Three- and Four-component Lenses

Triplet
The simplest design that is reasonable for 35mm cameras is the triple configuration. Once the glasses are chosen, there are just enough variables (three powers, three shapes, and two separations) to correct all seven Seidel aberrations, as well as the focal length. The basic method used to flatten the Petzval field curvature in all anastigmats is the longitudinal separation of positive power from negative power. Increasing the separation lowers the relative ray height on the negative, thus will reduce the (negative) power contribution. The result is an effective net positive power without the undesirable excess of inward Petzval curvature. All anastigmats make use of this principle. Cooke triplet can be viewed as the trunk of the family tree of the airspaced anastigmats, while older anastigmats such as the Dagor, Sonnar, and the Biotar (double-Guass) types use thick meniscus components to separate positive convex surfaces from negative concave surfaces.
The choice of glass types is an important degree of freedom and has a significant effect on the characteristics of the triplet design. The glass for the positive elements should be a dense barium or lanthanum crown type; an index of 1.6 or more is almost essential. A triplet using an ordinary low-index crown glass (or acrylic plastic) for the positive elements is possible, but the result is poor unless the aperture or field (or both) is small. The other important factor in glass choice is the use of the relative difference in V values between the crown and flint elements as a means to adjust the vertex length of the triplet to its optimum value.

Tessar
The Tessar, invented in 1902 by Paul Rudolph, is similar to the triplet. The difference is the cemented doublet replacing the rear singlet. Although Tessar is a modification of the Cooke triplet, it is actually a descendant of the double-meniscus anastigmat form. The Tessar performs better than the triple in COM3, AST3, PETZ3, and DIST3. The substitution of a new achromat doublet for a crown in the triplet is the equivalent of using high-index, high-V-value glass, and allows the possibility of utilizing the cemented surface of the doublet to control coma and oblique spherical aberration.

The Tessar design patent was held by Zeiss for two decades, and licensed to Rollei in Germany, Bausch & Lomb in the United States and to Krauss in France.
Sonnar
Sonnar was introduced by Bertele (Zeiss) in 1931, at a time when antireflection coatings were not yet available. It can thought of as a triplet derivative with a meniscus component in the front air space. If the meniscus is made thick, it can have a field-flattening effect of its own. The Sonnar form makes use of this to eliminate the need for the center flint negative element. These lenses begin to look like the front half of a Biotar/double-Guass combined with the rear of a triple or Tessar. Given that the transmission of a single air-glass surface is typically about 95%, a lens with six air-glass surfaces can have a transmission of 0.95^6 = 0.73.

The Sonnar has proven incompatible in shorter focal lengths with SLR cameras due to the space taken up by an SLR's mirror. Therefore, the Sonnar type had a brief vogue as 35mm camera lenses of normal and long focal length, but they have been largely superseded by other constructions (e.g. double-Guass).

Two-component Lenses

Petzval lenses
When longitudinal chromatic aberration has been corrected, we normally find that the focal positions for three wavelengths still do not coincide. This effect is known as secondary spectrum and is due to the nonlinearity of the curve of refractive index as a function of wavelength. It stems from a difference between the nonlinearity of the positive (crown) element and the negative (flint) element of an achromat. The basic Petzval lens consists of two positive components, spaced apart so that the astigmatism is controlled to be either zero or slightly positive. Usually the two components are achromats, typically doublets. Secondary spectrum is reduced in Petzval lenses because the second doublet operates at a lower aperture than the first.
The original Petzval lens was designed by Joseph Petzvel of Slovakia in 1839 as a portrait lens for the Daguerrotype camera and, at a speed of about f/3.5. Its purpose was to reduce the extremely long exposure times of the daguerreotype camera from about 30 minutes to about 30 seconds, using a wide aperture. The Petzval lens was optimized for large apertures and indeed could only operate at one fixed aperture. Petzval lenses were the mainstay of camera and projection lenses for almost 40 years during the dawn of photography. They are very rare now.

