HFAN Rev.1; 04/08

Transcription

1 pplication Note: HFN-0.0. Rev.; 04/08 Laser Diode to Single-Mode Fiber Coupling Efficienc: Part - Butt Coupling VILBLE

2 Laser Diode to Single-Mode Fiber Coupling Efficienc: Part - Butt Coupling Introduction For fiber-optic transmitters, it is generall desirable to utilize the optical power generated b the laser diode as efficientl as possible. In practice, more than half of this power ma be lost at the interface between a laser diode and a single-mode optical fiber. The purpose of this application note is to analze the efficienc of butt coupling. Other coupling methods will be addressed in future application notes. Butt coupling is the most basic method of coupling the optical output from a laser diode into an optical fiber. This simpl consists of placing the cleaved end of the fiber as close as possible to the output aperture of the laser diode. In addition to butt coupling, there are other (more comple) methods of coupling, but these are outside the scope of this application note. There are man tpes of optical fiber in use toda, including multimode, graded-inde, dispersionshifted, etc., and each has advantages for specific applications. For purposes of this analsis, we will use the published characteristics of Corning SMF-8 step-inde fiber, which is one of the most widel used single-mode optical fibers. For this fiber, the numerical aperture (equivalent to the acceptance angle in units of radians) is 0.4, the core inde of refraction is.650, the cladding inde of refraction is.644, the core diameter is 8.µm, and the cladding diameter is 5µm. Reflection and Refraction at the ir-glass Boundar When light is incident on the boundar between one medium and another, part of it will be reflected awa from the boundar and the rest will be transmitted across the boundar and into the new medium. The light that is transmitted into the new medium will generall eperience a change in diion due to refraction. This is illustrated in Figure. pplication Note HFN-0.0. (Rev.; 04/08) Reflected Incident θ r θ i ir n =.0 The incident, reflected, and transmitted light components shown in Figure all travel in the same two-dimensional plane. The angle of incidence, θ i, and the angle of reflection, θ r, are equal. The angle of the transmitted light, θ t, relative to the normal at the boundar, is related to the angle of incidence b Snell s law: sinθ n sin n = θ () where n and n are the indices of refraction for the incident and transmitting media, respectivel, θ = θ i = θ r, and θ = θ t. The fraction of the incident light amplitude that is reflected can be calculated using the Fresnel equations: n cosθ n cosθ R TE = () n cosθ + n cosθ Transmitted θ t n cosθ n cosθ R TM = (3) n cosθ + n cosθ where R TE and R TM represent the amplitude reflection coefficients (i.e., the ratios of the reflected to the incident amplitudes) for the transverse electric and transverse magnetic polarizations of the incident Core n =.65 Cladding n = µm 5µm Figure. Light incident on an air-glass boundar Page of 7

3 light relative to the plane of the boundar. The amplitude reflection coefficients for an air-glass boundar are plotted in Figure. Optical Power Reflection When coupling light into the single-mode optical fiber illustrated in Figure, the incident angles of interest are limited to θ i < 0 (for reasons that will be discussed in the net section). For this range of angles the reflection is approimatel constant and independent of polarization. From equations (), (), and (3) we can calculate R TE R TM 0.4 for θ i < 0. Since we are interested in the power of the reflected light (instead of the amplitude) we can calculate the power reflection coefficient for θ i < 0 as follows: TE TM n =.0, n =.65 Γ = R = R = 0.4 = (4) where Γ is the power reflection coefficient and represents the ratio of the reflected power to the incident power. The transmitted power is just the difference between the incident power and the reflected power, so, for the boundar in Figure, approimatel 94% of the incident power is transmitted across the air-glass boundar. 3 Total Internal Reflection For the 94% of the light power that is transmitted across the air-glass boundar, there is et another boundar that is encountered. This is the boundar between the core and the cladding of the optical fiber. The equations of the previous section appl equall to the core-cladding boundar, but with different indices of refraction. The inde of Γ TE Γ TM Incident ngle (degrees) Figure. Power reflected versus incident angle refraction for the incident medium (the core) is.65 versus.644 on the other side of the boundar (the cladding). This is illustrated in Figure 3. Cladding n =.644 Incident Core n =.65 Figure 3. Light incident on core-cladding boundar θ i Transmitted Reflected Since the cladding inde of refraction is greater than the core inde of refraction, the angle of transmitted light (θ t ) is greater than the angle of the incident light (θ i ). s the incident angle is increased, there is a point where the transmitted angle is 90 and no light is transmitted into the cladding. The incident angle that results in a transmitted angle of 90 is called the critical angle. The critical angle can be computed mathematicall using equation () (Snell s law) as follows: n n θ c = sin sin 90 = sin n (5) n where θ c is the critical angle. For n =.65 and n =.644, equation (5) gives a critical angle of 85.. The power reflection coefficients for these indices of refraction can be calculated using the Fresnel equations [() and (3)]. These reflection coefficients are plotted in Figure 4. From Figure 4 we can see that for incident angles greater than the critical angle 00% of the incident light is reflected, resulting in a condition called total internal reflection. Since the reflected angle is equal to the incident angle, the propagating light will repeatedl bounce back and forth between the corecladding boundaries with the same angles. The process is repeated indefinitel, and the light propagates unimpeded along the length of the fiber. θ t θ r pplication Note HFN-0.0. (Rev.; 04/08) Page 3 of 7

