Achieving Gaussian outputs from large-mode-areahigher-order-mode fibers
Norbert Lindlein,1,* Gerd Leuchs,1 and Siddharth Ramachandran2
1Institute of Optics, Information and Photonics (Max Planck Research Group) of the Friedrich Alexander University ofErlangen–Nuremberg, Staudtstrasse 7�B2, 91058 Erlangen, Germany
2OFS Laboratories, 25 Schoolhouse Road, Somerset, New Jersey 08873, USA
Higher-order modes (HOMs) in specially designed fi-bers have recently been shown to offer the means toobtain robust light propagation in record-large effec-tive areas �Aeff� with negligible mode-coupling or benddistortions [1]. Such beams can be generated withgreater than 99% coupling-efficiencies using in-fibergratings [1,2]. However, this conversion techniquemay not be suitable at the output of a high-powerfiber-laser or amplifier comprising an HOM fiber be-cause converting back to the (small-Aeff) fundamentalmode of a fiber would lead to high intensities at thefiber output. This could then, lead to nonlinear dis-tortions and dielectric breakdown, the very phenom-ena one would try to avoid by using HOM fibers.
In this paper, we investigate alternatively, free-space techniques to convert the output of a HOM fiberinto a fundamental Gaussian beam because, ulti-mately, for most applications of high-power lasersand amplifiers, a Gaussian output would be desired.
Note that although the M2 beam quality values [3]of HOMs are very high, the output comprises a single,spatially coherent mode�beam. This implies that theHOM can be collimated or otherwise relayed througha series of lenses from the tip of the fiber to somedesired location in free space. Hence, the mode con-verter may be deployed further downstream, andeven at the physical location where the focusing isimplemented. Indeed, one can envisage a complex“lensing system” for HOM beams where the opticaldevice at any desired location both converts the beaminto a Gaussian shape and subsequently focuses it.
There are a lot of different methods to converthigher-order beams into (nearly) fundamental modebeams [4–7]. Especially, the use of binary phaseplates to correct the phase of the sidelobes of thehigher-order beam, so that they can interfere posi-tively, is well-known [4,6]. However, it is also well-known that a binary phase plate alone cannotimprove the laser beam quality [8]. The reason is thatthere are some secondary maxima quite far awayfrom the center, in addition to a small central maxi-mum, which nearly looks like a fundamental mode, in
the far field. Nevertheless, in several cases it mightbe more useful to remove some of these secondarymaxima by losing some light power, if on the otherside the factor M2 is then decreased to the value of anearly fundamental Gaussian mode.
The higher-order fiber modes, which are regardedin this paper, are described in [1]. They are namedLP0m modes (LP stands for linearly polarized and m isthe radial mode number) and are in the near fieldquite similar to the Bessel beams that are truncatedin the radial direction. It is well-known that Besselbeams contain in each ring the same amount of lightpower [9]. Therefore, it is very important that thelight of all rings will positively interfere in the farfield. However, the regarded modes are not idealBessel beams, but they show some small deviationsand are truncated. This can also be seen by looking atthe far field. An ideal Bessel beam would form asingle ring in the far field, whereas in the case of theLP0m modes there are, besides a bright ring, alsosome darker rings. Figure 1 shows an experimentalresult of the intensity distribution in the near and inthe far fields of an LP08 mode and the correspondingsimulated data, which were obtained from a modelusing experimentally fitted data. In comparison, itshould be taken into account that the intensity Iwas measured with an infrared sensitive Vidiconcamera, which is extremely nonlinear. Therefore,the simulated results are displayed by applying a�-correction with � � 0.5 for the intensity I, i.e., thequantity I0.5 � �I is displayed in the simulations.
2. Design of the Phase Plate
A Bessel beam has many zeros of the intensity, and ateach of these zeros the complex amplitude or electricfield changes sign and therefore the phase jumps by�. It is well-known [4,6] that a binary phase plate,which just corrects these phase jumps of �, so thatthere is a uniform phase in the near field behind thephase plate, can form a quite narrow and high inten-sity maximum in the far field.