Telephoto lens
When moderately long focal lengths are required, design similar to normal 50mm local length systems are often use. The basic power arrangement of a positive component followed by a negative component, can produce a compact system with an effective focal length F which is longer than the overall length L of the lens. The ratio of L/F is called the telephoto ratio, and a lens for which this ratio is less than unity is classified as a telephoto lens. By definition, many camera lenses which are sold as telephoto lenses are simply long-focal-length lenses and are not true telephotos. Since the system is unsymmetrical, each component must be individually achromatized if both axial and lateral color are to be corrected. The aperture stop is usually at the front member or part way toward the rear. Since a telephotos lens covers only a relatively small angular field, coma, distortion, and lateral color (which in many lenses are reduced by an approximate symmetry about the stop) are not as troublesome as they would be with a wider field. Simple telephoto lenses are formed by two cemented or closely airspaced achromatic doublets. Additional degree of freedom can be gained by splitting the doublets into widely airspaced components.
Telephoto lenses

Long-focal-length lenses

Wide-angle lenses for SLR camera
In modern SLR camera, a back focus of about 38mm to 40mm is required for mirror mechanism. This means the earlier forms of wide-angle lens are not acceptable, and the inverted telephoto construction is necessary. The arrangement of a negative component following a positive component has a back focal length which is longer than the effective focal length. Although the lens is much larger than the Biogon, correction for field curvature is very good because of the separated negative power at the front of the lens. The simplest fully corrected form is a pair of achromatic components. For the negative front component, the crown glasses (used for the negative elements) should be high-index and the flint glasses (in the positive elements) should be low-index; this reduces the overcorrected Petzval contribution from this component. For the positive rear component, the ordinary glasses should be low-index crowns and high-index flint. This combination increases its negative, inward-curving contribution to the Petzval sum. Usually the aperture stop is at the rear component; the natural shape for the front negative achromat is then that of a meniscus, concave toward the stop.

Double-meniscus anastigmat (Hypergon, Topogon, Dagor, Dogmar)
The Hypergon lens consists of two identical meniscus elements, symmetrical about a central stop. The concave and convex radii differ by less than 0.7 percent so that the Petzval contributions of the convex surfaces are almost completely offset by the Petzval of the concave surfaces. The astigmatism is controlled by the distance of the lens from the stop, and the symmetrical construction almost completely eliminates the coma, distortion, and the lateral color. The lens covers an astonishing field of 135 degree. Rapid rectilinear lens is a double meniscus system in which two achromatized meniscus lenses are arranged symmetrically on either side of the aperture stop, reducing or eliminating distortion, coma and lateral color.
The obvious way to improve the Hypergon is to add negative flint elements to correct the spherical aberration and axial chromatic aberration. The Topogon covers a field of about 100 degree at a speed of f/6.3, using dense barium crown and extra-dense flint glasses, retaining the symmetrical construction and the strong meniscus configuration for all the elements.

The Dagor combines both the old achromat and the new achromat into a cemented triplet construction. If we visualize the central negative element of the triplet split into two parts, then the outer high-index element and the outer part of the middle (medium-index) element can be seen to make up a new achromat. The inner low-index crown element and the other part of the middle element make up the old achromat. The symmetrical construction about the stop minimizes the coma and distortion, while the spacing from the stop and the cemented surfaces control the astigmatism.

The Dogmar lens is also a member of the double-meniscus family. It can be realized if one considers each half to be a triplet with a center air lens. The Dogmar form is used as an excellent general-purpose camera lens, and its symmetry and stability of correction make it eminently suitable for an enlarger lens.

Third-order Aberrations and Lenses Components

The third-order (Seidel) aberration coefficients form the basis of lens design.

Seven Seidel coefficients (effectively wavefront aberration terms)
SA3- spherical aberration (S1)
COM3 - coma (S2)
AST3 - astigmatism (S3)
PETZ3 - Petzval (field curvature) (S4)
DIST3 - distortion (S5)
LCOLOR - lateral color (C1)
TCOLOR - longitudinal color (C2)

For lenses at higher numerical apertures and field sizes, third-order aberrations are insufficient for describing all of the aberrations. High order aberrations are induced because a surface is operating in an aberrated beam. Fifth-order and even the seventh-order coefficients need to include in the design.

