A simplified analysis for high-energy laser propagation*James Wallace

Far Field, Inc., Sudbury, Massachusetts 01776

Cynthia WhitneyCharles Stark Draper Laboratory, Inc., Cambridge, Massachusetts 02139

(Received 29 June 1977; revision received 2 February 1978)

A simplified propagation model is presented that provides a useful balance between accuracy,flexibility, completeness and ease of computation. The atmosphere is modeled by a combination ofaberrated lenses that includes the phase front distortions generated by the device, optical train,atmospheric turbulence, and thermal blooming. When atmospheric heating occurs away from thefocal plane the method gives results that are virtually identical to the exact, distributed lens analysis.The method applies especially to vertical propagation through the atmosphere and horizontal propa-gation of repetitively pulsed lasers. Numerical results are presented that illustrate the analysis.

I. INTRODUCTION

The inclusion of the major sources of optical phase frontdistortion creates special demands for modeling atmosphericpropagation. A recent article1 by Smith gives a comprehen-sive review of atmospheric propagation and compares thetheoretical predictions to laboratory-simulation experiments.The major sources of phase front distortion are generated bythe laser, the optical train, atmospheric turbulence, andthermal blooming. Assessment of the combined effects re-quires a large number of lengthy computations. Particularlytroublesome is the determination of atmospheric heatingwhen the random effects of atmospheric turbulence and me-chanical vibration (jitter) are included. There are severalnumerical programs 2 8 available that do include the combinedeffects of turbulence and heating. These programs haveheated, random-phase sheets distributed along the propaga-tion path and determine the irradiance at the focal plane bypropagating from the aperture to the focal plane. To deter-mine meaningful statistical averages as many as a hundredseparate calculations have to be performed. Because of thelarge number of calculations required several investigators 9 "have published analytical formulas summarizing the nu-merical results. Generally, the focal plane irradiance is de-termined for thermal blooming alone and the effects of tur-bulence and jitter added by root-sum-square (RSS) additionof the individual effects.10 This is adequate for the peak ir-radiance but not for the irradiance distribution or beam dis-placement. A simplified analysis that gives this additionalinformation, without the complexity of the exact codes, wouldbe a useful development. In the hierachy of models thisanalysis lies between the exact, heated, random phase sheetmodel and the simple RSS approach.

The approach is based on concepts that originated inscattering theory and accounts of it are given in several scat-tering texts.12 Important theoretical developments for opticalpropagation in inhomogeneous media have been obtained byKeller and Levy.13 The method uses geometrical optics todetermine the amplitude and phase shifts caused by the in-homogeneous media and substitutes these results into an exactintegral expression for the scattering cross section. The ap-proach is called the high-energy approximation in scatteringtheory and the high-frequency (short-wavelength) approxi-mation in optics. Scattering theory further assumes a smallperturbation in the index of refraction whereas the analysisof Keller and Levy is valid for arbitrary perturbations. In

atmospheric propagation the approach has also been used byZeiders14 to correlate the exact cw thermal blooming results.More recent developments have been reported byBreaux.1 5

The addition of turbulence and jitter increases the com-plexity of the geometrical optics solution and the resultspresented in this paper require small perturbations in theindex of refraction. In this limit the phase shifts for the in-dividual effects can be added linearly and statistical averagesdetermined analytically. The approach developed in thispaper is not severely limited by the assumption of a smallperturbation in the index of refraction. The requirement thatthe phase shifts be adequately given by geometrical optics ismore fundamental. In physical terms the simplified approachis virtually exact if atmospheric effects occur where geomet-rical optics is valid. This occurs for sea-level propagation ofrepetitively pulsed lasers when the density gradients from theprevious pulses do not overlap in the depth of focus and ver-tical propagation through the atmosphere.