4 Optical Power Reflection n =.65, n = Incident ngle (degrees) Figure 4. Power reflected versus incident angle For incident angles less than the critical angle, some of the light is lost through transmission into the cladding and the rest is reflected back into the core of the fiber. The process is repeated, but each time the light hits the core-cladding boundar a fraction of the power is lost. Thus, for incident angles less than the critical angle, light propagates onl a short distance in the core before it is totall lost to the cladding. When coupling light into the optical fiber, we are generall interested onl in the light that propagates, i.e. light that strikes the core-cladding boundar at an angle greater than the critical angle. Working back to the air-glass boundar, we see that the angle of the light entering the glass relative to the normal at the boundar (i.e., the angle labeled θ t in Figure ), must be less than 90 - θ c for total internal reflection. pplication of Snell s law to this condition ields the following equation for the acceptance angle of the fiber: θ = sin [ ncore sin(90 θ c )] Γ TM Γ TE pplication Note HFN-0.0. (Rev.; 04/08) (6) Using this equation for the eample fiber gives an acceptance angle of 8.. (Note that the numerical aperture (N) of the fiber is equivalent to the acceptance angle in units of radians.) n acceptance angle of 8. means that light striking the air-glass boundar at the core of the fiber will onl propagate if the angle of incidence is less than 8. (0.4 radians). 4 ngular Divergence of the Laser Diode Output Power In order to determine how much of the light output power from a laser diode will couple into the optical fiber, we need to know the angular divergence of its output power. This can be determined from the dimensions of the output aperture of the laser through the use of Fresnel/Fraunhofer diffraction theor and Fourier optics. For tpical laser diodes, the active region (resonant cavit) has dimensions of µm in length, 0-0µm in width, and µm in height 3,4. For eample purposes, we will assume the output aperture is 0µm wide (in the horizontal or - dimension) and 0.µm high (in the vertical or - dimension). Fraunhofer diffraction theor is based on specific approimations that result in a convenient Fourier transform technique for determining the effect of an aperture on a beam of light. The Fraunhofer approimation becomes increasingl valid as the distance from the aperture becomes greater than five to ten times the square of the maimum radius of the aperture divided b the wavelength (this is called the Fraunhofer distance). To use this technique we compute the product of the output aperture and the spatial intensit distribution of the light beam just inside the aperture. We then take the Fourier transform of the product and square the result 5. This procedure can be written mathematicall as: [ a(, ) b(, )] P( f, f ) = FT (7) where, P(f,f ) is the angular distribution of the output power as a function of the spatial frequencies f and f, FT[ ] signifies the Fourier transform, a(,) is the aperture function, and b(,) is the spatial intensit distribution of the light beam. The light output of a laser can be modeled as a Gaussian beam. For a Gaussian beam, the angular intensit profile can be represented b the Gaussian function ep[-π(/w ) ]ep[-π(/w ) ], where w represents the aperture width in the and dimensions. This function has two unique properties: it s Fourier transform is also a Gaussian function; and a Gaussian beam maintains the same angular intensit profile in the Fresnel and Fraunhofer Page 4 of 7

