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Thick optical films for use as interference filters
Surfaceand Coatings Technology99 ( 1998)60-G?
Thick optical films for use as interference filters Thomas R. Moore ‘+*, Jonathan B. Redmond a7Steven J. Frederiksen a, George R. Gray b, Ming W. Pan b, Peter Guernsey ’ a Deppnrtwzent of Physics, Phoforzics Research Cmter’, United States Military Acndenzy, West Point, NY 10996, USA b Department of Electrical Engineering, Uuivemit)! of Uth, Salt Lake City, UT 44112, I;ISA ’ Gmmsey Coating Laboratories, Ventwa, CA 93003, USA
Received11October 1996;accepted10June1997
Abstract
The design:manufactureand useof optically transparentfilms with thicknesses exceedingthe coherencelength of the incident light is discussed.An analysisof the reflection and transmissionof spatially and temporally incoherentlight by transparentthick filmsis presentedusingboth classicaland quantummechanicalarguments.The analysisdemonstratesthat, contrary to frequently statedassertions,films with thicknesses greatly exceedingthe coherencelengthof the incident light can be usedto constructoptical interferencefilters. The resultsof experimentsare presentedwhich confirm this conclusion.Finally, a discussionof the dengn, constructionand useof interferencefilters madefrom thick optical films is presented.Designsspecificallysuitedto eyeand sensor protection from laserradiation are emphasized.The manufactureof a prototype thick-film optical filter. usefulfor eyeand sensor protection againstthe radiation from severallasers,is reported. 0 1998ElsevierScienceS.A. K~~IV~IZ’KThick films; Sensorprotection; Narrow-band filters; Coherence
1. Introduction It is common to use thin optical films as interference
filters in the visible portion of the spectrum for various applications. However, optical films with thicknesses exceeding an optical wavelength are not commonly used. The lack of use of thick optical films appears to be due to two factors: the difficulty in manufacturing thick films that are transparent in the visible portion of the spectrum, and the often-quoted fact that interference phenomena will not be present when the film thickness exceeds the coherence length of the incident radiation [l-9]. The latter appears to have stifled research into manufacturing methods capable of overcoming the former. In the following sections? we analyze the propagation characteristics of radiation through dielectric films under conditions where the thickness of the dielectric exceeds the coherence length of the radiation. We show that
interference phenomena associated with radiation passing through a thick film is independent of the state of coherence of that radiation. Additionally, we show that * Corresponding author. 0X7-8972/98/$19.00 0 1998ElsevierScience B.V. ALIri&ts reserved. PII so257-8972(97)00414-3
spatial incoherence is not a barrier to using thick optical iilms as interference filters. This latter result is quite surprising since it has been shown that it is possible to discriminate between spatially coherent and spatially incoherent light of arbitrarily narrow bandwidth by observing the interference between the light diffracted by two separated slits [IO]. Finally, we report the design, manufacture and testing of thick optical filters for use as multiple-line interference filters. These titers can block up to five narrow spectral regions, pass over 60% of visible light and still allow white light to be perceived as white to the eye. We propose that this type of filter be used as protection for eyes and sensors against damaging laser radiation, In what follows, the temporal and spatial coherence properties of electromagnetic radiation will be considered separately. While it is well known that these two properties are intimately related [ 111, the relation&ip may often be suppressed in the interest of clarity and we will do so here. 2. Temporal coherence
Consider light with intensity 1, normally incident upon an optically transparent medium of thickness d, as
61
shown in Fig. 1. If the thickness of the medium greatly exceeds the coherence length of the incident radiation, the transmitted intensity, Zr, may be calculated by subtracting the sum of the intensities of the reflections from each interface from the incident intensity. IT = Z,- (Z,, + ZR,). This is because there is no correlation between the electromagnetic field entering the medium and the electromagnetic field that has been reflected from the second interface, and it indicates the inability to use films with thicknesses exceeding the coherence length of the light as interference filters. If. however, the coherence length of the incident radiation exceeds twice the thickness of the medium, the amplitudes of the fields must be summed to determine the transmissivity and interference phenomena clearly will occur. In order to investigate the system shown in Fig. l> we will consider only a single reflection from each of the surfaces of the dielectric. This simplification is useful to explore the physics of the problem and the results are easily extended to the real case of an infinite number of reflections. Assuming normally incident radiation, neglecting dispersion and assuming linear polarization, after the light has had time to completely traverse the medium twice, the reflected electromagnetic field is the sum of the amplitudes of the reflection from the first interface and the reflection from the second interface. Therefore. the electric field component of the refIected radiation from the surface is given by: EJt, T) = pE(t) + a2p’E(r - T),
(1) where E(t) is the analytic signal representation of the electric field incident upon the medium at time t. p is the amplitude reAectivity of the first interface, 0 is the amplitude transmissivity of the first interface. p’ is the amplitude reflectivity of the second interface and r is the time taken for the light to traverse the medium twice: r =2&c, where tl is the thickness of the medium. II is the index of refraction and c is the speed of light in vacuum. The intensity of the reflected light is given to within
a constant by Z,= lE,12 and therefore:
+:po’p’E(t)E*(t-T)+c.c.).
where C.C. refers to the complex conjugate. The average intensity of the light reflected from the medium 7, can be found by integrating over a time period long compared with the fluctuations in the electric field, and therefore:
Ir=r,+z2+-
pdp 2T
T is -.,-
E(r)E’(t-r)
dt-+c.c.
,
(3)
where I, and Zz are the time-averaged values of the intensities reflected from the two interfaces by themselves, p21E(t)12and a”p”lE(t - r)12 respectively and T is long compared with the time scale of the fluctuations in the electric field. It is evident from Eq. (3) that the average intensity of the light reflected from the medium is determined by the intensity of the incident light and the value of the integral: E(t)E*(t-T)
dt.
(4)
This integral is often termed the self-coherence function.[ 111 From Eqs. (3) and (4) one arrives at a form for the intensity of the reflected radiation which demonstrates that the reflected intensity is directly dependent upon the state of coherence of the incident radiation: I, =I, +I2 +2/xPp’Re[r(?)].
(5)
When the incident electromagnetic field is completely coherent within the time 2T, for example when the field is perfectly monochromatic, Eq. (5) becomes: I, =I, +I2 +2r(r)m?
(6)
where a(t) is a real constant with a value (between - 1 and 1) that is associated with the time delay 7’. In the case of a monochromatic wave a(r)=cos(~,~), where w, is the angular frequency of the wave. If the incident electromagnetic field is not coherent with itself over the time interval 2T. then r(t)-0 since for a completely incoherent field E(f) and E(t -r) are completely uncorrelated. In this case Eq. (5) becomes: I,=I,
Fig. 1. Schematic representation of radiation incident on a dielectric film with index of refraction II and a physical thickness a’.Only one reflection from each surpdce is shown.
(2)
+z,.
