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Imaging characteristics of a confocal cavity
Volume 28, number 2
OPTICS COMMUNICATIONS
February 1979
IMAGING CHARACTERISTICS OF A CONFOCAL CAVITY A. RIAZI, O.P. GANDHI and D.A. CHRISTENSEN Department of Electrical Engineering, University o f Utah, Salt Lake City, Utah 84112, USA
Received 31 October 1978
Mode decomposition method is employed to determine the nature of image distortion o n passage through a confocal cavity. It is shown that output image possesses either an even or an odd spatial symmetry, depending on resonance conditions. Possible methods of preventing image distortion are also discussed.
In various applications where an optical cavity is employed to resonate an image, such as for increasing the efficiency of a coherent upconverter [1 ] or spectral analysis of images of luminous bodies [2,3], it is important to know the degree of image deterioration due to multiple transits inside the cavity. The changes in image profile which occur upon passage through a piano-piano interferometer have been reported previously [ 1]. In this paper we wish to report on the nature of image distortions due to passage through a confocal resonator configuration. The confocal resonator possesses numerous advantages over a piano-piano interferometer such as: a. Insensitivity to angular misalignments [4]. b. Higher temporal resolving power [2]. c. Lower diffraction losses [5]. In our analysis we employ the mode decomposition technique [1,6] which is an efficient method for comparing the input and output images. In this method first the input image is decomposed into the cavity modes, then each mode is multiplied by an appropriate transfer function, and finally they are recombined to obtain the output image. For an input image given by Uin(X ) = ~ b n f n ( X ) , n=0
the output image is
(1)
oo
Uout(X) =
n=O
bnrnf.(x),
(2)
where bn =
f
Uin(X)fn*(x ) d x ,
(3)
and fn(X) represents the normal modes of the cavity (normalized Hermite gaussian funcLiuns [7] ). Tn is the transfer function which is determined by taking into account the losses which a mode suffers as it makes a transit inside the resonator. For a high finesse optical cavity, this transfer function is given by the expression [1,8] t2(1 - An)l/2 exp { - j (27rL/;k + An) }
Tn
1-r2(1-An)
exp(-2j(27rL/?~+An)}
(4)
where t is the amplitude transmissivity of each mirror, r is the amplitude reflectivity of each mirror, L is the cavity length - mirror radii of curvature for a symmetric confocal cavity, A n is the power loss per transit of the nth mode, An is the additional phase shift per transit of the nth mode, ;k is the illumination wavelength. In our analysis of image transformation through a plano-plano interferometer [ 1], it was emphasized that for the purpose of image enhancement it is advantageous to use interferomet, ers of large Fresnel number (N = a2/L;~ for an interferometer of mirror aperture 163
Volume 28, number 2
OPTICS COMMUNICATIONS
diameter 2a) where diffractional power losses are negligible. Such losses are even smaller [9] for corresponding confocal cavities. Hence even for moderate values of Fresnel number, it is valid to assume that An ~ 0 .
February 1979
focal cavity and those of a piano-piano interferometer. A confocal resonator has a transfer function with a magnitude that is one or zero for all modes and is insensitive to the Fresnel number (assuming that Fresnel
(5)
Also, from the work o f F o x and Li [9], we have An = (n + 1)zr/2,
n = 0, 1 , 2 .
(6) a
The condition for the resonance of the nth mode is given by
2rrL/X
+ A n = rrl~
(7)
,
where m is an integer. From eqs. (7) and (6) it follows that when the cavity spacing is properly selected to resonate an even mode, all the even modes will be in resonance while all the odd modes will be out o f resonance. The converse will be true when the cavity length is appropriately adjusted to resonate an odd mode and consequently the output image will always possess a spatial symmetry. An example o f image transfer through a confocal cavity o f Fresnel number N = 10 is illustrated in fig. 1. The input image is
\ i
-i.0
- 0 . ~
0
0,5
i 1.0
0.5
1.0
0.5
I 1.0
x/a
(a)
Uin(X ) = 0.95 f o ( x ) + 0.65 f l ( x ) , as plotted in fig. 1a. When condition for the resonance of even modes is satisfied, we have
!
