DECLARE SUB PAGER () REM REM GE Copeland REM Department of Physics REM Old Dominion University REM Feb. 1999 REM Simulation of Fabry-Perot Interferometer REM Tranmission Function REM REM 2/20/2001 100 DIM R(3) Pi = 3.141592654# 300 DIM I(700), a(700) 400 S1 = 5 REM This IS the step size used below 410 P1 = Pi PRINT CHR$(12) PRINT " Skip the Intro and go directly to the calculations? " INPUT a1$ IF a1$ = "yes" OR a1$ = "Yes" OR a1$ = "yes" THEN GOTO 666 415 PRINT CHR$(12) 500 PRINT "F A B R Y - P E R O T INTERFEROMETER" 600 PRINT 700 PRINT "MULTIPLE BEAM INTERFERENCE FRINGES from a plane" 800 PRINT "parallel plate are observed when illuminated by incident light." 900 PRINT "If the Fabry-Perot is used in the visible region of the" 1000 PRINT "E&M spectrum, the two plates are made from VERY FLAT [L/50] " 1100 PRINT "QUARTZ whose facing sides are coated to be HIGHLY RELECTING." 1200 PRINT 1300 CALL PAGER 1400 PRINT "Examine the diagram below:" 1500 PRINT 1600 FOR I = 1 TO 7 1700 PRINT "--->", "I", "I" 1800 NEXT I 1900 PRINT "O.A.---------------------------------------------------O.A." 2000 FOR I = 1 TO 7 2100 PRINT "--->", "I", "I" 2200 NEXT I 2300 PRINT , "I<--ETALON--> I" 2500 CALL PAGER 2600 PRINT " Let a BEAM be INCIDENT on the plate at an incident angle, 0," 2700 PRINT "as measured from the NORMAL . r is the coefficient" 2720 PRINT "of REFLECTION & t the coefficient of TRANSMISSION of the 1st" 2900 PRINT "surface. If the ELECTRIC VECTOR of the INCOMING WAVE is E0," 3000 PRINT "then the transmitted wave has a field of E= t*E." 3100 PRINT "At the 2nd surface the REFLECTED" 3200 PRINT "wave has E=E0*r*t and the TRANSMITTED E=E0*t*t, etc." 3300 CALL PAGER 3400 PRINT "The GEOMETRIC PATH DIFFERENCE between any 2 succesive TRANSMITTED" 3500 PRINT "RAYS is = 2*D*COS(0), where D is the plate spacing." 3600 PRINT "The corresponding PHASE DIFFERENCE is given by:" 3700 PRINT , "P = 2*K*D*COS(0) " 3800 PRINT "where K=the WAVE VECTOR MAGNITUDE =2*Pi/LAMBDA." 3900 PRINT "In order to get an idea of the OBSERVED INTENSITY, we must" 4000 PRINT "do a VECTOR SUM of all the ELECTRIC VECTORS TRANSMITTED and recall" 4100 PRINT "that INTENSITY = E^2 ." 4300 CALL PAGER 4400 PRINT "Let E(t)= the AMPLITUDE of the TRANSMITTED WAVES and remember" 4500 PRINT "the PHASE DIFFERENCE for each of wave is related to EXP(i*P)." 4600 PRINT "then E(t)=E0*t^2 + E0*t^2*r^2*EXP(i*P) + E0*t^2*r^4*EXP(2*iP)+..." 4700 PRINT " 1ST +(1ST)*r^2*E^iP + (1ST*2ND)*r^2*E^iP +..." 4800 PRINT "This INFINITE SUM just happens to be a GEOMETRIC PROGRESSION." 4900 PRINT "It's SUM is the 1ST TERM over ONE minus the COMMON RATIO. Thus" 5000 PRINT , "E(t)= E0*t^2/(1-r^2E^iP)" 5100 PRINT 5200 PRINT "The INTENSITY I(t) is given by:" 5300 PRINT "I(t)= ABS(E(t))^2 = I0 * ABS(t)^4" 5400 PRINT " ---------------" 5500 PRINT " [ABS(1-r^2*EXP(i*P))]^2" 5600 PRINT "WHERE i is SQR(-1)." 5700 CALL PAGER 5800 PRINT "We have seen from a study of Fresnel"; s; formulas; "" 5900 PRINT "(SEE PROGRAM FRESNL),that a PHASE CHANGE may occur on REFLECTION" 6000 PRINT "and in general r will be a COMPLEX NUMBER. If we DEFINE R and T" 6100 PRINT "such that they are REFLECTANCE and transmittance of a SURFACE, we" 6200 PRINT "have: R = ABS( r ) ^2 T = ABS( t )^2" 6205 PRINT 6300 PRINT "Now the INTENSITY FUNCTION can be rewritten as:" 6400 PRINT 6500 PRINT "I(TOTAL) = I0 T^2 1" 6600 PRINT " ------------- -----------------------------" 6700 PRINT " (1-R )^2 1 + 4*R SIN^2(DELTA/2)" 6800 PRINT " --------" 6900 PRINT " (1-R)^2" 6905 PRINT 7000 PRINT "where DELTA = P + P REF/2=PHI+N*K0*D*COS(PSI)" 7100 PRINT , "K0=2*Pi/LAMBDA ,PHI=ANGLE of REFRACTION at 2ND " 7200 PRINT "SURFACE and PSI=INCIDENCE ANGLE at 2ND SURFACE" 7300 PRINT "AND P REF/2= PHASE CHANGE on ONE REFLECTION. " 7500 CALL PAGER 7600 PRINT "Many times a new quantity F=4R/(1-R)^2 is defined and is a FIGURE" 7700 PRINT "OF MERIT for the