(7)
The difference between Eqs. (6) and (7) illustrates that the time-averaged intensity of the light reflected from the medium differs depending on the state of the second-order coherence of the incident radiation. Depending upon the value of X(T): coherent light may experience increased or decreased reflectivity compared with that of incoherent light. Note that the definition of incoherence in this case demands only that there be no
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deterministic relationship between the electric field at times I and t--z. Normally, the value of T necessary such that T(z) - 0 is given by I>> l/As, where AY is the bandwidth of the radiation. In general, the radiation incident on the medium will be neither completely coherent nor completely incoherent. Eq. (5) shows, however? that in all cases, the intensity of the reflection depends upon the state of coherence of the incident radiation. In terms of the individual frequency components of the incident radiation, in the case of coherent radiation described by Eq, (6) certain frequencies will have enhanced reflectivity from the medium due to constructive interference. Therefore, the intensity of the reflected radiation will differ depending upon the frequency of the incident field. In the case described by Eq. (7), there is no mechanism for enhanced reflectivity for certain frequency components and it appears that the reflected radiation will not have a different spectrum than that of the incident radiation, That is, no frequency component of the incident radiation will see enhanced reflectivity. This implies the inability of an optically thick medium to act as an interference titer when used in incoherent light. This is an especially intuitively pleasing analysis when the transmission of thermally generated light is considered since the emission of a thermal source is stochastic in nature. one may easily conclude that any correlation between the radiation at two widely separated times is vanishingly small, Thus, if the transit time of the light through the medium exceeds the correlation time of the source, one expects no interference phenomena on average. On the other hand, one expects that light with a coherence length that is long compared with the film thickness will exhibit strong interference phenomena. The lack of interference phenomena within dielectric films having a thickness exceeding the coherence length of stochastically generated radiation has been frequently cited [l-9]. Indeed, several researchers have proposed using thick optical films to discriminate between coherent laser radiation and incoherent thermal radiation. However, an alternative analysis of the situation described by Fig. 1 may be developed by noting that Eq. ( 1) may be written as: E,(r, ~)=E(r)*{pS(t)+~2p’6(t-r)),
(8)
where the symbol * represents the convolution of the two terms and S represents the Dirac delta function. Performing a Fourier transformation of both terms in Eq. (8) and multiplying them, results in an equation for the reflected field in terms of the frequency components of the incident field: E,(s, ~)=E(ll){p+0’p’exp(i2~zv)j, where 1’ represents a frequency electric field.
component
(9) of the
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The intensity of this field may be found, as before, by I,(v) = ~E,(Y)J~.Therefore, the intensity of the reflected field is given by: Ir(v, z)=IE(v)12{p2+G4p’2+2pG2p’COS(27E~~’)).
(10)
In the same manner in which the time average is taken in the time domain, the average intensity is found in the frequency domain by integrating over all frequencies. Defining the average intensities of the two individual reflections from the interfaces as I, and Z,, we may write the average intensity of the total reflected radiation as: r, =I, +I, +2pdp’
lE(s)12 cos(2m)
dv,
(11)
where the integral is over all frequencies present in the incident radiation. As with the analysis in the time domain, there is a difference in the nature of the reflected intensity for incoherent and coherent radiation. Perfectly coherent radiation is described by a single frequency component 11, and therefore for coherent radiation Eq. (11) becomes:
which is identical to Eq. (6). If the incident field exhibits some aspects of incoherence, the value of the integral in Eq. (11) will begin to approach zero. For completely incoherent radiation, the final term on the right hand side of Eq. (11) is equal to zero. In this case, Eq. (11) reduces to Eq. (7), demonstrating again a correlation between analysis in the tin& and frequency domains. However, Eq. (IO) clearly demonstrates some frequency dependence. So, while the value of the integral in Eq. ( 11) rnay begin to approach zero for incoherent radiation. it is clear from Eq. (10) that interference effects affect the spectrum of the transmitted radiation. This observation is in contrast to the conclusion that is sometimes drawn from the analysis in the time domain. Unfortunately. the above analysis is physically unsound in that it requires the Fourier transform of the function E(t) to exist. The fields from thermal sources fluctuate in a random manner. hence E(r) is a random variable. Under most conditions E(r) is stationary, implying that the probability distribution that characterizes the fluctuations is time independent. This being the case, E(I) does not possess a Fourier transform. Reference [12] contains an excellent discussion of the problems encountered when a.ssociating a Fourier spectrum with a statistically stationary field. On a more intuitive level. if ,the field of a thermal radiator were to possess a Fouriler transform, it would be possible to determine the value of the field at any given time, once the field was known at some previous time. As such, once the field was known at some time
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t,, it would be theoretically possible to determine the field at all times t # t,. This implies a level of complete coherence, albeit complicated, that cannot exist. If this level of coherence did exist in thermally generated fields, the radiation, which is the product of stochastic processes, would be chaotic and not stochastic in nature. Since the analytic signal of thermally generated light is a stochastic variable, it is impossible to take the Fourier transform of E(t): therefore, the simple analysis in the frequency domain presented above is not valid. To circumvent this problem, it is common to define the power spectrum as the Fourier transform of the autocorrelation of the field, where the autocorrelation function is defined by an ensemble average of realizations of the analytic signal of the field, i.e.:
broad bandwidth and the detector can detect a broad bandwidth, then little or no interference effects will be observed. However, they are still present and can be seen with correct spectral filtering at the detector. Unfortunately, this analysis using the power spectrum to determine the correlation function can add confusion to the case being considered here. The confusion is due to the fact that, in the example considered above, the physical process of the interference physically produces the autocorrelation function (Eq. (4)). Therefore, decomposing the incident radiation in terms of the process itself creates the intuitively unappealing case of apparent circular logic. Additionally, it can be shown that the power spectrum can also be written as [ 121:
T(z)=(E”(t)E(T-T)).
S(0) = T-rm lim &
(13)
In this manner, instead of decomposing the incident radiation into sinusoidally oscillating components, one decomposes the autocorrelation function? defined through an ensemble average, into sinusoidally varying components. Having correctly decomposed the field shows, however, that it is indeed possible to have interference phenomena present regardless of the state of coherence of the radiation. One can see this by examining the complex degree of self coherence defined now in terms of ensemble averages as: y(r) = -T(T) =
(14)
The function Y(T) is referred to as the complex degree of temporal coherence and its value is bounded by zero and unity. The magnitude of ~(7) will be equal to the visibility of interference phenomena observed when interfering the two fields E(r) and E(t - 7) [ 111. Now consider that, through the use of a suitably narrow filter placed in front of the detector, we can observe a very narrow portion of the spectrum, such that to a good approximation the power spectrum is given by: S(o)=6(0-0,).
(15)
[
1 >
(17)
where T v(w; T) = L E(t) eiWtdt. 2n l -y-
(18)
Note the importance of the ensemble average, as well as the necessity for the time interval to approach iniinity. Truncation of the time interval leads to an incorrect result and therefore it is difficult to work with the power spectrum in the case being considered here. To overcome the difficulty of not being able to Fourier transform stationary fields, while still attempting to maintain an intuitive hold on the subject matter, one may use a quantum mechanical analysis. From a quantum mechanical perspective, light is coherent and will exhibit time-averaged interference effects if the photons are intrinsically indistinguishable from one another. Note that this means at the point of detection and not at the point where the photons first occupy the same neighborhood (i.e. the dielectric interface in Fig. 1). We denote the distance along the direction of propagation within which photons are physically indistinguishable from one another as AZ. This distance may be defined in terms of the Heisenberg uncertainty relation:
Since the power spectrum is de~led as the Fourier transform of the autocorrelation function. the autocorrelation function of the signal at the detector is given by: m r(T) = S(w) exp(-iwt) do=exp(-io,t). (16) j^0
where I? is Plank’s constant divided by 2~ and ApZ is the uncertainty in the longitudinal component of the photon’s momentum ( pZ = h/i.):
Thus the magnitude of the complex degree of self coherence is unity I?(r)] = l? regardless of the value of T. Since Iy(r)l is equal to the visibility of the interference phenomena, clearly some interference effects will be observed. It is important to note here that if the source has a
where Ai. represents the span of wavelengths in the radiation about the center wavelength 17. Eq. (19) can be rewritten in terms of the bandwidth of the incident
(19)
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radiation: c AZAW > - , 2
(21)
clearly showing the bandwidth dependence in detemlining the ability to see interference effects, regardless of the source of the photons. It is important to emphasize that since the uncertainty principle is a statement of the limitations of the detector, it is the bandwidth of the radiation at the det’tectorthat is of importance and not the bandwidth of the source, unless the bandwidth response of the detector exceeds the bandwidth of the source. In conclusion to this section, we note that although many believe that when the time delay of the radiation exceeds the coherence length, interference phenomena will be absent: this is clearly not the case. Interference phenomena may always be detected if the bandwidth of the detector is limited, regardless of the state of temporal coherence of the source. To demonstrate this is the case, we have performed an experiment proving that one can detect interference phenomena due to broadband radiation in a dielectric slab with a thickness greatly exceeding the coherence length of the incident radiation. In the experiment. light from a high-intensity incandescent lamp was incident upon an uncoated etalon with a thickness of 100 pm. The incident radiation had a spectral width of approximately A/l-25O nm, leading to a coherence length inside the etalon of b 1.5 ,um. The etalon was placed normal to a series of apertures that allowed only light within a narrow divergence angle to pass. The light was passed through a polarizer oriented such that only the s-polarized light passing through the etalon was sampled and then it was directed into a grating spectrometer with a resolution of +0.04 nm, The transmission of the etalon over a portion of the spectrum is shown in Fig. 2. The presence of interference phenomena is clearly observable. The theoretical transmission of a 100 pm thick etalon for perfectly coherent radiation at normal incidence is also plotted in Fig. 2. Note that the theoretical transmission of the etalon varies from T=0.87 to 1. while the experimental transmissivity varies from T=O.89 to 0.99. This slight difference is attributable to the resolution of the spectrometer.