11. THEORETICAL FORMULATION

In the optical-frequency range, Maxwell's equations can beapproximated by the paraxial approximation to the scalarwave equation. The electric field is of the form

E = A(x,y,z) exp(ikn~z - az/2), (1)

where x and y are the coordinates transverse to the propaga-tion direction z, a is the absorption coefficient, n- is the indexof refraction in the atmosphere, and k is the wave number.The governing equation, in normalized variables X/Rm, ylRm,Z/Zf, is

2iFn-Az + V2A + 2k 2(n- - 1)R2 pA = 0, (2)

where Rm is the radius of the aperture, p is the density per-turbation caused by turbulence and atmospheric heating, Zfis the distance to the focal plane, and F is the Fresnel numberkR2 lzf. For a beam focused at Zf, the boundary condition atz = O is

A = Ao(x,y) exp[iF(lo(x,y) - 1/ 2 (x2 + y 2 ))], (3)

where Fo is the contribution to the optical phase from thelaser and the optical train. The formal solution to Eq. (2) canbe determined by propagating point sources1 6 from the focal

750 J. Opt. Soc. Am., Vol. 68, No. 6, June 1978 0030-3941/78/6806-0750$00.50 � 1978 Optical Society of America 750

plane to the aperture and is given1 7 by

A = 2 i X Ao(xy)a(Qn;xiyi)2iriJJ

X exp[iF(4(,77;xi,yi) + 4o(xy)+ 1/2(X2 + y2) - X1X-YY) dx dy, (4)

where

~X-xi, n=Y-Yi. (5)

In Eq. (4) x and y are the coordinates of the aperture plane,x 1 and y 1 are the coordinates of the focal plane and a (Q,-;x 1,y 1)and 4,7m;xi,y1) are the distortions in the amplitude and phaseexperienced by a point source in propagating from the focalplane to the aperture. The amplitude and phase satisfy, ina coordinate system converging to the point x1, Y1 at z = zf,

-gZ + 1/2[V/2 + t2]/(1 - z)2

= Z(n, - 1)E[p(xi + 4(1 - z), yi+ n(l - z),z)]/R2, (6)

-(1-z) 2az + (Va) - (V{) + 1/2a(V24') - i(V 2a)/2F = 0, (7)

where p is the normalized density perturbation due to atmo-

I

spheric turbulence and heating and f is a measure of thestrength of the density perturbation.

The equations governing the amplitude and phase in aturbulent and heated atmosphere are almost as complicatedas the original equation. However, for small perturbationsin the index of refraction Eqs. (6) and (7) can be solved by apower series expansion in the distortion parameter, N = z 2(n-.

-1)E/R'. The leading terms in the expansion are

4' = N r p(xi + (x - x1)(1 - )Yi + (y - y)(1 - z'), z') dz'; a = 1.0, (8)

and is an acceptable approximation if

N 4r p(0,0,') dz' < 1.0. (9)

Under the small perturbation assumption the total opticalphase is the sum of the individual components. This is animportant simplification because the spatial statistics of theirradiance can be determined analytically. Multiplying A byA* in Eq. (4) and averaging gives the mean irradiance at thefocal plane as

(I) = (F/27r) 2 exp(-azf) 4' 4f 4' J Mt(uv)Mj(u,v)Ao(x,y)Ao(x - u,y - v)

X exp [IiF (Nh ' Ph(XlZ' + X(I-Z'),YZ'+Y y(1-z'),z') dz'i+ 4q(x~y)- q(x-uy-v)

- Nh 5 Ph (XlZ'+ (x -u)(1 -z'), ylz' + (y -v)(1 -z'), z') dz' -xlu - ylv)] du dv dx dy- (10)

In Eq. (10), Nh and Ph are the distortion number and densitydistribution caused by thermal blooming, 4'q is the contribu-tion to the optical phase from the laser cavity and Mt and Mjare the long-term mutual coherence functions for atmosphericturbulence and jitter. For most cases of interest the mutualcoherence functions are given by

Mt = exp [-(1.45k 2zfR 5/3

x C2(Z')(1-Z) 5/3 dz') (u2 + v2)5/6], (11)

Mj = exp[-1/2 (02U2 + 02V 2)/02], (12)

where C2 is the refractive index structure constant, Ox and 0yare the variances of the jitter angles and Od is the diffractionangle. The analysis for the turbulent coherence function isdue to Lutomirski and Yura1 8 and Eq. (10), in the absence ofjitter and thermal blooming, is identical to their result. Theassumption of small perturbations in the index of refractionis not overly restrictive in atmospheric propagation. In Eq.(10) determining the dependence of the optical phase on thecoordinates x 1, Y1 in the focal plane is the difficult aspect ofthe calculation and we shall assume that Ph is independent ofthe coordinates in the focal plane. This requirement will besatisfied if there is no heating in the depth of focus. Thus Eq.(10), with x1 = yi = 0 in Ph, can be applied whenever

Nh J'Z Ph(0,0,Z') dz' < 1.0

and (13)

Rs/Rm < (1 - Zh)/Zh,

where Zh is the normalized thickness of the heated zone andR, is the spot size at the focal plane.