5 regions (near and far field). Using the properties of the Gaussian beam output of the laser diode along with equation (7), we can calculate the angular intensit profiles of the laser output power in the - ais and -ais. Figures 5 and 6 are plots of the laser diode output power as a function of the divergence angle. These figures were generated using the Fourier transform technique described above. The mathematical derivation of these figures is detailed in the appendi at the end of this application note. The ke point to observe is the inverse relationship between the angular divergence and the aperture dimensions. For eample, the larger horizontal dimension results in a tighter angular intensit profile (Figure 5) while the smaller vertical dimension results in a wider angular intensit profile (Figure 6). Recalling that the acceptance angle for the eample optical fiber is 8., we can see from Figures 5 and 6 that a butt-coupled optical fiber will accept most of the laser output power in the horizontal diion, but onl a fraction of the power in the vertical dimension. We can calculate a reasonable estimate of the output power that will be accepted, b integrating the two angular intensit patterns over the range of acceptance angles and computing the product of the results, i.e. P θ θ θ I ( θ ) dθ I ( θ ) dθ (8) θ where P is the accepted power and I(θ) represents the angular intensit pattern. pplication of equation (8) to the angular intensit patterns of Figures 5 and 6 (using numerical integration methods) results in P 3% of the total output power. 5 Total Butt-Coupling Losses For the eample analsis of this application note, we have calculated a reflection loss of 5.8% and an acceptance angle loss of 69%. Combining these losses, we can calculate the total fiber-coupled power as ( )( 0.69) = 9% of the total laser output power Figure 5. Gaussian angular intensit profile horizontal dimension (-ais) In addition to the reflection and acceptance angle losses, there are a number of other possible losses that have not been addressed. These include losses due to: () imperfect cleaving of the optical fiber, () misalignment of the optical fiber, (3) laser output aperture dimensions larger than the fiber core (mismatch loss), and (4) intentional angling of the face of the optical fiber to reduce back reflections into the laser diode. ll of these losses tpicall occur to some etent with butt coupling and will therefore reduce the fiber-coupled power beond the figure calculated above. For eample, some references report practical butt-coupling efficiencies as low as 0% Figure 6. Gaussian angular intensit profile vertical dimension (-ais) pplication Note HFN-0.0. (Rev.; 04/08) 8 6 Conclusions We have demonstrated through analsis that the tpical efficienc of butt coupling will be less than 9%. Clearl it is desirable to improve on this figure. There are man techniques that can be used to realize an improvement, including: () antireflection coatings and/or inde-matching gels to Page 5 of 7

6 reduce reflection losses, () improved alignment techniques, (3) lensed optical fibers, where the end of the fiber is speciall shaped to improve the acceptance angle, and (4) lens sstems that reshape the laser output for a better match to the fiber core. Using these and other techniques, coupling efficiencies as high as 87% have been reported 7. Unfortunatel these techniques can be complicated and epensive. Economical assembl to support competitive pricing ma rule out all but the simplest methods. Thus, in man cases, butt coupling ma be the onl feasible method. ppendi: Mathematical Details for Calculating the ngular Divergence of the Laser Output Power For the maimum radius of the eample aperture (5µm) and a wavelength of.3µm, the Fraunhofer distance is approimatel 95 to 90µm. This means that equation (7) begins to be a good approimation at a distance of 95µm, and becomes a ver accurate approimation for distances greater than 90µm. In practical butt-coupling configurations, the cleaved end of the optical fiber is placed as close as possible to the output aperture of the laser diode, which is tpicall 0 to 0µm. Even though the distance between the aperture and the fiber does not meet the criteria for the Fraunhofer approimation, the criteria will be met after the light propagates a short distance down the fiber. lso, if we assume the laser output is a Gaussian beam then the Fraunhofer distance is irrelevant, since a Gaussian beam has the unique propert that it maintains the same shape at all propagation distances. Thus, for practical purposes, we can assume that equation (7) is valid for a butt-coupled fiber. centered at = 0 and it s width is equal to w. Using this function, we can write the following epression for the output aperture: a(, ) = (9) 0µ m 0.µ m where λ is the wavelength of the light and d is the distance from the aperture. The spatial distribution of the emitted laser light depends on the geometr of the resonator and on the shape of the active medium. If we assume that the mirrors at the ends of the resonant cavit are planar and perfectl parallel, then the laser output can be modeled as a plane wave, which we can epress as b(,) =. Because of the difficult in achieving sufficient alignment accurac with planar mirrors, man laser designs use spherical mirrors. In this case, the laser output takes the form of a Gaussian beam 8, which we can epress as b(,) = Gauss(/w,/w ) = ep[-π(/w ) ]ep[-π(/w ) ]. Regardless of whether we model the laser output as a plane wave or a Gaussian beam, the calculated angular divergence of the output power resulting from equation (7) will be approimatel the same in the region of interest for the fiber coupling analsis. To illustrate this point, we will use equation (7) to calculate the result for both a plane wave and a Gaussian beam output. For a plane wave, equation (7) can be solved as follows: P( f, ) = FT f f 0. f 0 0. = sinc sinc where sinc(f) is defined as sin(πf)/πf. (0) We can write an epression for the two dimensional output aperture function in terms of the angular function (/w). This function has been defined to have a value of zero for /w > 0.5, 0.5 for /w = 0.5, and for /w < 0.5. We appl this function to the aperture b letting zero represent no light transmission and one represent full light transmission. Note that the (/w) function is pplication Note HFN-0.0. (Rev.; 04/08) Page 6 of 7