3. Spatial coherence Having addressed the issue of temporal coherence. one may reasonably ask if the spatial incoherence of stochastically generated radiation invalidates the above analysis for many practical applications. There are clearly differences in the interference properties of spatially coherent and spatially incoherent radiation,
theory
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wavelength
(nm)
Fig. 2. Theoretical (dashed) and measured (solid) transmission of a 100 pm thick etalon at normal incidence over a portion of the visibk spectrum.
Indeed, it has been shown that if one observes the intensity distribution or the spectrum of scattered radiation, there is an observable dependence upon the state of spatial coherence [lo? 13-171. However, here we are interested in the difference in the interference phenomena of spatially coherent and spatially incoherent radiation without introducing either diffraction or scattering. In order to investigate the effects-associated with transverse coherence, we introduce the spectral degree of coherence at frequency W: dr,, r2, 0) =
Wrl, r2, (0) [W(r,, rl, w)]“2[JW3, r2, CO)]“~’
(221
where JV(r,v r2, CO)is the cross-spectral density defined in terms of the mutual coherence function by 30 T(r,, r2, r) exp(iwz) dr. FVr,, r2, QJ)= r (23) J-CC
The mutual coherence function is simply the two-point version of the self-coherence function introduced in Eq. (13) above! i.e. : T(r,, r2, T.)=(Err(rl,
f)E(ra, f-z)).
t2.4)
As noted in Ref. [ 101: when spatially incoherent light illuminates two slits separated by a distance that exceeds the transverse coherence length, there will be no timeaveraged interference pattern observed on a screen placed behind the two slits. This lack of interference will be evident even if the light is atered at the slits, such that it has an arbitrarily narrow bandwidth. The physical situation illustrated in Fig. 3 is a thick film of dielectric material tilted at an angle 4 relative to the incident radiation. A ray of light enters the film at point (a), is partially reflected from the back surface and partially reflected again from the front surface at point (b). If the thickness of the dielectric is chosen
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Fig. 3. Schematic representation of radiation incident at an angle 4 on a dielectric film with index of refraction n. Only one reflection from each surface is shown. Interference phenomena can be detected at point 0.
such that the distance between points (a) and (b) in Fig. 3 exceeds the transverse coherence length of incident radiation, then no interference phenomena will be observed by a detector at point (0) when the observation time exceeds the inverse of the bandwidth of the radiation. We can analyze this situation in the same manner as described in Ref. [lo] by separating the spatial and temporal coherence within the complex degree of coherence y. We will separate the two coherence phenomena by assuming that the radiation is filtered through a narrow bandpass filter of center frequency w, prior to being detected. This filtration is important in our analysis in order to eliminate the issue of temporal coherence, which we have already addressed in Section 3. Assuming that the cross-spectral density of the light is a continuous, slowly varying function of Q, the complex degree of coherence may be written as: y(r,, r2, T.)=P(~~, r2, wo)4(Q,
m IT(w)~I exp(-icj)r) 4(9=
O
s0
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Eq. (25) shows that the spectral degree of coherence LL(rl, r7, oO), which is a measure of the transverse coherence of the light, is insensitive to spectral filtering. It also shows that the complex degree of coherence y(rl, r2, T), which is a measure of the light’s ability to produce interference phenomena. can be less than unity (even zero) even when the light has an arbitrarily narrow bandwidth [i.e. t(z) approaches unity]. Therefore, one may think that if the radiation at points (a) and (b) in Fig. 3 are spatially incoherent, an observer at point (0) would not detect any interference effects as long as the sampling time was significantly longer than the inverse of the bandwidth of the light. One is then left with the impression that it is possible to have a lack of interference effects after the light has traversed the tilted film. If, however, one limits the field of view of the detector, then rl and r2 in Eq. (25) become different only in the longitudinal dimension. Since we must assume that the power spectrum of the source is independent of the source point, when the transverse coordinates of the two sources become the same, the issue of spatial coherence is lost and the problem becomes one of temporal coherence. Therefore, in the case of spectrally filtered light considered here, the spectral degree of coherence approaches unity as the field of view of the detector is narrowed. Note that this is true even though the radiation reaching the detector may have originated from many widely separated, independently radiating points on the source. As with the case of temporal coherence. we may also view the spatial coherence in terms of the Heisenberg uncertainty relation. However, in this case it is the transverse component of the photon momentum that must be considered. Defining 23: as the acceptance angle of the detector, the uncertainty in the momentum of a photon reaching the detector in one transverse dimension is given by:
where we have ignored any uncertainty in R. Therefore, the transverse coherence length of the light striking the detector is approximately given by:
(25)
where t(7) is the normalized Fourier transform of the squared modulus of the transmission function of the filter. That is:
j
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IUw)121dw
where T(G) is the transmission function of the filter.
Note that, as expected, limiting the angle of acceptance of the detector increases the transverse coherence length of the light as seen by the detector. This occurs even when much of the radiation reaching the detector has originated from points outside the detector’s view (e.g., that radiation being brought to the detector via multiple reflections within the film). These analyses demonstrate
that interference phenomena will occur in a thick film
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regardless of the state of coherence of the incident radiation. We have experimentally demonstrated the ability to observe interference phenomena in a tilted dielectric using the apparatus described in Section 2. The dielectric used was the same etalon described above, but tilted such that the transverse distance between points (a) and (b) in Fig. 3 exceeded the transverse coherence limit set by the van Cittert-Zernike theorem [IS]. We then determined the presence or absence of interference phenomena by measuring the transmission as a function of wavelength with the spectrometer. In our experiment, the etalon was 100 urn thick and the angle of incidence was 4 = 70.0 2 0.5”. The transverse coherence length of the incident radiation at the etalon as calculated by the van Cittert-Zernike theorem was approximately 0.5 urn. The separation between the incident radiation and the reflected radiation at the interface was approximately 125 urn? and therefore approximately 250 times the transverse coherence length. The acceptance angle of the spectrometer was less than 3 mrad. The transmission of the etalon over a portion of the visible spectrum is shown in Fig. 4. Note the obvious presence of interference phenomena. The theoretical transmissivity for perfectly coherent radiation is also shown in Fig. 4. The difference in absolute transmission between the predicted and the measured value is attributable to the finite acceptance angle and resolution of the spectrometer.
4. Thick optical films for useas interference filters
As is evident from the study detailed in Sections 2 and 3. interference effects will be present regardless of the thickness of the optical medium. This being the case,
theory
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500 Wavelength
502 (nm)
Fig. 4. Theoretical (dashed) and measured (solid) transmission of a 100 pm thick etalon tilted at 70” relative to the center of the source of radiation over a portion of the visible spectrum.