The irradiance distribution at the focal plane is computedvery efficiently by noting that the integration over x and y isa convolution integral and is (1/27r)2 times the squared mod-ulus of the transform of the individual function. The irra-diance distribution is determined by the following sequenceof transforms:

(I) = T- 1 (Mt(u,v)Mj(u,v)H(u,v))/(27r) 2, (14)

with

H(u,v) = T- 1 { T (Ao(x,Y) exp [iF (iq(xy)

+ Nh 4' Ph (X( -z'), y( -z'), z') dz')]) 21, (15)

where the symbol T denotes the two-dimensional Fouriertransform, T-1 the inverse Fourier transform and the barsdenote the modulus of the transform. The transforms arecomputed by the FFT algorithm. The main results of theanalysis are expressed by Eqs. (14) and (15) and are validproviding Eq. (13) is satisfied.

751 J. Opt. Soc. Am., Vol. 68, No. 6, June 1978 J. Wallace and C. Whitney 751

x* x - Uwtsn/Rm.

UE HEATING ANDTURBULENCE t

LTURBULENCE ONLY

FIG. 1. Typical situation in atmospheric propagation for application ofthe simplified analysis. The laser is repetitively pulsed with a beam motionlarge enough to displace, near the focal plane, the column of air heated bythe previous pulses.

111. PHASE DISTRIBUTION FOR THERMALBLOOMING

When applying Eq. (14) we must accurately determine thethermal blooming contribution to the optical phase. Theassumption of small perturbations in the index of refractionrequires a weak coupling between the effects of turbulence,jitter, and thermal blooming. For repetitively pulsed lasersthis occurs for the conditions shown in Fig. 1. The heatedcolumn of air from the previous pulses is displaced by U t,at the laser and Utt. at the focal plane where t, (= 1/PRF) isthe interval between pulses. The next pulse experiences thedensity gradients from the previous pulses and atmosphericturbulence. For velocities at the focal plane much larger thanthe velocity at the laser, the region where turbulence andheating are simultaneously occurring is confined to a relativelysmall volume near the laser. In this region the irradiance isdetermined primarily from the distribution at the aperture.The turbulent fluctuations in the optical phase have not in-troduced amplitude distortions that significantly affect theheating calculation. Outside the heated volume the beampropagates in a turbulent atmosphere and Eq. (10) does thisaspect of the calculation exactly.

For repetitively pulsed lasers the density dependence onthe normalized irradiance is6

Ph = -exp(-az) (y - 1)a1at8TP -

X i I(x - U(z)tsn/Rm,y,z), (16)n=1

where -y = 1.4 is the ideal gas constant, p = 0.1 J/cm3 is thepressure and U(z) is the transverse flow velocity. We assumethe irradiance propagates geometrically in the heated columnof air that affects the next pulse

I = Io(x/(1 - z), y/(l - z))/(1 -Z)2

In Eq. (18) Io is the normalized irradiance distribution at theaperture and Ia is the average irradiance. The requirementof no overlapping of the density gradients from the previouspulses at distances greater than Zh determines Zh as

Zh = (1-Uwts/2Rm)/1[1 + (Ut - Uw)ts/2Rm]< (1 + Rs/Rm)-1 . (20)

For cw propagation the density dependence on the irradianceis

p =-exp(-az) ,U)(-y)al j x I(u,y,z) du, (21)-yp -U(Z) E.,

and gives, after a calculation equivalent to that of Eq. (17), thecw thermal blooming phase as

F11h = -kzf(n- - 1) (y - 1)aIaRmyP ,

X(SZh exp(-az')d )S Io(u,y) du (22)

The cw result has been given previously by Zeiders.14 We cannow substitute either Eq. (18) or Eq. (22) for the thermalblooming term in Eq. (15) and determine the irradiance in thefocal plane from Eq. (14).

