ARTICLE IN PRESS

0030-3992/$ - se

doi:10.1016/j.op

�CorrespondE-mail addr

Optics & Laser Technology 39 (2007) 1033–1039

www.elsevier.com/locate/optlastec

Mathematical modeling of tunable TEA CO2 lasers

Jin Wu�, Changjun Ke, Donglei Wang, Rongqing Tan, Chongyi Wan

Institute of Electronics, Chinese Academy of Sciences, Beijing 100080, China

Received 2 September 2005; received in revised form 5 May 2006; accepted 6 May 2006

Available online 23 June 2006

Abstract

A mathematical model, based on the Landau–Teller equations of six-temperature model for the CO2–N2–He–CO system, to describe

the process of dynamic emission in tunable TEA CO2 lasers is introduced. In this model, the Landau–Teller equations are rewritten with

regard to fine longitudinal mode frequencies in the laser resonator. These revised equations can be utilized to estimate the laser output

spectra as well as other laser output pulse parameters. Examples are given to show the modeling results of non-tunable, grating tuned or

injection-locking TEA CO2 lasers.

r 2006 Elsevier Ltd. All rights reserved.

Keywords: Modeling; Tea CO2 laser; Tunable

1. Introduction

Pulsed TEA CO2 lasers are widely used in scientific andindustrial fields. There are several models appearing inliterature describing the kinetic process of TEA CO2 laserswith gas mixture CO2–N2–He–CO, i.e., four-, five- and six-temperature mode [1–6]. Among them, four or five-temperature model is a special case of the six-temperaturemodel which will be considered in this paper. Fig. 1 is aschematic diagram showing the excitation energy levels ofthe six-temperature model in a CO2–N2–He–CO lasersystem [5]. Generally, these models can be well applied tomodeling non-tunable TEA CO2 lasers. As to tunable TEACO2 lasers utilizing grating, Fabry–Perot etalon, injectionlocking, etc., papers concerning their mathematical model-ing are seldom printed out in literature. However, tunableTEA CO2 lasers are frequently applied in civil and defensefields [7,8], it is helpful to set up a mathematical model tosimulate the kinetic process occurring in frequency-agileTEA CO2 lasers.

In this paper, a generalized mathematical model basedon six-temperature mode is introduced, which can be well

e front matter r 2006 Elsevier Ltd. All rights reserved.

tlastec.2006.05.003

ing author.

ess: jwu909@263.net (J. Wu).

applied to tunable TEA CO2 lasers with CO2–N2–He–COgas mixtures for numerically predicting their performancecharacteristics including the output laser spectra of finelongitudinal mode frequency.

2. Mathematical model

To set up mathematical model of CO2–N2–He–COsystem for tunable TEA CO2 lasers, the followingassumption is accepted, which is generally true in the caseof a TEA CO2 laser:

(1)

All vibration–rotational transitions are homogeneouslybroadened. In fact, they are pressure broadened due tohigh gas pressure in a pulsed TEA CO2 laser.

(2)

Two lower laser energy levels (1010 and 0210) areregarded as one energy level due to Fermi resonance.

For a CO2 molecule, the energy levels (1010) and (0210)belonging to different vibrations have nearly the sameenergy, the same radiative lifetimes and there exists veryfast energy transfer between them. Such a resonance leadsto strong perturbation (first recognized by Fermi in 1931,thus called Fermi resonance). As a consequence of thisperturbation, a strong mixing of the eigenfunctions of these

ARTICLE IN PRESS

Fig. 1. Schematic of excitation energy levels in a six-temperature model CO2 laser system [5].

J. Wu et al. / Optics & Laser Technology 39 (2007) 1033–10391034

two levels occurs so that the two observed levels can nolonger be unambiguously designated as (1010) and (0210).Each actual level is a mixture of two. Therefore, it is asound approximation for simplicity to treat these two levelsas one.