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one can use very thick optical films to filter very narrow wavelength regions, contrary to what is often stated in the literature. In this section, we will discuss the design and manufacture of thick-film filters that pass a large portion of the visible spectrum while reflecting several narrow wavelength regions. This type of jilter may be useful in providing protection from laser radiation from multiple sources for eyes and sensors and can be especially useful for this application when the filter is designed to appear colorless to the eye. To our knowledge, this has not been achieved successfully using standard thinfilm technology. III this section, we refer to a thick-film filter as one in which a single layer within the film stack has an optical thickness that exceeds a few wavelengths of visible light. One may think that increasing the thickness of multilayer dielectric films would also decrease their usefulness for two reasons. First, because increasing the thickness of the individual films increases the number of transmission minima across any given spectral region, one niBy conclude that increasing the thickness of the individual films would decrease the overall transmissivity. In addition, as noted above, it is thought by many that when the thickness of the film exceeds the coherence length of the incident light, the effects attributable to interference between the film layers will be diminished. In the previous section we have shown that interference effects are present regardless of the film thickness. We will demonstrate here that, under many circumstances, thickfilm filters can be designed that have very high overall transmissivity across the visible spectrum while having significant reflectivity within many narrow wavelength regions. There are two challenges in making thick-film optical filters. First, the ability to manufacture film stacks that are transparent in the visible portion of the spectrum, where each layer within the stack is many wavelengths thick, has to our knowledge, not been successfully demonstrated. The internal stresses produced by coating such thick layers usually result in the cracking of the layers themselves and the separation of contiguous layers. The second problem in making thick-film filterscomes in devising a design methodology. Design methods for thin films are the subject of numerous articles and books and there are many commercially available software programs to assist the thin-film designer. However, there are special considerations when designing filters using thick films and we have found most available thin-film design assistance to be of little value. Overcoming the challenge of manufacturing thick, optical films was accomplished at Guernsey Coating Laboratories, CA, using ion-beam deposition to bond the individual layers to the substrate and to each other.We have successfully coated several extremely thick layers of SiOz and TiO, to a fused-silica substrate using
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ion-beam deposition and selective doping of the TiO,. To date, we have made several thick-film filters, with the thickest being made of nine alternating layers of SiO, and Ti02, each 75 quarter-wave optical thickness (QWOT), using 514 run as a reference wavelength. To our knowledge, these are the thickest single layers ever manufactured that are transparent in the visible portion of the spectrum. It has been thought by many that very thick filters would be fragile and would not withstand normal use even if the technology could be developed to fabricate them. However, the environmental stability of our filters was tested by the Technical Application Branch of the US Army Soldier Systems Command, Natick Research, Development and Engineering Center. The filters showed no detrimental effects when subjected to the standard environmental tests required for military applications. The challenge of designing a filter with several narrow transmission minima was overcome by developing a design methodology that assumes a priori the necessity of using thick optical films. By recognizing that thick optical films provide high overall transmission over very broad wavelength regions, appear almost colorless, and have a significant number of transmission minima: we begin the design process by demanding that only thick optical films be used in the filter stack. The results of our design efforts have been a series of designs that all have transmission minima at multiple laser wavelengths while maintaining a high overall transmission when integrated across the visible portion of the spectrum. As a proof-of-principal exercise, we have manufactured a filter designed to transmit less than 1% of the incident radiation at 514.5 nm (Ar’), 532 nm (doubled Nd:YAG), 694.3 nm (ruby) and 1064 nm (Nd:YAG), while transmitting approximately 55% of the total radiation across the visible spectrum. Additionally, white light transmitted through the filter appears almost white to the eye. The actual transmission of the filter across the visible spectrum is shown in Fig. 5. This filter is composed of 19 layers of alternating SiO, and TiO,, each with a thickness between 15 and 16 QWOT of 500 nm light. Since the manufacturing process is still in the experimental stage, there were some slight errors in the uniformity of the individual film layers. These errors caused a small shift in the transmission minima away from being centered on the desired wavelengths and an increase in the value of the transmission minima. However, the measured total transmission (integrated across the visible spectrum) of 0.505 i 0.003, combined with a transmission between 0.02 to 0.05 for the three minima within the visible portion of the spectrum, clearly demonstrates the feasibility of manufacturing such filters. As expected, the filter is almost colorless to the eye when white light is viewed in transmission.
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E
‘; 0.6 .E g 0.4 c 0.2 -
400
500 Wavelength
600 (nm)
700
Fig. 5. Actual transmission of a thick-film interference filter designed to transmit greater than 50% of the visible spectrum while reflecting light at four common laser wavelengths. White light appears white when viewed through the filter.
The above-described filter demonstrates the possibility of manufacturing a thick-film filter; however, for practical uses it is usually desirable for the transmission at the minima to be approximately OD3 or less. We have shown that the design of such a filter is clearly possible. For example, Fig. 6 shows the predicted transmission spectrum for a thick-film filter designed to satisfy the requirements for eye and sensor protection. This design uses films with thicknesses up to 1.5 urn. The overall power transmission exceeds 0.60 and white light transmitted through the filter would appear white to the
100 80 s ..-ii E s
60 40
c 20
600 700 Wavelength (nm)
800
Fig. 6. Predicted transmission spectrum of a thick-film filter designed to reflect light at five laser wavelengths and transmit greater than 60% of the light in the visible portion of the spectrum. White light will appear white when viewed through the filter.
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eye ([.338,.321] on the CIE color chart). The optical density at five common laser wavelengths (488, 514.5, 633, 694 and 1064 nm) is greater than OD4. The acceptance angles of the filter for each wavelength, defined as the angle where the optical density exceeds 002, ranges from 13 to 30”. Clearly a filter made from this design would be effective for protection of a sensor needing access to the visible portion of the spectrum.
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Acknowledgement
The authors wish to thank E. Wolf and M. Kowar? for many helpful discussions. This work was supported by the US Army Ballistic Missile Defense Office, the Us Army Natick Research: Development and Engineering Center and the US Army Research Office.
References 5. Conclusions
In the previous sections, we have shown that. contrary to what many believe, the transmission characteristics of a dielectric film are independent of the state of coherence of the incident radiation. Passing radiation through dielectric films, be they thick or thin, changes the spectrum of the radiation, regardless of the source. A similar conclusion holds when one is considering the spatial coherence of radiation propagating through dielectric films. We have also shown that it is possible to design and manufacture a filter that has a minimum transmission within several narrow wavelength regions while transmitting a majority of the light within the visible spectrum. These filters are environmentally stable and do not significantly alter the perceived color of the transmitted radiation. These filters can be used to protect eyes and sensors while providing an almost true color rendition of the environment and minimal reduction in irradiance.
[I] P. Baumeister, Optical Coating Technology, Sholem Press, Sebastopol, CA, 1994. [2] C.G. Granqvist, Spectrally Selective Surfaces for Heating and Cooling Applications, SPIE Press, Bellingham, WA, 1989. [3] J.D. Rancourt, Optical Thin Films, McGraw-Hill. New York, 1987.
[5] [6] [7] [Sj
M. Young, Optics and Lasers, Springer, Berlin, 1986. Z. Knittl, Optics of Thin Films, Wiley, New York, 1976. P. Baumeister. R. Hahn, D. Harrison, Opt. Acta 14 (1972) 853. C.J. Gabriel, A. Nedoluha, Opt. Acta 18 (1971) 415, J.M. Stone, Radiation and Optics, McGraw-Hill, New York,
[9j
A. Vdsicek, Optics of Thin Films, North-Holland,
[4]
1963.
Amsterdam,
1960.