Application of these results is more restrictive for cw lasersthan for repetitively pulsed lasers. For cw lasers atmosphericheating occurs all the way to the focal plane and the constraint,Eq. (13), on Zh will not be satisfied. For this case the opticalphase also depends upon the coordinates in the focal plane anda geometrical optics calculation of the phase will not be ade-quate. The effects of turbulence, jitter, beam quality, anddiffraction strongly affect the irradiance distribution near thefocal plane and the thermal blooming phase will not be de-termined just by the irradiance distribution at the aperture.However these results do apply to the important case of ver-tical propagation through the atmosphere. For this case theabsorption scale height is only a few kilometers and Zh typi-cally be less than 0.2.

IV. RESULTS

We now present computations for a 10.6 pm repetitivelypulsed laser focused at 2.5 km with an aperture of 1 m, anabsorption coefficient of 0.2 per km, a transverse wind velocityat the laser of 2.5 m/s and an angular slew rate of 0.08 rad/s.

-, N0. 03

(17)

and substitute Eq. (17) into Eq. (16) and integrate in thepropagation direction. This gives, after a change in variablefrom z to x, the thermal blooming contribution to the opticalphase as

FP1 = -kzf(n- - 1) exp(-azh/ 2 )

X (ry - 1)aIYRU

'yp - Ut

J 1 roX dF - f Io(u~y) du,

n=l n __X(18)

I

S=.783 S=.805

/' - -I /' \ II / ' '

II I '-- I~

I , -\ II-

0.05 s269 \\ ,IS=.268

FIG. 2. Comparison of the Strehl ratios and the bloomed isoirradiancecontours for a 10.6 Am, repetitively pulsed laser focused at 2.5 km. Thecontours determined by the analysis are shown as solid lines and the exactresults by the dashed lines.

752 J. Opt. Soc. Am., Vol. 68, No. 6, June 1978

where

(19)

/

J. Wallace and C. Whitney 752

0 0 0 o .05

.025

.05

FIG. 3. Effects of turbulence on the isoirradiance contours for a 10.6 Am,repetitively pulsed laser focused at 2.5 km. The lower contours are forthermal blooming alone and the upper contours are for moderate turbulenceand thermal blooming.

In Fig. 2 we compare the Strehl ratio and the irradiance dis-tributions determined from the simplified analysis with anexact calculation. The product FNh of the Fresnel numberand geometrical distortion parameter Nh was 0.9 and 1.8 rad.The energy per pulse was constant for both cases and the in-crease in distortion number is due to the increase in thenumber of pulses. The pulses per flow time, 2Rm/Uwtsj, at thelaser were 20 and 40 at 0.9 and 1.8 rad, respectively. Theagreement with the exact results, shown as the dashed linesin Fig. 2, are remarkably good for both the Strehl ratio and thedetailed irradiance distribution. Application of the simplifiedanalysis for repetitively pulsed lasers with high slew ratesrepresents the ideal application of this analysis and gives re-sults that are virtually identical to the exact, distributed lenscodes.

In Fig. 3 we compare the mean irradiance distribution whenturbulence is added. The beam parameters are identical tothose of Fig. 2. The lower contours are for thermal bloomingalone and the upper contours are for moderate turbulence andthermal blooming. The refractive index structure constantCn is 3 X 10-14 m 21 3 and the FNh parameter varies from 0.9at the smallest value to 2.25 at the largest value. From thedistributions in Fig. 3 turbulence dominates thermal bloomingand, except at the highest distortion number, there is verylittle effect of atmospheric heating on the irradiance.

In Fig. 4 we compare the isoirradiance contours determinedfrom Eq. (14) with the exact results for a horizontal cw beamfocused at 2 and 3 km. The atmospheric conditions areidentical to those of Fig. 2. The distortion parameters are 4.90at 2 km and 5.90 at 3 km. The smaller spot sizes are for thebeam focused at 2 km. The solid contours have been com-puted from Eq. (14) and the dashed contours are the resultsof an exact calculation. In these calculations atmospheric

62 , I/I, / ,

I I / l q, /

\ ll

\ II

FIG. 4. Comparison of the bloomed isoirradiance contours for horizontalpropagation of a cw laser beam focused at 2 and 3 km. Because heatingoccurs near the focal plane the results are less accurate than those in Fig.

heating occurs up to the focal plane and we have exceeded thelimits of the analysis. As expected, the method gives resultsfor the shape of the isoirradiance contours that are less accu-rate when compared to the exact results. To obtain betteragreement the dependence of the optical phase on the coor-dinates in the focal plane must be added. However, the Strehlratios compare more favorably. The exact Strehl ratios at 2and 3 km are 0.194 and 0.123 and those obtained from Eq. (14)are 0.215 and 0.142.