Fifth-order aberrations
SA5 - spherical aberration
COM5 - coma
AST5 - astigmatism
PETZ5 - Petzval
DIST5 - distortion
TOBSA - tangential oblique spherical aberration
SOBSA - sagittal oblique spherical aberration
LCOMA - Elliptical coma

Singlets
plano-convex, plano-concave, equi-convex, or equi-concave. Most are made of BK7 lenses may have large aberrations. They are mostly used when apertures are quite small or when aberrations are not important.

Field flatteners
Although plano-concave singlets for field flattener do not correct astigmatism, the improvement in field curvature is significant.

Doublet and flattener


Meniscus singlets
The earliest anastigmats flattened the field by using a think meniscus, which separated the positive and negative outer surfaces. Approximating an aplanatic condition on the front surface and a rear surface that is almost concentric with the marginal ray when used in the converging beam after a doublet. The combination has significant better resolution than the equivalent doublet. However, meniscus lens is not color corrected. At the time there were no antireflection coatings, designers tried to minimize the number of air-glass surfaces. For this reason, most of the early anastigmats were limited to two cemented meniscus components.

Doublet and meniscus


Doublets and triplets
Spherical aberration, coma, and chromatic aberration are corrected.

Cemented triplets
Symmetrical triplets have zero coma, distortion, and lateral color at unit magnification. When it is used at infinity conjugates, the aberrations may be small enough for some applications.

Single lens for disposable camera
They use a single plastic lens, which can be aspheric, and a film (ASA 400-800) that is curved around a cylindrical surface. Although it is possible to correct coma, the astigmatism is then quite large. Moreover, field curvature, axial and lateral color are not corrected, but at an aperture of f/16 the performance is just acceptable.

VL4% 126-dma & NC6% 20-dma

Be fearful when others are greedy and greedy only when others are fearful.
-- Warren Buffett

Almost all Exchange-Trade Funds are listed on the American Stock Exchange (AMEX). The official definition of ETFs is "registered investment companies under the Investment Company Act of 1940, which have received certain exemptive relief from the SEC to allow secondary market trading in the ETF shares. ETFs are index-based products, in that each ETF holds a portfolio of securities that is intended to provide investment results that, before fees and expenses, generally corresponding to the price and yield performance of the underlying benchmark index."

Some of the heavily traded ETFs are:
SPY - the Standard & Poor's Depositary Receipt (SPDR) Trust Series securities, tracks the price and yield of the S&P 500 Index. The S&P 500 Index which is calculated as the number of common shares outstanding multiplied by the stock's price, is market-capitalization-weighted. Therefore, the big market cap stocks exert a larger influence on the index.

QQQ - the Nasdaq-100 Trust Series I, tracks the price and yield of the Nasdaq 100 Index.

DIA - Diamonds Trust Series I, Tracks the price and yield of the DJIA. The DJIA is price-weighted by each of the 30 components. Therefore, the highest-priced stocks have the most impact on the average.

iSHARES - pooled-securities with an open-ended investment structure issued by Barclays Global Investors. There are over 50 different iShares index funds that trade like stocks. Each share tracks a specific portfolio.

Value Line Composite Index
The Value Line Composite Index (VLCI), also known as the Value Line Geometric Index, is a geometrically weighted index of 1700 stocks in 90 industries. It was first appeared in 1961 to provide the "median" performance of all the stocks tracked. About three-fourths of the stocks are traded on the New York Stock Exchange, 20 percent are traded on the Nasdaq, and the remainder on the AMEX and in Canada.

On February 1, 1998, Value Line published a new index dubbed the Value Line Arithmetic Index (VLAI). The index provides the average performance of all stocks in the index. It is based on summing up the percentage price increase in every stock each day and dividing that total by the number of stocks in the index. Therefore, it is an equally weighted index of 1700 stocks.

VL4% Strategy
The VL4% strategy uses a daily chart of the VLCI. If the VLCI rises 4% from its last market low, based solely on its weekly Friday closing price, that is a buy signal. On the other hand, if VLCI declines from its last market top by 4% based on the weekly Friday closing price, that is sell signal. You cannot buy or sell VLCI because it is an index. Therefore, you should invest in an index fund or ETF instead.