[lo] E. Wolf, Opt. Lett. 8 (1983) 250. [ 111 L. Mandel, E. Wolf, Optical Coherence and Quantum Optics, Cambridge, New York, 1995. 1121 E. Wolf, J. Opt. Sot. Am. 72 ( 1982) 343. [13] W. Wang, E. Wolf, Opt. Comm. 79 (1990) 131. [ 141 A.T. Friberg, E. Wolf, Opt. Lett. 20 (1995) 623. [ 151 W. Wang, R. Simon, E. Wolf, J. Opt. Sot. Am. A9 (1992) 287, [ 161 F. Gori, C. Palma, M. Santarsiero, Opt. Cotimun. 74 ( 1990) 353. [ 171 J. Jannson, T. Jannson, E. Wolf, Opt. Lett. 13 (1988) 106K [ 181 J. Goodman, Statistical Optics, Wiley, Nebv-York, 1985.
Thick optical films for use as interference filters Thomas R. Moore ‘+*, Jonathan B. Redmond a7Steven J. Frederiksen a, George R. Gray b, Ming W. Pan b, Peter Guernsey ’ a Deppnrtwzent of Physics, Phoforzics Research Cmter’, United States Military Acndenzy, West Point, NY 10996, USA b Department of Electrical Engineering, Uuivemit)! of Uth, Salt Lake City, UT 44112, I;ISA ’ Gmmsey Coating Laboratories, Ventwa, CA 93003, USA
Received11October 1996;accepted10June1997
Abstract
The design:manufactureand useof optically transparentfilms with thicknesses exceedingthe coherencelength of the incident light is discussed.An analysisof the reflection and transmissionof spatially and temporally incoherentlight by transparentthick filmsis presentedusingboth classicaland quantummechanicalarguments.The analysisdemonstratesthat, contrary to frequently statedassertions,films with thicknesses greatly exceedingthe coherencelengthof the incident light can be usedto constructoptical interferencefilters. The resultsof experimentsare presentedwhich confirm this conclusion.Finally, a discussionof the dengn, constructionand useof interferencefilters madefrom thick optical films is presented.Designsspecificallysuitedto eyeand sensor protection from laserradiation are emphasized.The manufactureof a prototype thick-film optical filter. usefulfor eyeand sensor protection againstthe radiation from severallasers,is reported. 0 1998ElsevierScienceS.A. K~~IV~IZ’KThick films; Sensorprotection; Narrow-band filters; Coherence
1. Introduction It is common to use thin optical films as interference
filters in the visible portion of the spectrum for various applications. However, optical films with thicknesses exceeding an optical wavelength are not commonly used. The lack of use of thick optical films appears to be due to two factors: the difficulty in manufacturing thick films that are transparent in the visible portion of the spectrum, and the often-quoted fact that interference phenomena will not be present when the film thickness exceeds the coherence length of the incident radiation [l-9]. The latter appears to have stifled research into manufacturing methods capable of overcoming the former. In the following sections? we analyze the propagation characteristics of radiation through dielectric films under conditions where the thickness of the dielectric exceeds the coherence length of the radiation. We show that
interference phenomena associated with radiation passing through a thick film is independent of the state of coherence of that radiation. Additionally, we show that * Corresponding author. 0X7-8972/98/$19.00 0 1998ElsevierScience B.V. ALIri&ts reserved. PII so257-8972(97)00414-3
spatial incoherence is not a barrier to using thick optical iilms as interference filters. This latter result is quite surprising since it has been shown that it is possible to discriminate between spatially coherent and spatially incoherent light of arbitrarily narrow bandwidth by observing the interference between the light diffracted by two separated slits [IO]. Finally, we report the design, manufacture and testing of thick optical filters for use as multiple-line interference filters. These titers can block up to five narrow spectral regions, pass over 60% of visible light and still allow white light to be perceived as white to the eye. We propose that this type of filter be used as protection for eyes and sensors against damaging laser radiation, In what follows, the temporal and spatial coherence properties of electromagnetic radiation will be considered separately. While it is well known that these two properties are intimately related [ 111, the relation&ip may often be suppressed in the interest of clarity and we will do so here. 2. Temporal coherence
Consider light with intensity 1, normally incident upon an optically transparent medium of thickness d, as
61
shown in Fig. 1. If the thickness of the medium greatly exceeds the coherence length of the incident radiation, the transmitted intensity, Zr, may be calculated by subtracting the sum of the intensities of the reflections from each interface from the incident intensity. IT = Z,- (Z,, + ZR,). This is because there is no correlation between the electromagnetic field entering the medium and the electromagnetic field that has been reflected from the second interface, and it indicates the inability to use films with thicknesses exceeding the coherence length of the light as interference filters. If. however, the coherence length of the incident radiation exceeds twice the thickness of the medium, the amplitudes of the fields must be summed to determine the transmissivity and interference phenomena clearly will occur. In order to investigate the system shown in Fig. l> we will consider only a single reflection from each of the surfaces of the dielectric. This simplification is useful to explore the physics of the problem and the results are easily extended to the real case of an infinite number of reflections. Assuming normally incident radiation, neglecting dispersion and assuming linear polarization, after the light has had time to completely traverse the medium twice, the reflected electromagnetic field is the sum of the amplitudes of the reflection from the first interface and the reflection from the second interface. Therefore. the electric field component of the refIected radiation from the surface is given by: EJt, T) = pE(t) + a2p’E(r - T),
(1) where E(t) is the analytic signal representation of the electric field incident upon the medium at time t. p is the amplitude reAectivity of the first interface, 0 is the amplitude transmissivity of the first interface. p’ is the amplitude reflectivity of the second interface and r is the time taken for the light to traverse the medium twice: r =2&c, where tl is the thickness of the medium. II is the index of refraction and c is the speed of light in vacuum. The intensity of the reflected light is given to within
a constant by Z,= lE,12 and therefore:
+:po’p’E(t)E*(t-T)+c.c.).
where C.C. refers to the complex conjugate. The average intensity of the light reflected from the medium 7, can be found by integrating over a time period long compared with the fluctuations in the electric field, and therefore:
Ir=r,+z2+-
pdp 2T
T is -.,-
E(r)E’(t-r)
dt-+c.c.
,
(3)
where I, and Zz are the time-averaged values of the intensities reflected from the two interfaces by themselves, p21E(t)12and a”p”lE(t - r)12 respectively and T is long compared with the time scale of the fluctuations in the electric field. It is evident from Eq. (3) that the average intensity of the light reflected from the medium is determined by the intensity of the incident light and the value of the integral: E(t)E*(t-T)
dt.
(4)
This integral is often termed the self-coherence function.[ 111 From Eqs. (3) and (4) one arrives at a form for the intensity of the reflected radiation which demonstrates that the reflected intensity is directly dependent upon the state of coherence of the incident radiation: I, =I, +I2 +2/xPp’Re[r(?)].
(5)
When the incident electromagnetic field is completely coherent within the time 2T, for example when the field is perfectly monochromatic, Eq. (5) becomes: I, =I, +I2 +2r(r)m?
(6)
where a(t) is a real constant with a value (between - 1 and 1) that is associated with the time delay 7’. In the case of a monochromatic wave a(r)=cos(~,~), where w, is the angular frequency of the wave. If the incident electromagnetic field is not coherent with itself over the time interval 2T. then r(t)-0 since for a completely incoherent field E(f) and E(t -r) are completely uncorrelated. In this case Eq. (5) becomes: I,=I,
Fig. 1. Schematic representation of radiation incident on a dielectric film with index of refraction II and a physical thickness a’.Only one reflection from each surpdce is shown.
(2)
+z,.