Since tunable TEA CO2 lasers are frequency agile, theLandau–Teller equations of six-temperature model shouldbe revised to cover all lasing frequencies, in our considera-tion, all longitudinal mode frequencies. As a consequence,the population inversion densities of the four transitionbranches can be described by the following equations:

DN10PðJÞ ¼ N00�1P10PðJ � 1Þ �2J � 1

2J þ 1N10�0P10PðJÞ, (1)

DN10RðJÞ ¼ N00�1P10RðJ þ 1Þ �2J þ 3

2J þ 1N10�0P10RðJÞ, (2)

DN9PðJÞ ¼ N00�1P9PðJ � 1Þ �2J � 1

2J þ 1N02�0P9PðJÞ, (3)

DN9RðJÞ ¼ N00�1P9RðJ þ 1Þ �2J þ 3

2J þ 1N02�0P9RðJÞ, (4)

where, in Eqs (1)–(4), the rotational distribution functionP(J) of the four branches is expressed as

PðaÞðJÞ ¼2hcBðaÞ

kT

� �ð2J þ 1Þ exp

�hcBðaÞJðJ þ 1Þ

kT

� �ða ¼ 10P; 10R; 9P; 9RÞ. ð5Þ

Since every vibration–rotational transition line (denotedby the rotational quantum number J of the lower laserlevel) is composed of many longitudinal mode frequenciesdue to pressure broadening, the Landau–Teller equationsof six-temperature mode related to the upper or lower laserlevel are thus specifically modified to cover all longitudinalmode frequencies and all the population inversion densitiesin the four branches as

dE1

dt¼ NeðtÞNCO2

hv1X 1 �E1 � Ee

1ðTÞ

t10ðTÞ�

E1 � Ee1ðT2Þ

t12ðT2Þ

þhn1hn3

� �E3 � Ee

3ðT ;T1;T2Þ

t3ðT ;T1;T2Þþ

hn1hn5

� �E5 � Ee

5ðT ;T1;T2Þ

t5ðT ;T1;T2Þ

þ hn1FX

J

Xi

DN10PðJÞðliÞ

2

8ptð10PÞsp ðJÞ

f ðni; nð10PÞ0 ðJÞÞ

IðniÞ

hni

( )

þ hn1FX

J

Xj

DN10RðJÞðljÞ

2

8ptð10RÞsp ðJÞ

f ðnj ; nð10RÞ0 ðJÞÞ

IðnjÞ

hnj

( )

þ hn1FX

J

Xk

DN9PðJÞðlkÞ

2

8ptð9PÞsp ðJÞ

f ðnk; nð9PÞ0 ðJÞÞ

IðnkÞ

hnk

( )

þ hn1FX

J

Xl

DN9RðJÞðllÞ

2

8ptð9RÞsp ðJÞ

f ðnl ; nð9RÞ0 ðJÞÞ

IðnlÞ

hnl

( ),

ð6Þ

dE3

dt¼ NeðtÞNCO2

hv3X 3 �E3 � Ee

3ðT ;T1;T2Þ

t3ðT ;T1;T2Þ

þE4 � Ee

4ðT3Þ

t43ðTÞþ

hn3hn5

� �E5 � Ee

5 T ;T3ð Þ

t53 T ;T3ð Þ

� hn3FX

J

Xi

DN10PðJÞðliÞ

2

8ptð10PÞsp

1

hni

�f ðni; nð10PÞ0 ðJÞÞIðniÞ � hn3F

XJ

Xj

DN10RðJÞ

�ðljÞ

2

8ptð10RÞsp

1

hnj

f ðnj ; nð10RÞ0 ðJÞÞIðnjÞ

� hn3FX

J

Xk

DN9PðJÞðlkÞ

2

8ptð9PÞsp

1

hnk

�f ðnk; nð9PÞ0 ðJÞÞIðnkÞ � hn3F

XJ

Xl

DN9RðJÞ

�ðllÞ

2

8ptð9RÞsp

1

hnl

f ðnl ; nð9RÞ0 ðJÞÞIðnlÞ, ð7Þ

where, in Eq. (6) and (7),P

J is to sum up all the rotationalquantum numbers in each branch,

Pi;P

j ;P

k orP

l is tosum up all the longitudinal mode frequencies in 10P, 10R,9P, or 9R branch, respectively.