V. CONCLUSION

We have presented a simplified analysis for high-energylaser propagation that includes contributions to the opticalphase from the device, atmospheric turbulence, and thermalblooming. The method requires small perturbations in theindex of refraction and uses geometrical optics to determinethe phase shifts caused by atmospheric heating and turbu-lence. The mean irradiance distribution at the focal planeis then determined from this optical phase by a sequence ofFourier transforms. Numerical results are presented andcompared to the exact calculation. Excellent agreement isobtained for repetitively pulsed lasers if the density gradientsfrom the previous pulses do not overlap near the focal plane.For cw lasers the method is less accurate because atmosphericheating occurs near the focal plane and geometrical opticscannot adequately determine the phase.

*Work was supported by the U.S. Army Missile Research and De-velopment Command under Contract No. DAAH01-75-C-0991.

'D. C. Smith, "High-power laser propagation: thermal blooming,"Proc. IEEE 65, 1679-1714 (1977).

2J. N. Hayes, P. B. Ulrich, and A. H. Aitken, "Effects of the atmo-sphere on the propagation of 10.6,am laser beams," Appl. Opt. 1 1,257-260 (1972).

3L. C. Bradley and J. Herrmann, "Numerical calculation of lightpropagation in a nonlinear medium," J. Opt. Soc. Am. 61, 668(1971).

4C. B. Hogge, "Propagation of high-energy laser beams in the atmo-sphere," in High Energy Lasers and Their Applications, editedby S. Jacobs (Addison-Wesley, Reading, MA, 1974).

5H. J. Breaux, "An analysis of mathematical transformations and acomparison of numerical techniques for computation of high-energycw laser propagation in an inhomogeneous medium," BallisticResearch Laboratories Rep. BRLR 1723 (June 1974).

6J. Wallace and J. Lilly, "Thermal blooming of repetitively pulsedlaser beams," J. Opt. Soc. Am. 64, 1651-1655 (1974).

7W. P. Brown, Jr., "Computer simulation of adaptive optical systems,"Hughes Research Laboratory Report N60921-74-C-0249 (Sep-tember 1975).

8J. A. Fleck, Jr., J. R. Morris, and M. J. Feit, "Time dependentpropagation of high energy laservbeams through the atmosphere,"UCRL Rep. 51826 (June 1975).

9E. H. Takken and D. M. Cordray, "Simplified analytical formulasfor thermal blooming," Appl. Opt. 13, 2753-2755 (1974).

10 F. G. Gebhardt, "High power laser propagation," Appl. Opt. 15,1479-1493 (1976).

*"J. A. Lilly, "Simplified calculation of laser beam propagationthrough the atmosphere," U.S. Army Missile Command Tech. Rep.RH-76-8 (1976).

12R. Glauber, "High-energy collision theory" in Lectures in Theo-retical Physics, Vol. 1, edited by W. Brittin and L. Dunham (In-terscience, New York, 1959).

1 3J. B. Keller and B. R. Levy, "Scattering of short waves," in Elec-tromagnetic Scattering, edited by M. Kerker (Pergamon, NewYork, 1963).

14G. W. Zeiders, "A study of wave characteristics influences on laserselection for applications: propagation analysis," W. J. SchaferAssociates Rep. WJSA-TR-74-18(1974).

1 5H. J. Breaux, "A methodology for development of simple scaling

753 J. Opt. Soc. Am., Vol. 68, No. 6, June 1978 J. Wallace and C. Whitney 753

A,_,",

41,11,li"�.1,

,1-11:.I'll

laws for high-energy cw laser propagation," Ballistic ResearchLaboratories Rep. BRLR 02039 (Jan. 1978).

' 6 G. S. S. Avila and J. B. Keller, "High-frequency asymptotic fieldof a point source in an inhomogeneous medium," Commun. PureAppl. Math. 16, 363-381 (1963).