(7)
The difference between Eqs. (6) and (7) illustrates that the time-averaged intensity of the light reflected from the medium differs depending on the state of the second-order coherence of the incident radiation. Depending upon the value of X(T): coherent light may experience increased or decreased reflectivity compared with that of incoherent light. Note that the definition of incoherence in this case demands only that there be no
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deterministic relationship between the electric field at times I and t--z. Normally, the value of T necessary such that T(z) - 0 is given by I>> l/As, where AY is the bandwidth of the radiation. In general, the radiation incident on the medium will be neither completely coherent nor completely incoherent. Eq. (5) shows, however? that in all cases, the intensity of the reflection depends upon the state of coherence of the incident radiation. In terms of the individual frequency components of the incident radiation, in the case of coherent radiation described by Eq, (6) certain frequencies will have enhanced reflectivity from the medium due to constructive interference. Therefore, the intensity of the reflected radiation will differ depending upon the frequency of the incident field. In the case described by Eq. (7), there is no mechanism for enhanced reflectivity for certain frequency components and it appears that the reflected radiation will not have a different spectrum than that of the incident radiation, That is, no frequency component of the incident radiation will see enhanced reflectivity. This implies the inability of an optically thick medium to act as an interference titer when used in incoherent light. This is an especially intuitively pleasing analysis when the transmission of thermally generated light is considered since the emission of a thermal source is stochastic in nature. one may easily conclude that any correlation between the radiation at two widely separated times is vanishingly small, Thus, if the transit time of the light through the medium exceeds the correlation time of the source, one expects no interference phenomena on average. On the other hand, one expects that light with a coherence length that is long compared with the film thickness will exhibit strong interference phenomena. The lack of interference phenomena within dielectric films having a thickness exceeding the coherence length of stochastically generated radiation has been frequently cited [l-9]. Indeed, several researchers have proposed using thick optical films to discriminate between coherent laser radiation and incoherent thermal radiation. However, an alternative analysis of the situation described by Fig. 1 may be developed by noting that Eq. ( 1) may be written as: E,(r, ~)=E(r)*{pS(t)+~2p’6(t-r)),
(8)
where the symbol * represents the convolution of the two terms and S represents the Dirac delta function. Performing a Fourier transformation of both terms in Eq. (8) and multiplying them, results in an equation for the reflected field in terms of the frequency components of the incident field: E,(s, ~)=E(ll){p+0’p’exp(i2~zv)j, where 1’ represents a frequency electric field.
component
(9) of the
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The intensity of this field may be found, as before, by I,(v) = ~E,(Y)J~.Therefore, the intensity of the reflected field is given by: Ir(v, z)=IE(v)12{p2+G4p’2+2pG2p’COS(27E~~’)).
(10)
In the same manner in which the time average is taken in the time domain, the average intensity is found in the frequency domain by integrating over all frequencies. Defining the average intensities of the two individual reflections from the interfaces as I, and Z,, we may write the average intensity of the total reflected radiation as: r, =I, +I, +2pdp’
lE(s)12 cos(2m)
dv,
(11)
where the integral is over all frequencies present in the incident radiation. As with the analysis in the time domain, there is a difference in the nature of the reflected intensity for incoherent and coherent radiation. Perfectly coherent radiation is described by a single frequency component 11, and therefore for coherent radiation Eq. (11) becomes:
which is identical to Eq. (6). If the incident field exhibits some aspects of incoherence, the value of the integral in Eq. (11) will begin to approach zero. For completely incoherent radiation, the final term on the right hand side of Eq. (11) is equal to zero. In this case, Eq. (11) reduces to Eq. (7), demonstrating again a correlation between analysis in the tin& and frequency domains. However, Eq. (IO) clearly demonstrates some frequency dependence. So, while the value of the integral in Eq. ( 11) rnay begin to approach zero for incoherent radiation. it is clear from Eq. (10) that interference effects affect the spectrum of the transmitted radiation. This observation is in contrast to the conclusion that is sometimes drawn from the analysis in the time domain. Unfortunately. the above analysis is physically unsound in that it requires the Fourier transform of the function E(t) to exist. The fields from thermal sources fluctuate in a random manner. hence E(r) is a random variable. Under most conditions E(r) is stationary, implying that the probability distribution that characterizes the fluctuations is time independent. This being the case, E(I) does not possess a Fourier transform. Reference [12] contains an excellent discussion of the problems encountered when a.ssociating a Fourier spectrum with a statistically stationary field. On a more intuitive level. if ,the field of a thermal radiator were to possess a Fouriler transform, it would be possible to determine the value of the field at any given time, once the field was known at some previous time. As such, once the field was known at some time
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63
t,, it would be theoretically possible to determine the field at all times t # t,. This implies a level of complete coherence, albeit complicated, that cannot exist. If this level of coherence did exist in thermally generated fields, the radiation, which is the product of stochastic processes, would be chaotic and not stochastic in nature. Since the analytic signal of thermally generated light is a stochastic variable, it is impossible to take the Fourier transform of E(t): therefore, the simple analysis in the frequency domain presented above is not valid. To circumvent this problem, it is common to define the power spectrum as the Fourier transform of the autocorrelation of the field, where the autocorrelation function is defined by an ensemble average of realizations of the analytic signal of the field, i.e.:
broad bandwidth and the detector can detect a broad bandwidth, then little or no interference effects will be observed. However, they are still present and can be seen with correct spectral filtering at the detector. Unfortunately, this analysis using the power spectrum to determine the correlation function can add confusion to the case being considered here. The confusion is due to the fact that, in the example considered above, the physical process of the interference physically produces the autocorrelation function (Eq. (4)). Therefore, decomposing the incident radiation in terms of the process itself creates the intuitively unappealing case of apparent circular logic. Additionally, it can be shown that the power spectrum can also be written as [ 121:
T(z)=(E”(t)E(T-T)).
S(0) = T-rm lim &
(13)
In this manner, instead of decomposing the incident radiation into sinusoidally oscillating components, one decomposes the autocorrelation function? defined through an ensemble average, into sinusoidally varying components. Having correctly decomposed the field shows, however, that it is indeed possible to have interference phenomena present regardless of the state of coherence of the radiation. One can see this by examining the complex degree of self coherence defined now in terms of ensemble averages as: y(r) = -T(T) =
(14)
The function Y(T) is referred to as the complex degree of temporal coherence and its value is bounded by zero and unity. The magnitude of ~(7) will be equal to the visibility of interference phenomena observed when interfering the two fields E(r) and E(t - 7) [ 111. Now consider that, through the use of a suitably narrow filter placed in front of the detector, we can observe a very narrow portion of the spectrum, such that to a good approximation the power spectrum is given by: S(o)=6(0-0,).