i
-i,0
-0.5
T O = t2/(1 - r 2) = t2/(t 2 + A ) ~ 1 , (b)
assuming power absorption A can be neglected. Also T 1 = 0. Then the output image is
0.7
Uout(X ) = 0.95 f o ( x ) ,
C
0.5
which is shown in fig. lb. When the odd modes are in resonance, it follows that
T1
=t2/(t2 + A ) ~ l
,
and T O ---0, so that Uout(X ) = 0.65 f l ( x ) , and this is plotted in fig. 1 c. It can be seen that fig. l b is the even symmetric component o f fig. I a and fig. 1c is the odd symmetric component o f the input image. Significant differences exist between the image transfer properties o f a con164
L -I.0
t - 0 . 5 ~
q
-0.5
(c)
Fig. 1. Passage of an image through a confocal resonator. (a) Input image; (b) output image when cavity spacing is set for resonance of even modes; (c) output image for resonance of odd modes.
Volume 28, number 2
OPTICS COMMUNICATIONS Voltage generator
Piezoelectric crystal
Input image
~ /~ Matching lens
,,, / ~ V " - - - - - ~ Ob ..... tldn A/~ ~_~ ! screeft Collimating lens
Fig. 2. A possible configuration for retaining both the odd and even symmetric components of an image.
number is sufficiently large so that A n ~ 0 without any variation). A piano-piano interferometer has a transfer function which resembles a low-pass spatial filter with a cutoffmode number [1 ] that is proportional to the square root of Fresnel number. As shown in fig. 1, an image profile can change dramatically as it propagates through a confocal resonator. Such high levels of image deterioration are often undesirable and therefore it is essential to design a method o f reducing image distortion. One such method consists of using a confocal cavity whose length can be adjusted by means of a piezoelectric crystal as shown in fig. 2. The cavity length is initially set to resonate the even modes, then it is changed by ~/4 to resonate the odd modes, then changed back again, and the process is repeated at a sufficiently fast rate. Fig. 2 also shows that lenses of proper focal lengths are required in order to match [8] the input image to the cavity and to collimate the output image. Another possible means to preserve both image components is to use a configuration consisting of two confocal cavities [8]. This latter method is less economical than the former. In addition, both methods suffer from the drawback of losing half of the power.
February 1979
In those applications where a resonator is to be employed for enhancing a weak image such as in an infrared image to visible upconverter [8] or in a millimeter image converter [6], the conversion efficiency is very important, hence the necessity of sacrificing power in order to improve the output image quality can be a crucial factor in determining the value of using a confocal cavity as compared to a piano-piano interferometer of large Fresnel number. In conclusion, it has been shown that the asymmetry of an image is destroyed upon passage through a confocal resonator so that the output image is always spatially symmetric. There are possible methods of eliminating such distortions; however, they require a sacrifice in power. Due to the one-dimensional treatment o f our approach, the results are directly applicable only to separable images.
References [1] A. Riazi, D.A. Christensen and O.P. Gandhi, J. Opt. Soc. Am., to be published. [2] S.A. Clark, Optics Comm. 5 (1972) 163. [3] J.V. Ramsay, Applied Optics 8 (1969) 569. [4] J.R. Johnson, Applied Optics 7 (1968) 1061. [5] W. Culshaw, IRE Transactions on Microwave Theory and Technique, September 1962, pp. 331-339. [6] D.H. Ulmer, O.P. Gandhi and R.W. Grow, Electronic Letters, Vol. 10, 1974. [7] A.E. Siegman and E.A. Sziklas, Applied Optics 13 (1974) 2775. [8] A. Riazi, doctoral dissertation, Electrical Engineering Department, University of Utah, Salt Lake City, Utah 1978. [9] A.G. Fox and T. Li, Bell System Technical Journal 40 (1961) 453.