To be sure that this is one of the optimum solu-tions we also used an iterative Fourier transformalgorithm [10], where first the complex amplitudein the far field is calculated, then the intensity inthe far field is set to the desired value, i.e., a Gauss-ian intensity distribution, by keeping the phase asit is, and afterward this new complex amplitude ispropagated back to the near field of the fiber. There,the intensity is again set to the intensity of the LP0m
mode by keeping the phase, and then the next itera-tion can start. The result is that after a few iterations,the phase in the near field of the fiber converges to aconstant phase value. This is equivalent to the solu-tion with a binary phase plate, which corrects theoriginal phase jumps of � so, that in the end, there isa constant phase behind the phase plate.
To correct the phase jumps at a wavelength � anda refractive index n of the phase plate, the depth d ofthe binary levels of the phase plate have to be d� ���2�n � 1��, and the concentric rings of the phaseplate have to start and end at the zeros of the inten-sity distribution of the respective LP0m mode. Table 1shows the radii of the zeros of the intensity for thecase of a LP08 mode at � � 1080 nm. These radii arealso the radii of the phase transition points of thephase plate if the phase plate is directly mounted atthe fiber end. If there is an additional imaging systembetween the fiber end and the phase plate to magnifythe mode, the radii of the phase plate are just theproduct of the scaling factor of the imaging systemand the radii of the zeros of the intensity of the mode.A scheme of the cross-section of a phase plate (madeof fused silica with a refractive index of n � 1.4495 at� � 1080 nm) for converting a LP08 mode at � �1080 nm into a Gaussian mode is shown in Fig. 2.
3. Simulation of the Mode Conversion of the LP0m
Modes with the Help of Phase Plates
In the following, the intensity distributions I(r) in thenear and in the far field of some LP0m modes will beshown, whereby for the switching between near andfar fields, the well-known Fraunhofer diffraction in-tegral is used, which is numerically implemented byusing a fast Fourier transformation. The near fielddata, which are the basic data for the simulations,
Fig. 1. Intensity distributions of an LP08 mode at a 1080 nmwavelength in the near field and at a distance of 15 mm behindthe fiber, i.e., nearly in the far field: (a) Experimental result ofthe near field, (b) simulation data of the near field, (c) experi-mental data of the far field, (d) simulated data of the far field.The simulated data are displayed with a gamma corrected scalewith output � I� (here: � � 0.5) to imitate the nonlinear responseof the Vidicon camera.
Table 1. Radii, at Which the Phase of the Phase Plate Has to Changebetween 0 and �, Respectively, for the Case of an LP08 Mode at
� � 1080 nm, if the Phase Plate Is Directly Mountedat the Fiber End
coincide very well with the experimental results [1].The field of complex amplitudes that is used for thesimulation has 2048 � 2048 samples with a diameterof 0.4 mm, although the intensity in the near field isonly different from zero in a central region of lessthan 100 �m. Therefore the sampling in the Fourierdomain will also be quite dense, and aliasing effectsare avoided due to the still high number of samples inthe central region. For the graphical display only thecentral part of the intensity is shown. Figure 3 showsthe near field and the far field of a LP02 mode of a fiberdesigned for a wavelength of � � 1080 nm. The beamquality factor M2 [3] is calculated numerically fromthe intensity distribution in the near field I(x, y) andthe intensity distribution in the far field I�x, y�,where x and y are direction cosines. For small val-
ues, the direction cosines are nearly identical to theangles in the far field:
Mx2 �
4
��x�x,
with
�x2 �
���
����
�
x2I�x, y�dxdy
���
����
�
I�x, y�dxdy
,
�x2 �
���
����
�
x2I�x, y�dxdy
���
����
�
I�x, y�dxdy
. (1)
Fig. 4. Far field intensity distribution of a LP02 mode at � �
1080 nm plus phase plate immediately behind the fiber.
Fig. 5. Sketch of the mode conversion system including the tele-scope for optical filtering.