(15)
[
1 >
(17)
where T v(w; T) = L E(t) eiWtdt. 2n l -y-
(18)
Note the importance of the ensemble average, as well as the necessity for the time interval to approach iniinity. Truncation of the time interval leads to an incorrect result and therefore it is difficult to work with the power spectrum in the case being considered here. To overcome the difficulty of not being able to Fourier transform stationary fields, while still attempting to maintain an intuitive hold on the subject matter, one may use a quantum mechanical analysis. From a quantum mechanical perspective, light is coherent and will exhibit time-averaged interference effects if the photons are intrinsically indistinguishable from one another. Note that this means at the point of detection and not at the point where the photons first occupy the same neighborhood (i.e. the dielectric interface in Fig. 1). We denote the distance along the direction of propagation within which photons are physically indistinguishable from one another as AZ. This distance may be defined in terms of the Heisenberg uncertainty relation:
Since the power spectrum is de~led as the Fourier transform of the autocorrelation function. the autocorrelation function of the signal at the detector is given by: m r(T) = S(w) exp(-iwt) do=exp(-io,t). (16) j^0
where I? is Plank’s constant divided by 2~ and ApZ is the uncertainty in the longitudinal component of the photon’s momentum ( pZ = h/i.):
Thus the magnitude of the complex degree of self coherence is unity I?(r)] = l? regardless of the value of T. Since Iy(r)l is equal to the visibility of the interference phenomena, clearly some interference effects will be observed. It is important to note here that if the source has a
where Ai. represents the span of wavelengths in the radiation about the center wavelength 17. Eq. (19) can be rewritten in terms of the bandwidth of the incident
(19)
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radiation: c AZAW > - , 2
(21)
clearly showing the bandwidth dependence in detemlining the ability to see interference effects, regardless of the source of the photons. It is important to emphasize that since the uncertainty principle is a statement of the limitations of the detector, it is the bandwidth of the radiation at the det’tectorthat is of importance and not the bandwidth of the source, unless the bandwidth response of the detector exceeds the bandwidth of the source. In conclusion to this section, we note that although many believe that when the time delay of the radiation exceeds the coherence length, interference phenomena will be absent: this is clearly not the case. Interference phenomena may always be detected if the bandwidth of the detector is limited, regardless of the state of temporal coherence of the source. To demonstrate this is the case, we have performed an experiment proving that one can detect interference phenomena due to broadband radiation in a dielectric slab with a thickness greatly exceeding the coherence length of the incident radiation. In the experiment. light from a high-intensity incandescent lamp was incident upon an uncoated etalon with a thickness of 100 pm. The incident radiation had a spectral width of approximately A/l-25O nm, leading to a coherence length inside the etalon of b 1.5 ,um. The etalon was placed normal to a series of apertures that allowed only light within a narrow divergence angle to pass. The light was passed through a polarizer oriented such that only the s-polarized light passing through the etalon was sampled and then it was directed into a grating spectrometer with a resolution of +0.04 nm, The transmission of the etalon over a portion of the spectrum is shown in Fig. 2. The presence of interference phenomena is clearly observable. The theoretical transmission of a 100 pm thick etalon for perfectly coherent radiation at normal incidence is also plotted in Fig. 2. Note that the theoretical transmission of the etalon varies from T=0.87 to 1. while the experimental transmissivity varies from T=O.89 to 0.99. This slight difference is attributable to the resolution of the spectrometer.
3. Spatial coherence Having addressed the issue of temporal coherence. one may reasonably ask if the spatial incoherence of stochastically generated radiation invalidates the above analysis for many practical applications. There are clearly differences in the interference properties of spatially coherent and spatially incoherent radiation,
theory
’
498
502
500
wavelength
(nm)
Fig. 2. Theoretical (dashed) and measured (solid) transmission of a 100 pm thick etalon at normal incidence over a portion of the visibk spectrum.
Indeed, it has been shown that if one observes the intensity distribution or the spectrum of scattered radiation, there is an observable dependence upon the state of spatial coherence [lo? 13-171. However, here we are interested in the difference in the interference phenomena of spatially coherent and spatially incoherent radiation without introducing either diffraction or scattering. In order to investigate the effects-associated with transverse coherence, we introduce the spectral degree of coherence at frequency W: dr,, r2, 0) =
Wrl, r2, (0) [W(r,, rl, w)]“2[JW3, r2, CO)]“~’
(221
where JV(r,v r2, CO)is the cross-spectral density defined in terms of the mutual coherence function by 30 T(r,, r2, r) exp(iwz) dr. FVr,, r2, QJ)= r (23) J-CC
The mutual coherence function is simply the two-point version of the self-coherence function introduced in Eq. (13) above! i.e. : T(r,, r2, T.)=(Err(rl,
f)E(ra, f-z)).
t2.4)
As noted in Ref. [ 101: when spatially incoherent light illuminates two slits separated by a distance that exceeds the transverse coherence length, there will be no timeaveraged interference pattern observed on a screen placed behind the two slits. This lack of interference will be evident even if the light is atered at the slits, such that it has an arbitrarily narrow bandwidth. The physical situation illustrated in Fig. 3 is a thick film of dielectric material tilted at an angle 4 relative to the incident radiation. A ray of light enters the film at point (a), is partially reflected from the back surface and partially reflected again from the front surface at point (b). If the thickness of the dielectric is chosen
T. R. Alowe
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Fig. 3. Schematic representation of radiation incident at an angle 4 on a dielectric film with index of refraction n. Only one reflection from each surface is shown. Interference phenomena can be detected at point 0.
such that the distance between points (a) and (b) in Fig. 3 exceeds the transverse coherence length of incident radiation, then no interference phenomena will be observed by a detector at point (0) when the observation time exceeds the inverse of the bandwidth of the radiation. We can analyze this situation in the same manner as described in Ref. [lo] by separating the spatial and temporal coherence within the complex degree of coherence y. We will separate the two coherence phenomena by assuming that the radiation is filtered through a narrow bandpass filter of center frequency w, prior to being detected. This filtration is important in our analysis in order to eliminate the issue of temporal coherence, which we have already addressed in Section 3. Assuming that the cross-spectral density of the light is a continuous, slowly varying function of Q, the complex degree of coherence may be written as: y(r,, r2, T.)=P(~~, r2, wo)4(Q,
m IT(w)~I exp(-icj)r) 4(9=
O
s0
do, >
o3
60-68
65
Eq. (25) shows that the spectral degree of coherence LL(rl, r7, oO), which is a measure of the transverse coherence of the light, is insensitive to spectral filtering. It also shows that the complex degree of coherence y(rl, r2, T), which is a measure of the light’s ability to produce interference phenomena. can be less than unity (even zero) even when the light has an arbitrarily narrow bandwidth [i.e. t(z) approaches unity]. Therefore, one may think that if the radiation at points (a) and (b) in Fig. 3 are spatially incoherent, an observer at point (0) would not detect any interference effects as long as the sampling time was significantly longer than the inverse of the bandwidth of the light. One is then left with the impression that it is possible to have a lack of interference effects after the light has traversed the tilted film. If, however, one limits the field of view of the detector, then rl and r2 in Eq. (25) become different only in the longitudinal dimension. Since we must assume that the power spectrum of the source is independent of the source point, when the transverse coordinates of the two sources become the same, the issue of spatial coherence is lost and the problem becomes one of temporal coherence. Therefore, in the case of spectrally filtered light considered here, the spectral degree of coherence approaches unity as the field of view of the detector is narrowed. Note that this is true even though the radiation reaching the detector may have originated from many widely separated, independently radiating points on the source. As with the case of temporal coherence. we may also view the spatial coherence in terms of the Heisenberg uncertainty relation. However, in this case it is the transverse component of the photon momentum that must be considered. Defining 23: as the acceptance angle of the detector, the uncertainty in the momentum of a photon reaching the detector in one transverse dimension is given by:
where we have ignored any uncertainty in R. Therefore, the transverse coherence length of the light striking the detector is approximately given by:
(25)
where t(7) is the normalized Fourier transform of the squared modulus of the transmission function of the filter. That is:
j
Technolog~~ 99 (1998)
(26)
IUw)121dw
where T(G) is the transmission function of the filter.
Note that, as expected, limiting the angle of acceptance of the detector increases the transverse coherence length of the light as seen by the detector. This occurs even when much of the radiation reaching the detector has originated from points outside the detector’s view (e.g., that radiation being brought to the detector via multiple reflections within the film). These analyses demonstrate
that interference phenomena will occur in a thick film
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regardless of the state of coherence of the incident radiation. We have experimentally demonstrated the ability to observe interference phenomena in a tilted dielectric using the apparatus described in Section 2. The dielectric used was the same etalon described above, but tilted such that the transverse distance between points (a) and (b) in Fig. 3 exceeded the transverse coherence limit set by the van Cittert-Zernike theorem [IS]. We then determined the presence or absence of interference phenomena by measuring the transmission as a function of wavelength with the spectrometer. In our experiment, the etalon was 100 urn thick and the angle of incidence was 4 = 70.0 2 0.5”. The transverse coherence length of the incident radiation at the etalon as calculated by the van Cittert-Zernike theorem was approximately 0.5 urn. The separation between the incident radiation and the reflected radiation at the interface was approximately 125 urn? and therefore approximately 250 times the transverse coherence length. The acceptance angle of the spectrometer was less than 3 mrad. The transmission of the etalon over a portion of the visible spectrum is shown in Fig. 4. Note the obvious presence of interference phenomena. The theoretical transmissivity for perfectly coherent radiation is also shown in Fig. 4. The difference in absolute transmission between the predicted and the measured value is attributable to the finite acceptance angle and resolution of the spectrometer.