ARTICLE IN PRESSJ. Wu et al. / Optics & Laser Technology 39 (2007) 1033–1039 1035

The equation on the laser intensity inside the laserresonator is also revised according to each longitudinalmode frequency, that is

dIðniÞ

dt¼ �

IðniÞ

tcðniÞþ chni

�FDN ðaÞðJÞðliÞ

2IðniÞ

8phnitðaÞsp ðJÞ

f ðni; lðaÞ0 ðJÞÞ

" #þ chni

�½N001PðaÞðJÞSðniÞ� ða ¼ 10P; 10R; 9P; 9RÞ: ð8Þ

when there is injection locking, the injection laser powermust be taken into consideration, thus Eq. (8) is modifiedas

dIðniÞ

dt¼ �

IðniÞ

tcðniÞþ chni

�FDN ðaÞðJÞðliÞ

2IðniÞ

8phnitðaÞsp

f ðni; lðaÞ0 ðJÞÞ

" #þ chni

� N001PðaÞðJÞSðniÞ� �

þ cI injectðnÞdðn� ninjectÞ

ða ¼ 10P; 10R; 9P; 9RÞ; ð9Þ

where, in Eq. (9), the injection laser intensity I injectðnÞ isconsidered simply in a way mentioned in Ref. [9] that onlythe effective injection laser intensity on certain longitudinalmode frequency is considered.

The spontaneous item of each longitudinal modefrequency in Eqs. (8) or (9) is written as [5]

SðniÞ ¼0:58

tðaÞsp ðJÞ

l2iA

f ðni; nðaÞ0 ðJÞÞdn

ða ¼ 10P; 10R; 9P; 9RÞ ð10Þ

and the Lorentz profile of pressure broadening is written as

f ðn; n0Þ ¼DnL=2p

ðn� n0Þ2þ ðDnL=2Þ

2. (11)

The broadened width in Eq. (10) can also be found inRef. [7] as

DnLðTÞ ¼X

i

NiQi

p8kT

pmi

� �1=2" #

, (12)

mi ¼MCO2

Mi

MCO2þMi

, (13)

where, in Eqs. (12) and (13), MCO2;Mi is molecule masses,

Ni is molecule density, Qi is collision cross-section betweenmolecules and i ¼ CO2–N2–He or CO.

The other Landau–Teller equations keep the same formsas those given in Refs. [5,6], namely:

dE2

dt¼ NeðtÞNCO2

hn2X 2 �E2 � Ee

2ðTÞ

t20ðTÞþ

E1 � Ee1ðT2Þ

t12ðT2Þ

þhn2hn3

� �E3 � Ee

3ðT ;T1;T2Þ

t3ðT ;T1;T2Þ

þhn2hn5

� �E5 � Ee

5ðT ;T1;T2Þ

t5ðT ;T1;T2Þ, ð14Þ

dE4

dt¼ NeðtÞNN2

hn4X 4 �E4 � Ee

4ðT3Þ

t43ðTÞ

þhn4hn5

� �E5 � Ee

5ðT ;T4Þ

t54ðT ;T4Þ, ð15Þ

dE5

dt¼ NeðtÞNCOhn5X 5 �

E5 � Ee5ðT ;T3Þ

t53ðT ;T3Þ

�E5 � Ee

5ðT ;T1;T2Þ

t5ðT ;T1;T2Þ�

E5 � Ee5ðT ;T4Þ

t54ðT ;T4Þ, ð16Þ

dE

dt¼

E1 � Ee1ðTÞ

t10ðTÞþ

E2 � Ee2ðTÞ

t20ðTÞ

þ 1�v1

v3�

v2

v3

� �E3 � Ee

3ðT ;T1;T2Þ

t3ðT ;T1;T3Þ

þ 1�v1

v5�

v2

v5

� �E5 � Ee

5ðT ;T1;T2Þ

t53ðT ;T1;T3Þ

þ 1�v4

v5

� �E5 � Ee

5ðT ;T4Þ

t54ðT ;T4Þ

þ 1�v3

v5

� �E5 � Ee

3ðT ;T3Þ

t53ðT ;T3Þ. ð17Þ

The initial conditions for all the differential equationsare set as those given in Ref. [6]:

Eiðt ¼ 0Þ ¼ hviNi

1

expðhvi=kTÞ � 1, (18)

IðviÞ��t¼0¼ c� 10�10

J

m2

� �¼ 3� 108 � 10�10 ¼ 0:03ðWm�2Þ; ð19Þ

Tðt ¼ 0Þ ¼ 300K: (20)

The unstated physical meanings and expressions of allthe symbols appearing in the above Eqs. (1)–(20) arereferred to Refs. [5] and [6]. Data of all the parametersappearing above can also be found in Refs. [5,6,10].The equations given above can be utilized to describe the

dynamic emission of tunable TEA CO2 lasers as well asnon-tunable TEA CO2 lasers. By solving these equationswith necessary data of the laser construction, detailedperformance characteristics can be estimated numerically.