Fig. 2. Scheme of the central cross section of a phase plate madeof fused silica for converting an LP08 mode at a 1080 nm wave-length into a Gaussian mode. The height of the plate at the rim(i.e., for |r| � 45 �m), where the intensity of the fiber mode is zero,can also be at the high level. Then, the structure is really etchedinto a plate and is mechanically more stable.
Fig. 3. Intensity distribution of an LP02 mode at � � 1080 nm in(a) the near field and (b) the far field.
Analogous equations are valid in the y direction. Be-cause of the rotational symmetry of our problem, wecan just take one of the values M2 :� Mx
2 � My2. The
LP02 mode of Fig. 3 has a numerical value of M2
� 2.03.Please note that in the following figures the in-
tensity distributions of the original modes in thenear field are always normalized to a maximumvalue of 1. The intensity distributions in the farfield are displayed as functions of the sine of the farfield angle, which is in the figures abbreviated asNA (numerical aperture). Additionally, the valuesof the intensity in the far field are definitely con-nected to the near field intensity values, althoughthe values as such have arbitrary units. So, it ispossible to compare the values of the near fieldintensities of each mode before and after filtering,and the values of the far field intensities with andwithout the phase plate. But, it is not directly pos-sible to compare the intensity values in the nearand far fields crosswise.
By introducing a phase plate immediately behindthe fiber end, the far field intensity distribution ofFig. 4 results. In this special case of an LP02 mode,where most of the light is in a ring with identicalphase [see Fig. 3(a)], the phase plate does not changea lot, and the far field distribution in Figs. 3(b) and 4
is nearly identical. Both figures show that there aresome secondary maxima outside of the high centralpeak that contain due to the large area of the rings�13.5% of the total light power. By blocking the lightof these sidelobes the M2 value can be decreased.Optically, this means that behind the fiber plus phaseplate a 4f-telescope is introduced, and a stop in thefocal plane of the first lens of the 4f-system allowsfiltering the “far field” intensity distribution (see Fig.5). In practice, it may also be necessary to include abeam expander, i.e., another telescope, but with ascaling factor of |�| � 1, between the fiber and thephase plate. This would first of all mean more coarsestructures for the phase plate, i.e., an easier fabrica-tion, and additionally the lateral adjustment require-ments of the phase plate relative to the fiber would beless severe. On the other hand, the beam expandermeans additional costs. So, in the following simula-tions and in Fig. 5 the beam expander telescope isskipped, but it will be investigated more in detail inSection 5.
Figure 6 shows the near and the far fields of theconverted LP02 mode after passing the phase plateand the optical filtering system. The stop in the fil-tering system was adjusted to a radius correspondingto a far field angle of 0.019 rad. The far field in Fig.6(b) is identical to that of Fig. 4 if the sidelobes aretruncated. In this converted and filtered beam thereis 86.5% of the total light power of the original LP02
mode (losses due to reflections at optical surfaces arenot taken into account), and the beam quality factoris M2 � 1.07. So, the M2 factor is nearly that of anideal Gaussian beam by losing only �13.5% of thelight power.
Whereby in this special case of the LP02 mode thephase plate did not have much influence, the phaseplate is necessary for higher-order modes. There-fore, the same simulations were also done for allmodes up to a LP08 mode at � � 1080 nm, and themost important parameters of these simulationsare shown in Table 2. This table also gives theeffective area Aeff of the fiber modes in the near field,which is defined as
Fig. 6. Intensity distribution of a converted LP02 mode at �
� 1080 nm in (a) the near field and (b) the far field after using aphase plate plus an optical filtering system.
Table 2. Numerically Simulated Beam Parameters for Different Modes and Different Measures to Convert Them for the Fibers Made for aWavelength of � � 1080 nma
aP�P0 is the conserved amount of the total light power after filtering. In all cases the stop in the filtering telescope has a radiuscorresponding to a far field angle of 0.019 rad.
There, I is the intensity in the near field and theintegrals (or numerical sums) are made over thewhole fiber area.