4. Thick optical films for useas interference filters
As is evident from the study detailed in Sections 2 and 3. interference effects will be present regardless of the thickness of the optical medium. This being the case,
theory
/
498
500 Wavelength
502 (nm)
Fig. 4. Theoretical (dashed) and measured (solid) transmission of a 100 pm thick etalon tilted at 70” relative to the center of the source of radiation over a portion of the visible spectrum.
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one can use very thick optical films to filter very narrow wavelength regions, contrary to what is often stated in the literature. In this section, we will discuss the design and manufacture of thick-film filters that pass a large portion of the visible spectrum while reflecting several narrow wavelength regions. This type of jilter may be useful in providing protection from laser radiation from multiple sources for eyes and sensors and can be especially useful for this application when the filter is designed to appear colorless to the eye. To our knowledge, this has not been achieved successfully using standard thinfilm technology. III this section, we refer to a thick-film filter as one in which a single layer within the film stack has an optical thickness that exceeds a few wavelengths of visible light. One may think that increasing the thickness of multilayer dielectric films would also decrease their usefulness for two reasons. First, because increasing the thickness of the individual films increases the number of transmission minima across any given spectral region, one niBy conclude that increasing the thickness of the individual films would decrease the overall transmissivity. In addition, as noted above, it is thought by many that when the thickness of the film exceeds the coherence length of the incident light, the effects attributable to interference between the film layers will be diminished. In the previous section we have shown that interference effects are present regardless of the film thickness. We will demonstrate here that, under many circumstances, thickfilm filters can be designed that have very high overall transmissivity across the visible spectrum while having significant reflectivity within many narrow wavelength regions. There are two challenges in making thick-film optical filters. First, the ability to manufacture film stacks that are transparent in the visible portion of the spectrum, where each layer within the stack is many wavelengths thick, has to our knowledge, not been successfully demonstrated. The internal stresses produced by coating such thick layers usually result in the cracking of the layers themselves and the separation of contiguous layers. The second problem in making thick-film filterscomes in devising a design methodology. Design methods for thin films are the subject of numerous articles and books and there are many commercially available software programs to assist the thin-film designer. However, there are special considerations when designing filters using thick films and we have found most available thin-film design assistance to be of little value. Overcoming the challenge of manufacturing thick, optical films was accomplished at Guernsey Coating Laboratories, CA, using ion-beam deposition to bond the individual layers to the substrate and to each other.We have successfully coated several extremely thick layers of SiOz and TiO, to a fused-silica substrate using
T. R. Moore
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ion-beam deposition and selective doping of the TiO,. To date, we have made several thick-film filters, with the thickest being made of nine alternating layers of SiO, and Ti02, each 75 quarter-wave optical thickness (QWOT), using 514 run as a reference wavelength. To our knowledge, these are the thickest single layers ever manufactured that are transparent in the visible portion of the spectrum. It has been thought by many that very thick filters would be fragile and would not withstand normal use even if the technology could be developed to fabricate them. However, the environmental stability of our filters was tested by the Technical Application Branch of the US Army Soldier Systems Command, Natick Research, Development and Engineering Center. The filters showed no detrimental effects when subjected to the standard environmental tests required for military applications. The challenge of designing a filter with several narrow transmission minima was overcome by developing a design methodology that assumes a priori the necessity of using thick optical films. By recognizing that thick optical films provide high overall transmission over very broad wavelength regions, appear almost colorless, and have a significant number of transmission minima: we begin the design process by demanding that only thick optical films be used in the filter stack. The results of our design efforts have been a series of designs that all have transmission minima at multiple laser wavelengths while maintaining a high overall transmission when integrated across the visible portion of the spectrum. As a proof-of-principal exercise, we have manufactured a filter designed to transmit less than 1% of the incident radiation at 514.5 nm (Ar’), 532 nm (doubled Nd:YAG), 694.3 nm (ruby) and 1064 nm (Nd:YAG), while transmitting approximately 55% of the total radiation across the visible spectrum. Additionally, white light transmitted through the filter appears almost white to the eye. The actual transmission of the filter across the visible spectrum is shown in Fig. 5. This filter is composed of 19 layers of alternating SiO, and TiO,, each with a thickness between 15 and 16 QWOT of 500 nm light. Since the manufacturing process is still in the experimental stage, there were some slight errors in the uniformity of the individual film layers. These errors caused a small shift in the transmission minima away from being centered on the desired wavelengths and an increase in the value of the transmission minima. However, the measured total transmission (integrated across the visible spectrum) of 0.505 i 0.003, combined with a transmission between 0.02 to 0.05 for the three minima within the visible portion of the spectrum, clearly demonstrates the feasibility of manufacturing such filters. As expected, the filter is almost colorless to the eye when white light is viewed in transmission.
Technology
99 (1998)
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67
E
‘; 0.6 .E g 0.4 c 0.2 -
400
500 Wavelength
600 (nm)
700
Fig. 5. Actual transmission of a thick-film interference filter designed to transmit greater than 50% of the visible spectrum while reflecting light at four common laser wavelengths. White light appears white when viewed through the filter.
The above-described filter demonstrates the possibility of manufacturing a thick-film filter; however, for practical uses it is usually desirable for the transmission at the minima to be approximately OD3 or less. We have shown that the design of such a filter is clearly possible. For example, Fig. 6 shows the predicted transmission spectrum for a thick-film filter designed to satisfy the requirements for eye and sensor protection. This design uses films with thicknesses up to 1.5 urn. The overall power transmission exceeds 0.60 and white light transmitted through the filter would appear white to the
100 80 s ..-ii E s
60 40
c 20
600 700 Wavelength (nm)
800
Fig. 6. Predicted transmission spectrum of a thick-film filter designed to reflect light at five laser wavelengths and transmit greater than 60% of the light in the visible portion of the spectrum. White light will appear white when viewed through the filter.
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eye ([.338,.321] on the CIE color chart). The optical density at five common laser wavelengths (488, 514.5, 633, 694 and 1064 nm) is greater than OD4. The acceptance angles of the filter for each wavelength, defined as the angle where the optical density exceeds 002, ranges from 13 to 30”. Clearly a filter made from this design would be effective for protection of a sensor needing access to the visible portion of the spectrum.
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Acknowledgement
The authors wish to thank E. Wolf and M. Kowar? for many helpful discussions. This work was supported by the US Army Ballistic Missile Defense Office, the Us Army Natick Research: Development and Engineering Center and the US Army Research Office.
References 5. Conclusions
In the previous sections, we have shown that. contrary to what many believe, the transmission characteristics of a dielectric film are independent of the state of coherence of the incident radiation. Passing radiation through dielectric films, be they thick or thin, changes the spectrum of the radiation, regardless of the source. A similar conclusion holds when one is considering the spatial coherence of radiation propagating through dielectric films. We have also shown that it is possible to design and manufacture a filter that has a minimum transmission within several narrow wavelength regions while transmitting a majority of the light within the visible spectrum. These filters are environmentally stable and do not significantly alter the perceived color of the transmitted radiation. These filters can be used to protect eyes and sensors while providing an almost true color rendition of the environment and minimal reduction in irradiance.
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