3. Simulation results

In order to calculate the output characteristics of a TEACO2 laser, the specific parameters of the laser are required.In this paper, the geometrical dimensions of the TEA CO2

laser are set as: cavity length L ¼ 1:7m, effective gainlength l ¼ 1:0m, gap of the electrode pair 5 cm andeffective discharge width 2.5 cm. The resonator is com-posed of one total reflective concave copper mirror andone partial transmission plane mirror or a Littrowconfigured grating (d�1 ¼ 120mm�1, Brazed angleyB ¼ 30�, Number of grooves Ng ¼ 5000) with the zerothdiffraction order as laser output. The laser gas is a mixture

ARTICLE IN PRESS

0.0 2.0 4.0 6.0 8.0 10.0 12.00.0

10.0

20.0

30.0

40.0

50.0

60.0

70.0

Lase

r O

utpu

t Pow

er (

MW

)

Time (us)

2.80000 2.90000 3.00000 3.10000 3.200000.0

0.5

1.0

1.5

2.0

2.5

10P(14)

10P(18)

10P(16)

Lase

r O

utpu

t Ene

rgy

(J)

Frequency (1013Hz)

2.83590 2.83592 2.83594 2.83596 2.83598 2.83600 2.83602 2.83604

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

Pul

se E

nerg

y (J

)

Frequency (1013Hz)

(a)

(b)

(c)

Calculated laser output pulse profile

Calculated laser output spectra

Calculated fine longitudinal modes of 10P(18) line

Fig. 2. Numerical results of a non-tunable TEA CO2 laser.

J. Wu et al. / Optics & Laser Technology 39 (2007) 1033–10391036

ARTICLE IN PRESSJ. Wu et al. / Optics & Laser Technology 39 (2007) 1033–1039 1037

of 10%CO2:10%N2:77%He:3%CO with total gas pressureof 101.325 kPa.

The pumping electron density Ne(t) is empirically writtenas [6]

NeðtÞ ¼ 6:75� 1019 expð�t=0:5� 10�6Þ

�ð1� expð�t=1� 10�6ÞÞ. ð21Þ

Data of line center frequencies, the spontaneous emis-sion lifetimes, rotational constants, etc., can be found inRefs. [5,6,10].

A C-language computer program, based on Runge–Kutta method is developed to solve the differentialequations given above. Fig. 2 is the numerical results of a

2.80000 2.90000 3.0000

1

2

3

4

5

10P(20)

Lase

r O

utpu

t Ene

rgy

(J)

Frequency

Calculated laser

2.83055 2.83057 2.83060 2.830

1

2

3

4

5

Lase

r O

utpu

t Ene

rgy

(J)

Frequency

Calculated fine longitudin

(a)

(b)

Fig. 3. Numerical results of a gra

non-tunable TEA CO2 laser with output coupler reflectiv-ity R ¼ 50%. Fig. 2(a) is the output pulse profile andFig. 2(b) is frequency spectra of the laser pulse. It can beseen that the laser output spectra consist of several branchlines, each line with fine structure of several longitudinalmode frequencies. Fig. 2(c) shows the longitudinal modecomposition of 10P(18) line.Fig. 3 is the numerical results of a tunable TEA CO2

laser with the aforementioned Littrow configuration.Fig. 3(a) shows that if the grating is tuned exactly at thecenter frequency of 10P(20) line, only one 10P(20) lineappears in the output laser spectra. This only line also hasseveral fine longitudinal mode frequencies as shown inFig. 3(b). This result shows that with only one grating with

00 3.10000 3.20000(1013Hz)

output spectra

062 2.83065 2.83067 2.83070 (1013Hz)

al modes of 10P(20) line

ting tunable TEA CO2 Laser.

ARTICLE IN PRESSJ. Wu et al. / Optics & Laser Technology 39 (2007) 1033–10391038

Littrow configuration, it is unlikely to obtain SingleLongitudinal Mode (SLM) operation in this tunable TEACO2 laser.