For the LP08 mode Figs. 7, 8, and 9 are analogous toFigs. 3, 4, and 6 in which the intensity distributionsin the near and far fields are shown without thephase plate (Fig. 7), with phase plate but withoutfiltering (Fig. 8), and with the phase plate and opticalfiltering using a stop radius corresponding again to0.019 rad (Fig. 9). The M2 parameters are (see alsoTable 2): M2 � 13.99 for the original LP08 mode, M2
� 13.86 for the LP08 mode plus the phase plate, and
M2 � 1.04 for the converted and filtered beam byconserving 75.3% of the total light power. The simu-lation verifies, as it was generally shown in [8], thata binary phase plate alone cannot improve the beamquality factor M2, because now there are some smallsecondary maxima in the far field far away from thecenter (see Fig. 8). Although the height of these sec-ondary maxima is quite small, they are responsiblefor the high M2 value because they are far away fromthe center and therefore considerably contribute to�x of Eq. (1). But, by filtering out the sidelobes of thefar field distribution, the M2 value is nearly that of afundamental Gaussian beam. Of course, this is at theexpense of losing some light power.
All simulations were also repeated for a second setof modes for a fiber that was designed for a wave-length of � � 1550 nm. The results, which have asimilar behavior as for the fiber at 1080 nm wave-length, are given in Table 3.
The results of Tables 2 and 3 both show that theLP02 mode can be converted to a nearly fundamentalGaussian beam with the highest efficiency concern-ing the conserved amount of light power P�P0. How-ever, all higher-order modes �m � 2� can be convertedto a fundamental Gaussian beam with nearly thesame efficiency P�P0. Only the LP03 mode has a littlebit higher efficiency P�P0 than the others but is con-siderably smaller than that of the LP02 mode. Thetables also show that the effective areas Aeff of theconverted modes after filtering are nearly constantfor all mode orders with some exception of the LP02mode. Only for the higher-order modes at � �1550 nm is there a small increase in the effectiveareas for the mode orders 6–8. Moreover, the effec-tive areas of the converted modes are a little bitlarger than the effective area of the original LP02mode.
4. Tolerance Analysis
The sensitivity of the phase plate against misalign-ments and fabrication errors has to be investigated.For this purpose three tolerances are regarded:
(a) Lateral misalignment of the phase plate rela-tive to the fiber core.
Fig. 7. Intensity distribution of an LP08 mode at � � 1080 nm in(a) the near field and (b) the far field.
Fig. 8. Far field intensity distribution of a LP08 mode at � �
1080 nm plus a phase plate immediately behind the fiber.
Fig. 9. Intensity distribution of a converted LP08 mode at � �
1080 nm in (a) the near field and (b) the far field after using aphase plate plus an optical filtering system.
(b) Axial misalignment of the phase plate relativeto the fiber end (simulation of the free-space propaga-tion between the fiber and phase plate using the an-gular spectrum of the plane waves method).
(c) Variation of the phase shift of the phase plate,i.e., either variation of the etching depth of the fiber(or the refractive index) or variation of the wave-length of the mode.
Table 4 lists the M2 factors and the amount of con-served light power for the different tolerances in thecase of the LP08 mode at � � 1080 nm. The LP08 modeis taken because it should be the most sensitive onebecause of the smallest feature sizes in the intensitydistribution compared to modes with a lower order.Hereby, it is assumed that the stop in the filteringtelescope and the telescope itself are exactly alignedrelative to the fiber. This means that only the phaseplate has some tolerances, either in the positioning orin the fabrication.
