiding*Expensive

Current ConfinementOptical Confinement

Reε(x)Stable single mode

Refractive index step >0.1

*Very expensive

y

x

Lecture 9/12

Measurements of the laser near field

F

A B ABM

F1

B1

A1

=

=+

General approach is to amplify image of the laser output facet and project it on video camera.

Magnification of 10-100 times or more is required for well resolved image

* Near field microscopy is another option: fiber tip is scanned with submicron resolution along laser output facet. This technique is accurate and free from aberrations that could be introduced by imaging optics.

Lecture 9/13

Measurements of the laser far field

Scanning of the single detector Detector array (CCD) or Vidicon camera

High resolution true far field for all anglesSlow and only one dimension at a time

Fast image of the 2D far field patternEasy alignment and adjustmentSpecial optics required due to limited size of the photosensitive matrix

Lecture 9/14

Typical diode laser transverse (Y) far field pattern

-90 -60 -30 0 30 60 900.0

0.5

1.0λ

0=1.5µ m

d=260nmn1=3.17n2=3.33

FWHM ~ 400

Nor

mal

ized

Inte

nsity

Transverse Angle (Degree)

Approximate expression for full angle at half intensity for small d

( ) ( )0

21

22 λ

dnn4radΘ ⋅−⋅≈⊥

Single mode operation (diffraction limited beam)

Beam divergence is current independent

Lecture 9/15

Lateral (X) far field pattern of wide stripe (200µm) gain guided laser

-20 -10 0 10 200.0

0.5

1.0

170

λ=1.5µmSTRIPE200µm

Nor

mal

ized

Pow

er

Lateral Angle (Degree)

Multimode operation

Beam divergence is current dependent and orders of magnitude higher than diffraction limited

Lecture 9/16

Ridge waveguide lasers and effective index technique

Isolator n~2

Etch depth

etched region etched regionridge

Metal contact 1. Find transverse effective indexes in ridge and etched sections, nridge and netched

2. Use nridge and netched to find lateral field distribution with effective index nlateral

3. Use nridge for transverse near/far field calculations and nlateral for lateral near/far field calculations.*Current spreading and gain guiding are usually important and should be taken into account.

* For CMBH devices lateral and transverse waveguide dimensions are comparable and exact 2D waveguide problem should be solved numerically.

n1

n1

n2

Lecture 9/17

Design of single mode ridge waveguide laser

Design parameters:

1. Ridge width – w

2. Etching depth - t

t defines netched – transverse effective index of the etched region

nridge and netched define lateral effective index nlateral

Lateral single mode condition

Isolator n~2

etched region etched regionridge

n1

n1

n2

wt

nridgenetched netched ( )2λtnnw 02

etched2ridge <−⋅

Lecture 9 app1/1

Laser near field E(x,y)y

xε2

ε1

z

( ) ( ) ( )2

2

02

tt,rErεµt,rE

∂∂

⋅=∇

Consider TE modes in 3 layer symmetric waveguide

ε1

ε1

ε2

y

z

d/2

-d/2

For TE modes EZ=EY=0 and d/dx=0

Let's look for the solution in the form

( ) ( ) ( )( )kz-ωtjexpyEEtz,y,E X0X ⋅⋅=

( ) 0Ekεµωy

EX

20

22

X2

=⋅−+⎟⎟⎠

⎞⎜⎜⎝

⎛∂

∂

APPENDIX 1

Laser near field E(x,y)

n1

n1

n2

y

z

d/2

-d/2

( ) 0Ennkx

EX

2eff

2i

202

X2

=⋅−+⎟⎟⎠

⎞⎜⎜⎝

⎛∂

∂

20

22effi

2i

00 k

kn ;εn ;λ2πk ===

Solution for guided modes (n1<neff<n2)

( ) ( ) ( ) ( )( ) ( )

( ) ( ) ( ) 20

21

2eff

2X3

X2

20

2eff

22

2OEX1

knn γ;γyCexpyE ;2dy

γyBexpyE : 2dy 2.

knnβ ;βysinAβycosAyE

:2dy

2d 1.

⋅−==−<

−=>

⋅−=+=

<<−y

z

n2k0

neffk0

β

Lecture 9 app1/2APPENDIX 1

Laser near field E(x,y)

Mode intensity distribution is defined by values of AE, AO, B, C and neff

They can be found from boundary and normalization conditions

1. Boundary condition:

Tangential components of the E and H should be continuous at theinterfaces.

For TE modes it means:

2d

X3

2d

X1

2dX3

2dX1

2d

X2

2d

X12

dX22

dX1

dydE

dydE and EE

dydE

dydE and EE

−−−− ==

==

2. Normalization condition:

Total area under the envelope

curve should be equal to unity.( ) 1dyyE

-

2X =∫

∞

∞

Lecture 9 app1/3APPENDIX 1

Laser near field E(x,y)

1. Observation of the boundary conditions obtains the equation:

Even TE modes (AO=0):

Odd TE modes (AE=0):

2γd

2βdtan

2βd

=⎟⎠⎞

⎜⎝⎛⋅

2γd

2βdcot

2βd

−=⎟⎠⎞

⎜⎝⎛⋅

2. Observation of the relation between k, γ and β gives:

( )21

22

20

22

nn2

dk2dγ

2dβ

−⋅⎟⎠⎞

⎜⎝⎛=⎟

⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛

Numerical (graphical) solution of these equations gives discrete values of neff for guided modes

Lecture 9 app1/4APPENDIX 1

Laser near field E(x,y)

0 1 2 3 4 5 6 70

1

2

3

4

5

6

7

ππ /2 m=3

m=2

m=1

m=0

λ=1.5µm, d=2µmn1=3.6, n2=3.4red - TE evenblue - TE odd

γd/2

βd/2

Example of graphical solution

0 1 20.0

0.5

1.0

m=3m=2

m=1

E Xm2

y, µ m

m=0

Mode intensity distributions

Condition of the single mode operation:2λnnd 02

122 <−⋅

* In full analysis, mode loss and confinement should be taken into account

Lecture 9 app1/5APPENDIX 1