165
OPTICS COMMUNICATIONS
February 1979
IMAGING CHARACTERISTICS OF A CONFOCAL CAVITY A. RIAZI, O.P. GANDHI and D.A. CHRISTENSEN Department of Electrical Engineering, University o f Utah, Salt Lake City, Utah 84112, USA
Received 31 October 1978
Mode decomposition method is employed to determine the nature of image distortion o n passage through a confocal cavity. It is shown that output image possesses either an even or an odd spatial symmetry, depending on resonance conditions. Possible methods of preventing image distortion are also discussed.
In various applications where an optical cavity is employed to resonate an image, such as for increasing the efficiency of a coherent upconverter [1 ] or spectral analysis of images of luminous bodies [2,3], it is important to know the degree of image deterioration due to multiple transits inside the cavity. The changes in image profile which occur upon passage through a piano-piano interferometer have been reported previously [ 1]. In this paper we wish to report on the nature of image distortions due to passage through a confocal resonator configuration. The confocal resonator possesses numerous advantages over a piano-piano interferometer such as: a. Insensitivity to angular misalignments [4]. b. Higher temporal resolving power [2]. c. Lower diffraction losses [5]. In our analysis we employ the mode decomposition technique [1,6] which is an efficient method for comparing the input and output images. In this method first the input image is decomposed into the cavity modes, then each mode is multiplied by an appropriate transfer function, and finally they are recombined to obtain the output image. For an input image given by Uin(X ) = ~ b n f n ( X ) , n=0
the output image is
(1)
oo
Uout(X) =
n=O
bnrnf.(x),
(2)
where bn =
f
Uin(X)fn*(x ) d x ,
(3)
and fn(X) represents the normal modes of the cavity (normalized Hermite gaussian funcLiuns [7] ). Tn is the transfer function which is determined by taking into account the losses which a mode suffers as it makes a transit inside the resonator. For a high finesse optical cavity, this transfer function is given by the expression [1,8] t2(1 - An)l/2 exp { - j (27rL/;k + An) }
Tn
1-r2(1-An)
exp(-2j(27rL/?~+An)}
(4)
where t is the amplitude transmissivity of each mirror, r is the amplitude reflectivity of each mirror, L is the cavity length - mirror radii of curvature for a symmetric confocal cavity, A n is the power loss per transit of the nth mode, An is the additional phase shift per transit of the nth mode, ;k is the illumination wavelength. In our analysis of image transformation through a plano-plano interferometer [ 1], it was emphasized that for the purpose of image enhancement it is advantageous to use interferomet, ers of large Fresnel number (N = a2/L;~ for an interferometer of mirror aperture 163
Volume 28, number 2
OPTICS COMMUNICATIONS
diameter 2a) where diffractional power losses are negligible. Such losses are even smaller [9] for corresponding confocal cavities. Hence even for moderate values of Fresnel number, it is valid to assume that An ~ 0 .
February 1979
focal cavity and those of a piano-piano interferometer. A confocal resonator has a transfer function with a magnitude that is one or zero for all modes and is insensitive to the Fresnel number (assuming that Fresnel
(5)
Also, from the work o f F o x and Li [9], we have An = (n + 1)zr/2,
n = 0, 1 , 2 .
(6) a
The condition for the resonance of the nth mode is given by
2rrL/X
+ A n = rrl~
(7)
,
where m is an integer. From eqs. (7) and (6) it follows that when the cavity spacing is properly selected to resonate an even mode, all the even modes will be in resonance while all the odd modes will be out o f resonance. The converse will be true when the cavity length is appropriately adjusted to resonate an odd mode and consequently the output image will always possess a spatial symmetry. An example o f image transfer through a confocal cavity o f Fresnel number N = 10 is illustrated in fig. 1. The input image is
\ i
-i.0
- 0 . ~
0
0,5
i 1.0
0.5
1.0
0.5
I 1.0
x/a
(a)
Uin(X ) = 0.95 f o ( x ) + 0.65 f l ( x ) , as plotted in fig. 1a. When condition for the resonance of even modes is satisfied, we have
!