It can be seen that in all cases the M2 parameterdoes not change significantly. This is due to the
filtering in the telescope. But, the conservedamount of light power changes dramatically insome cases because in the case of misalignmentssome light power is absorbed at the filtering stop.Therefore, the dependence of the conserved lightpower on the different tolerances is shown graphi-cally in Fig. 10. There is in all cases a nearly par-abolic dependence between the tolerance parameterand the conserved light power. In the case of thevariation in the phase shift of the phase plate it hasto be mentioned that there are in practice threeparameters (or in practice two because the smallvariation of the refractive index n can be neglectedin most cases), which can generate a variation inthe phase shift. The phase shift �� of the phaseplate has the following dependence:
�� �2
� �n � 1�d. (3)
Here, n is the refractive index of the phase plate,which is assumed to have air as the surrounding
Table 3. Numerically Simulated Beam Parameters for Different Modes and Different Measures to Convert Them for the Fibers Made for aWavelength of � � 1550 nma
aP�P0 is the conserved amount of the total light power after filtering. In all cases the stop in the filtering telescope has a radiuscorresponding to a far field angle of 0.024 rad.
Table 4. M2 Factors and Conserved Amount of Total Light Power for Different Tolerances of the Phase Plate for the Case of the FilteredLP08 Mode at � � 1080 nma
aThe tolerances are: lateral misalignment of the phase plate relative to the fiber core, axial misalignment of the phase plate relative tothe fiber end, and variation of the phase shift of the phase plate, i.e., either variation of the etching depth of the phase plate or variationof the wavelength of the mode.
medium. d is the etching depth of the binary struc-tures of the phase plate. Since the phase plate shouldideally generate a phase shift of �, there is for a givenwavelength � an ideal etching depth, dideal � ���2�n� 1��. In practice, the wavelength of the higher-ordermodes is allowed to vary by approximately �50 nmaround the design wavelength because of the broadallowed bandwidth of these fibers [2]. For a designwavelength of � � 1080 nm this would mean a vari-ation in the phase shift of the phase plate between�0.95 and 1.05. So, according to Fig. 10(c) or Table4, in the “Influence of Phase Shift of the Phase Plate”section, less than 0.5% of the total light power will getlost by varying the wavelength between 1030 and
1130 nm. Of course, a change in the wavelengthwould also mean a lateral stretching or compressionof the far field intensity distribution by a factor of���� if the near field intensity distribution is notchanged. So, first of all the diameter of the stop in thefiltering telescope needs to be changed accordingly bya factor of ����. However, since the stop in the fil-tering telescope has a radius that is identical to theradius of the minimum of the central intensity peak[see Figs. 8 and 9(b)], a small variation in the size ofthe central intensity peak will have no influence. So,the results of Table 4, in the “Influence of Phase Shiftof the Phase Plate” section, and Fig. 10(c) will also beobtained without a variation in the stop size in thefiltering telescope, with good approximation if thewavelength is changed by only �50 nm.
In total, it can clearly be seen that the most criticalmisalignment parameter is a lateral misalignment ofthe fiber core relative to the phase plate. The amountof conserved light power is very sensitive to lateralmisalignments of the phase plate, so that just a lat-eral misalignment of less than 1 �m can be accepted.Therefore, it is either necessary to have a very accu-rate positioning stage for the phase plate, or it isnecessary to form a magnified image of the wave frontat the fiber end with the help of a beam expandertelescope or another optical device. Other misalign-ments, such as, e.g., a tilt in the phase plate relativeto the fiber end, will not have a significant effect. Atilt by an angle causes, for example, a locally de-pendent axial misalignment by r (r is the lateraldistance from the optical axis), a change in the effec-tive etching depth by a factor of 1�cos , and an el-liptic deformation of the effect of the phase plate by afactor of cos . But, for a typical value of � 1° all ofthese misalignments can be neglected.