Leslie N. Masonson, in the back-test, found that performance can be improved by adding the 20-dma (day-move-average) to the VL4% VLAI daily strategy. With the 20-dma, we add:
A buy signal occurs when the VLAI pierces its 20-dma to the upside, from below.
A sell signal occurs when the VLAI drops below its 20-dma to the downside, from above.
If the other strategy issued a buy/sell signal while the current buy/sell signal was in place, no action would be necessary.

Nasdaq Composite Index (COMPX) NC6% and Nasdaq 100 Index (NDX) NDX6% Strategy
The COMPX contains about 3500 companies. It tends to be more volatile and has wider prices swing than the VLCI and VLAI. Based on the back-test results, Leslie N. Masonson found that the weekly 6% signals are superior to the daily 6% signals as the latter are much more frequent because of the volatility of their respective indexes.

Back-test results in choosing the best dma
Michael McDonald, Predict Market Swings with Technical Analysis (Wiley, 2002)
S&P 500 Index - 72-dma and 132-dma.

Robert W. Colby, The Encyclopedia of Technical Market Indicators (McGraw-Hill, 2003)
DJIA - 66-dma and 126-dma

Leslie N. Masonson, All About Market Timing (McGraw-Hill, 2004)
Nasdaq Composite - 20-dma and 25-wma

US Presidential Cycle and ETFs

In US, there is a four-year cycle in the stock market accompanies by the presidential election years. Usually, the market rises more in the pre-election year (95, 99, 03) than in the election year (96, 00, 04). The election year is the second best of those four years. The worst performance year is the post-election year (97, 01, 05) and the next worst is the midterm year (98, 02, 06).

The investment vehicles to use for both the seasonal and presidential cycle strategies are index funds, leverage funds and ETFs (such as the SPY and QQQs).

Mark Vakkur done a research in 1996 to analyze the optimum return of investment of the presidential cycles. He tested a number of different strategies to determine their profitability and had shown the comparison of Buy-and-hold to Presidential Election Cycles scenarios Jan 1950 - Dec 1995, in the descending order, are shown as below (value of $10,000 at the beginning):

Optimal Months: $5,189,384
Leverage Best Months: $2,183,257
2:1 Leverage: $1,272,369
Pre-election and election years: $733,605
Buy-and-Hold: $372,388 (8.4% ave annual return, 14.4% std dev)
100% Cash: $110,905 (5.5% ave annual return, 2.8% std dev)
Post-election and midterm years: $56,297

Strategies explained
Pre-election and election years - investing only in these two years of the presidential cycle and remaining in cash for the other two years.

Post-election and midterm years - the opposite of above.

2:1 Leverage - 50% margin in the pre-election year and 100% invested in election year.

Leverage Best Months - 2:1 leveraged (Nov - April) in the pre-election and election years; 100% invested during these same months in the post-election and midterm election years, and 100% invested during the May - Oct period during the same two years.

Optimal Months - 2:1 leverage in pre-election and election years in the following months in those years: Jan to April, July, Nov and Dec; 100% cash in May and Sept of all 4 years, and Jun and Aug of the post-election and midterm years; and 100% invested in all the other months not mentioned: Jan to April, July, Nov and Dec in the post-election/midterm years, and Jun and Aug in the pre-election years.

Vakkur found that the best seven months in the pre-election year and election years were Jan - April, July, Nov and Dec. The worst months in all the four election cycle year were May and Sept. Two more worse months occured in Jun and Aug of the post-election and midterm election years. All the other months not mentioned in these two years were considered average.

Balloon Lumpfish

中文稱為太平洋刺獅子魚，英文俗稱 Balloon Lumpfish。體形細小，只有數mm大。冷海魚類，生活於水深90米至150米的日本海及鄂霍次克海一帶。



攝於日本北海道網走流冰館。

物性

對於人來說，動物的可愛和可怕都可能在於其率性而為。北極熊 - 陸上最大的食肉動物，水中玩球。

攝於日本北海道旭川動物園。