i
-i,0
-0.5
T O = t2/(1 - r 2) = t2/(t 2 + A ) ~ 1 , (b)
assuming power absorption A can be neglected. Also T 1 = 0. Then the output image is
0.7
Uout(X ) = 0.95 f o ( x ) ,
C
0.5
which is shown in fig. lb. When the odd modes are in resonance, it follows that
T1
=t2/(t2 + A ) ~ l
,
and T O ---0, so that Uout(X ) = 0.65 f l ( x ) , and this is plotted in fig. 1 c. It can be seen that fig. l b is the even symmetric component o f fig. I a and fig. 1c is the odd symmetric component o f the input image. Significant differences exist between the image transfer properties o f a con164
L -I.0
t - 0 . 5 ~
q
-0.5
(c)
Fig. 1. Passage of an image through a confocal resonator. (a) Input image; (b) output image when cavity spacing is set for resonance of even modes; (c) output image for resonance of odd modes.
Volume 28, number 2
OPTICS COMMUNICATIONS Voltage generator
Piezoelectric crystal
Input image
~ /~ Matching lens
,,, / ~ V " - - - - - ~ Ob ..... tldn A/~ ~_~ ! screeft Collimating lens
Fig. 2. A possible configuration for retaining both the odd and even symmetric components of an image.
number is sufficiently large so that A n ~ 0 without any variation). A piano-piano interferometer has a transfer function which resembles a low-pass spatial filter with a cutoffmode number [1 ] that is proportional to the square root of Fresnel number. As shown in fig. 1, an image profile can change dramatically as it propagates through a confocal resonator. Such high levels of image deterioration are often undesirable and therefore it is essential to design a method o f reducing image distortion. One such method consists of using a confocal cavity whose length can be adjusted by means of a piezoelectric crystal as shown in fig. 2. The cavity length is initially set to resonate the even modes, then it is changed by ~/4 to resonate the odd modes, then changed back again, and the process is repeated at a sufficiently fast rate. Fig. 2 also shows that lenses of proper focal lengths are required in order to match [8] the input image to the cavity and to collimate the output image. Another possible means to preserve both image components is to use a configuration consisting of two confocal cavities [8]. This latter method is less economical than the former. In addition, both methods suffer from the drawback of losing half of the power.
February 1979
In those applications where a resonator is to be employed for enhancing a weak image such as in an infrared image to visible upconverter [8] or in a millimeter image converter [6], the conversion efficiency is very important, hence the necessity of sacrificing power in order to improve the output image quality can be a crucial factor in determining the value of using a confocal cavity as compared to a piano-piano interferometer of large Fresnel number. In conclusion, it has been shown that the asymmetry of an image is destroyed upon passage through a confocal resonator so that the output image is always spatially symmetric. There are possible methods of eliminating such distortions; however, they require a sacrifice in power. Due to the one-dimensional treatment o f our approach, the results are directly applicable only to separable images.
References [1] A. Riazi, D.A. Christensen and O.P. Gandhi, J. Opt. Soc. Am., to be published. [2] S.A. Clark, Optics Comm. 5 (1972) 163. [3] J.V. Ramsay, Applied Optics 8 (1969) 569. [4] J.R. Johnson, Applied Optics 7 (1968) 1061. [5] W. Culshaw, IRE Transactions on Microwave Theory and Technique, September 1962, pp. 331-339. [6] D.H. Ulmer, O.P. Gandhi and R.W. Grow, Electronic Letters, Vol. 10, 1974. [7] A.E. Siegman and E.A. Sziklas, Applied Optics 13 (1974) 2775. [8] A. Riazi, doctoral dissertation, Electrical Engineering Department, University of Utah, Salt Lake City, Utah 1978. [9] A.G. Fox and T. Li, Bell System Technical Journal 40 (1961) 453.
165