5. Imaging Systems between Fiber End and PhasePlate
As mentioned before, the allowed tolerances of thephase plate can be increased by using an optical im-aging system between the fiber end and the phaseplate so that the wavefront is laterally magnified bya factor of |�| � 1. By using a telescope, a magnifiedimage of the wavefront can be formed without chang-ing the shape of the wavefront itself. However, it hasto be assured that the telescope can image the wholewavefront without any aberrations. In Figs. 3(b) and7(b) the far field intensity distributions of the LP02and LP08 modes are shown without a phase plate. So,it can be seen that most of the light is in a ring withan angle corresponding to an NA of �0.1, and alsothat some light is outside of this ring up to an NA of�0.2. So, the telescope has to image an object with aninput NA of at least 0.2 without aberrations. So, itmay not be sufficient to use a standard beam ex-pander or to build a telescope by using standard ach-romatic doublets (which are not used because of theirachromatic correction, but because they also fulfillthe sine condition) as it is usually done in optics. But,by using a microscope objective with an NA � 0.2 asa first telescope lens and an achromatic doublet as
Fig. 10. Tolerance analysis for the LP08 mode at � � 1080 nm; (a)conserved light power as function of a lateral misalignment of thephase plate; (b) conserved light power as function of an axial mis-alignment of the phase plate; and (c) conserved light power asfunction of the phase shift of the phase plate.
the second lens, a telescopic system with the requiredspecifications can be built. A sketch of the completemode conversion system is shown in Fig. 11.
If the telescope magnifies the wavefront by a factorof � ���� � 1�, the lateral misalignment of the phaseplate can also be increased by a factor of |�| in orderto have the same values for M2 and the conservedlight power as for the miniaturized phase plate di-rectly spliced to the fiber end without using a beamexpander telescope. The allowed axial misalignmenttolerance is even increased by a factor of �2 by usingthe telescope. Of course, when using a telescope witha magnification of the lateral extension of the ringsof the phase plate has also to be magnified by |�|. So,the radii values of the phase plate for converting, forexample, the LP08 mode (see Table 1) have to be mag-nified by |�|. Therefore, the fabrication of the phaseplate should also be easier. On the other hand, thebeam expander telescope may also be a source oferror if, for example, the scaling factor is in practicedifferent from the value that is used in the design ofthe phase plate. Then, there is a kind of radial shearbetween the wavefront and the phase plate.
It should also be mentioned that the requirementsof the NA on the filtering telescope are not as severeas on the magnifying telescope, even if no magnifyingtelescope is used to save optical elements or money(on the costs of a very tight lateral misalignmenttolerance of the phase plate). This can clearly be seenby looking at Figs. 4 and 8. There, the central peak isquite narrow corresponding to an NA of only �0.02.The sidelobes, which are responsible for ensuringthat the M2 factor does not decrease by using only abinary phase plate without spatial filtering, will becut off anyway by the filtering telescope. So, a NA ofless than 0.05 for the filtering telescope is sufficient.If a magnifying telescope is used between the fiberand the phase plate the requirements on the NA ofthe filtering telescope are decreased again by a factorof 1�||.
One can also think about other possibilities formagnifying the wavefront to have more tolerance forthe phase plate. First of all, it is not really necessaryto have a telescopic imaging of the wavefront, but it isalso possible to take a “normal” lens for imaging. Ofcourse, there is then an additional spherical wave-front curvature in the image plane that has to becompensated by the optics of the filtering telescope or
some additional optics. So, a telescopic imaging ispreferable, but not really necessary.
A gradient index (GRIN) lens could, for example, bespliced to the fiber, and then the phase plate can bespliced to the GRIN lens, so that a very compactdevice is generated. However, a simple paraxial anal-ysis of GRIN lenses shows that using a single GRINlens as imaging device without a free-space distancein front of and behind the GRIN lens is only possiblefor a scaling factor of �1. The argument just uses theparaxial ray transfer matrix M of a GRIN lens [11],which connects a paraxial ray with height x and angle� directly at the front plane of the GRIN lens with theparaxial ray with height x� and angle �� directly atthe back plane of the GRIN lens with length z andrefractive index distribution n�r� � n0 � n1r
2:
x�
��� Mx�,
with
M � A BC D� cos � f sin �
�sin �
fcos � ,
� � �2n1
n0z; f �
1
�2n0n1
. (4)
Here, n0 is the refractive index of the GRIN lens inthe center, and n1 describes the change of the refrac-tive index with the radial coordinate r. f is the focallength of the GRIN lens.