17J. Wallace and J. Pasciak, "Theoretical aspects of thermal bloomingcompensation," J. Opt. Soc. Am. 67, 1569-1575 (1977).

18R. F. Lutomirski and H. T. Yura, "Propagation of a finite opticalbeam in an inhomogeneous medium," Appl. Opt. 10, 1652-1658(1971).

Contrast reduction of broadband optical signals by theatmosphere

L. J. PinsonAuburn University, Auburn, Alabama 36830

C. E. KulasMissile Research and Development Command, Redstone Arsenal, Alabama 35809

(Received 3 October 1977)

The theory for atmospheric contrast reduction is extended to include the effects of spectral depend-ence for broadband optical signals. A result with the same form as the monochromatic theory isachieved and the broadband extinction coefficient is defined. A comparison of atmospheric transmit-tance for the broadband model with that for the single wavelength average transmittance indicateslittle error except for applications where the slant range is larger than the meteorological range.

INTRODUCTION

Three separate but related phenomena contribute to theeffects of the atmosphere on optical radiation: absorption,scattering, and refractive index fluctuations. Absorption andscattering are typically related to the effects of atmosphericgas and particle constituents; whereas refractive index fluc-tuations are typically related to turbulence effects (i.e., densitygradients, temperature, and pressure differences). Absorp-tion and scattering effects on contrast between an object andits background are based on work by Koschmieder,l which waslater simplified in a paper by Duntley 2 to give the "two-con-stant" theory. That is, the reduction of contrast by the at-mosphere is adequately represented by two parameters formost "seeing" conditions. The two parameters are not in factconstants but depend on optical wavelength and on the spe-cific geometry of the observation scenario. Middleton 3 pre-sents an easy to read development of the Koschmieder theoryand of Duntley's "two-constant" theory. In addition hepresents arguments for eliminating certain restrictive as-sumptions made by Duntley. The work by Duntley and byMiddleton still is the basis for contrast reduction formulascited in more recent treatises on the optical effects of the at-mosphere.

In evaluating the usefulness of theoretical expressions forcontrast reduction by the atmosphere in terms of applicabilityfor determining performance of imaging seekers or otherbroadband optical systems, it becomes necessary to examinethe effect of spectral dependence. Some method for recon-ciling a monochromatic theory with a wideband, spectrallynontrivial application such as is true for visible and near-infrared imaging seekers must be found. To the knowledgeof the authors no mathematically concise development hasbeen reported for the broadband case.

This paper gives a development of contrast reduction effectsof the atmosphere for a finite spectral band. The result is ina form that retains much of the simplicity found in themonochromatic result.

BROADBAND MODEL

From the work of Duntley and Middleton, the spectral ra-diance transfer characteristic of the atmosphere is given formonochromatic radiation by

LX(R) = [La,x(0)/ox(0)](1 - Tx) + Lx(0)Tx, (1)

where LX(R) is the spectral radiance at slant range R due toa source radiance LX(0) at zeio, ux(O) is the atmospheric ex-tinction coefficient at zero (including scattering and absorp-tion) for a given wavelength, LaX (0) is the spectral radianceof the air light, and TX is the atmospheric spectral transmit-tance given by

TX = exp[-cx(O)Rx], (2)

where Rx is the optical path length from zero to R for wave-length X. In Eqs. (1) and (2) the subscript X has been addedto emphasize wavelength dependence for the various pa-rameters.

For a finite band of wavelengths the total radiance LA\(R)is the summation over AX of Eq. (1):

LAX(R) = X (1 - Tx) + Lx(0)Tx dX. (3)

Similarly, for the background radiance,

Li,(R) = f (PLA.x(O) (1 - Tx) + LA(O)Tx) dx. (4)

Then for contrast defined as (L - L')/L' we get the broadbandcontrast at range R,

CAX(R) = f {[LX(O)-LX(O)IT,1 dX/Lix(R).. X

(5)

Further we have the broadband inherent contrast at rangezero

CAX(O) = f {LX(O) - LX(O)} dX/Lx(O),.,Ax

(6)

754 J. Opt. Soc. Am., Vol. 68, No. 6, June 1978 0030-3941/78/6806-0754$00.50 � 1978 Optical Society of America 754