Fig. 4 is the numerical results of an injection-lockingTEA CO2 laser. The laser structure is the same asmentioned in Fig. 2 but with a well-matched injectionpower of 0.1mW existing in the resonator at a centerfrequency of 10P(20) line. Fig. 4(a) shows the laser outputspectra. Fig. 4(b) shows the fine longitudinal modefrequencies of this spectral line. Evidently, though thelaser itself is non-tunable, 0.1mW effective injection powerin the resonator at center frequency of 10P(20) line couldlead to SLM operation.

More simulation results by means of the above Land-au–Teller equations on tunable TEA CO2 lasers can befound in Refs. [11,12].

2.80000 2.90000 3.000000

2

4

6

8

10

12

14

16

Lase

r O

utpu

t Ene

rgy

(J)

Frequency (1013

Calculated laser outpu

2.83055 2.83057 2.83060 2.830

2

4

6

8

10

12

14

16

Lase

r O

utpu

t Ene

rgy

(J)

Frequency

SLM output o

(a)

(b)

Fig. 4. Numerical results of an inj

4. Conclusion

A mathematical model is given to calculate theperformance characteristics of both non-tunable TEACO2 lasers and tunable TEA CO2 lasers (includinginjection locking). This model is set up by rewriting theLandau–Teller equations of six-temperature model in theform of fine longitudinal mode frequencies of the lasingtransitions in a pulsed TEA CO2 laser containing gasmixtures of CO2–N2–He–CO. The numerical results givenas illustrations show very good agreement with those well-known features in (tunable) TEA CO2 lasers. Of course, thekinetic process occurring in a tunable TEA CO2 laser isvery complicated; however, this simple mathematicalmodel will be kind of helpful in numerically predictingthe output characteristics (such as pulse energy, pulse

3.10000 3.20000

Hz)

t spectra

062 2.83065 2.83067 2.83070

(1013Hz)

f 10(20) line

ection-locking TEA CO2 laser.

ARTICLE IN PRESSJ. Wu et al. / Optics & Laser Technology 39 (2007) 1033–1039 1039

profile, lasing spectra, etc.) of any given (tunable) TEACO2 laser in detail.

References

[1] Vlases GC, Money WM. Numerical modeling of pulsed electrical

CO2 lasers. J Appl Phys 1972;43(4):1840–4.

[2] Gilbert J, Lachambre JL, Rheault F, Fortin R. Dynamics of the CO2

atmospheric laser with transverse pulse excitation. Can J Phys

1972;50:2532–5.

[3] Andrews KJ, Dyer PE, James DJ. A rate equation model for the

design of TEA CO2 oscillators. J Phys E 1975;8:493–7.

[4] Manes KR, Seguin HJ. Analyses of the CO2 TEA laser. J Appl Phys

1972;32(12):5073–8.

[5] Smith K, Thomson RM. Computer modeling of gas lasers.

New York: Plenum Press; 1978. p. 1–78.

[6] Soukieh M, Abdul Ghani B, Hammadi M. Mathematical modeling of

CO2 TEA laser. Opt Laser Technol 1998;30(8):451–7.

[7] Karapuzikov AI, Ptashnik IV, Sherstov IV, et al. Modeling of

helicopter-borne tunable TEA CO2 DIAL system employment for

detection of methane and ammonia leakages. Infrared Phys Technol

2000;41(2):87–96.

[8] Hasson Victor. Review of recent advancements in the development of

compact high power pulsed CO2 laser radar systems. SPIE 1999;3707:

499–512.

[9] Okida T, Manabe Y, Muraoka K, Akazaki M. Off-line center

operation of an injection-locked single mode TEA CO2 laser. Appl

Opt 1981;20(23):2176–8.

[10] Wittemann WJ. The CO2 Lasers. Berlin: Springer; 1987. p. 22–60.

[11] Jin Wu. Theoretical model on calculating grating tuned TEA CO2

laser. Acta Opt Sin 2004;24(4):472–6.

[12] Jin Wu, Chongyi Wan, Rongqing Tan, et al. High repetition rate

TEA CO2 laser with randomly coded wavelength selection. Chin Opt

Lett 2003;10(1):601–3.