7 For a Gaussian beam, equation (7) can be solved as: π π P( f, ) = FT 0 e e 0. f 0 0. intensit patterns are calculated b observing that the radial epansion of the intensit pattern divided b the propagation distance is equivalent to the tangent (in radians) of the divergence angle. For eample, for the /w point on the intesit pattern the propagation distance is d, and therefore tan θ = = λ () w d w where θ is the angle of divergence f f 0 0. π π e e () Note that the result in equation () is obtained b observing that the angle function almost entirel contains the Gaussian function, resulting in a slightl truncated Gaussian function that closel approimates the original. Figure 7 is a plot of both the sinc ( ) function and the Gauss ( ) functions. The plots are normalized to a spatial frequenc of /w, where λ is the wavelength of the light, d is the distance from the aperture, and w is the width of the aperture. In the region between ±/w the results are similar for both functions. For purposes of this analsis, we will use the Gauss( ) function from this point forward in order to be consistent with results published in the literature Gauss (wf X/ sinc (wf X/) /w -/w 0 /w /w Figure 7. Normalized sinc ( ) and Gauss ( ) functions Figures 5 and 6 are plots of the angular output intensit patterns along the and ais as calculated in equation (). The divergence angles of the B.E.. Saleh and M.C. Teich, Fundamentals of Photonics, New York, NY: John Wile & Sons, Inc., 99, pp J.W. Goodman, Introduction to Fourier Optics, San Francisco, C: McGraw-Hill, 968, pp J.C. Palais, Fiber Optic Communications, 4 th Ed., Upper Saddle River, New Jerse: Prentice Hall, 998, pp P.E. Green, Fiber Optic Networks, Englewood Cliffs, New Jerse: Prentice Hall, 993, pp J.D. Gaskill, Linear Sstems, Fourier Transforms, and Optics, New York, NY: John Wile & Sons, 978, pp J.M. Senior, Optical Fiber Communications: Principles and Practice, nd edition, New York, NY: Prentice Hall, 99, pp H. Yoda and K. Shiraishi, New Scheme of a Lensed Fiber Emploing a Wedge-Shaped Graded-Inde Fiber Tip for the Coupling Between High-Power Laser Diodes and Single-Mode Fibers, IEEE Journal of Lightwave Technolog, vol. 9, pp , December B.E.. Saleh and M.C. Teich, Fundamentals of Photonics, New York, NY: John Wile & Sons, Inc., 99, pp Ogura, S. Kuchiki, K. Shiraishi, K. Ohta, and I. Oishi, Efficient Coupling Between Laser Diodes With a Highl Elliptic Field and Single-Mode Fibers b Means of GIO Fibers, IEEE Photonics Technolog Letters, vol. 3, pp. 9-93, November 00. pplication Note HFN-0.0. (Rev.; 04/08) Page 7 of 7