FABRY-PEROT. F is related to the FINESSE and is a" 7800 PRINT "MEASURE of the SHARPNESS of FRINGES. The INTENSITY varies with" 7900 PRINT "DELTA according to the FUNCTION 1" 8000 PRINT " ------------------------" 8100 PRINT " 1 + F * SIN^2( DELTA/2)" 8200 PRINT "This function is called the AIRY FUNCTION. If the SINE ARGUMENT" 8300 PRINT "is an integral multiple of Pi , then AI(DELTA)=1" 8400 PRINT "for ALL values of R. If R is very small (<0.2) then fringes are" 8500 PRINT "broad. If R->1, we have very SHARP FRINGES. You will be given" 8600 PRINT "an opportunity below to calculate this function." 8700 CALL PAGER 8800 PRINT " There are 3 parameters that are useful when discussing an" 8900 PRINT "ETALON. Thet are: CONTRAST, C; WIDTH, G; and FINESSE, F" 8910 PRINT 9000 PRINT "The CONTRAST is defined as the RATIO OF THE MAXIMUM" 9100 PRINT "TO THE MINIMUM TRANSMITTANCE. C=1+F" 9190 PRINT 9200 PRINT "The WIDTH, G, is the distance BETWEEN HALF-MAXIMUM" 9300 PRINT "POINTS. If DELTA is close to an INTEGER *Pi, then SIN^2 is " 9400 PRINT "almost equal to (DELTA-M*PI)^2 where M is the INTERFERENCE ORDER" 9500 PRINT "Then the TRANSMISSION function has a RESONANCE DEMOMINATOR" 9600 PRINT " T/TMAX = G^2/4" 9700 PRINT " ----------------------" 9800 PRINT " [DELTA - M*PI]^2 -G^2/4" 9900 PRINT " where the WIDTH = G = 2/SQR(F)" 9910 CALL PAGER 10000 PRINT "Lastly, the FINESSE F is defined as the RATIO of the SEPARATION" 10100 PRINT "of SUCESSIVE MAXIMA to the WIDTH G. SO" 10200 PRINT , "F = PI / G = PI * SQR(F) / 2 " 10300 CALL PAGER 666 REM calculations 10400 PRINT "Now we calculate the intensity RATIO I/IO for a given special case." 10500 PRINT "We must know the chatactersitics of the ETALON. This is where" 10600 PRINT "the ETALON REFLECTIVITIES are involked." PRINT "Example:" 10700 PRINT "For an etalon that passes the IR (wavelength L is 8 to 12 MICRONS)," 10780 PRINT "we use a single piece (etalon)" 10800 PRINT "of the element SILICON or GERMANIUM. For Ge the INDEX OF REFRACTION" 10900 PRINT "is 4.05 at 10.6 microns. For NEAR NORMAL INCIDENCE, we can use" 11000 PRINT "the FRESNEL FORMULAS to determine the value of R." PRINT "Use Ge or select another N? "; INPUT ntype$ IF ntype$ = "yes" OR ntype$ = "YES" OR ntype$ = "Yes" THEN GOTO 11100 PRINT "Input the Index of Refraction of the etalon"; INPUT N GOTO 11300 11100 PRINT " R=[(1-N)/(1+N)]^2 ="; 11200 N = 4.05 11300 R = (1 - N) / (1 + N) 11400 R = R * R 11500 PRINT R 11600 F = 4 * R / (1 - R) ^ 2 11700 PRINT "The function F="; F 11800 PRINT "The CONTRAST C="; 1 + F 11900 PRINT "The WIDTH G="; 2 / SQR(F) 12000 PRINT "The FINESSE ="; Pi * SQR(F) * .5 12100 PRINT "Now the AIRY FUNCTION will be evaluated over three PERIODS of " 12200 PRINT "the PATH DIFFERENCE=DELTA = 3 * 2*PI for this value of R="; R 12300 PRINT "and for 2 other cases where R=.6 and .8" 12400 REM DEFINE REFLECTIVITIES 12500 R(1) = R 12600 R(2) = .6 12700 R(3) = .8 OPEN "PLTTEK.DAT" FOR OUTPUT AS #2 12800 PRINT "We now proceed to generate the plotting file." 13800 FOR R1 = 1 TO 3 13900 R9 = R(R1) 14000 F = 4 * R9 / (1 - R9) ^ 2 14100 k = 0 14200 FOR D = 0 TO 3 * 360 STEP S1 14300 D1 = D * Pi / 180! 14400 k = k + 1 14500 I(k) = 1 / (1 + F * SIN(D1) ^ 2) 14600 a(k) = D REM IF R1 = 2 OR R1 = 3 THEN PRINT k, I(k), A(k) 14700 NEXT D 14800 REM TEKTRONIX OUTPUT OF DATA REM TEKTRONIX HEADER IF R1 > 1 THEN GOTO 15000 PRINT #2, 1 PRINT #2, 1 k = 0 FOR D = 0 TO 3 * 360 STEP S1 k = k + 1 NEXT D PRINT #2, k * 3 PRINT #2, " Transmittance" PRINT #2, "Total Phase Difference" PRINT #2, "Fabry-Perot Transmittance n= "; N PRINT #2, "R= "; R(1); ", "; R(2); " & "; R(3); "" 15000 FOR I = 1 TO k 15100 PRINT #2, a(I), I(I) 15200 NEXT I 15900 NEXT R1 CLOSE #2 PRINT "Now go do the plot = using WPLOT." CHAIN "TEK2WPLT.BAS" STOP END 19100 SUB PAGER 19110 PRINT , "Push RETURN to continue:" 19112 INPUT DUM$ 19120 PRINT CHR$(12) END SUB