To have imaging between the back and front planeof the GRIN lens, the matrix coefficient B has to bezero because then the position x� of the rays contrib-uting to the image point is independent of the rayangle �. In this case, A is the lateral magnification ofthe imaging A � �. But, B � 0 requires sin � � 0 andthen it is automatically � � A � cos � � �1. SuchGRIN lenses with sin � � 0, i.e., � � m and aninteger number m, are called “half pitch” lenses or“full pitch” lenses because the light in the GRIN lens,which follows a sinusoidal path, then makes half of aperiod �� � � or a full period �� � 2�.
But, we need a scaling factor |�| � 1. So, there hasto be free-space propagation in front of and behindthe GRIN lens if it is used as magnifying lens. Thiscan also be done in glass so that, for example, a smallglass rod can be spliced to the fiber, and then theGRIN lens spliced to the glass rod, and so on. Ofcourse, the diameter of the glass rod has to be muchlarger than the fiber diameter to have real free-spacepropagation without the guidance of the modes. So, intotal the solution with a single GRIN lens is also notso easy because a quite accurate adjustment of the airgaps or lengths of the glass rods is required.
Fig. 11. Sketch of the complete mode conversion system includinga magnifying telescope between the fiber and the phase plate anda filtering telescope behind the phase plate. The first lens symbol-izes a microscope objective, whereas the other lenses should beachromatic doublets or other lenses that fulfill the sine condition.
Besides, aberrations of a GRIN lens for off-axispoints have to be taken into account. Some simula-tions were made using a GRIN lens with a refractiveindex distribution of [12]:
n�r� � n0 sech�gr� �n0
cosh�gr��
2n0
exp�gr� exp��gr�,
with
g � �2n1
n0, (5)
which is the ideal refractive index distribution fornonparaxial imaging in the case of the scaling factor|�| � 1. Here, n0 and g are the two material param-eters of the GRIN lens.
A scalar beam propagation method implemented inour in-house tool WAVESIM was used to simulate thepropagation of the LP08 mode at � � 1080 nmthrough a GRIN lens with a length of z � 2.4 mm,n0 � 1.6289, and n1 � 0.3484 mm�2. This type ofGRIN lens with � � �2 is called a “quarter pitch”lens. The scaling factor was � � �5 in this case,which means that the distance between the fiber andthe front plane of the GRIN lens has to be 0.188 mm(in air). The distance between the back plane of theGRIN lens and the image plane is then 4.69 mm (alsoin air). Figure 12(a) shows the intensity distribution
in the image plane, and Fig. 12(b) shows the phase inthe image plane. Finally, Fig. 12(c) illustrates howthe light propagates along the optical axis. It can beseen that the LP08 mode is imaged quite well by theGRIN lens by comparing the results with Fig. 7(a). Bycomparing the peak-to-valley values (P�V) of bothplots, it is verified that the P�V of the intensity of theimage is decreased by a factor of 1��2 � 0.04, asexpected. But, looking at Fig. 12(b) it can also be seenthat there is now a parabolic phase term added,whereas the original LP08 mode just had phase jumpsof � at the zeros of the intensity and besides this aplane wavefront.
It should also be mentioned that, if we use only oneGRIN lens by itself, only nontelescopic imaging canbe obtained if a scaling factor of |�| � 1 is required.A telescopic imaging with a GRIN lens is only possi-ble with a scaling factor of � � �1 as can also be
easily shown using the GRIN lens matrix of Eq. (4),and by multiplying it with two free-space propagationmatrices:
Here, d1 is the propagation distance from the objectplane (fiber end) to the GRIN lens and d2 is the dis-tance from the GRIN lens to the image plane (phaseplate). For telescopic imaging both the coefficients B�and C� have to be zero. But, C� � 0 requires againthat sin � � 0 (or the trivial solution 1�f � 0 meaningno GRIN lens at all and therefore no imaging), andtherefore � � A� � cos � � �1.
It is possible to build a telescope by two quarterpitch GRIN lenses with different focal lengths f1 andf2, and therefore different material parameters n1 ofthe GRIN lenses. Then, the object plane (fiber end)can be directly in front of the first GRIN lens and theimage plane (phase plate) directly behind the secondGRIN lens, so that all can be spliced together withoutpropagation in air in between. Figure 13(a) shows theintensity distribution in the image plane of this typeof GRIN lens telescope using two GRIN lenses with
M� � A� B�
C� D���cos � �d2
f sin � d1 cos � f sin � d2 cos � �d1d2
f sin �
�1f sin � cos � �
d1
f sin � �. (6)
Fig. 12. Nontelescopic imaging of the LP08 mode at � �
1080 nm with a single GRIN lens and free-space propagation inair; (a) intensity distribution in the image plane; (b) phase in theimage plane; (c) intensity plot (logarithmic scale) showing the lightpropagation from the object plane (bottom) to the image plane(top).
z1 � 2.4 mm, n0,1 � 1.6289, n1,1 � 0.3484 mm�2, andz2 � 12 mm, n0,1 � 1.6289, and n1,2 � 0.0140 mm�2.According to Eq. (4) the focal lengths are thereforef1 � 0.94 mm and f2 � 4.68 mm, so that a scalingfactor of � � �5 results for the telescopic imaging.Figure 13(b) illustrates that now the phase is nearlyplane besides the phase jumps of � at the zeros of theintensity. Finally, Fig. 13(c) shows the light propaga-tion along the optical axis in this case. Again, the LP08mode is imaged very well and magnified laterally bya factor of |�| � 5. Additionally, the phase is nearlyunchanged besides the lateral magnification by a fac-tor of 5, and there is no free-space propagation that isnecessary in air. So, a quite compact telescopic imag-ing system can be built by splicing the two GRINlenses to the fiber and finally the phase plate to theGRIN lenses. The grooves of the phase plate have tobe filled with air and not with optical glue, so that thephase plate works properly. In practice the GRINlenses may have some aberrations because the refrac-tive index distribution in a real GRIN lens may de-viate from Eq. (5).
6. Conclusion
It has been shown by simulations that binary phaseplates can be taken to obtain a narrow bright cen-tral maximum in the far field of a fiber that emits ahigher-order LP0m mode, which is similar to a trun-cated Bessel beam. The beam quality factor M2 can-not be decreased by doing this alone but by filtering
the sidelobes in the far field, (i.e., experimentally inthe focal plane of a lens), the beam quality factorcan be decreased to nearly unity by losing only�22%–25% of the total light power for m � 3 and only14%–16% for the LP02 mode. In practice there will beadditional losses due to the auxiliary optics, whichwas here assumed to be ideal. The simulation alsoshow that it is a little bit easier to convert a lowerorder beam than a higher order beam; the LP02
mode is especially easier to convert. But, for modeorders m � 3 the differences are quite small (seeTables 2 and 3) and should not be important inpractice.
A tolerance analysis indicated that a lateral dis-placement of the phase plate relative to the modepattern is the most critical misalignment, so that amagnification of the mode pattern preferably by atelescopic imaging system or by another nontele-scopic imaging system may be necessary in practiceto avoid very accurate alignment devices for thephase plate. Such a telescopic imaging system caneither be built by conventional refractive optics,such as microscope objective and achromatic dou-blet, or it can be built by two quarter-pitch GRINlenses with different focal lengths. The advantageof the GRIN lens telescope is that the GRIN lensescan be spliced to the fiber and the phase plate canfinally be spliced to the second GRIN lens, so that avery compact system without air gaps results. Thetolerance analysis of the phase plate also showedthat a variation of the wavelength of the mode byapproximately �50 nm (for a central wavelength of� � 1080 nm) will be allowed without a significantloss of light power (less than 0.5%). Of course, thelenses of the filtering telescope and magnifying tele-scope have to be achromatic in this case. In thefuture, the phase plates will be manufactured, andexperimental results will be presented in a laterpublication.
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